Would you like to relate your idea of an example of an extreme review of your ebook?

In "Nuts & Bolts" the author claims to show that Einstein's special relativity theory is incorrect. Next, he'll be telling us that miniature pieces of driftwood form inside decaying white birch logs, which have fallen to the forest floor. That twenty to forty slugs will crawl into an empty beer can and then are lethally poisoned by the last sip of beer that is left inside the beer can by the litterbugs who discard them along our roadsides. That moles can get fatally trapped inside glass beer bottles that have sunk into the ground. That wind can slow the flow of a mountain stream so that it flows with a pulsing rhythm. That groundhogs sometimes climb up saplings, which bend under their weight. That pine cones open and close long after they have fallen off the tree. That the negative terminal of a car battery is really the positive terminal and vice versa. That a car's starter motor never really breaks; it just gets dirty because small debris get in through the vents (this only applies to starter motors that have vents). That very large lightning bolts shoot up from the earth and small segments of them start to fall back to earth just before they disappear. That designs of black ice that are embedded into the ice of lakes and ponds that look like an abstract spider are caused by rainwater finding its way through the solid ice and not by the warmer water that lies under the ice. That when it turns bitter cold there's a rusty orange, mushy ice that forms on the bottom of streams and that if you scoop it out of the water it turns bright white. That because of all the detergents found in suburban streams a new kind of brittle ice can form, which uses detergent bubbles as a template. That when a film of detergent lies on the surface of a roiling stream you can see the shadow of the line where the edge of the film meets the water though you can't see the line itself. That in the suburbs the trees are too close together because they're planted when they're small and never thinned out. Doesn't the author know there are examples of relativity in everyday life? Take a tape measure for instance: as they get old, the flange at the free end gets loose and slides backward so that an older tape measure will measure the same object as shorter than a new tape measure.

What associations have been most helpful in advancing your literary career?

Without a doubt the SFABC, the Science Fiction Association of Bergen County, helmed by Philip DeParto has been truly advantageous to my writing endeavors. I have been able to overlook the fact that the group's acronym omits an "o" for the word "of" with ease because of the fellowship provided by the group.

Who are your favorite authors?

Jorge Luis Borges is one of my favorite authors. I also like Michael Piazza, Charles Garofolo and Jonathan Wood. I still find it distressing that authors who write about the sciences and mathematics have seemingly failed to write skeptical books on the following topics: the Michelson-Morley experiment, Bell's inequality, stellar aberration, Gödel's incompleteness theorems and Einstein's theories of relativity.

What motivated you to become an indie author?

Sadly, the nature of my ebook, "Nuts & Bolts: Taking Apart Special Relativity," is such that it will never be published by a traditional publishing house. I have found this out through the experience of having my ebook rejected by traditional publishers as well as fringe publishers. Also, I have been told this by the resident expert at Elderberry Press. The idea that the most significant dilemma of our time (and, in fact, of all times and places) is hidden knowledge seems to have gone out of fashion around the time of the Garden of Eden. Today, society in general believes what the wise Okies believe in "The Grapes of Wrath." They believe that problems arise when deep emotional bonds are broken such as the bonds of family, the bonds of traditional culture or a spiritual bond to the land, i.e., nature.

How has Smashwords contributed to your success?

It is difficult to say. I don't know why people are downloading my free ebook. I've had only one radio interview, and that was on the show Night People on WFMU. Soon after that Night People was demoted to a pod cast. I've had a few press releases published, but that was several years ago. My book promoting strategy was to write letters to anyone whom I thought might be interested in the ebook, provided I could obtain their mailing adress. These were mostly people I read about in newspapers and magazines. I have a revised version of this ebook and another ebook, "Bell's Inequality Untwisted," available for free at lulu.com. I have been using pinterest to promote these ebooks, and I have garnered less than 1% of the downloads I have at Smashwords. Perhaps, the downloads of my ebook are taking place because of word of mouth.

What is the greatest joy of writing for you?

I have the idea that some force is trying to keep certain questions from being asked. I see that force at work in the movies. Why are only certain types of movies made? Some say that it's the audience that demands only certain types of movies. But, then the question becomes after years and years why doesn't the audience develop a yearning for something different. I see that force at work in popular music. There was a time when Art Rock was popular. There were groups such as Yes, King Crimson and Henry Cow. These groups were difficult for many to understand, but afterwards why did popular music decline so rapidly? Strangely, I think it might have something to do with humor. If the humor of an era is typified by "what's huge and purple and lives at the bottom of the sea? (answer: Moby Grape)" as opposed to an era where Wayne and Garth make quips about whom "blew chunks" then the former era may have a chance for growth that the latter era lacks.

What do your fans mean to you?

I have two of them on right now. I've always believed in an attic fan that you could leave on for hours and hours on very hot days. Although, when the hot, summer sun beats down on the roof all day long, it's questionable how much good an attic fan can do. Unfortunately, the fans inside air conditioners seem to develop mold and mildew. I once made a homemade filter for my air conditioner, and it got so fouled that it let only a small amount of air through. During a very severe heat wave, this clogged filter may have been the main cause of my air conditioner's failure. Strangely, the heat wave was so prolonged that for the first time in my life I became very ill from its effects. I was not able to sleep at night for many days and did not feel comfortable during the day. I developed a painful skin rash. So my fans are very important to me.

What are you working on next?

I am working on a long essay in which I analyze Kurt Gödel's incompleteness theorem, formula by formula. I reach the conclusion that Gödel's incompleteness theorem is invalid. Douglas Hofstadter's award winning book, "Gödel, Escher, Bach" is a good example of a force that keeps certain questions from being asked. Because of its extraordinary length and complexity, apparently, no one has noticed that at the critical moment in his explanation of Gödel's incompleteness theorem he employs a clever deception. He lets the variable a' represent two distinct equations. If we carefully inspect the use Hofstadter makes of these two distinct equations, we will see that the conclusions he draws, through his use of the equations, are invalid. And, thus, his entire proof of Gödel's incompleteness theorem is invalid.

What inspires you to get out of bed each day?

My inspiration comes from the fact that there appears to be a limit to the success that can be obtained through social engineering that is based on dogmatic principles, i.e., things can only get so bad.

Do you remember the first story you ever read, and the impact it had on you?

Not really. But, I do think we can make boring books serve an important function. For instance, I find Jules Verne's "Mysterious Island" to be a boring book. All the characters are responsible and intelligent so with only a minimum of difficulty they transform their desert island (wild and uninhabited island as opposed to a dry, barren, sandy region often extremely hot) into an utopia. Now, our society has a great deal of responsibility and intelligence, yet no utopia has been produced. It may be that novels use an overabundance of irresonsibility to disguise the fact that the wisdom and intelligence displayed within the novel's pages is only the apearance of widom and intelligence. If we mentally correct all the irresponsible behavior that occurs in a novel, will that correction make the wisdom and intelligence displayed within the novel seem invalid and self-serving?

How do you approach cover design?

My covers have a distinctly homemade appearance. I try to break the rules. If the rules say the book's title and the author's name must appear on the cover, I would produce the title and name as say a square block of letters containing six rows and six columns (provided, of course, that the sum of the number of letters in the book's title and author's name was 36) so that the traditional breaks between words would be confounded.

When you're not writing, how do you spend your time?

I often collect the trash that accumulates along our roadsides. Then I carry it home and throw it away unless I find a homemade bong, which I keep. A homemade bong is a plastic bottle that has had several holes melted into it probably with a cigarete lighter. One hole is for the homemade pipe stem and the other is for carburetion. When I collect five or so of these bongs, I take a picture of them and send it to the local newspapers. I claim to have discovered new and desultory evidence of the frightening effects of global warming and climate change.

Any final thoughts about your fans?

Yes, I once had a fan that had a polarized plug, but the outlet I wanted to plug it into was made in the old style and would not accept the one widened prong of the polarized plug. The outlet was in a difficult location in which to do electrical work. So I decided to file down the widened prong of the polarized plug. I never had any trouble with the fan. It leads one to question if there is any real value to polarized plugs.

How can we gauge the intelligence level of our society?

I don't really know. A temptation that must be avoided is to overemphasize the minor foibles that are present in society. That said, I've always been amazed by the fact that people continue to put out garbage bags for pickup by the garbageman that contain scraps of food in them. Don't they know the crows will peck holes in the bags and make a mess searching for the food scraps. Maybe they think crows are migratory and go south for the winter.

What would you have to have to consider yourself a success as a writer?

A squadron of tanks.

If you could change one thing in the world, what would it be?

I'd rename the kingfisher the jackass bird.

What's the most esoteric riddle you've come across in nature?

That the plant chrysanthemum and the plant mums are one and the same. It's hidden in plain sight. That's what makes it so difficult to figure out.

As an American, what is it hard for you to understand?

That Vince screwed Bret ! ! ! Vincent Kennedy McMahon used trickery to force Bret "the Hitman" Hart to lose the WWE title belt in Montreal, Canada. Bret Hart is one of the most famous Canadians, and the one thing he didn't want to do was lose the title belt in Canada.

How does your knowledge of science influence you as a writer?

Instead of saying an ebook's sales are miniscule, I can say they are in the horizontal limb of the exponential curve.

Can scientific problem solving be used to solve political problems?

I've always felt that the images of Jesus that believers find on pancakes, potato chips and grilled cheese sandwiches etc. serve as an important societal ameliorant. I think a group of experts should study the phenomenon. They should find out which objects most often bear these images. Then they should should search those objects for images of other religious icons such Buddha, Krishna and Mohammed. If for instance an image of Mohammed could be found on a grilled cheese sandwich this could serve as an important bridge between Christianity and Islam.

What surprises you about the internet?

I find it amazing that the internet is not flooded with photos of the mushrooms Ravenel's Stinkhorn (Phallus ravenelii) and Netted Stinkhorn (Phallus duplicantus). As the first term of the binomial nomenclature suggests these mushrooms bear a resemblance (a resemblance that so approaches subtle caricature as to make an argument for intelligent design) to the nominative case of the binomial nomenclature.

What is your favorite figure of speech?

Zeugma is one of my favorite figures of speech because I’m not exactly sure what it means. According to the dictionary it is a single word which is syntactically related to two or more words though having a different sense in relation to each.

What changes have you seen recently in your local community?

For several years I noted that the storm drains alongside roads that were lined with tall, leafy trees were completely clogged with leaves. During a heavy rain the water that was unable to enter the storm drains would erode the sides of the road. I would try to keep the storm drains clear by scraping the leaves away with the side of my shoe. As I pushed the leaves aside, I would always see a large number of earthworms. The only ones working to keep the storm drains unclogged were me and the earthworms; we both were in over our heads. Today I saw a specially designed truck with a powerful vacuum hose that was unclogging the storm drains.

Is one of the keys to solving problems recognizing design flaws that society doesn't recognize as design flaws?

I wrote a letter-to-the-editor of the Bergen Record suggesting that Bergen County adopt the phrase, “Goose Toilet Bowl” as their official nickname. The Bergen Record couldn’t be bothered to publish my letter. It’s this kind of blinkered provincialism that is hindering progress on a number of fronts. If it’s true, and it may well be, that you can’t solve certain problems until it is acceptable to make fun of them, we are in difficult straits.

As a conservative, who would you designate as one of the least perspicacious conservative leaders?

Although he would probably use a greater sockdolager than perspicacious in conversation, I find William F. Buckley to be among the least perspicacious conservative leaders. If we acknowledge that for a political movement to have power, it must appeal to the average voter, we can see that W.F. Buckley did almost nothing to endear himself to the average voter. In fact, his persona seemed to exude a disdain, if not contempt, for the average person. His musical instrument: the harpsichord. His philosophical influences: the instrumentalism of William James. His penchant for the obscure doesn't fully capture his elitist character.

When W.F. Buckley was the face of conservatism, it had waning popular appeal. When R. Reagan, a figure that connected to the masses, was the face of conservatism, its popular support waxed.

When W.F. Buckley was the face of conservatism, it had waning popular appeal. When R. Reagan, a figure that connected to the masses, was the face of conservatism, its popular support waxed.

Can you think of a suitable replacement for the lackluster words "emoticon" and "emoji"?

All my engagements I will construe to thee,

All the charactery of my sad brow.

"Julius Caesar" Act II Scene 1 lines 308 & 309

All the charactery of my sad brow.

"Julius Caesar" Act II Scene 1 lines 308 & 309

How can you tell when a rapidly flowing stream is about to freeze over?

Ponds and the slow flowing sections of streams freeze over many weeks before the rapidly flowing sections of streams freeze over. When the level of a stream drops a few inches overnight, this is a sign that much of the ground under the stream and alongside the stream has frozen. This greatly limits the flow of ground water into the stream. During the next cluster of days where the temperature dips well below freezing the rapidly flowing sections of the stream should freeze over and that ice should link up with the ice of the already frozen slow flowing sections of the stream.

What experiments could facebook friends perform?

If an individual had a large number of followers, he could ask his followers to collect data. That data could be analyzed to shed light on particular questions. For instance, the followers could walk along the roads and streets near their homes for a week. They could note whether any lost gloves they came across were lost singly or in pairs. The common sense notion is that gloves are lost one at a time. I have been taking note of lost gloves for some time, and it is remarkable how many gloves are lost as pairs.

What magazine articles have you read recently?

I read an interesting article in “Popular Science.” There are several types of new cement that are vastly superior to traditional cement in strength. These new cements are used in some modern skyscrapers as a partial replacement for steel girders. Yet, in the article describing these new stronger cements, no mention was made of using these cements to pave roads and thus reduce or eliminate the pothole problem. We’re trained to accept progress in one field and not to wonder about the lack of progress in another closely related field.

What have you learned about splitting logs into firewood?

Almost every depiction I’ve seen of splitting logs into firewood has been idealized whether in literature, TV or movies. Since it is often depicted as a manly activity, you often see a muscular man splitting log after log with only an ax. Actually an ax would easily get stuck in a log of moderate size because of the presence of knots that resist splitting along the grain. I find that splitting logs for firewood requires at least three log splitting wedges and a sledge hammer of moderate weight. Often two log splitting wedges will get stuck in the process 0f splitting a log of a large diameter. The artist Andrew Wyeth, who is noted for his realistic depictions of rural life, portrayed a log splitter, and his tools were a sledge hammer and one log splitting wedge; an unlikely occurrence. Log splitting is often depicted as a well-ordered and valued activity as it may have been in the past. Today because it is anachronistic and noisy it is often relegated to a messy corner.

What sight gags are you working on for your video?

I want to make some extravagant claim about wormholes such as their production locally for purposes of scientific study shouldn't be that difficult. Then I want to rake away about two inches of my compost heap hopefully revealing several large earthworms that will disappear into wormholes. The problem is that during the summer months the earthworms don't seem inclined to disappear into wormholes.

What are your thoughts regarding the roles of native plants and invasive plants in local ecosystems?

The idea of invasive plants may need to be rethought. What we may be witnessing is the ecological principle of homeostasis in operation. Here in New Jersey the local ecosystems may be solving the whitetail deer overpopulation problem. They are rebalancing themselves. As the deer population boomed they consumed many of the native plants that grew in the woods and open fields; the Cardinal flower being the most spectacular. What seems like an invasion by certain plants is actually the growth of plants that the deer will not eat. With less food available, the deer population will decline.

As a recent newspaper article stated, “native plants are preferred by wildlife to invasive plants.” Replace the word “wildlife” with “deer” and we see that a successful campaign against invasive plants allows the deer population to boom, which is exactly what we don’t want because the deer have consumed so many native plants. The article continued, “With more artificial habitats, parks and gardens, growing native plants rather than non-native varieties provides an easier transition for wildlife. . . .” Again replace the word “wildlife” with “deer” and we see that encouraging native plants means allowing the deer population to boom.

Are the environmentalists deliberately trying to preserve our local eco-systems in a destabilized state to further their own agenda?

As a recent newspaper article stated, “native plants are preferred by wildlife to invasive plants.” Replace the word “wildlife” with “deer” and we see that a successful campaign against invasive plants allows the deer population to boom, which is exactly what we don’t want because the deer have consumed so many native plants. The article continued, “With more artificial habitats, parks and gardens, growing native plants rather than non-native varieties provides an easier transition for wildlife. . . .” Again replace the word “wildlife” with “deer” and we see that encouraging native plants means allowing the deer population to boom.

Are the environmentalists deliberately trying to preserve our local eco-systems in a destabilized state to further their own agenda?

Is the Bible Code empirical evidence that Gödel’s Proof is correct?

At first glance, the Bible Code’s failure to use every letter of the Bible seems to distinguish it from the kind of mapping Gödel was concerned with. We can overcome this flaw by assigning all the unused letters and punctuation some kind of null symbol. Now, with this alteration, since all the letters and punctuation are used, the Bible Code appears more in line with Gödel’s Proof. Must the Bible Code be predictive to fulfill the demands of Gödel’s Proof? How meaningful is it to find unexpected and unconventional groups of letters that form meaningful phrases in the Bible? Another interesting point is that we could seemingly disallow the Bible Code with a few simple rules such as traditional words breaks must be observed and the formation of vertical words is not allowed. The claim that Gödel makes that there is no way to rule out his mapping has always seemed quite spurious. The suspicion that for the Bible Code to be meaningful it must be predictive hints at the supernatural aura that surrounds Gödel’s Proof.

How do you regard the art movement known as abstract expressionism?

If you study it for some time, I think you will come to the conclusion that it is unintentionally hilarious. The pretentious verbiage that surrounds the paintings themselves is both amusing and amazing not to mention annoying. An artist who drew human shadows in some of his later paintings is credited with reintroducing the human form into western art. The paintings themselves are, as time goes by, summed up best by a line the architect Doug Roberts (Paul Newman) says to his girlfriend Susan (Faye Dunaway) in the movie “The Towering Inferno” referring to the burned out skyscraper. “May be they should leave it as it is; a kind of shrine to all the bullshit in the world.” Abstract expressionism seems to have ushered in the worldview that the opinion of the cognoscenti and/or clever advertising establishes the merit of an art object or an idea regardless of any intrinsic value it has. It is somewhat mysterious that the paintings made by today’s artists can sell for astronomical prices. Of course, the explanation may be that the patrons simply like the artwork. But, if the patrons simply liked the artwork, it is odd that they show no apparent interest in the modern poetry that is imbued with the same kind of worldview. You would think there would be gatherings where the poets (often referred to as language poets) could read their poems in which they used their literary device of choice, a modern version of parataxis, surrounded by the paintings of today’s foremost artists.

What do you think of the concept of the multi-verse?

I think there is only one universe and, of course the Leon verse, which is the second verse of “Beware of Darkness” that Leon Russel sings in The Concert for Bangladesh.

What is your analysis of the meth-amphetamine problem?

I find it interesting that meth-amphetamine is a terribly destructive drug both to the user’s health and to those caught up in the violence of the drug gangs. Yet, meth-amphetamine is no more powerful than caffeine; a drug that is seldom characterized as having a deleterious effects either on the individual or on society. Is it wrong to think that there is some sort of mass irrationality involved in drug use. If drug users made an effort to buy their drugs from the least violent dealers over time it would seem that the violent drug dealers would go out of business.

For a long time I’ve believed that people who drink and drive or people who use drugs and drive do so because they want to drive while they’re high. It is more difficult than we imagine finding interesting things to do whether or not we are high. In an attempt to make life exciting I think people will drive when they are high. This notion has seldom been examined by those fighting the drug problem.

For a long time I’ve believed that people who drink and drive or people who use drugs and drive do so because they want to drive while they’re high. It is more difficult than we imagine finding interesting things to do whether or not we are high. In an attempt to make life exciting I think people will drive when they are high. This notion has seldom been examined by those fighting the drug problem.

What is your favorite bumper sticker?

Years ago, I recall often seeing a bumper sticker consisting of two words “Question Authority.” My intuition is that the people who displayed the bumper sticker never questioned ideas that are associated with the liberal worldview such as global climate change. If my intuition is correct, this would be an example of irony. But, my unquestionably favorite bumper sticker that you see currently is the one in which the word "co-exist" is spelled with the familiar iconography of the various mainstream religions along with the peace sign and a symbol denoting harmony between male and female..

Can you recall any line of poetry?

“I swallowed the pain of my childhood like a tree swallows barbed wire.” What I recall is the criticism of the simile “like a tree swallows barbed wire.” The criticism was that the simile was terrible because trees don’t have mouths, and if they did have mouths they wouldn’t eat barbed wire, they would consume minerals dissolved in water. The flaw with that line of reasoning is that the poet is referring to another characteristic that trees have. As a tree grows it will very slowly engulf things like wire, rope or nails that are fastened to it. If a person never sees trees to which barbed wire has been nailed in order to construct a fence, there is no reason they would be aware of this phenomenon.

What can you say about the relativity of time?

Let’s say I took the clear, plastic face off a clock and glued a wing onto the minute hand. Then when the minute hand was traveling from 12 to 6, I placed a fan below the 6 so that the flow of air impeded the progress of the minute hand. Next when the minute hand was traveling from 6 to 12, I moved the fan so that it was above the 12, and now the flow of air from the fan again impeded the progress of the minute hand. Would my simple experiment have slowed the passage of time, at least in my local vicinity? Or, would I merely have slowed a particular device that measures the passage of time?

There have been a number of experiments in which two atomic clocks have been set side by side and synchronized then one of the atomic clocks has been flown around the earth. When the two clocks are set side by side again the clock that has been flown around the earth has fallen behind the clock that was stationary with regard to the earth. The claim is that time itself slowed in the local vicinity of the moving clock and not that a particular device that measures time has slowed.

The triumph of relativity has been to take the common sense notion that measuring devices such as clocks will behave differently in different environments and incorporate it as a proof of the relativity of time. This is accomplished by redefining clocks as devices that define time instead of measure time. Defining clocks as devices that define time leads to certain problems such as does the deliberate slowing of a clock by using fans or other electromechanical devices mean that time itself is slowed. These problems are overcome by several strategies: downplaying the notion that relativity requires that clocks define time instead of measuring it, ignore the problem of deciding which clocks and under which environmental conditions the flow of time is altered, focus on the results of atomic clocks in gravity fields of different strengths and atomic clocks traveling a different speeds. Atomic clocks seemingly produce incontestable results with their ability to measure time to the trillionth of a second and perhaps with greater accuracy than that.

There have been a number of experiments in which two atomic clocks have been set side by side and synchronized then one of the atomic clocks has been flown around the earth. When the two clocks are set side by side again the clock that has been flown around the earth has fallen behind the clock that was stationary with regard to the earth. The claim is that time itself slowed in the local vicinity of the moving clock and not that a particular device that measures time has slowed.

The triumph of relativity has been to take the common sense notion that measuring devices such as clocks will behave differently in different environments and incorporate it as a proof of the relativity of time. This is accomplished by redefining clocks as devices that define time instead of measure time. Defining clocks as devices that define time leads to certain problems such as does the deliberate slowing of a clock by using fans or other electromechanical devices mean that time itself is slowed. These problems are overcome by several strategies: downplaying the notion that relativity requires that clocks define time instead of measuring it, ignore the problem of deciding which clocks and under which environmental conditions the flow of time is altered, focus on the results of atomic clocks in gravity fields of different strengths and atomic clocks traveling a different speeds. Atomic clocks seemingly produce incontestable results with their ability to measure time to the trillionth of a second and perhaps with greater accuracy than that.

What do you think of the notion that pot isn't a gateway drug?

It seems completely false from my experience. If it is true according to sociological data, what does it say about the curiosity I’d like to believe is inherent in human nature? Perhaps people will say that humans are still by nature curious, it’s just that pot smokers aren’t curious about other drugs that will alter their perceptions. It seems to be another example of cognitive dissonance.

Is part of the difficulty with general relativity that its mathematical foundations are problematic?

Yes, there is a problem with the differentiation of tensors. If we differentiate a covariant tensor of rank two sometimes we will get a tensor as the result and sometimes we do not get a tensor as the result. In general the differentiation of a tensor does not give a tensor. This posed a problem because some mathematicians wanted a way to differentiate a tensor so that the result was always a tensor. They invented a method of differentiation they called covariant differentiation. This method of differentiation of a tensor always produces a tensor. But, is covariant differentiation a valid method of differentiation? Or is it merely an arbitrary method designed to get out of the dilemma posed by the fact that the traditional method of differentiation when applied to a tensor does not always result in a tensor? The traditional method of differentiation has substance and meaning if it doesn’t give the results that certain mathematicians desired when applied to tensors, can it be valid to arbitrarily design a new method of differentiation to gets the results they desired?

Do you have any directions for your readers?

You are halfway through to finding your hat. Go past the road that goes toward Deer Lake. After turn onto Kingsland Road and go to the first pond you see on that road and the clue will be on a railing. . .

What hobbies do you have?

I want to collect all the catalytic converter covers I see lying along the roadsides and turn them into some kind of sculpture.

Is it true that in general relativity the mass of the sun is given as 1.47 kilometers.

I have not found that value used by Einstein, but one of his foremost interpreters Arthur S. Eddington uses that value. Also, Lillian R. Lieber who wrote “The Einstein Theory of Relativity,” which is an attempt to make the very complicated math of the relativity theories accessible to a wide audience, gives the mass of the sun as 1.47 km. Both Eddington and Lieber use that value to calculate the deflection of a ray of light as it passes close to the edge of the sun. Lieber uses that value to calculate the displacement of the Fraunhofer lines (spectral lines). The use of that value is implied in Eddington’s calculation of the displacement of the Fraunhofer lines.

Also, in Lieber's interpretation of general relativity mass is introduced as a constant of integration, which seems odd for a quantity that seems to be a variable, but it may be mathematically correct.

Also, in Lieber's interpretation of general relativity mass is introduced as a constant of integration, which seems odd for a quantity that seems to be a variable, but it may be mathematically correct.

What is a ghost tool?

Well, I’ve only come across one, unless we count socket wrenches with a hole in the center of the socket end that allows long bolts to pass through the socket in order to reach the nut. Such socket wrenches, I assume actually exist.. A ghost tool is a tool that appears in a store's online catalog, but it seems the tool doesn’t really exist. Since I split logs for firewood with a sledge hammer and three wood splitting wedges, I was interested in finding some inexpensive hand-tool that would help me perform the task with less effort. The Home Depot online catalog carried a log splitting tool that consisted of a weight that traveled along a rod and along with some other devices I don’t recall it was claimed that this tool could slit logs into firewood. I don’t see how a weight traveling along a rod could develop enough force to split a log. My local Home Depot had the price tag for the tool on the shelf that contained sledge hammers and other tools, but it seemed there was no tool that ever went along with the price tag. Perhaps, because people believe the internet can solve all our problems, the internet somehow generates ghost tools in order to sustain this illusion.

Have you seen anyhing recently that caught your interest?

As I was collecting roadside trash along Powerville Road, I came across several plastic bottles with curious holes burned in them. Perhaps this is a new sign of global climate change. There is a lot of decaying matter along certain stretches of that road. This could cause low level micro-clouds of the very potent greenhouse gas methane. These micro-clouds could act as a heat lens and focus the heat waves much as a magnifying glass does, thus burning holes in the plastic bottles.

Are there any aspects of quantum mechanics that you find troubling?

It is interesting to compare “one of the strangest features of quantum spin” as described by John Gribbin in his book "Q is for Quantum" with certain properties of complex variables described by Murray Spiegel in his book "Theory and Problems of Complex Variables." J. Gribbin describes the very weird world of quantum physics where the quantum particle “is able to discern a difference between two identical copies of the universe.” M. Spiegel describes a complex mathematical calculation that depends on the properties we ascribe to the square root of negative 1, which we designate as the imaginary number i.

John Gribbin writes, “One of the strangest features of quantum spin is shown by the behavior of fermions. If an object like the earth turns in space through 360 degrees, it returns where it started. But if a fermion rotates through 360 degrees, it arrives at a quantum state which is measurably different from its starting state. In order to get back to where it started, it has to rotate another 360 degrees, making 720 degrees, a double rotation, in all. One way of picturing this is that the quantum particle “sees” the universe differently from how we see it. What we see if we turn through 360 degrees twice are two identical copies of the universe, but the quantum particle is able to discern a difference between the two copies of the universe.”

I will paraphrase M. Spiegel calculations to avoid complex equations. Suppose we are given the function w = the square root of z, in which z is a term that includes the imaginary number i. Suppose further that we allow z to make a complete circuit (counterclockwise) around the origin starting from point A. After a complete circuit back to A we have w=the negative of the square root of z. Thus we have not achieved the same value of w with which we started. However, by making a second complete circuit back to point A we obtain w= the square root of z, which is the same value of w with which we started.

I believe the strange features of quantum spin are merely the result of using certain equations that contain the imaginary number i as an exponent of e along with other complex mathematics such as polar coordinates.

John Gribbin writes, “One of the strangest features of quantum spin is shown by the behavior of fermions. If an object like the earth turns in space through 360 degrees, it returns where it started. But if a fermion rotates through 360 degrees, it arrives at a quantum state which is measurably different from its starting state. In order to get back to where it started, it has to rotate another 360 degrees, making 720 degrees, a double rotation, in all. One way of picturing this is that the quantum particle “sees” the universe differently from how we see it. What we see if we turn through 360 degrees twice are two identical copies of the universe, but the quantum particle is able to discern a difference between the two copies of the universe.”

I will paraphrase M. Spiegel calculations to avoid complex equations. Suppose we are given the function w = the square root of z, in which z is a term that includes the imaginary number i. Suppose further that we allow z to make a complete circuit (counterclockwise) around the origin starting from point A. After a complete circuit back to A we have w=the negative of the square root of z. Thus we have not achieved the same value of w with which we started. However, by making a second complete circuit back to point A we obtain w= the square root of z, which is the same value of w with which we started.

I believe the strange features of quantum spin are merely the result of using certain equations that contain the imaginary number i as an exponent of e along with other complex mathematics such as polar coordinates.

What do you think of individuals who accept all the mainstream versions of all the major scientific theories?

How is it different from brainwashing?

What do you think of the latest fashion trends?

For a long time, I tried to patch the holes in my jeans. The patches didn’t work. Either they came loose in the wash or a new tear would form between two patches. It leads me to speculate that patches only seemed to work when people ironed their clothes after every wash thus resealing any loosened edges of the patch. Ironing seems like a boring and unnecessary task. Was it ever really necessary? Were the wrinkles so pronounced and the creases so valuable? Creases don’t seem important today. Was ironing something people were programmed to do? It’s unfortunate there’s no time travel movie in which a modern character goes back to the ‘50s and notices that all the clothes are ironed. It’s odd that a revolutionary feminist movement began about the same time that women and society in general were questioning the value of ironing.

Do you have any thougths on education?

We should try to teach deer to swim. If we could teach a few deer perhaps the habit would spread through the entire species. The advantage would be that, if the deer stayed in the water for a significant length of time, the deer ticks they carry, which cause Lyme disease, would perish. From an evolutionary perspective deer may have avoided lakes and streams because when they were climbing the slippery banks they would be vulnerable to predators. Since for the most part man is the only predator of deer today, if we couldn’t teach deer to swim, there doesn’t seem to be a downside to genetically modifying deer so they would want to swim. Perhaps a group of swimming deer would discourage the Canadian geese from using our lakes.

Where will you end up if you take a random drive consisting of alternate right and left turns?

There is no specific ending place, but you will probably encounter places that you have unwittingly avoided in your normal travels. You may encounter a localized low income neighborhood. It is not that you consciously avoided this neighborhood or these kinds of neighborhoods, but in your local travels you were more concerned going to stores, movies, restaurants and other places not located near low income neighborhoods.

Do flying saucers have brakes now?

Apparently they have a dual breaking system that is spring loaded. There are two plastic handles at opposite points on the circumference of the saucer. When the hinged handles are raised the bottom of each handle, which is a three inch plastic wedge, is lowered into the snow. When the handles are released a spring withdraws the wedge from the snow. And much like their science fiction counterparts these new saucers are made of a plastic that proved indestructible under repeated blows from my sledge hammer. I was breaking the saucer apart so it would easily fit into my garbage can. The repeated blows of my sledge hammer only served to make a strikingly decorative design appear on the plastic. A decorative design would probably have appeared on the flying saucers that were been subjected to intense investigation in science fiction novels if the authors had had a more literary bent.

Often the entire cumulus cloud is bright white while other times they are bright white except for their bottom layer, which is grey. Why is this?

I’m not sure but I think the bottom grey layer occurs when there are many cumulus clouds in the sky. When there are many cumulus clouds in the sky, the sun will often be covered by a large cloud; the shadow of the cloud will cover a locally large swath of the earth. If you climb to the top of a moderately high hill, and view the surrounding landscape, you may see the surrounding countryside dappled with cloud shadows. When the direct sunlight is cut off, clouds are illuminated by the blue light of the sky instead of the bright yellow of the sun. As the blue light of the sky passes through the upper layers of a cumulus cloud it diffuses. When the blue light of the sky reaches the base of the cloud, it has diffused to such an extent that the base of the cloud appears grey. The problem with this argument is the following: what do I mean when I say the direct sunlight is cut off? If the sun is overhead, can a cloud cut off direct sunlight to the surrounding clouds? The argument makes more sense when the sun is lower in the sky. If the sun is sinking in the west and is covered by a cloud, this could cut off direct sunlight to a great many clouds in the eastern portion of the sky.

What's the relation between the "Curse of the Bambino" and global warming?

It’s interesting that thoughtful people, who believe in manmade global warming, are not in the least troubled by some well known aberrations to which no unnatural cause is ascribed. Two that come to mind are: the failure of the Chicago Cubs to win a World Series championship since 1908 and the failure of the Boston Red Sox to do the same for 86 years from 1818 to 2004. It would seem that a strong argument can be made that there is no possible natural explanation for the failure of these two teams to win a World Championship in 104 years and 86 years, respectively. In fact, it may be possible to make a stronger argument that the World Series drought suffered by both these teams has no natural explanation than to argue that there is no natural explanation for global warming.

An operational description of the game of professional baseball, as system with many interlocking levels organization, lays the groundwork for forming the opinion that there is no natural explanation for such extended World Series championships droughts. One level of professional baseball’s organization that we may overlook is the simple fact that professional baseball has rules that are stringently and fairly enforced. Recently, professional baseball introduced the use of the instant reply to review and change incorrect calls made by the umpires. In 2014 season, the Miami Marlins introduced a new sculpture into their stadium. When in the beginning of the season, they won more games at home than on the road there was speculation that the sculpture had a hidden camera in it that allowed the Marlins to steal signals from the visiting teams. This is evidence of the kind of regulation that is possible of the professional baseball system. The concept of Money Ball is another indication of the way that any advantage that one team may possess can be negated by another team employing a more comprehensive strategy.

The fact that professional baseball can draft players from a myriad of colleges and high schools ensures that there is a super abundance of talented players for the teams to choose from. There are a great many reasons to suspect that professional baseball teams are evenly matched.

Many people find the game of professional baseball somewhat boring. By listening to and analyzing the commentary of the baseball announcers one can build up a large stock of information that indicates long World Series droughts are unnatural. Appreciating the game through this perspective may make it more interesting.

An operational description of the game of professional baseball, as system with many interlocking levels organization, lays the groundwork for forming the opinion that there is no natural explanation for such extended World Series championships droughts. One level of professional baseball’s organization that we may overlook is the simple fact that professional baseball has rules that are stringently and fairly enforced. Recently, professional baseball introduced the use of the instant reply to review and change incorrect calls made by the umpires. In 2014 season, the Miami Marlins introduced a new sculpture into their stadium. When in the beginning of the season, they won more games at home than on the road there was speculation that the sculpture had a hidden camera in it that allowed the Marlins to steal signals from the visiting teams. This is evidence of the kind of regulation that is possible of the professional baseball system. The concept of Money Ball is another indication of the way that any advantage that one team may possess can be negated by another team employing a more comprehensive strategy.

The fact that professional baseball can draft players from a myriad of colleges and high schools ensures that there is a super abundance of talented players for the teams to choose from. There are a great many reasons to suspect that professional baseball teams are evenly matched.

Many people find the game of professional baseball somewhat boring. By listening to and analyzing the commentary of the baseball announcers one can build up a large stock of information that indicates long World Series droughts are unnatural. Appreciating the game through this perspective may make it more interesting.

Why do you say, "Nothing happens when the wind blows"?

It’s a strange idea I have about insects and small animals. I note that humans enjoy and are most active on days when there is a vigorous breeze. A strong, steady breeze keeps away annoying insects, and it keeps you cool if you’re laboring. Does it have the same invigorating effect on insects and small animals? A strong steady breeze may make it impossible for a mosquito to find nourishment. Such a breeze may be detrimental to insects, small birds and other small animals. They may be inclined to seek shelter from a strong breeze. If they are secreting themselves, all the other animals that prey on them would find no easy prey to feed on and so they might retire from activity as well. This trend could continue along the food chain.

Does the Riemann-Christoffel tensor distinguish between curved space and flat space?

Einstein’s law of gravity takes the form of a tensor equation, G_st=0, where s and t are subscripts of G. It is derived from another tensor equation in which the Riemann-Christoffel curvature tensor B_stp^a is set equal to zero, where s, t and p are subscripts of B and a is a superscript of B. Through the tensor calculus operation known as contraction, the curvature tensor is transformed into Einstein’s law of gravity. It would give Einstein’s law of gravity added merit if the curvature tensor could tell us something profound about the geometry of space. Lillian Lieber in her book "The Einstein Theory of Relativity" claims that the curvature tensor does just that. She writes, “Given a Euclidean space the curvature tensor will be zero, whatever coordinate system is used, and conversely, given this tensor equal to zero, then we know that the space must be Euclidean.” I have investigated this claim, and I cannot falsify it. The only question I can raise is about the validity of the tensor calculus operation known as contraction. This operation is used throughout tensor calculus, but it does appear to be based on a misuse of a technique used to find derivatives known as the chain rule.

Is Einstein's gravity equation, Gst =0 (where s and t are subscripts) misleading?

It may be possible to produce a version of the Einstein tensor Gst (where s and t are subscripts), which is also known as the Ricci tensor, that is equal to zero. But, it does not seem correct to automatically set the Christoffel symbols and the partial derivatives of the Christoffel symbols that make up the tensor equal to zero. Lillian Lieber writes in her book The Einstein Theory of Relativity, “Since Gst =0 does not necessarily imply that the B’s [components of the Riemann Christoffel curvature tensor] are zero hence Gst =0 can be true even if the space is non Euclidean.” “Gst =0 can be true even if the space is non Euclidean” is the important phrase. It can be true, but it is not automatically true. It seems that the coefficients that Einstein chooses for his version of a non Euclidean, four dimensional distance equation would make it difficult for Gst to equal zero because the coefficients consist of two variables x1 and x2, (where 1 and 2 are subscripts) and the variable x2 is used only once which would make it difficult to cancel out the variable with its negative counterpart.

What previous answer has proven to be incorrect as these preposterous self-interview questions drag on?

The answer to the question about the Riemann curvature tensor appears to be in error; at the time I didn’t understand how to correctly calculate the Riemann curvature tensor. Now, that I more fully understand how to work with the calculations that produce the Riemann curvature tensor, I believe that if we calculate the Riemann curvature for Einstein’s version of ds as given in his theory of general relativity and then contract Riemann curvature tension to produce Einstein’s equation for his law of gravity we will not obtain his law of gravity, which is Gst=0 (where the subscript s could be replaced by the Greek letter sigma and the subscript t could be replaced with the Greek letter tau to make the equation more formal). Instead I believe we would obtain Gst= some value other than zero. Einstein’s value for ds as employed in his theory of general relativity can be found in "The Einstein Theory of Relativity" by Lillian R. Lieber. The methods for calculating the Riemann curvature tensor and contracting the Riemann curvature tensor can be found in "Schaums Outline of Tensor Calculus" by David Kay. The calculations would seem to be quite lengthy in nature.

What are your thoughts on the various intake duct filters sold for forced hot air furnaces?

I purchased the expensive, high tech, microbial trapping filters. I could never tell when they were so clogged that it was time to replace them So as an experiment, I didn't replace the filters for a long time. One cold, winter morning my forced, hot-air furnace came on and its run seemed to extend for 3 or 4 times longer than normal. Then quite soon after it had shut off, it came on again. I knew something was amiss. It was highly likely that the filters were so clogged they were preventing an adequate amount of air to come through them. I replaced the clogged filters with the inexpensive, low tech filters. The furnace returned to its nominal status. Here I use the word "nominal" as used in Ron Howard's movie "Apollo 13," which I suspect is an accurate reflection of NASA terminology. Thus, one of NASA's greatest accomplishments may be clouding the meaning of the word "nominal." Perhaps "nominal" is short for "nominal fluctuations," but then it would seem more accurate to say "normal fluctuations."

What is it about Taoism that you find interesting?

There seems to be some sort of enigmatic force and mysterious understanding that pervades Taoism.

How do you scare a snake?

In the morning, take a walk in the woods and sit under a tree. Wait until you hear the sound of a large, wood boring bumble bee drying its wings of the rain that fell during the night. If you recall that no rain fell during the night, you are probably hearing the sound of a snake tasting the air with its tongue. Snakes seem to sense heat radiation with their tongues. If you see the snake, hide behind a large tree; this will disguise your heat signature. Next, you should move out from your hiding place for a short amount of time, then move back. Do this a number of times. You can consider moving behind another large tree that is near by and repeating your actions, or you could move from large tree to large tree circling around the snake. These actions should scare the snake. It may head back to its home, which could be a small hole in an old stone wall. The stone wall may have been made by farmers generations ago. You could tap the snake on its tail as it disappears into its hole.

Do you have any rules of thumb regarding newspapers, magazines and books?

In the past, I regarded newspapers and magazines as unreliable sources accurate information. I found that books offered more accurate information with the caveat that many books were no more accurate than newspapers and magazines.

The elemental and elementary mathematical concept of apples and oranges is both elegant and obvious; its meaning being that only like quantities can be added together, subtracted from each other, multiplied together or divided into one another.

You can be counted on to somehow muddy these waters. Right?

I don’t know if I like the tone of that question. Lillian Lieber writes in “The Einstein Theory of Relativity,” on page 315, the following: “. . . the ‘dimensionality’ of velocity is Length / Time; the ‘dimensionality’ of acceleration is Length / Time Squared . . .” If we say the acceleration of an airplane is 3 miles per minute per minute, we can rewrite it as 3 miles / minute / minute. We treat this quantity as a complex fraction that is similar to the complex fraction 100 / 5 / 4. This complex fraction we rewrite as (100 / 5) x (1 / 4) using the principle that to divide a complex fraction we invert the denominator and then multiply the numerator times the inverted denominator. We notice that we have rewritten the denominator of the complex fraction as 4 / 1 instead of merely 4. This is acceptable since we are dividing by one, which leaves the value 4 unchanged. We are also dividing like by like (a number by another number). When we perform a similar operation with the acceleration 3 miles / minute / minute, we are not dividing like by like. We are dividing a number, 1, by the unit of time measure, minutes.

We rewrite 3 miles / minute / minute as (3 miles / minute) x (1 / minute) and obtain 3 miles / minute squared. But, is it correct to rewrite the denominator of the “complex fraction,” which is minutes, as minutes divided by one with one being a number and not a minute? Is it correct to denote the term 3 miles / minute / minute as a complex fraction?

I don’t know if I like the tone of that question. Lillian Lieber writes in “The Einstein Theory of Relativity,” on page 315, the following: “. . . the ‘dimensionality’ of velocity is Length / Time; the ‘dimensionality’ of acceleration is Length / Time Squared . . .” If we say the acceleration of an airplane is 3 miles per minute per minute, we can rewrite it as 3 miles / minute / minute. We treat this quantity as a complex fraction that is similar to the complex fraction 100 / 5 / 4. This complex fraction we rewrite as (100 / 5) x (1 / 4) using the principle that to divide a complex fraction we invert the denominator and then multiply the numerator times the inverted denominator. We notice that we have rewritten the denominator of the complex fraction as 4 / 1 instead of merely 4. This is acceptable since we are dividing by one, which leaves the value 4 unchanged. We are also dividing like by like (a number by another number). When we perform a similar operation with the acceleration 3 miles / minute / minute, we are not dividing like by like. We are dividing a number, 1, by the unit of time measure, minutes.

We rewrite 3 miles / minute / minute as (3 miles / minute) x (1 / minute) and obtain 3 miles / minute squared. But, is it correct to rewrite the denominator of the “complex fraction,” which is minutes, as minutes divided by one with one being a number and not a minute? Is it correct to denote the term 3 miles / minute / minute as a complex fraction?

You made an error in your last question: 3 apples can be multiplied by the number 3 and 3 oranges can be divided by the number 6, but your notions about complex fractions were intriguing..

In the "Mathematics Dictionary" 5th ed. by Glenn James and Robert James part of the definition for the term fraction reads, "A simple fraction (or common fraction or vulgar fraction) is a fraction whose numerator and denominator are both integers, as contrasted to a complex fraction which has a fraction for the numerator or denominator or both." Note that it doesn't say a complex fraction has a simple fraction for the numerator . . . , but merely a fraction. In the "Mathematics Dictionary" the broad definition for fraction is given as an indicated quotient of two quantities so the prohibition of dividing apples into oranges could prohibit the division of minutes into miles, hence 3 miles / minute could not be considered a fraction.

The question arises have we been lied to with impunity for decades by intellectuals or are some of us making mountains out of mole hills?

I don’t know. Lillian Lieber in “The Einstein Theory of Relativity,” makes the following statement, “any high school student knows that if x represents the length of an arc, and θ is the number of radians in it, then x / θ = 2π r / 2π . . .” Why does she say, “any high school student knows” ? Upon reflection, it seems there is no way the statement could be true either now or in the 1940s when the book was written. Is Lieber exercising her ability to inveigle at a fundamental level or is it an example of something less sinister?

Can you give another example of disingenuousness by an intellectual?

Somewhere in the Norton Critical Edition of "The Waste Land" by T. S. Eliot a critic writes that a certain line in the poem is a listing of the poet's personal synecdoches in order of increasing terror. I do not believe there can be personal synecdoches since according to "Webster's New World College Dictionary" a synecdoche is a figure of speech in which a part is used as a whole, an individual for a class, a material for a thing, or the reverse of any of these. (Ex: bread for food, the army for a soldier, or a copper for a penny). For a synecdoche to be understood by a reader there must be a widely known and accepted notion such as the notion that the words "pig skin" can be used for the word football.

How do you characterize the deleterious effects of the crumb-making ability of young children who are allowed to eat crackers in the car?

It is worse than the allied bombing of Germany during W.W.II. The only respite is calculating how long you could survive on the treasure trove of cracker crumbs secreted in every crevice of the car.

Can you make a mean spirited observation?

Appreciating nature --the plants and wildlife-- is like being surrounded by people who refuse to use a toilet when defecating and convincing yourself overtime and through compartmentalization that it is beautiful.

Your on a roll, make another such observation.

Because no one questions whether Kepler's laws can actually be derived from Newtonian mechanics and because no one questions quantum mechanics, I'm willing to entertain the notion that physics is a kind of intellectual blather.

You like Fig Newtons, but you are skeptical of the Newtonian derivation of Kepler's three laws of planetary motion. This is a singular confirmation of the saying, "a cobbler should stick to his last."

If you peruse the chapter/section 12.5 of "Calculus with Analytic Geometry" 4th ed. by C.H. Edwards, Jr. and David E. Penny entitled "Orbits of Planets and Satellites," you will find that their derivation of Kepler's three laws from Newtonian mechanics is suspect. I believe other derivations of the three laws from Newtonian mechanics are suspect, as well. We read in the sentence preceding Eq. (7), "We drop the factor 1/r . . . ." It cannot be correct to drop the factor 1/r especially since r=r(t), and therefore is an important function. This invalidates the derivation of Kepler's second law. Since Eq. (7), which is obtained by dropping the factor 1/r, is used in the derivation of Kepler's first and third laws, their derivations are invalid, as well.

There is also some confusing use of cos(theta) as both a constant and a variable in conjunction with unit vectors in the derivation of Kepler's second law. We are told that we must choose the coordinate axes so that at time t=0 the planet is on the polar axis and is at its closest approach to the sun. Under these conditions we are told that two unit vectors are equal: unit vector u(subscript theta)= unit vector j. While these initial conditions are still presumably in force, we are told that the dot product of: unit vector u(subscript theta) (dot) unit vector j equals cos(theta), and not one. It is true that under these initial conditions the dot product of the unit vectors equals cos(theta), but under these initial conditions cos(theta) is not a variable as is required by the derivation, instead it is a constant. Theta is equal to an angle of zero degrees. The cos(theta) under these initial conditions equals one. The dot product of the two unit vectors equals one under all conditions, and it does not equal cos(theta) as a variable as required by the derivation under any conditions.

In the derivation of Kepler's third law, Eq. (16) is confusing. The left side of the equation comes from the equation for an ellipse that uses polar coordinates. The ellipse has only one focus and a directrix p. The focus is at the pole of the polar coordinate system. The term (major semiaxis a) on the right side of the equation comes from the Cartesian coordinate equation of an ellipse in which there are two foci. The origin of the Cartesian coordinate system does not coincide with the pole of the polar coordinates. However, the equation is correct.

There is also some confusing use of cos(theta) as both a constant and a variable in conjunction with unit vectors in the derivation of Kepler's second law. We are told that we must choose the coordinate axes so that at time t=0 the planet is on the polar axis and is at its closest approach to the sun. Under these conditions we are told that two unit vectors are equal: unit vector u(subscript theta)= unit vector j. While these initial conditions are still presumably in force, we are told that the dot product of: unit vector u(subscript theta) (dot) unit vector j equals cos(theta), and not one. It is true that under these initial conditions the dot product of the unit vectors equals cos(theta), but under these initial conditions cos(theta) is not a variable as is required by the derivation, instead it is a constant. Theta is equal to an angle of zero degrees. The cos(theta) under these initial conditions equals one. The dot product of the two unit vectors equals one under all conditions, and it does not equal cos(theta) as a variable as required by the derivation under any conditions.

In the derivation of Kepler's third law, Eq. (16) is confusing. The left side of the equation comes from the equation for an ellipse that uses polar coordinates. The ellipse has only one focus and a directrix p. The focus is at the pole of the polar coordinate system. The term (major semiaxis a) on the right side of the equation comes from the Cartesian coordinate equation of an ellipse in which there are two foci. The origin of the Cartesian coordinate system does not coincide with the pole of the polar coordinates. However, the equation is correct.

Why does Wikipedia state that absolute zero is the temperature at which molecular motion ceases?

Perhaps, because if Wikipedia stated that absolute zero was the temperature at which the vibrational motion of atoms ceases, it might possibly allow the reader to develop the notion that atomic clocks slow as the temperature approaches absolute zero. In Einstein's view, clocks do more than measure the passage of an absolute time (with varying degrees of accuracy), and instead they define time. This leads to an interesting possibility, if atomic vibrational motion slows as the temperature approaches absolute zero. We could imagine a thought experiment that employs a series of many atomic clocks. Each atomic clock is isolated from the others in its own temperature-controlled chamber. As the temperature decreased in the long row of atomic clocks to near absolute zero, we could see the flow of time gradually slow and come to a near standstill as the temperature in an isolated chamber approached absolute zero. Would Einstein adopt the view that time was flowing at a different rate in each of the isolated chambers? Or would he adopt the more common sense notion that the environment affects all measuring devices and clocks are merely a kind of measuring device. Since this is a thought experiment, perhaps we should add a second atomic clock to each isolated chamber. We can refer to the second clock as clock B. The series of B clocks could be modified in some way to precisely compensate for the lowered temperature. This would give us a long row of B clocks that all told the same time. You would then have a series of isolated chambers in which the flow of time was both gradually decreasing as measured by one series of clocks and flowing at a constant rate as measured by another series of clocks.

How do other text books present the derivation of Kepler's three laws of planetary motion?

Calculus and Analytic Geometry by George B. Thomas and Ross L. Finney provides the reader with a complex derivation of Kepler’s three laws of planetary motion. In chapter 14 “Vector Functions and Motion” we find section 14.4 “Planetary Motion and Satellites,” where they write, “We introduce polar coordinates in this plane and use polar coordinate equations to derive Kepler’s three laws of planetary motion. The derivation, while lengthy in comparison to what we’ve seen so far, is one of the triumphs of vector calculus and the calculations in polar coordinates are typical of work done in orbital mechanics today.” The derivation depends heavily on the cross multiplication of vectors. Cross multiplication of two vectors A and B gives us vector C whose length is the product of the lengths of A and B and the sine of the angle between them (the angle from the first to the second). Vector C is perpendicular to the plane of the given vectors and directed so that the three vectors in order A, B, C form a positively oriented trihedral. This leads to the interesting conclusion that when the vector representing the velocity of the earth is cross multiplied by itself the product is zero. This is so because the angle between the two vectors is zero degrees and the sine of zero degrees is zero. The mathematical fact that cross multiplication of two vectors in which the angle between them is zero gives us a product of zero is used several times in the derivation of Kepler’s three laws. It is a mathematical fact, but is it reasonable? It would seem to indicate that if one man were pulling a heavy cart by means of a rope and another man joined him in his effort by tugging on the same rope the cross product of the vectors representing their efforts would be zero. Also, it would seem to indicate that if one large mass were attracting a distant and much smaller mass and then another large mass was introduced beside the original large mass (such that the two large masses and the much smaller distant mass formed a straight line), the cross multiplication of the vectors representing the attractive force of the two large masses would give us a product of zero. It seems cross multiplication of vectors is introduced into the derivation so that the product of vector forces can be set to zero although it is unlikely that this is a realistic outcome.

Have people learned anything from smoking marijuana?

Is society different in anyway that is attributable to many people smoking marijuana? I think one of the first notions you would learn from smoking marijuana is that taste and smell are subjective. When you first start smoking marijuana the taste, smell and sensation of drawing dense smoke into your lungs is unpleasant. But as you learn to appreciate the effects of marijuana the taste, smell and sensation become quite enjoyable. Strangely, I don't think the notion that taste and smell are subjective has entered society in any significant way. The reason for this is unclear, except perhaps for the difficulty people have in dealing with abstract concepts.

Is there a section of road in the outer suburbs where horns honk constantly?

Yes, I came across it putting up lawn signs for Mayor Steve Lonegan. It is off Route 23 in the area where Route 23 boarders the Newark reservoir. There is a curved single lane tunnel on a back road. A car will come to a stop before the entrance to the tunnel and honk its horn to warn oncoming cars that it is entering the tunnel. Every car does this.

Do you have any further thoughts on marijuana?

Another notion marijuana smokers could have learned is the value of frugality. It takes very little marijuana to get high. Often smokers will continue smoking their expensive marijuana well after the point of greatly diminished returns has been passed. This is another example of the difficulty in dealing with abstract concepts. Interestingly, I often read about teenagers and young adults who are stopped by the police for some driving infraction. The police smell the odor of burning marijuana, and they use this as a reason to increase their scrutiny of the occupants of the car, which often leads to an arrest. A possible solution to this problem would be the use of a strongly scented essential oil such as lavender to mask the odor of marijuana. The failure of marijuana smokers to refrain from driving while high and the failure of marijuana smokers to disguise the fact that they are getting high while driving represents more difficulties in dealing with abstract concpts. The failure of marijuana smokers to realize that the movie "Reefer Madness" presents valid insights into problems surrounding smoking marijuana is another example.

When using a handsaw to cut a wooden plank, is it possible to make the cut along the vertical axis precisely perpendicular to the horizontal plane?

I don't know, for years making a precisely vertical cut has stymied me. I think it would be possible if your sawhorses were precisely level with each other and precisely level to a flat section of the ground. If those requirements were met and you pushed and pulled the handsaw quite lightly, the vertical cut might be perpendicular to the horizontal surface of the board. The idea is hat you are letting gravity pull the saw blade down along a line perpendicular to the surface of the board.

How is a portion of Godel's first incompleteness theorem like the ingredients list of Stove Top Stuffing Mix?

In subsection 2.4 Expressing metamathematical concepts, function 32 of his first incompleteness theorem he skillfully uses comas and parentheses to make certain terms seem to mean notions that they do not actually mean. Stove Top Stuffing uses two kinds of parentheses and the attendant comas to make it seem high fructose corn syrup is the eighth ingredient in terms of the amount present in the stuffing mix while actually it is the second.

Any further thoughts on the Newtonian derivation of Kepler's three laws?

There seems to be a problem with the Newtonian derivation of Kepler’s three laws as presented in various textbooks of calculus and analytic geometry such as Calculus with Analytic Geometry 4th ed. by Earl W. Swokowski. The problem arises with the derivative of the cross product of certain vector functions. The vector functions in question are the vector functions that 1.describe the path of the earth (or any planet) and 2.describe the velocity of the earth (or any planet). The derivation of the vector cross product of these two vector functions is zero. For instance, in section 15.6 “Kepler’s Laws” of his textbook Swokowski writes, “ d/dt (r x v)=0.” The mathematical operation follows the rules for vector functions, but for all other types of functions the derivative of a constant is zero not the derivative of the product of two functions with variables. The path of the earth is a variable since the distance from the earth to the sun varies. The earth is actually closer to the sun during winter (in the northern hemisphere) than during the summer. Kepler’s second law which states that the vector from the sun to a moving planet sweeps out area at a constant rate coupled with the fact that the orbit of the earth is an ellipse indicates the velocity of the earth is a variable. How can the derivation of the product of two variables equal zero? In Cartesian coordinates you could have an equation y=x2(x-1)½. (The superscripts have been lowered unfortunately.) The derivation of the product of the two variables x2 (x squared) and (x-1)½ (the square root of (x-1)) is dy/dx = 2x (x-1)½ + x2(x-1)-½ . It is not zero. This may explain why vector functions are merged into polar coordinates without there ever being formal transformation equations presented. If formal transformation equations were given, the vector functions could be transformed into polar coordinates before the derivative was taken, and thus, perhaps when the derivative was taken it wouldn’t equal zero.

Do bread crumbs in a toaster behave as a recursive function?

Employing a minimum of sophistry, we could argue that with every slice of bread that is toasted more bread crumbs are deposited on the tray, which is located at the bottom of the toaster. That is at least until the toaster is cleaned. If we could make the argument that the increasing number of crumbs deposited on the crumb catching tray of the toaster somehow influenced the number of bread crumbs produced by each slice of toast, it seems we could describe the behavior of the bread crumbs by a recursive function. We could assume that the bread crumbs deposited on the tray act as a layer of insulation, and thus they trap heat around the slices of toast. We might also assume the layer of crumbs acting as insulation doesn't trap a significant amount of heat around the bi-metal device that regulates when the slice or slices of toast are done. We could assume that each slice of toast is now being exposed to a greater amount of heat while the duration of the exposure remains constant. This exposure to increasing heat could cause more bread crumbs to form, and thus we may have the elements necessary for the behavior of the bread crumbs to be defined by a recursive function.

What is your view of the Wikipedia entry for the Michelson-Morley experiment?

Studying the Wikipedia article is instructive. At first glance, the description of the experiment is quite plausible. All the equations are mathematically sound.

There are two typos in the section entitled “1881 and 1887 experiments.” In the 2nd paragraph of that section, “.04 fringes” is written instead of “.40 fringes,” and in the 3rd paragraph “.02 fringes” is written instead of “.20 fringes.”

If you ponder the section “Light path analysis and consequences,” a curious thought may occur to you. It seems the beam of light perpendicular to the earth’s motion is completely dragged by the aether, and the aether is in turn completely dragged by the earth's motion. In contrast, the beam of light parallel to the earth’s motion travels through an apparent aether wind generated by the earth's motion through a stationary aether. An explanation of the complete dragging of the aether by the earth is called for since the Michelson-Morley experiment assumes a stationary aether. The explanation for the “inclined travel path” of the beam of light (i.e. the forward motion of the light beam as though it is being dragged by the aether) perpendicular to the earth’s motion is given in the section of the article mentioned above. The article states, “This inclined travel path follows from the transformation from the interferometer rest frame to the aether rest frame.” So, the article doesn’t actually claim the aether is completely dragged by the earth with regard to the light beam perpendicular to the earth’s motion.

In the sub-section “Mirror reflection” of the section mentioned above, we find this statement, “For an apparatus in motion, the classical analysis requires that the beam splitting mirror be slightly offset from an exact 45 degrees if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed.” Offsetting the mirror from 45 degrees would alter subtly the constant L. But not where the constant L represents the length of the arm of the interferometer perpendicular to the aether wind. It would alter L when L represents a hypothetical vector. This hypothetical vector L is the path the light beam would travel if it were at rest with respect to the aether in other words if there was no aether wind. This hypothetical vector L is the vector used in the calculations to determine the duration of the light path perpendicular to the earth’s motion. This vector is added to the vector that represents the earth's motion to produce the “inclined travel path” of the beam of light perpendicular to the earth’s motion. Offsetting the beam splitting mirror would seem to lengthen the hypothetical path the light beam travels in the arm of the interferometer before it is acted upon by either the earth’s motion (apparent aether wind?) or the transformation to the rest state of the aether. The parallelogram of forces rule could still be used to add hypothetical vector L to the vector that represents the earth’s motion to obtain the “inclined travel path.” But, the Pythagorean Theorem could no longer be used to add the vectors together because the vectors would no longer be at right angles to each other since the beam splitting mirror has been offset from an exact 45 degrees. When the beam splitting mirror is at exactly 45 degrees the hypothetical beam of light (vector L) is perpendicular to the velocity of the earth.

The sub-section “Mirror reflection” continues with this statement, “In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely the amount necessary to compensate for the angle discrepancy of the two beams.” This is difficult to visualize. Let us imagine some object traveling at a very great velocity, many times greater than the velocity of the earth. Can we imagine that the contraction of the horizontal component of the beam splitting mirror will change the angle of the mirror by 20 degrees, by 30 degrees or perhaps more than 30 degrees?

Interestingly, assuming that both beams of light are dragged along by the aether that is in turn dragged along by the earth doesn’t alter the final outcome of the experiment because the experimental results are around 40 times smaller than the theoretically expected results. The theoretically expected result is a fringe shift of .44 fringes and the experimental result is an average fringe shift of .01 fringes. My calculation of the theoretical result obtained when both beams of light are dragged along by the aether which is in turn completely dragged along by the earth is a fringe shift of about .20 fringes, which is still 20 times larger than the experimental result.

There appears to be a problem with the theoretical calculation of the fringe shift. In theory the amount of fringe shift is a ratio. The numerator (top) of the ratio is the maximum total difference in the path lengths between the light beam that is perpendicular to the earth’s (continued)

There are two typos in the section entitled “1881 and 1887 experiments.” In the 2nd paragraph of that section, “.04 fringes” is written instead of “.40 fringes,” and in the 3rd paragraph “.02 fringes” is written instead of “.20 fringes.”

If you ponder the section “Light path analysis and consequences,” a curious thought may occur to you. It seems the beam of light perpendicular to the earth’s motion is completely dragged by the aether, and the aether is in turn completely dragged by the earth's motion. In contrast, the beam of light parallel to the earth’s motion travels through an apparent aether wind generated by the earth's motion through a stationary aether. An explanation of the complete dragging of the aether by the earth is called for since the Michelson-Morley experiment assumes a stationary aether. The explanation for the “inclined travel path” of the beam of light (i.e. the forward motion of the light beam as though it is being dragged by the aether) perpendicular to the earth’s motion is given in the section of the article mentioned above. The article states, “This inclined travel path follows from the transformation from the interferometer rest frame to the aether rest frame.” So, the article doesn’t actually claim the aether is completely dragged by the earth with regard to the light beam perpendicular to the earth’s motion.

In the sub-section “Mirror reflection” of the section mentioned above, we find this statement, “For an apparatus in motion, the classical analysis requires that the beam splitting mirror be slightly offset from an exact 45 degrees if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed.” Offsetting the mirror from 45 degrees would alter subtly the constant L. But not where the constant L represents the length of the arm of the interferometer perpendicular to the aether wind. It would alter L when L represents a hypothetical vector. This hypothetical vector L is the path the light beam would travel if it were at rest with respect to the aether in other words if there was no aether wind. This hypothetical vector L is the vector used in the calculations to determine the duration of the light path perpendicular to the earth’s motion. This vector is added to the vector that represents the earth's motion to produce the “inclined travel path” of the beam of light perpendicular to the earth’s motion. Offsetting the beam splitting mirror would seem to lengthen the hypothetical path the light beam travels in the arm of the interferometer before it is acted upon by either the earth’s motion (apparent aether wind?) or the transformation to the rest state of the aether. The parallelogram of forces rule could still be used to add hypothetical vector L to the vector that represents the earth’s motion to obtain the “inclined travel path.” But, the Pythagorean Theorem could no longer be used to add the vectors together because the vectors would no longer be at right angles to each other since the beam splitting mirror has been offset from an exact 45 degrees. When the beam splitting mirror is at exactly 45 degrees the hypothetical beam of light (vector L) is perpendicular to the velocity of the earth.

The sub-section “Mirror reflection” continues with this statement, “In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely the amount necessary to compensate for the angle discrepancy of the two beams.” This is difficult to visualize. Let us imagine some object traveling at a very great velocity, many times greater than the velocity of the earth. Can we imagine that the contraction of the horizontal component of the beam splitting mirror will change the angle of the mirror by 20 degrees, by 30 degrees or perhaps more than 30 degrees?

Interestingly, assuming that both beams of light are dragged along by the aether that is in turn dragged along by the earth doesn’t alter the final outcome of the experiment because the experimental results are around 40 times smaller than the theoretically expected results. The theoretically expected result is a fringe shift of .44 fringes and the experimental result is an average fringe shift of .01 fringes. My calculation of the theoretical result obtained when both beams of light are dragged along by the aether which is in turn completely dragged along by the earth is a fringe shift of about .20 fringes, which is still 20 times larger than the experimental result.

There appears to be a problem with the theoretical calculation of the fringe shift. In theory the amount of fringe shift is a ratio. The numerator (top) of the ratio is the maximum total difference in the path lengths between the light beam that is perpendicular to the earth’s (continued)

What is your view of the Wikipedia entry for the Michelson-Morley experiment? (continued)

motion and the light beam that is parallel to the earth’s motion. The denominator (bottom) of the ratio is a particular wavelength of visible light namely 500 nanometers. Since the light source used in the experiment is white light whose wavelengths range from 400 nanometers to 700 nanometers, how can the choice of one particular wavelength of visible light be justified? Why not divide maximum total difference in path lengths by 600 nanometers or 700 nanometers? Why does dividing the maximum total difference in light path lengths by that particular wavelength give us the amount of fringe shift?

The maximum, total, path length, difference between the two light beams can be calculated. The maximum length difference is only 220 nanometers. Light waves can interfere with each other in two ways. (1) Where the crest of one wave meets the crest of another wave or where the trough of one wave meets the trough of another, the two waves combine and form a bright spot of light. (2) Where a crest meets a trough, the two waves cancel, leaving a dark spot. With a maximum path length difference of only 220 nanometers, it seems only light beams with a wavelength in the range of 440 nanometers could interfere with the cancelling type of interference (trough meets crest type of interference). The cancelling interference would occurr because the light beams would be ½ a wavelength out of phase. To obtain the reinforcing type of interference (crest meets crest or trough meets trough interference) would we need wavelengths in the ultra-violet spectrum in the range of 220 nanometers?

Another startling claim regards the interference fringes themselves. The interference fringes themselves change or shift under the influence of the changing of the total difference in the path length between the light beam that is perpendicular to the earth’s motion and the light beam that is parallel to the motion of the earth. The total difference changes between a maximum and a minimum. While the interference patterns change, the article maintains that a part of the interference pattern does not change. A bright center fringe in the interference pattern doesn’t change and is used as the zero point from which the changes in the surrounding fringes of the changing interference patterns can be measured. The article provides us with a quotation from Dayton Miller who writes, “White light fringes were chosen for the observations because they consist of a small group of fringes having a central, sharply defined black fringe which forms a permanent zero reference mark for all the readings.” How is it that the center fringe of the changing interference patterns doesn’t change while the surrounding fringes do change?

It is difficult to understand a portion of the lengthy caption for figure 5. The difficult portion reads, “c is a compensating plate so that both the reflected and transmitted beams travel through the same amount of glass (important since experiments were run with white light which has an extremely short coherence length requiring precise matching of optical path lengths for fringes to be visible; monochromatic sodium light was used only for initial alignment)”

This statement is confusing. At first glance, compensating glass plate doesn’t appear necessary. First, we must acknowlegde that both the transmitted and reflected beam travel through 30 widths of glass as they make their along the light path dictated by the 4 mirrors located at each end of each arm of the interferometer. Each arm has a total of 8 mirrors. There are 4 mirrors at each end of each arm. On the outward journey, a light beam passes through 16 widths of glass as it is reflected from the mirrors located at the ends of each arm. On the return journey a light beam passes through 14 widths of glass as it is reflected from the mirrors located at the ends of each arm. For the notion of the need for a compensating glass to make sense we must acknowledge that the beam of light from the source is not split into two beams (the transmitted and the reflected) until it reaches the backside of the beam splitter b. I assume beam splitter b is a half-silvered mirror. The transmitted beam isn’t formed until the precise moment that it leaves the beam splitter b so at that point it has traveled through no widths of glass. The reflected beam by virtue of being reflected travels through one width of glass at the beginning of its journey. It is the un-split beam that enters the beam splitter b and travels to the very back of beam splitter b, a half-silvered mirror. The returning reflected beam passes through the beam splitter b as it heads toward the telescope. So it passes through one width of glass at the beginning of its journey plus 30 widths of glass as it makes its way up and down the arm of the interferometer plus one final width of glass as it passes through the half-silvered mirror as it heads to the telescope for a total of 32 widths of glass. The transmitted beam passes

The maximum, total, path length, difference between the two light beams can be calculated. The maximum length difference is only 220 nanometers. Light waves can interfere with each other in two ways. (1) Where the crest of one wave meets the crest of another wave or where the trough of one wave meets the trough of another, the two waves combine and form a bright spot of light. (2) Where a crest meets a trough, the two waves cancel, leaving a dark spot. With a maximum path length difference of only 220 nanometers, it seems only light beams with a wavelength in the range of 440 nanometers could interfere with the cancelling type of interference (trough meets crest type of interference). The cancelling interference would occurr because the light beams would be ½ a wavelength out of phase. To obtain the reinforcing type of interference (crest meets crest or trough meets trough interference) would we need wavelengths in the ultra-violet spectrum in the range of 220 nanometers?

Another startling claim regards the interference fringes themselves. The interference fringes themselves change or shift under the influence of the changing of the total difference in the path length between the light beam that is perpendicular to the earth’s motion and the light beam that is parallel to the motion of the earth. The total difference changes between a maximum and a minimum. While the interference patterns change, the article maintains that a part of the interference pattern does not change. A bright center fringe in the interference pattern doesn’t change and is used as the zero point from which the changes in the surrounding fringes of the changing interference patterns can be measured. The article provides us with a quotation from Dayton Miller who writes, “White light fringes were chosen for the observations because they consist of a small group of fringes having a central, sharply defined black fringe which forms a permanent zero reference mark for all the readings.” How is it that the center fringe of the changing interference patterns doesn’t change while the surrounding fringes do change?

It is difficult to understand a portion of the lengthy caption for figure 5. The difficult portion reads, “c is a compensating plate so that both the reflected and transmitted beams travel through the same amount of glass (important since experiments were run with white light which has an extremely short coherence length requiring precise matching of optical path lengths for fringes to be visible; monochromatic sodium light was used only for initial alignment)”

This statement is confusing. At first glance, compensating glass plate doesn’t appear necessary. First, we must acknowlegde that both the transmitted and reflected beam travel through 30 widths of glass as they make their along the light path dictated by the 4 mirrors located at each end of each arm of the interferometer. Each arm has a total of 8 mirrors. There are 4 mirrors at each end of each arm. On the outward journey, a light beam passes through 16 widths of glass as it is reflected from the mirrors located at the ends of each arm. On the return journey a light beam passes through 14 widths of glass as it is reflected from the mirrors located at the ends of each arm. For the notion of the need for a compensating glass to make sense we must acknowledge that the beam of light from the source is not split into two beams (the transmitted and the reflected) until it reaches the backside of the beam splitter b. I assume beam splitter b is a half-silvered mirror. The transmitted beam isn’t formed until the precise moment that it leaves the beam splitter b so at that point it has traveled through no widths of glass. The reflected beam by virtue of being reflected travels through one width of glass at the beginning of its journey. It is the un-split beam that enters the beam splitter b and travels to the very back of beam splitter b, a half-silvered mirror. The returning reflected beam passes through the beam splitter b as it heads toward the telescope. So it passes through one width of glass at the beginning of its journey plus 30 widths of glass as it makes its way up and down the arm of the interferometer plus one final width of glass as it passes through the half-silvered mirror as it heads to the telescope for a total of 32 widths of glass. The transmitted beam passes

What is your view of the Wikipedia entry for the Michelson-Morley experiment? (final thoughts)

passes through no widths of glass at beam splitter b, and it passes through one width of glass at compensating plate c on its outward journey. It passes through 30 widths of glass as it makes its way up and down the arm of the interferometer. It passes through another width of glass at compensating plate c on its return journey. The total widths of glass the transmitted beam passes through is 32.

The transmitted beam and the reflected beam both pass through 32 widths of glass. This forces us to conclude that the transmitted beam on its return journey doesn’t enter into beam splitter b, but instead it is reflected from the backside of beam splitter b. The backsides of mirrors aren’t usually associated with the reflection of light beams. How is a beam of light that impinges on a mirror from the rear face reflected at all?

The transmitted beam and the reflected beam both pass through 32 widths of glass. This forces us to conclude that the transmitted beam on its return journey doesn’t enter into beam splitter b, but instead it is reflected from the backside of beam splitter b. The backsides of mirrors aren’t usually associated with the reflection of light beams. How is a beam of light that impinges on a mirror from the rear face reflected at all?

Do you have any comments on Newton's proof of Kepler's second law?

Reading portions of Newton’s original text of the “Principia,” which appear in Colin Pask’s “Magnificent Principia: Exploring Isaac Newton’s Masterpiece” gives an idea of how difficult Newton’s language is for us today. In Chapter 8, C. Pask gives an overview of Newton’s proof of Kepler’s second law. Newton’s proof “uses only the simple geometry of triangles.” Interestingly, the force of gravity, which Newton refers to as centripetal force, switches on and off repeatedly. Newton writes, “But when the body [planet] arrived at B, suppose that the centripetal force acts at once with a great impulse, and, turning aside the body [planet] from the right line Bc compels it afterwards to continue its motion along the right line BC. . . . By the like argument, if the centripetal force acts successively in C, D,E, etc., and makes the body, [planet] in each single particle of time, to describe the right lines CD, DE, EF, etc. . . . . The word “successively” is different from the word “continuously.” It implies one gravitational impulse coming after the other in time. At point B gravity acts and changes the course of the planet, but while the planet is moving along the straight line BC gravity is not acting on the planet. If gravity were acting on the planet it would not continue on its straight line path. It would be diverted from that straight line path as it will be when it reaches point C. When the planet reaches point C gravity acts again and changes the course of the planet to line CD, but while the planet is traveling along the line CD gravity doesn’t act on the planet. Newton writes that the length of the lines BC and CD etc. diminish “in infinitum,” but does that overcome the difficulty of gravity being switched on and off repeatedly?

Do you have any further comments on Newton's proof of Kepler's second law?

In Newton’s proof of Kepler’s second law a planet sweeps out triangles of equal areas in equal amounts of time. The various lengths of the bases of these triangles of equal areas represent the varying velocities of a planet. From this we obtain the notion that the greater the length of the base of a triangle then the greater the velocity of the planet at a certain point in its orbit. In a triangle in which the base of a triangle is formed by the line AB, the line AB represents the velocity of the planet at point A. The bases of the triangles that is to say the velocity measuring lines vary in length as the velocity of the planet varies. Interestingly, these velocity measuring lines or perhaps we could refer to them as velocity vectors are not perpendicular to the position lines. The position lines are the lines connecting the force center (the sun) to the planet; they could also be refered to as the position vectors. The velocity measuring lines cannot be perpendicular to all the position lines because they are the sides of the triangles of equal area. This point is shown in the original diagrams from Newton’s “Principia” reproduced in C. Pask’s “Magnificent Principia” and in C. Pask’s own diagrams. The lines that measure the velocity of a planet, that is the base of the triangles, vary in length as the velocity of the planet varies. Why isn’t the velocity measuring line, the velocity vector, perpendicular to the position line, the position vector? Perhaps, the reason is because the area of a triangle is as follows: A= ½ bh. The area of the triangle equals one-half times the base times the height of the triangle. If the base represents the velocity, the velocity varies inversely with the height of the triangle, b= 2A/h. The velocity varies inversely with the heights of the triangles of equal areas; it doesn’t vary inversely with the lengths of the sides of the triangles.

This poses a problem for the modern versions of Newton’s proof of Kepler’s second law. In the modern versions the acceleration vector is parallel to (superimposed on) the position vector. When the position vector is cross multiplied with the acceleration vector the product is zero. The acceleration vector is the derivative of the velocity vector and perpendicular to the velocity vector. If the velocity vector is not perpendicular to the position vector, the acceleration vector cannot be paralell to (superimposed on) the position vector. A modern version of Newton’s proof of Kepler’s second law is presented in “Calculus and Analytic Geometry” by George B. Thomas and Ross L. Finney. In diagram 14.23 the velocity vector is not perpendicular to the position vector. In the same diagram one of the two components of the velocity vector is perpendicular to the position vector. In diagram 14.25 “The force of gravity is directed along the line joining the centers of mass.” That is to say the acceleration vector is parallel (superimposed on) the position vector.

Apparently, the difficulty presented by the velocity vector not being perpendicular to the position vector can be overcome in Newton’s original proof by diminishing the areas of the equal area triangles “in infinitum,” since as the bases of the triangles decrease in length the sides of the triangle becomes closer to the height of the triangle, which is to say closer to a line that is perpendicular to the velocity measuring line (the base of the triangle).

This poses a problem for the modern versions of Newton’s proof of Kepler’s second law. In the modern versions the acceleration vector is parallel to (superimposed on) the position vector. When the position vector is cross multiplied with the acceleration vector the product is zero. The acceleration vector is the derivative of the velocity vector and perpendicular to the velocity vector. If the velocity vector is not perpendicular to the position vector, the acceleration vector cannot be paralell to (superimposed on) the position vector. A modern version of Newton’s proof of Kepler’s second law is presented in “Calculus and Analytic Geometry” by George B. Thomas and Ross L. Finney. In diagram 14.23 the velocity vector is not perpendicular to the position vector. In the same diagram one of the two components of the velocity vector is perpendicular to the position vector. In diagram 14.25 “The force of gravity is directed along the line joining the centers of mass.” That is to say the acceleration vector is parallel (superimposed on) the position vector.

Apparently, the difficulty presented by the velocity vector not being perpendicular to the position vector can be overcome in Newton’s original proof by diminishing the areas of the equal area triangles “in infinitum,” since as the bases of the triangles decrease in length the sides of the triangle becomes closer to the height of the triangle, which is to say closer to a line that is perpendicular to the velocity measuring line (the base of the triangle).

What are your thoughts on the Wikipedia article "Romer's determination of the speed of light"?

The Wikipedia article “Romer’s determination of the speed of light” is confusing and instructive. The section 5.2 entitled “Cumulative effect” deserves special attention. The difference between the predicted (calculated) time of the emergence of Jupiter’s moon Io from Jupiter’s shadow and the observed time of the emergence of Jupiter’s moon Io from Jupiter’s shadow is given as 15 minutes for one particular observation on April 29, 1672 at 10:30:06 am of a series of observations. The April 29, 1672 observation is the 30th and last in this series of observations. This series of observations lasts from March 7, 1672 to April 29, 1672. The observed delay in the appearance of Io is 15 minutes that is to say Io appears 15 minutes later than the calculations made at the beginning of this series of observations suggested it should appear. The observed delay in the appearance of Io is explained by the fact that the Earth has been moving further away from Io for the entire period of March 7 through April 29. The Wikipedia article doesn’t give a figure denoting how far the Earth has moved away from Io during this time, but my calculations suggest the Earth has moved about 85 million miles away from Io during this time. If we divide 85 million miles by the speed of light, we obtain 457 seconds or 7.6 minutes. Thus, the observed appearance of Io should have been delayed only 7.6 minutes. How can we explain a delay of 15 minutes? Perhaps, if the solar system itself along with the entire Milky Way galaxy were traveling at about slightly more than half the speed of light or approximately 95,000 miles per second, this would explain the delay of 15 minutes. This notion seems fanciful. It should be noted the Wikipedia article states that the delay is 15 minutes; my calculations suggest the delay is slightly more than 16 minutes (16 minutes 5 seconds).

Let’s examine how the apparent orbital period of Jupiter’s moon Io is calculated. It is denoted as the apparent orbital period because the Earth is either moving toward Jupiter or away from Jupiter when such calculations are being made. And, that unaccounted for movement of the Earth introduces error into the calculations. The Earth was moving away from Jupiter during the period of March 7 to March 14 when the calculation of an apparent orbital period of 42 hours 28 minutes 31.25 seconds was made. If the Earth wasn’t moving toward or away from Jupiter, an exact as opposed to an apparent orbital period could be determined. We would note the time Io emerged from Jupiter’s shadow then we would note the time when Io next emerged from Jupiter’s shadow. The duration of that time period would give us the orbital period of Io, provided, of course, that the observation of an intervening occurrence of Io’s emergence from Jupiter’s shadow had not been blocked by cloudy weather or some other factors. This is because the time it would take light from Io to reach the Earth would be the same for both the first noted observation of Io’s emergence from Jupiter’s shadow and the second noted observation of Io’s emergence from Jupiter’s shadow. For example, when Jupiter is at its closest approach to the Earth, it would take light from Io about 35 miutes to reach the Earth. So when we first noted Io’s emergence from Jupiter’s shadow it actually would have occurred about 35 minutes before our observation denoted it occurred. If the Earth was not moving, when we made our second observation of Io’s emergence it would again take light from Io about 35 minutes to reach the Earth. So when we made our second observation of Io’s emergence it would have again actually occurred about 35 minutes before our observations denoted that it occurred. The delay of about 35 minutes for each observation would cancel each other out and we could determine the actual orbital period of Io.

But, the Earth is moving, and in the case we are examining the Earth is moving away from Jupiter so for each observation of Io’s emergence it take light longer to reach the Earth than it did on the previous observation. To calculate the apparent orbital period of Io this fact is neglected for purposes of calculation. We begin (as Romer did) with our first observation of Io’s emergence on March 7 at 7:58:25 am and for calculation purposes we conclude (as Romer did) with the fourth emergence of Io on March 14 at 9:52:30 am. This gives us a total of 169 hours 54 minutes 5 seconds for four orbits of Io. We divide by four and obtain 42 hours 28 minutes 31.25 seconds as the apparent orbital period of Io.

We assume that each orbit of Io (from one emergence to the next emergence) will last 42 hours 28 minutes and 31.25 seconds. From March 7 at 7:58:25 am to April 29 at 10:30:06, Io makes thirty orbits of Jupiter. Therefore, it should emerge from Jupiter’s shadow 30(42 hours +28 minutes + 31.25 seconds) or 1,274 hours 15 minutes 36 seconds after the date and time of March 7 at 7:58:25. But, when Io makes its 30th emergence from Jupiter’s

Let’s examine how the apparent orbital period of Jupiter’s moon Io is calculated. It is denoted as the apparent orbital period because the Earth is either moving toward Jupiter or away from Jupiter when such calculations are being made. And, that unaccounted for movement of the Earth introduces error into the calculations. The Earth was moving away from Jupiter during the period of March 7 to March 14 when the calculation of an apparent orbital period of 42 hours 28 minutes 31.25 seconds was made. If the Earth wasn’t moving toward or away from Jupiter, an exact as opposed to an apparent orbital period could be determined. We would note the time Io emerged from Jupiter’s shadow then we would note the time when Io next emerged from Jupiter’s shadow. The duration of that time period would give us the orbital period of Io, provided, of course, that the observation of an intervening occurrence of Io’s emergence from Jupiter’s shadow had not been blocked by cloudy weather or some other factors. This is because the time it would take light from Io to reach the Earth would be the same for both the first noted observation of Io’s emergence from Jupiter’s shadow and the second noted observation of Io’s emergence from Jupiter’s shadow. For example, when Jupiter is at its closest approach to the Earth, it would take light from Io about 35 miutes to reach the Earth. So when we first noted Io’s emergence from Jupiter’s shadow it actually would have occurred about 35 minutes before our observation denoted it occurred. If the Earth was not moving, when we made our second observation of Io’s emergence it would again take light from Io about 35 minutes to reach the Earth. So when we made our second observation of Io’s emergence it would have again actually occurred about 35 minutes before our observations denoted that it occurred. The delay of about 35 minutes for each observation would cancel each other out and we could determine the actual orbital period of Io.

But, the Earth is moving, and in the case we are examining the Earth is moving away from Jupiter so for each observation of Io’s emergence it take light longer to reach the Earth than it did on the previous observation. To calculate the apparent orbital period of Io this fact is neglected for purposes of calculation. We begin (as Romer did) with our first observation of Io’s emergence on March 7 at 7:58:25 am and for calculation purposes we conclude (as Romer did) with the fourth emergence of Io on March 14 at 9:52:30 am. This gives us a total of 169 hours 54 minutes 5 seconds for four orbits of Io. We divide by four and obtain 42 hours 28 minutes 31.25 seconds as the apparent orbital period of Io.

We assume that each orbit of Io (from one emergence to the next emergence) will last 42 hours 28 minutes and 31.25 seconds. From March 7 at 7:58:25 am to April 29 at 10:30:06, Io makes thirty orbits of Jupiter. Therefore, it should emerge from Jupiter’s shadow 30(42 hours +28 minutes + 31.25 seconds) or 1,274 hours 15 minutes 36 seconds after the date and time of March 7 at 7:58:25. But, when Io makes its 30th emergence from Jupiter’s

Do you have any final thoughts on the Wikipedia article "Romer's determination of the speed of light"?

shadow in this sequence of observation, the date and time are April 29 at 10:30:06 am. This gives us a total time for 30 orbits (from one emergence to the next emergence) of 1,274 hours 31 minutes 41 seconds. So the observed 30th emergence of Io from Jupiter’s shadow occurs 16 minutes 5 seconds after the predicted (calculated) time for the 30th emergence of Io from Jupiter’s shadow.

The explanation for the delay is the cumulative distance the Earth has moved away from Jupiter in the 52 days from March 7 to April 29. The cumulative extra distance the light from Io must travel before it reaches the Earth accounts for the delay. But, the cumulative extra distance seems to be about 85 million miles not the 180 million one would expect for a delay of 16 minutes. If the solar system and Milky Way galaxy were traveling between 95,000 miles per second and 100,000 miles per second this might explain the delay which is variously calculated as 15 or 16 minutes. Another explanation could be that the emergence times of Io from Jupiter’s shadow weren’t noted correctly. During the 52 day period of observation Jupiter traveled about 35 million miles, but since this is only about one per cent of the distance Jupiter travels in its slightly less than 12 year orbit, it doesn’t seem significant.

The explanation for the delay is the cumulative distance the Earth has moved away from Jupiter in the 52 days from March 7 to April 29. The cumulative extra distance the light from Io must travel before it reaches the Earth accounts for the delay. But, the cumulative extra distance seems to be about 85 million miles not the 180 million one would expect for a delay of 16 minutes. If the solar system and Milky Way galaxy were traveling between 95,000 miles per second and 100,000 miles per second this might explain the delay which is variously calculated as 15 or 16 minutes. Another explanation could be that the emergence times of Io from Jupiter’s shadow weren’t noted correctly. During the 52 day period of observation Jupiter traveled about 35 million miles, but since this is only about one per cent of the distance Jupiter travels in its slightly less than 12 year orbit, it doesn’t seem significant.

Can you find a flaw with the tensor calculus operation called contraction?

The tensor calculus operation called contraction is unique. There isn’t anything analogous to it in everyday mathematics such as addition or multiplication. Tensors are a difficult category of items to understand. They are grouped in ranks according to Lillian Lieber’s “The Einstein Theory of Relativity” while Glen and Robert James’s “Mathematics Dictionary” prefers the term “orders.” A tensor of rank 0 is a scalar. A tensor of rank 1 is a vector. A tensor of rank 2 is formed from the combination of two vectors in a certain way according to Lieber. The “Mathematics Dictionary” defines a tensor as “an abstract object having a definitely specified system of components in every coordinate system under consideration and such that, under transformation of coordinates the components of the object undergo a transformation of a certain nature.”

Tensors are made of components except for the tensor of rank 0 which is a scalar. A scalar is a number as distinguished from a vector or a tensor. Lieber gives temperature and age as examples of scalars. Tensors have indexes (indices). Indexes can be both superscripts and subscripts. Lieber favors using letters from the Greek alphabet to denote a tensors superscripts and subscripts. The “Mathematics Dictionary” favors the use of letters from the English alphabet to denote a tensors superscripts and subscripts. A tensor with both subscripts and superscripts is called a mixed tensor. A tensor with only superscripts is called a contravariant tensor. A tensor with only subscripts is called a covariant tensor.

The number of components that form a particular tensor is determined by two conditions. The first condition is the number of spatial dimensions under consideration. Lieber gives examples that are located in two, three and four dimensional space. As the number of spatial dimensions increases, the number of components of the tensor increase, as well. The more components a tensor has the more difficult the calculations with the tensor become. It less demanding to perform calculations with tensors located in only two spatial dimensions. The second condition that determines the number of components a tensor is formed from is the number of indexes (indices) that the tensor is denoted by that is to say the total number of subscripts and superscripts. Lieber writes, “In general in n-dimensional space, a tensor of rank two consists of n^2 (n squared) equations, each containing n^2 terms in the right-hand member.” The rank of a tensor is determined by calculating the total number of indexes. For instance, if a mixed tensor has two superscripts and one subscript the total number of indexes is three therefore its rank would be three. In two dimensional space the tensor would consist of 2^3 equations or eight equations. And each of the eight equations would have 2^3 terms on the right-hand side of the equation. Each equation is a component of the tensor so the tensor would have eight components in two dimensional space.

In two dimensional space we usually think of the components of a tensor of rank one that is to say a vector as the component along the x-axis and the component along the y-axis. In tensor calculus the x, y, and z axes are renamed. The x-axis is called the x1- axis. The y-axis is called the x2-axis. The z-axis is called the x3-axis. The two components of a tensor of rank one (vector) in two dimensional space would be referred to as the x1 component and the x2 component.

There is another feature of tensor calculus known as the summation convention. The “Mathematics Dictionary” defines it as “the convention of letting the repetition of an index (subscript or superscript) denote a summation with respect to that index over its range. E.g., if (1, 2, 3, 4, 5, 6) is the range of the index i, then aixi stands for the sum of aixi over the range 1 through 6, which equals a1x1 + a2x2 + a3x3 + a4x4 + a5x5 + a6x6. The superscript i in xi is not the ith power of the number x, but merely an index which denotes that xi is the ith object of the six objects x1, x2,. . . x6.” The summation convention seems straightforward, but with the repetition of more than one pair of indexes it can become confusing. Working out the details of the summation convention with two pairs of repeated indexes is challenging enough, but for the operation of contraction we are going to look at there are four pairs of repeated indexes on the right-hand side of the equation and one pair of repeated indexes on the left-hand side of the equation. The operation of contraction itself in the example we are going to examine causes the tensor on the right-hand side of the equation to generate a repeated index that didn’t exist at the beginning of the operation of contraction. Another difficulty with the operation of contraction is that on the right-hand side of the equation a repeated index exits between two of the partial derivatives. Very often a repeated index will exist between a partial derivative and the

Tensors are made of components except for the tensor of rank 0 which is a scalar. A scalar is a number as distinguished from a vector or a tensor. Lieber gives temperature and age as examples of scalars. Tensors have indexes (indices). Indexes can be both superscripts and subscripts. Lieber favors using letters from the Greek alphabet to denote a tensors superscripts and subscripts. The “Mathematics Dictionary” favors the use of letters from the English alphabet to denote a tensors superscripts and subscripts. A tensor with both subscripts and superscripts is called a mixed tensor. A tensor with only superscripts is called a contravariant tensor. A tensor with only subscripts is called a covariant tensor.

The number of components that form a particular tensor is determined by two conditions. The first condition is the number of spatial dimensions under consideration. Lieber gives examples that are located in two, three and four dimensional space. As the number of spatial dimensions increases, the number of components of the tensor increase, as well. The more components a tensor has the more difficult the calculations with the tensor become. It less demanding to perform calculations with tensors located in only two spatial dimensions. The second condition that determines the number of components a tensor is formed from is the number of indexes (indices) that the tensor is denoted by that is to say the total number of subscripts and superscripts. Lieber writes, “In general in n-dimensional space, a tensor of rank two consists of n^2 (n squared) equations, each containing n^2 terms in the right-hand member.” The rank of a tensor is determined by calculating the total number of indexes. For instance, if a mixed tensor has two superscripts and one subscript the total number of indexes is three therefore its rank would be three. In two dimensional space the tensor would consist of 2^3 equations or eight equations. And each of the eight equations would have 2^3 terms on the right-hand side of the equation. Each equation is a component of the tensor so the tensor would have eight components in two dimensional space.

In two dimensional space we usually think of the components of a tensor of rank one that is to say a vector as the component along the x-axis and the component along the y-axis. In tensor calculus the x, y, and z axes are renamed. The x-axis is called the x1- axis. The y-axis is called the x2-axis. The z-axis is called the x3-axis. The two components of a tensor of rank one (vector) in two dimensional space would be referred to as the x1 component and the x2 component.

There is another feature of tensor calculus known as the summation convention. The “Mathematics Dictionary” defines it as “the convention of letting the repetition of an index (subscript or superscript) denote a summation with respect to that index over its range. E.g., if (1, 2, 3, 4, 5, 6) is the range of the index i, then aixi stands for the sum of aixi over the range 1 through 6, which equals a1x1 + a2x2 + a3x3 + a4x4 + a5x5 + a6x6. The superscript i in xi is not the ith power of the number x, but merely an index which denotes that xi is the ith object of the six objects x1, x2,. . . x6.” The summation convention seems straightforward, but with the repetition of more than one pair of indexes it can become confusing. Working out the details of the summation convention with two pairs of repeated indexes is challenging enough, but for the operation of contraction we are going to look at there are four pairs of repeated indexes on the right-hand side of the equation and one pair of repeated indexes on the left-hand side of the equation. The operation of contraction itself in the example we are going to examine causes the tensor on the right-hand side of the equation to generate a repeated index that didn’t exist at the beginning of the operation of contraction. Another difficulty with the operation of contraction is that on the right-hand side of the equation a repeated index exits between two of the partial derivatives. Very often a repeated index will exist between a partial derivative and the

Can you find a flaw with the tensor calculus operation called contraction? (continued)

tensor that it multiplies. The operation of contraction begins in the example we are going to examine with the mixed tensor on the left-hand side of the equation. It is a primed tensor with indexes that aren’t repeated. Its indexes consist of two superscripts and one subscript. Lillian Lieber denotes her indexes using lower case Greek letters. The tensor itself is denoted by the uppercase letter A and it is further denoted as A prime because it has been formed in the primed coordinate system. We will denote the indexes using lowercase English letters. The mixed tensor has two superscripts a and b. The mixed tensor also has one subscript c. We can denote it as A prime with superscripts a, b and subscript c. The operation of contraction begins when the subscript c is replaced by an a thus forming a repeated index of superscript a and subscript a on the left-hand side of the equation. On the right-hand side of the equation replacing the subscript c with the subscript a produces an addition partial derivative with the index a thus we have a pair of partial derivatives with the repetition of the index a so we must sum on the partial derivatives that contain the index a. Plus between the tensor and the three partial derivatives that are multiplying the tensor we have three pairs of repeated indexes.

I had written this much before I came to the conclusion that contraction is a mathematically sound operation as long as certain transformation equations are used.

I had written this much before I came to the conclusion that contraction is a mathematically sound operation as long as certain transformation equations are used.

Do you have any thoughts on David Wick's book "The Infamous Boundary"?

David Wick’s book “The Infamous Boundary,” published in 1995, contains an interesting presentation of Bell’s inequality. His succinct description is provided to the reader in the chapter entitled “Bell’s Theorem.” Remarkably, his description seems to be brilliant propaganda so brilliant in fact I am not sure it is propaganda of the disinformation variety. He seems to be saying something very similar to the following: What is the probability of flipping three fair coins (each coin being flipped once) and obtaining a particular outcome. The particular outcome we want to know the probability of obtaining is three heads; that is each of the three coins landing heads up. The probability of obtaining this outcome must be less than one. The following is where I believe a fairly stark error creeps in: Since the probability of obtaining the outcome heads in the flipping of one fair coin is ½, all that we must do in order to obtain the probability of flipping three coins and obtaining three heads is add this probability for each of the three flips. Hence, ½ + ½ + ½ = 1 ½ < 1. Thus, in a very similar manner David Wick arrives at his version of Bell’s inequality. The actual probability of obtaining three heads in the flipping of three fair coins is 1/6. D. Wick seems to violate one of the basic tenets of probability which is the following: Determine the total number of possible outcomes and let this be the denominator of the fraction. Let the outcome you desire to know the probability of be the numerator of the fraction.

Do you have any thoughts on F. David Peat's book "Einstein's Moon: Bell's Theorem and the Curious Quest for Quantum Reality"?

F. David Peat’s book “Einstein’s Moon: Bell’s Theorem and the Curious Quest for Quantum Reality,” published in 1990, devotes several chapters to Bell’s inequality, yet his explanation is no more convincing than David Wick’s explanation in his book “The Infamous Boundary.” Peat introduces a formula for the hidden variable explanation of the correlation of paired particles that is too restrictive. His explanation is ambiguous at points, and it is only too restrictive in one out of four instances, but that restrictiveness is enough to cut a number (the sum of six “probabilities”) associated with the local-reality explanation of the correlation of paired particles from a value that should be close to two to zero. Local-reality is another term for the phrase: hidden variable explanation of the correlation of paired particles. The use of the term number refers to the value obtained from the summation of an inequality. It is an inequality that has six probability terms on the left-hand side of the less than or equal to inequality. Three of the terms are positive, and three of the terms are negative. Zero is on the right-hand side of the less than or equal to inequality. When the phrase probability term is used, it can be interpreted as a value that gives the correlation between the two detectors A and B. Interestingly, for the local reality explanation of the correlation of paired particles Peat employs a formula that is too restrictive while when he gives an example of a calculation of the correlation of paired particles using quantum theory, he has no difficulty letting all four angles equal 45 degrees. The four angles in the local-reality explanation all were required to be different and there was a further clever restriction, as well. There could only be four different angles used for the four different settings in question. Usually, there would be eight different angles employed for four different settings, that is to say two different angles for each setting (one angle for each detector). There would be an exception when both detectors were set to an angle of zero degrees.

Does the magnetic field of our sun bend a ray of light from a distant star that passes close to the edge of the sun?

The magnetic field of the sun is often described as disorganized so perhaps it would not affect a ray of light from a distant star. Interestingly, I recall reading somewhere that a beam from a sodium light source can be visibly bent by the introduction of a relatively weak magnetic field. One rarely, if ever, reads about magnetic fields bending rays of light. If magnetic fields do bend rays of light and this fact were widely known, it would have to be taken into consideration in the explanations given of the bending of a ray of light from a distant star that passes close to the edge of the sun during an eclipse and is thus photographed displaced from its normal position in the sky by the bending of its light rays under the influence of the gravitational field of the sun. The displacement from its normal position in the sky is very small. The effects of magnetic fields on starlight are never mentioned, as far as I know, in any of the descriptions of the famous eclipse experiments that are used to support Einstein’s theory of general relativity. These are experiments that use photographs taken of stars near the sun during an eclipse in order to detect a displacement of the stars from their normal positions in the sky due to the gravitational field of the sun. I wonder if the magnetic field of the earth could bend the light rays from a distant star.

Do you have any thoughts on the greenhouse gases?

“The Britannica Book of the Year 1970” contains an interesting statement in the “Astronomy” section. In the subsection entitled “Infrared Astronomy” we read, “In order to study physical processes in cool stars, observations are required at infrared wavelengths, where most of the energy is emitted. Progress in this field was retarded by the opacity of the earth’s atmosphere to much of this radiation . . . .” It is widely assumed that the greenhouse gases absorb light in the infrared wavelengths that is to say they absorb the radiation (heat) that the earth itself gives off. The earth gives off infrared radiation because it has been warmed by the sun which emits a great deal of radiation in the visible wavelengths of light. Presumably, the other gases in the atmosphere such as nitrogen, oxygen and argon which make up 99.93% of the gases in the atmosphere absorb infrared radiation to a much lesser extent than the greenhouse gases, if at all. Since the greenhouse gases make up less than .07% of the gases in the atmosphere, shouldn’t the earth’s atmosphere be transparent to infrared radiation? If we divided a large pane of glass into 10,000 equal sized squares and then blackened 7 of the squares, we would still consider the pane of glass transparent to visible light. Does the opacity of the earth’s atmosphere to infrared radiation suggest that the gases nitrogen, oxygen and argon absorb infrared radiation and perhaps to the same degree as the greenhouse gases?

Have you changed your mind about the tensor operation known as contraction?

After re-evaluating the tensor operation designated by either of the following two appellations contraction or the inner product, I have reached the conclusion that it is an invalid operation, but with the caveat that it does appear to be true for a mixed tensor of rank three, in a space consisting of two dimensions, when the coordinates are transformed by the rotation of the axes. If contraction is an invalid operation in most instances, it would undercut the mathematical foundations of Einstein’s theory of general relativity since it is the contraction of the curvature tensor (which is also known as the Riemann-Christoffel tensor) that generates Einstein’s law of gravity, i.e., the gravitational tensor. The curvature tensor is Bastr, and in Euclidean space, Bastr =0. If we replace the index r with a, through contraction we obtain Gst, the gravitational tensor, and Gst =0 is Einstein’s law of gravitation. The curvature tensor and the gravitational tensor are most often presented with lower case Greek letters as the indexes (subscripts and superscripts), but I am trying to use use only the symbols commonly available on a keyboard.

According to Lillian Lieber, one of the ways in which tensors are defined is by the transformation equations that are used to transfer the tensor from one coordinate system to another. The transformation equations presented by L. Lieber in her book “The Einstein Theory of Relativity” are rotational formulas for coordinates in a plane that is to say in a space of two dimensions. It can be confusing. We can begin with the primed coordinates (x/, y/) that belong to the old set of rectangular axes X/ and Y/. Next, we rotate a new set of rectangular axes, designated X and Y, through the angle theta with respect to the old set of rectangular axes, which we have designated X/ and Y/. The new unprimed coordinates are designated (x, y). The relationship between the new unprimed coordinates (x, y) and the old primed coordinates (x/, y/) is given by the following rotational formulas: x= x/cos (theta) – y/sin (theta) and y= x/sin (theta) + y/cos (theta). We can write the rotational formulas in another way in which the primed and unprimed x and y variables are replaced by primed and unprimed x1 and x2 variables, respectively. We refer to the variables as x subscript one and x subscript two; we append the terms primed and unprimed as the occasion calls for. The rotational formulas become the following: x1 = x1/cos (theta) – x2/sin (theta) and x2= x1/sin (theta) + x2/cos (theta). It is this version of the rotational formulas that are referred to in the complex and condensed notation used to define tensors.

There are two important items to note: 1. the use of both sin (theta) and cos (theta) and 2. the strategic placement of the negative sign. These two items will turn out to be essential for the operation of contraction to be performed validly on a mixed tensor of rank three in a space of two dimensions. In the operation of contraction on a mixed tensor in two dimensional space, we will repeatedly get the terms cos2 (theta) + sin2 (theta). Since cos2 (theta) + sin2 (theta) equals one, this is one of the key factors that allows contraction to lower or reduce the rank of a mixed tensor by two. The other key factor that allows us to reduce the rank of the mixed tensor by two is the strategic placement of the negative sign. In the operation of contraction on a mixed tensor in two dimensional space, we also repeatedly will get the terms cos (theta) sin (theta) – cos (theta) sin (theta). Since cos (theta) sin (theta) – cos (theta) sin (theta) equals zero this will allow half of the equations generated through the summation (the summation convention) called for in the operation of contraction of a mixed tensor in a space of two dimensions to cancel themselves through being multiplied by zero. The other half of the equations will be reduced in rank by two during the required process of summation through the use of the identity cos2 (theta) + sin2 (theta) =1.

The three following facts about the rotational formulas for coordinates in a space of three dimensions should make a convincing argument that the operation of contraction is invalid for tensors in a space of three dimensions. 1. The rotational formulas do not employ both of the terms sin (theta) and cos (theta); they only employ the term cos and it isn’t the cos of only one angle, theta. 2. The term cos does not refer to a single angle, theta, but instead is matched with nine different angles A1, A2, A3, B1, B2, B3, C1, C2, and C3. 3. There are no negative terms that would allow many terms and hence equations to cancel one another.

According to the “Mathematics Dictionary” by Glenn James and Robert C. James the formulas for rotation of axes in a space of three dimensions are the following:

“If the direction angles, with respect to the old axes, of the new x-axis (the x/-axis) are A1, B1, C1; of the y/-axis are A2

According to Lillian Lieber, one of the ways in which tensors are defined is by the transformation equations that are used to transfer the tensor from one coordinate system to another. The transformation equations presented by L. Lieber in her book “The Einstein Theory of Relativity” are rotational formulas for coordinates in a plane that is to say in a space of two dimensions. It can be confusing. We can begin with the primed coordinates (x/, y/) that belong to the old set of rectangular axes X/ and Y/. Next, we rotate a new set of rectangular axes, designated X and Y, through the angle theta with respect to the old set of rectangular axes, which we have designated X/ and Y/. The new unprimed coordinates are designated (x, y). The relationship between the new unprimed coordinates (x, y) and the old primed coordinates (x/, y/) is given by the following rotational formulas: x= x/cos (theta) – y/sin (theta) and y= x/sin (theta) + y/cos (theta). We can write the rotational formulas in another way in which the primed and unprimed x and y variables are replaced by primed and unprimed x1 and x2 variables, respectively. We refer to the variables as x subscript one and x subscript two; we append the terms primed and unprimed as the occasion calls for. The rotational formulas become the following: x1 = x1/cos (theta) – x2/sin (theta) and x2= x1/sin (theta) + x2/cos (theta). It is this version of the rotational formulas that are referred to in the complex and condensed notation used to define tensors.

There are two important items to note: 1. the use of both sin (theta) and cos (theta) and 2. the strategic placement of the negative sign. These two items will turn out to be essential for the operation of contraction to be performed validly on a mixed tensor of rank three in a space of two dimensions. In the operation of contraction on a mixed tensor in two dimensional space, we will repeatedly get the terms cos2 (theta) + sin2 (theta). Since cos2 (theta) + sin2 (theta) equals one, this is one of the key factors that allows contraction to lower or reduce the rank of a mixed tensor by two. The other key factor that allows us to reduce the rank of the mixed tensor by two is the strategic placement of the negative sign. In the operation of contraction on a mixed tensor in two dimensional space, we also repeatedly will get the terms cos (theta) sin (theta) – cos (theta) sin (theta). Since cos (theta) sin (theta) – cos (theta) sin (theta) equals zero this will allow half of the equations generated through the summation (the summation convention) called for in the operation of contraction of a mixed tensor in a space of two dimensions to cancel themselves through being multiplied by zero. The other half of the equations will be reduced in rank by two during the required process of summation through the use of the identity cos2 (theta) + sin2 (theta) =1.

The three following facts about the rotational formulas for coordinates in a space of three dimensions should make a convincing argument that the operation of contraction is invalid for tensors in a space of three dimensions. 1. The rotational formulas do not employ both of the terms sin (theta) and cos (theta); they only employ the term cos and it isn’t the cos of only one angle, theta. 2. The term cos does not refer to a single angle, theta, but instead is matched with nine different angles A1, A2, A3, B1, B2, B3, C1, C2, and C3. 3. There are no negative terms that would allow many terms and hence equations to cancel one another.

According to the “Mathematics Dictionary” by Glenn James and Robert C. James the formulas for rotation of axes in a space of three dimensions are the following:

“If the direction angles, with respect to the old axes, of the new x-axis (the x/-axis) are A1, B1, C1; of the y/-axis are A2

The tensor operation known as contraction continued.

B2, C2; and of the z/-axis, A3, B3, C3, then the formulas for rotation of axes in space are x= x/cosA1 + y/cosA2 + z/cosA3, y= x/cosB1 + y/cosB2 + z/cosB3, z= x/cosC1 + y/cosC2 + z/cosC3.” To write the formulas in the condensed form used in tensor operations both the primed and unprimed x, y, and z coordinates would need to be replaced with primed and unprimed x1, x2, and x3 coordinates, respectively.

L. Lieber in her book “The Einstein theory of Relativity” denotes tensors using lower case Greek letters as subscripts and superscripts, i.e., indexes. I will use lower case English letters for the subscripts and superscripts of tensors. Instead of the familiar symbol for partial derivative, which seems to be similar to the lower case Greek letter delta, I will use the letter p followed by a period, p. Since I am using the symbols commonly available on a keyboard, the symbol I am using to indicate that a variable is primed is the forward slash raised to a superscript. The symbol that I am using to separate the numerator from the denominator of a partial derivative is also the forward slash (so it can be confusing), but it is neither raised to a superscript nor lowered to a subscript. Using this nomenclature, the formula for a mixed tensor of rank three is the following: A/abc = (p.xn/p.x/c) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). We can read the formula as the following: The primed tensor A with superscripts a and b and subscript c equals the partial derivative of the equation x with subscript n with respect to the primed variable x with subscript c multiplied by the partial derivative of the primed equation x with subscript a with respect to the variable x with the subscript k multiplied by the partial derivative of the primed equation x with subscript b with respect to the variable x with the subscript m multiplied by the unprimed mixed tensor A with superscripts k and m and subscript n. We must further take into account the summation convention. It tells us that since the indexes k, m, and n occur twice on the right-hand side of the formula we must sum on these indexes. We should further note that since we are operating in a space of two dimensions the values of the indexes will range from one to two in other words a, b, c, k, m, and n take on either the value 1 or 2.

In the operation of contraction, the subscript c (or index c) is replaced with the index a on both the right-hand side and left-hand side of the formula. The formula that is generated is the following: A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). We can read the formula as saying the following: The primed tensor A with superscripts a and b and subscript a equals the partial derivative of the equation x with subscript n with respect to the primed variable x with subscript a multiplied by the partial derivative of the primed equation x with subscript a with respect to the variable x with the subscript k multiplied by the partial derivative of the primed equation x with the subscript b with respect to the variable x with the subscript m multiplied by the unprimed tensor A with the superscripts k and m and the subscripts n. We must further take into account that since the indexes k, m, n and now a occur twice on the right-hand side of the formula we must sum on these indexes. Also, the index a occurs twice on the left-hand side of the equation so we must sum on the index a on the left-hand side of the formula, as well.

L. Lieber draws our attention to two of the partial derivatives that occur on the right-hand side of the formula. She analyses these two partial derivatives when the subscripts n and k have different values, and she also analyses these two partial derivatives when the subscripts n and k have the same value.

When the subscripts n and k happen to have different values L. Lieber concludes that (p.xn/p.x/a) (p.x/a/p.xk) = (p.xn/p.xk) = 0. She tells us that because the x’s are not functions of each other (but only of the x/’s) and therefore there is no variation of xn with respect to a different x, namely xk. Thus coefficients of (Akmn) when n does not equal k will all be zero and this will make the terms drop out.

L. Lieber further tells us that when n=k, then (p.xn/p.x/a) (p.x/a/p.xk) = (p.xk/p.x/a) (p.x/a/p.xk) = (p.xk/p.xk) = 1. Thus formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn) becomes A/aba = (p.x/b/p.xm) (Akmk) in which we must still sum on the right for k and m. We must recall that mixed tensor (Akmn) becomes mixed tensor (Akmk) because n=k. Thus, a mixed tensor of rank three is reduced through the operation of contraction to a tensor of rank one.

This is L. Lieber’s argument, but it appears to be a misuse of the chain rule for partial differentiation.

Let’s examine the chain rule as it is used in ordinary differentiation that is to say the differentiation of a function with two variables x and y. The following is an example: y = (x2 +1)3. We should note that the chain rule seems

L. Lieber in her book “The Einstein theory of Relativity” denotes tensors using lower case Greek letters as subscripts and superscripts, i.e., indexes. I will use lower case English letters for the subscripts and superscripts of tensors. Instead of the familiar symbol for partial derivative, which seems to be similar to the lower case Greek letter delta, I will use the letter p followed by a period, p. Since I am using the symbols commonly available on a keyboard, the symbol I am using to indicate that a variable is primed is the forward slash raised to a superscript. The symbol that I am using to separate the numerator from the denominator of a partial derivative is also the forward slash (so it can be confusing), but it is neither raised to a superscript nor lowered to a subscript. Using this nomenclature, the formula for a mixed tensor of rank three is the following: A/abc = (p.xn/p.x/c) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). We can read the formula as the following: The primed tensor A with superscripts a and b and subscript c equals the partial derivative of the equation x with subscript n with respect to the primed variable x with subscript c multiplied by the partial derivative of the primed equation x with subscript a with respect to the variable x with the subscript k multiplied by the partial derivative of the primed equation x with subscript b with respect to the variable x with the subscript m multiplied by the unprimed mixed tensor A with superscripts k and m and subscript n. We must further take into account the summation convention. It tells us that since the indexes k, m, and n occur twice on the right-hand side of the formula we must sum on these indexes. We should further note that since we are operating in a space of two dimensions the values of the indexes will range from one to two in other words a, b, c, k, m, and n take on either the value 1 or 2.

In the operation of contraction, the subscript c (or index c) is replaced with the index a on both the right-hand side and left-hand side of the formula. The formula that is generated is the following: A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). We can read the formula as saying the following: The primed tensor A with superscripts a and b and subscript a equals the partial derivative of the equation x with subscript n with respect to the primed variable x with subscript a multiplied by the partial derivative of the primed equation x with subscript a with respect to the variable x with the subscript k multiplied by the partial derivative of the primed equation x with the subscript b with respect to the variable x with the subscript m multiplied by the unprimed tensor A with the superscripts k and m and the subscripts n. We must further take into account that since the indexes k, m, n and now a occur twice on the right-hand side of the formula we must sum on these indexes. Also, the index a occurs twice on the left-hand side of the equation so we must sum on the index a on the left-hand side of the formula, as well.

L. Lieber draws our attention to two of the partial derivatives that occur on the right-hand side of the formula. She analyses these two partial derivatives when the subscripts n and k have different values, and she also analyses these two partial derivatives when the subscripts n and k have the same value.

When the subscripts n and k happen to have different values L. Lieber concludes that (p.xn/p.x/a) (p.x/a/p.xk) = (p.xn/p.xk) = 0. She tells us that because the x’s are not functions of each other (but only of the x/’s) and therefore there is no variation of xn with respect to a different x, namely xk. Thus coefficients of (Akmn) when n does not equal k will all be zero and this will make the terms drop out.

L. Lieber further tells us that when n=k, then (p.xn/p.x/a) (p.x/a/p.xk) = (p.xk/p.x/a) (p.x/a/p.xk) = (p.xk/p.xk) = 1. Thus formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn) becomes A/aba = (p.x/b/p.xm) (Akmk) in which we must still sum on the right for k and m. We must recall that mixed tensor (Akmn) becomes mixed tensor (Akmk) because n=k. Thus, a mixed tensor of rank three is reduced through the operation of contraction to a tensor of rank one.

This is L. Lieber’s argument, but it appears to be a misuse of the chain rule for partial differentiation.

Let’s examine the chain rule as it is used in ordinary differentiation that is to say the differentiation of a function with two variables x and y. The following is an example: y = (x2 +1)3. We should note that the chain rule seems

More on the tensor operation known as contraction.

to be used when the variable x is raised to a power in more than one instance and in a somewhat complicated manner. The variable x is raised to the second power and then the sum (x2 +1) is raised to the third power. To use the chain rule we must form the composite function y = u3 where u = x2 +1. To find dy/dx, we say that dy/dx = (dy/du) (du/dx). We see that in our example dy/du = 3u2 and du/dx = 2x. We substitute x2 +1 = u into 3u2, and we obtain an answer to dy/dx as follows: dy/dx = 3(x2 +1)2 (2x). The striking notion to consider is that the u in the derivative dy/du is the same as the u in the derivative du/dx. This isn’t the case in L. Lieber’s argument. To be fair, we must point out that L. Lieber is dealing with the chain rule as it applies to partial derivatives where the requirement that the u in the derivative dy/du is the same as the u in the derivative du/dx or more precisely that the u in the derivative p.y/p.u is the same as the u in the derivative p.u/p.x can be relaxed. Well, it actually can’t be relaxed; perhaps I should say it can be expanded so that it can be employed when there is no apparent need for it to be employed to enable the derivation. What also must be called into question instead is L. Lieber’s false notion or at least overly simplistic notion that (p.x1/p.x2) equals zero since x1 doesn’t vary with x2, instead x1 varies with x/1 and x/2.

Let’s look at an example of L. Lieber’s argument. We will substitute numbers into the subscripts of the following formula: (p.xn/p.x/a) (p.x/a/p.xk) = (p.xn/p.xk). First, we will substitute values in which n doesn’t equal k, and we will see if the result is zero. Let n=1 and k=2 and a =2, thus the formula becomes the following: (p.x1/p.x/2) (p.x/2/p.x2) = (p.x1/p.x2). The formula reads as follows: the the partial derivative of the equation x with subscript 1 with respect to the primed variable x with subscript 2 multiplied by the partial derivative of the primed equation x with subscript 2 with respect to the variable x with subscript 2 equals the partial derivative of the equation x with subscript 1 with respect to the variable x with subscript 2.

The equations referred to are the rotation formulas for the primed and unprimed coordinates x and y in a space of two dimensions when the primed and unprimed coordinate x is replaced with the primed and unprimed coordinate x1, respectively and the primed and unprimed coordinate y is replaced by the primed and unprimed coordinate x2, respectively. The two rotation formulas we need to solve our example are the following: x1 = x1/cos (theta) – x2/sin (theta) and x/2 = – x1 sin (theta) + x2cos (theta). Using the rotation formula x1, we can find the first called for partial derivative. Thus, (p.x1/p.x/2) = –sin (theta). Using the rotational formula x/2, we can find the second called for partial derivative. Thus, (p.x/2/p.x2) = cos (theta). We have run into the dilemma: [–sin (theta)] [cos (theta)] should equal (p.x1/p.x2), in other words [–sin (theta)] [cos (theta)] should equal zero if the chain rule for partial differentiation isn’t being misused or at least confusingly used by L. Lieber. According to L. Lieber, (p.x1/p.x2) equals zero since as she falsely maintains or perhaps over simplistically maintains x1 doesn’t vary with x2, instead x1 varies with x/1 and x/2. But, in a circular fashion x/1 and x/2 vary x1 and x2.

The confusing use of the chain rule occurs in part because the x/2 in (p.x1/p.x/2) refers to the variable x/2 found in the equation x1 = x/1cos (theta) – x/2sin (theta) while the x/2 in (p.x/2/p.x2) doesn’t refer back to some form of the variable x/2 in the equation x1 = x1/cos (theta) – x2/sin (theta), but instead it refers to a different equation x/2 = – x1 sin (theta) + x2cos (theta). As we saw in our example of the chain rule in ordinary differentiation, the chain rule works when the u in (dy/du) and the u in (du/dx) are essentially the same and can be equated as in our example u = x2 +1. We could write the partial derivatives another way in which they didn’t resemble the chain rule. We could write the following: (p.f(x/1, x/2)/p.x/2) (p.f(x1, x2)/p.x2).

Now, let’s examine the manner in which, in many instances, but not all, (p.xn/p.x/a) (p.x/a/p.xk) generates zero in a space of two dimensions. We should recall that it is part of the larger formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn) in which we must sum on the indexes a, k, m, and n. We will concentrate on summing on the index a in two dimensional space. That means, we hold the other indexes constant and let the index a assume first the value 1 and then the value 2 then we add the two terms together. We have already determined the value of (p.xn/p.x/a) (p.x/a/p.xk) when a=2, k=2, n=1. Now, we can arbitrarily assign values to the other indexes: let b=1, and m=1. Since we have already determined the value of (p.x1/p.x/2) (p.x/2/p.x2) as [–sin (theta)] [cos (theta)] we can plug that

Let’s look at an example of L. Lieber’s argument. We will substitute numbers into the subscripts of the following formula: (p.xn/p.x/a) (p.x/a/p.xk) = (p.xn/p.xk). First, we will substitute values in which n doesn’t equal k, and we will see if the result is zero. Let n=1 and k=2 and a =2, thus the formula becomes the following: (p.x1/p.x/2) (p.x/2/p.x2) = (p.x1/p.x2). The formula reads as follows: the the partial derivative of the equation x with subscript 1 with respect to the primed variable x with subscript 2 multiplied by the partial derivative of the primed equation x with subscript 2 with respect to the variable x with subscript 2 equals the partial derivative of the equation x with subscript 1 with respect to the variable x with subscript 2.

The equations referred to are the rotation formulas for the primed and unprimed coordinates x and y in a space of two dimensions when the primed and unprimed coordinate x is replaced with the primed and unprimed coordinate x1, respectively and the primed and unprimed coordinate y is replaced by the primed and unprimed coordinate x2, respectively. The two rotation formulas we need to solve our example are the following: x1 = x1/cos (theta) – x2/sin (theta) and x/2 = – x1 sin (theta) + x2cos (theta). Using the rotation formula x1, we can find the first called for partial derivative. Thus, (p.x1/p.x/2) = –sin (theta). Using the rotational formula x/2, we can find the second called for partial derivative. Thus, (p.x/2/p.x2) = cos (theta). We have run into the dilemma: [–sin (theta)] [cos (theta)] should equal (p.x1/p.x2), in other words [–sin (theta)] [cos (theta)] should equal zero if the chain rule for partial differentiation isn’t being misused or at least confusingly used by L. Lieber. According to L. Lieber, (p.x1/p.x2) equals zero since as she falsely maintains or perhaps over simplistically maintains x1 doesn’t vary with x2, instead x1 varies with x/1 and x/2. But, in a circular fashion x/1 and x/2 vary x1 and x2.

The confusing use of the chain rule occurs in part because the x/2 in (p.x1/p.x/2) refers to the variable x/2 found in the equation x1 = x/1cos (theta) – x/2sin (theta) while the x/2 in (p.x/2/p.x2) doesn’t refer back to some form of the variable x/2 in the equation x1 = x1/cos (theta) – x2/sin (theta), but instead it refers to a different equation x/2 = – x1 sin (theta) + x2cos (theta). As we saw in our example of the chain rule in ordinary differentiation, the chain rule works when the u in (dy/du) and the u in (du/dx) are essentially the same and can be equated as in our example u = x2 +1. We could write the partial derivatives another way in which they didn’t resemble the chain rule. We could write the following: (p.f(x/1, x/2)/p.x/2) (p.f(x1, x2)/p.x2).

Now, let’s examine the manner in which, in many instances, but not all, (p.xn/p.x/a) (p.x/a/p.xk) generates zero in a space of two dimensions. We should recall that it is part of the larger formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn) in which we must sum on the indexes a, k, m, and n. We will concentrate on summing on the index a in two dimensional space. That means, we hold the other indexes constant and let the index a assume first the value 1 and then the value 2 then we add the two terms together. We have already determined the value of (p.xn/p.x/a) (p.x/a/p.xk) when a=2, k=2, n=1. Now, we can arbitrarily assign values to the other indexes: let b=1, and m=1. Since we have already determined the value of (p.x1/p.x/2) (p.x/2/p.x2) as [–sin (theta)] [cos (theta)] we can plug that

And even more on the tensor operation known as contraction.

into our formula along with the indexes we’ve chosen arbitrarily. We must recall we are summing on index a on both the left-hand and right-hand side of the equation. We should note that we can choose certain indexes arbitrarily with the caveat that we will cycle through all the possible combinations of the indexes all the while paying proper attention to the summation convention to arrive at the final answer. But, for now, we are examining only certain representative parts of the complete solution of the formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn).

With the index choices we have made we can now generate one of the partial representations of the formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). The partial representation is the following: A/111 + A/212 = (p.x1/p.x/2) (p.x/2/p.x2) (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x2) (p.x/1/p.x1) (A211). It seems confusing, but what is being done is that we are summing on the index a while holding all the other indexes at our arbitrarily chosen values. Also, I have begun the summation with the index a set at 2 and then proceeded to setting the index a at 1. This is only because I have already calculated the value we obtain when the index a equals 2. Thus we can rewrite the formula as A/111 + A/212 = [–sin (theta)] [cos (theta)] (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x2) (p.x/1/p.x1) (A211).

Now, we must determine the value of (p.x1/p.x/1) (p.x/1/p.x2). We need the rotation formulas for x1 and x/1. They are the following: x1= x/1cos (theta) – x/2sin (theta) and x/1= x1cos (theta) + x2sin (theta). We see that (p.x1/p.x/1) = cos (theta) and (p.x/1/p.x2)=sin (theta). Now, we can plug this value into the formula, and we obtain the following: A/111 + A/212 = [–sin (theta)] [cos (theta)] (p.x/1/p.x1) (A211) + [cos (theta)] [sin (theta)] (p.x/1/p.x1) (A211). This can be rewritten as the following: A/111 + A/212 = (p.x/1/p.x1) (A211) [–sin (theta) cos (theta) +cos (theta) sin (theta)] or A/111 + A/212 = (p.x/1/p.x1) (A211) [0] or A/111 + A/212 =0. This is an example of how summing on the index a when index k doesn’t equal index n allows many of terms and hence equations that determine the rank of the tensor to cancel each other. Let’s point out that [–sin (theta) cos (theta) +cos (theta) sin (theta)] = [cos (theta) sin (theta) – cos (theta) sin (theta)] =0. Let’s also point out that it is the number of partial derivatives that determine the rank of the tensor. Our formula has three partial derivatives, (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm), thus it is a tensor of rank three. That is to say it was a tensor of rank three before we began the operation of contraction. We have discovered that when the index k doesn’t equal n the partial derivatives equal zero. If this were the only possible outcome, the tensor would reduce to zero, which is a scalar which is also known as a tensor of rank zero.

Next, we will show how summing on index a when index k equals index n produces the value one where just previously we obtained zero. We will let index k equal one, and also we will let index n equal one. We will maintain the same values as we have previously for the values of the other indexes and we will sum on index a. The formula when subscript c has been replaced with subscript a is the following: A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). Next, we will replace the letter indexes with the number indexes that we indicated we would use, and the formula we obtain is the following: A/111 + A/212 = (p.x1/p.x/2) (p.x/2/p.x1) (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x1) (p.x/1/p.x1) (A211). As before, we will solve for (p.x1/p.x/2) (p.x/2/p.x1) and then we will solve for (p.x1/p.x/1) (p.x/1/p.x1).

To solve (p.x1/p.x/2) (p.x/2/p.x1), we need the rotational equation x1 and x/2. They are the following: x1= x/1cos (theta) – x/2sin (theta) and x/2 = – x1sin (theta) + x2cos (theta). Thus, (p.x1/p.x/2) = –sin (theta) and (p.x/2/p.x1) = –sin (theta) so (p.x1/p.x/2) (p.x/2/p.x1)= [–sin (theta)] [–sin (theta)] = sin2 (theta).

To solve (p.x1/p.x/1) (p.x/1/p.x1), we need the rotational formulas for x1 and x/1. They are the following: x1=x/1cos (theta) –x/2sin (theta) and x/1= x1cos (theta) + x2sin (theta). Thus, (p.x1/p.x/1)=cos (theta) and (p.x/1/p.x1)=cos (theta). Therefore, (p.x1/p.x/1) (p.x/1/p.x1)= cos2 (theta). We can plug these values back into our formula and we obtain the following: A/111 + A/212 = sin2 (theta) (p.x/1/p.x1) (A211) + cos2 (theta) (p.x/1/p.x1) (A211). This can be re-written as the following: A/111 + A/212 = (p.x/1/p.x1) (A211) [sin2 (theta) + cos2 (theta)]. This can be re-written as the following: A/111 + A/212 = (p.x/1/p.x1) (A211) [1].

Thus, we have encountered examples of the mechanism by which a mixed tensor of rank three in a space of two dimensions is reduced to a tensor of rank one through the operation of contraction. Every detail hasn’t been explained, for instance, the manner by

With the index choices we have made we can now generate one of the partial representations of the formula A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). The partial representation is the following: A/111 + A/212 = (p.x1/p.x/2) (p.x/2/p.x2) (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x2) (p.x/1/p.x1) (A211). It seems confusing, but what is being done is that we are summing on the index a while holding all the other indexes at our arbitrarily chosen values. Also, I have begun the summation with the index a set at 2 and then proceeded to setting the index a at 1. This is only because I have already calculated the value we obtain when the index a equals 2. Thus we can rewrite the formula as A/111 + A/212 = [–sin (theta)] [cos (theta)] (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x2) (p.x/1/p.x1) (A211).

Now, we must determine the value of (p.x1/p.x/1) (p.x/1/p.x2). We need the rotation formulas for x1 and x/1. They are the following: x1= x/1cos (theta) – x/2sin (theta) and x/1= x1cos (theta) + x2sin (theta). We see that (p.x1/p.x/1) = cos (theta) and (p.x/1/p.x2)=sin (theta). Now, we can plug this value into the formula, and we obtain the following: A/111 + A/212 = [–sin (theta)] [cos (theta)] (p.x/1/p.x1) (A211) + [cos (theta)] [sin (theta)] (p.x/1/p.x1) (A211). This can be rewritten as the following: A/111 + A/212 = (p.x/1/p.x1) (A211) [–sin (theta) cos (theta) +cos (theta) sin (theta)] or A/111 + A/212 = (p.x/1/p.x1) (A211) [0] or A/111 + A/212 =0. This is an example of how summing on the index a when index k doesn’t equal index n allows many of terms and hence equations that determine the rank of the tensor to cancel each other. Let’s point out that [–sin (theta) cos (theta) +cos (theta) sin (theta)] = [cos (theta) sin (theta) – cos (theta) sin (theta)] =0. Let’s also point out that it is the number of partial derivatives that determine the rank of the tensor. Our formula has three partial derivatives, (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm), thus it is a tensor of rank three. That is to say it was a tensor of rank three before we began the operation of contraction. We have discovered that when the index k doesn’t equal n the partial derivatives equal zero. If this were the only possible outcome, the tensor would reduce to zero, which is a scalar which is also known as a tensor of rank zero.

Next, we will show how summing on index a when index k equals index n produces the value one where just previously we obtained zero. We will let index k equal one, and also we will let index n equal one. We will maintain the same values as we have previously for the values of the other indexes and we will sum on index a. The formula when subscript c has been replaced with subscript a is the following: A/aba = (p.xn/p.x/a) (p.x/a/p.xk) (p.x/b/p.xm) (Akmn). Next, we will replace the letter indexes with the number indexes that we indicated we would use, and the formula we obtain is the following: A/111 + A/212 = (p.x1/p.x/2) (p.x/2/p.x1) (p.x/1/p.x1) (A211) + (p.x1/p.x/1) (p.x/1/p.x1) (p.x/1/p.x1) (A211). As before, we will solve for (p.x1/p.x/2) (p.x/2/p.x1) and then we will solve for (p.x1/p.x/1) (p.x/1/p.x1).

To solve (p.x1/p.x/2) (p.x/2/p.x1), we need the rotational equation x1 and x/2. They are the following: x1= x/1cos (theta) – x/2sin (theta) and x/2 = – x1sin (theta) + x2cos (theta). Thus, (p.x1/p.x/2) = –sin (theta) and (p.x/2/p.x1) = –sin (theta) so (p.x1/p.x/2) (p.x/2/p.x1)= [–sin (theta)] [–sin (theta)] = sin2 (theta).

To solve (p.x1/p.x/1) (p.x/1/p.x1), we need the rotational formulas for x1 and x/1. They are the following: x1=x/1cos (theta) –x/2sin (theta) and x/1= x1cos (theta) + x2sin (theta). Thus, (p.x1/p.x/1)=cos (theta) and (p.x/1/p.x1)=cos (theta). Therefore, (p.x1/p.x/1) (p.x/1/p.x1)= cos2 (theta). We can plug these values back into our formula and we obtain the following: A/111 + A/212 = sin2 (theta) (p.x/1/p.x1) (A211) + cos2 (theta) (p.x/1/p.x1) (A211). This can be re-written as the following: A/111 + A/212 = (p.x/1/p.x1) (A211) [sin2 (theta) + cos2 (theta)]. This can be re-written as the following: A/111 + A/212 = (p.x/1/p.x1) (A211) [1].

Thus, we have encountered examples of the mechanism by which a mixed tensor of rank three in a space of two dimensions is reduced to a tensor of rank one through the operation of contraction. Every detail hasn’t been explained, for instance, the manner by

Penultimate thoughts on the tensor operation known as contraction.

which A/111 + A/212 =C/1 hasn’t been explained. But, what has been made clear is that L. Lieber’s explanation of contraction involves the confusing use of the chain rule. By explaining the mechanism of contraction at the more generalized level of the partial derivatives, L. Lieber makes it appear that contraction would be a valid operation for all mixed tensors of at least rank three and above and in a space of two, three or more dimensions. As has been demonstrated contraction may only be valid in a space of two dimensions with a mixed tensor of rank three where the transformation of the coordinates involves the rotational formulas. It has also been demonstrated that it seems unlikely contraction would be a valid operation for spatial dimensions of three or more. This should undercut the mathematical foundations of Einstein’s theory of general relativity.

In closing, we should examine three items: the definition of the chain rule for partial differentiation, the curvature tensor, and contravariant and covariant vectors.

We should examine the formula given in the “Mathematics Dictionary” as the chain rule for partial derivatives. It is the following p.F/p.xp= Sigma (Summation from i=1 through n) (p.F/p.ui) (p.ui/p.xp). One of the examples given in Wikipedia under the heading chain rule should make this formula understandable. Wikipedia tells us that given u(x, y) = x2 + 2y where x(r, t) = r(sin(t)) and y(r, t) = sin2(t) then p.u/p.r = (p.u/p.x) (p.x/p.r) + (p.u/p.y) (p.y/p.r) = (2x) (sin(t)) + (2) (0) =2r (sin2(t)) and p.u/p.t = (p.u/p.x) (p.x/p.t) + (p.u/p.y) (p.y/p.t) = (2x) (rcos (t)) + (2) (2sin(t) cos(t)) = (2rsin(t)) (rcos(t)) + 4sin(t)cos(t) = 2(r2 +2) sin(t)cos(t) = (r2 +2) sin(2t).

In the example above, we could attempt to use L. Lieber’s argument and claim that both p.u/p.r and p.u/p.t equal zero since function u(x, y) varies with the variables x and y and not the variables r and t. But, we can see L. Lieber’s argument isn’t valid.

Let’s apply the formula for partial differentiation to a formula we have examined. Given u(x1, x2) = x1cos (theta) + x2sin (theta) or in its more familiar form x/1= x1cos (theta) + x2sin (theta) where x1= x/1cos (theta) – x/2sin (theta) and x2= x/1sin (theta) + x/2cos (theta). Now, using the formula for partial differentiation p.F/p.xp= Sigma (Summation from i=1 through n) (p.F/p.ui) (p.ui/p.xp) we obtain the following: p.x/1/p.x/2= (p.x/1/p.x1) (p.x1/p.x/2) + (p.x/1/p.x2) (p.x2/p.x/2) = [cos (theta)] [– sin (theta)] + [sin (theta)] [cos (theta)] =0 and p.x/1/p.x/ 1= (p.x/1/p.x1) (p.x1/p.x/1) + (p.x/1/p.x2) (p.x2/p.x/1) = [cos (theta)] [cos (theta)] + [sin (theta)] [sin (theta)] = cos2 (theta) + sin2 (theta) = 1. We can see that L. Lieber’s argument that p.x/1/p.x/2 =0 because function x/1 varies with the variables x1 and x2 and not the variable x/2, isn’t correct though it may superficially appear to be. We can see that L. Lieber’s argument that p.x/1/p.x/1 =1 for which surprisingly she doesn’t give an explanation also isn’t correct except under the circumstances where p.x/1/p.x/1 = cos2 (theta) + sin2 (theta). Using L. Lieber’s argument we could insist that p.x/1/p.x/1 =0 since function x/1 varies with the variables x1 and x2 and not the variable x/1.

It turns out the curvature tensor itself is worthy of examination. Since it is inner multiplication, i.e., the inner product form of contraction that allows the formation of the curvature tensor itself, the curvature tensor itself can be considered invalid. We recall that the curvature tensor is Bastr. The final step in the complex derivation of the curvature tensor is the following: Astr – Asrt = (Bastr) (Aa), which upon multiplication of the right-hand side of the equation gives us Caastr. Perhaps, the multiplication that gives us Caastr can be thought of as a kind of inner multiplication. Using the operation of contraction we obtain Cstr, and thus the rule for the addition and subtraction of tensors is maintained. The rule in this case is that if a tensor with three subscripts is subtracted from another tensor with three subscripts the answer must be another tensor with three subscripts. Thus, Astr – Asrt = Cstr.

It is interesting to note that L. Lieber maintains that the coefficient of a contravariant vector is the reciprocal of the coefficient of a covariant vector. That is to say p.x/m/p.xs is the reciprocal of p.xs/p.x/m. They do indeed seem to have the characteristics of a reciprocal unless we recall the numerator of a partial derivative refers to a function while the denominator refers to a variable. Actually, when using the rotational formulas in a space of two dimensions the same formulas are produced for both contravariant vectors and covariant vectors.

The formula for a contravariant vector is the following: A/m= (p.x/m/p.xs) As when we expand this formula using summation on the s index and using the rotational formulas for a space of two dimensions we obtain the following:

In closing, we should examine three items: the definition of the chain rule for partial differentiation, the curvature tensor, and contravariant and covariant vectors.

We should examine the formula given in the “Mathematics Dictionary” as the chain rule for partial derivatives. It is the following p.F/p.xp= Sigma (Summation from i=1 through n) (p.F/p.ui) (p.ui/p.xp). One of the examples given in Wikipedia under the heading chain rule should make this formula understandable. Wikipedia tells us that given u(x, y) = x2 + 2y where x(r, t) = r(sin(t)) and y(r, t) = sin2(t) then p.u/p.r = (p.u/p.x) (p.x/p.r) + (p.u/p.y) (p.y/p.r) = (2x) (sin(t)) + (2) (0) =2r (sin2(t)) and p.u/p.t = (p.u/p.x) (p.x/p.t) + (p.u/p.y) (p.y/p.t) = (2x) (rcos (t)) + (2) (2sin(t) cos(t)) = (2rsin(t)) (rcos(t)) + 4sin(t)cos(t) = 2(r2 +2) sin(t)cos(t) = (r2 +2) sin(2t).

In the example above, we could attempt to use L. Lieber’s argument and claim that both p.u/p.r and p.u/p.t equal zero since function u(x, y) varies with the variables x and y and not the variables r and t. But, we can see L. Lieber’s argument isn’t valid.

Let’s apply the formula for partial differentiation to a formula we have examined. Given u(x1, x2) = x1cos (theta) + x2sin (theta) or in its more familiar form x/1= x1cos (theta) + x2sin (theta) where x1= x/1cos (theta) – x/2sin (theta) and x2= x/1sin (theta) + x/2cos (theta). Now, using the formula for partial differentiation p.F/p.xp= Sigma (Summation from i=1 through n) (p.F/p.ui) (p.ui/p.xp) we obtain the following: p.x/1/p.x/2= (p.x/1/p.x1) (p.x1/p.x/2) + (p.x/1/p.x2) (p.x2/p.x/2) = [cos (theta)] [– sin (theta)] + [sin (theta)] [cos (theta)] =0 and p.x/1/p.x/ 1= (p.x/1/p.x1) (p.x1/p.x/1) + (p.x/1/p.x2) (p.x2/p.x/1) = [cos (theta)] [cos (theta)] + [sin (theta)] [sin (theta)] = cos2 (theta) + sin2 (theta) = 1. We can see that L. Lieber’s argument that p.x/1/p.x/2 =0 because function x/1 varies with the variables x1 and x2 and not the variable x/2, isn’t correct though it may superficially appear to be. We can see that L. Lieber’s argument that p.x/1/p.x/1 =1 for which surprisingly she doesn’t give an explanation also isn’t correct except under the circumstances where p.x/1/p.x/1 = cos2 (theta) + sin2 (theta). Using L. Lieber’s argument we could insist that p.x/1/p.x/1 =0 since function x/1 varies with the variables x1 and x2 and not the variable x/1.

It turns out the curvature tensor itself is worthy of examination. Since it is inner multiplication, i.e., the inner product form of contraction that allows the formation of the curvature tensor itself, the curvature tensor itself can be considered invalid. We recall that the curvature tensor is Bastr. The final step in the complex derivation of the curvature tensor is the following: Astr – Asrt = (Bastr) (Aa), which upon multiplication of the right-hand side of the equation gives us Caastr. Perhaps, the multiplication that gives us Caastr can be thought of as a kind of inner multiplication. Using the operation of contraction we obtain Cstr, and thus the rule for the addition and subtraction of tensors is maintained. The rule in this case is that if a tensor with three subscripts is subtracted from another tensor with three subscripts the answer must be another tensor with three subscripts. Thus, Astr – Asrt = Cstr.

It is interesting to note that L. Lieber maintains that the coefficient of a contravariant vector is the reciprocal of the coefficient of a covariant vector. That is to say p.x/m/p.xs is the reciprocal of p.xs/p.x/m. They do indeed seem to have the characteristics of a reciprocal unless we recall the numerator of a partial derivative refers to a function while the denominator refers to a variable. Actually, when using the rotational formulas in a space of two dimensions the same formulas are produced for both contravariant vectors and covariant vectors.

The formula for a contravariant vector is the following: A/m= (p.x/m/p.xs) As when we expand this formula using summation on the s index and using the rotational formulas for a space of two dimensions we obtain the following:

Final thoughts on the tensor operation known as contraction.

A/m= (p.x/m/p.xs) As when we expand this formula using summation on the s index and using the rotational formulas for a space of two dimensions we obtain the following: A/1= (p.x/1/p.x1) A1 + (p.x/1/p.x2) A2 from which we obtain A/1= cos (theta) A1 + sin (theta) A2 and A/2= (p.x/2/p.x1) A1 + (p.x/2/p.x2) A2 from which we obtain A/1= –sin (theta) A1 + cos (theta) A2.

The formula for a covariant vector is the following: A/m= (p.xs/p.x/m) As when we expand this formula using summation on the s index and using the rotational formulas for a space of two dimensions we obtain the following: A/1= (p.x1/p.x/1) A1 + (p.x2/p.x/1) A2 from which we obtain A/1= cos (theta) A1 + sin (theta) A2 and A/1= (p.x1/p.x/2) A1 + (p.x2/p.x/2) A2 from which we obtain A/1= –sin (theta) A1 + cos (theta) A2.

Thus, we can see that when using rotational formulas in a space of two dimensions a contravariant vector is essentially the same as a covariant vector. L. Lieber may want us to believe the relationship between the vectors is reciprocal in order to support her arguments concerning contraction.

The formula for a covariant vector is the following: A/m= (p.xs/p.x/m) As when we expand this formula using summation on the s index and using the rotational formulas for a space of two dimensions we obtain the following: A/1= (p.x1/p.x/1) A1 + (p.x2/p.x/1) A2 from which we obtain A/1= cos (theta) A1 + sin (theta) A2 and A/1= (p.x1/p.x/2) A1 + (p.x2/p.x/2) A2 from which we obtain A/1= –sin (theta) A1 + cos (theta) A2.

Thus, we can see that when using rotational formulas in a space of two dimensions a contravariant vector is essentially the same as a covariant vector. L. Lieber may want us to believe the relationship between the vectors is reciprocal in order to support her arguments concerning contraction.

What are your thoughts on the famous solar eclipse experiments of May 29, 1919 that measured the bending of the light rays from distant stars by the gravitational field of the sun?

The solar eclipse experiment was one of the three types of experiments that were suggested to provide rigorous tests for Einstein’s theory of general relativity. The results from the solar eclipse experiment suggested that general relativity was correct. The reporting of these results by many of the world’s leading newspapers helped catapult Albert Einstein into worldwide fame. The details of the experiment are difficult to understand.

It would be helpful to first to approximately calculate how large the disk of the sun is in the daytime sky. This approximate calculation is in degrees. We use the properties of triangles to make this calculation. We draw a line from the Earth to the center of the Sun. We say this distance is equal to 93,000,000 miles plus the radius of the Sun, which is 432,500 miles. We obtain 93,432,500 miles for the length of the longer leg of the right triangle we are constructing. The shorter leg of the right triangle is a line segment from the center of the sun to the edge of the solar disk that is perpendicular to the longer leg of the right triangle. Is length is 432,500 miles. Using the Pythagorean Theorem (a2 +b2 = c2) we can calculate the length of the hypotenuse of this triangle, which runs from the Earth to what we might view as the North Pole of the Sun. We have (93,432,500)2 + (432,500)2 = c2 = 8,729,819,112,250,000 and by taking the square root we obtain c = 93,433,501 miles. We now know all the lengths of the sides of the right triangle, and we know the degree measure of the right angle, which is 90 degrees. The acute angle that is the smallest angle of this right triangle gives us the degree measure of the radius of the solar disk as seen in the daytime sky, or in other words the degree measure of half the solar disk as seen in the daytime sky. Using the law of sines we can determine this acute angle. We should note that the angles A, B, and C are opposite of their respective sides so that angle C is opposite side c and angle B is opposite side b. Side c is the hypotenuse of the right triangle so angle C is the right angle of the right triangle. Side b is the radius of the Sun, and it is opposite the acute angle we are trying to calculate. The law of sines is a/sin A = b/sin B = c/sin C. We will use the portion c/sin C = b/sin B, which gives us 93,433,501/sin 90° = 432,500/sin B or since the sin 90° =1 we obtain 93,433,501 = 432,500/sin B. Therefore sin B = 432,500/93,433,501 or sin B = .004629. The angle with the sine of .004629 is the acute angle of .265°. This degree measure gives us the degree measure of half the solar disk as seen in the daytime sky so the entire solar disk would measure .53° in the daytime sky. This seems very small. I would have estimated the solar disk would be around 10°.

We can use a similar procedure to calculate the size of the moon in the sky. In this case, the sides of the right triangle are a = 239,937 and b = 1,080 and c = 239,939. So by the law of sines, we have 239,939/sin 90° = 1,080/sin B from which we obtain sin B = 1,080/239,939 = .0045. The acute angle whose sine is .0045 is an angle of .258°. That is the degree measure of half of the lunar disk as it appears in the sky so the entire lunar disk would measure .516° in the sky. The degree measure for the sun is .53° while the degree measure for the moon is .516°. So in this approximation, the apparent size of the Moon is .014° smaller than the apparent size of the Sun.

Can we convince ourselves that these approximations are accurate? Let’s take a paper disk with a radius of 1/8 inch. Let’s connect a very thin, rigid, steel rod, which is 30 inches long, perpendicularly to the center of this disk. The 30 inch long rod and a radius of the paper disk form the legs of a right triangle so by the Pathagorean Theorem the hypotenuse would be (30)2 + (1/8)2 = c2 = 900.0156 and therefore c = 30.0003. Using the portion of the law of sines, which we find useful, c/sin C =b/sin B we obtain 30.0003/sin 90° = (1/8)/sin B. Therefore sin B = (1/8)/30.0003 or sin B = .0042. The angle with the sine .0042 is .241° so the degree measure of the entire paper disk at a distance of 30 inches would be .482° which is slightly smaller than the degree measure for either the Sun or the Moon. This is interesting because if you go outside on the night of a full Moon you can make the following observation: close one eye and extend one arm then hold a slip of paper about ¼ of an inch wide in the hand of the arm you have extended. The slip of paper ¼ of an inch wide should cover the moon. This indicates that our approximations are roughly correct.

What use can we make of our approximation that the degree measure of the sun in the daytime sky is about .53°? It seems a typical measure of the degree of radial displacement of a ray of starlight by the gravitational field of the sun for the solar eclipse experiment of May 29, 1919 was .87 arc seconds. In viewing

It would be helpful to first to approximately calculate how large the disk of the sun is in the daytime sky. This approximate calculation is in degrees. We use the properties of triangles to make this calculation. We draw a line from the Earth to the center of the Sun. We say this distance is equal to 93,000,000 miles plus the radius of the Sun, which is 432,500 miles. We obtain 93,432,500 miles for the length of the longer leg of the right triangle we are constructing. The shorter leg of the right triangle is a line segment from the center of the sun to the edge of the solar disk that is perpendicular to the longer leg of the right triangle. Is length is 432,500 miles. Using the Pythagorean Theorem (a2 +b2 = c2) we can calculate the length of the hypotenuse of this triangle, which runs from the Earth to what we might view as the North Pole of the Sun. We have (93,432,500)2 + (432,500)2 = c2 = 8,729,819,112,250,000 and by taking the square root we obtain c = 93,433,501 miles. We now know all the lengths of the sides of the right triangle, and we know the degree measure of the right angle, which is 90 degrees. The acute angle that is the smallest angle of this right triangle gives us the degree measure of the radius of the solar disk as seen in the daytime sky, or in other words the degree measure of half the solar disk as seen in the daytime sky. Using the law of sines we can determine this acute angle. We should note that the angles A, B, and C are opposite of their respective sides so that angle C is opposite side c and angle B is opposite side b. Side c is the hypotenuse of the right triangle so angle C is the right angle of the right triangle. Side b is the radius of the Sun, and it is opposite the acute angle we are trying to calculate. The law of sines is a/sin A = b/sin B = c/sin C. We will use the portion c/sin C = b/sin B, which gives us 93,433,501/sin 90° = 432,500/sin B or since the sin 90° =1 we obtain 93,433,501 = 432,500/sin B. Therefore sin B = 432,500/93,433,501 or sin B = .004629. The angle with the sine of .004629 is the acute angle of .265°. This degree measure gives us the degree measure of half the solar disk as seen in the daytime sky so the entire solar disk would measure .53° in the daytime sky. This seems very small. I would have estimated the solar disk would be around 10°.

We can use a similar procedure to calculate the size of the moon in the sky. In this case, the sides of the right triangle are a = 239,937 and b = 1,080 and c = 239,939. So by the law of sines, we have 239,939/sin 90° = 1,080/sin B from which we obtain sin B = 1,080/239,939 = .0045. The acute angle whose sine is .0045 is an angle of .258°. That is the degree measure of half of the lunar disk as it appears in the sky so the entire lunar disk would measure .516° in the sky. The degree measure for the sun is .53° while the degree measure for the moon is .516°. So in this approximation, the apparent size of the Moon is .014° smaller than the apparent size of the Sun.

Can we convince ourselves that these approximations are accurate? Let’s take a paper disk with a radius of 1/8 inch. Let’s connect a very thin, rigid, steel rod, which is 30 inches long, perpendicularly to the center of this disk. The 30 inch long rod and a radius of the paper disk form the legs of a right triangle so by the Pathagorean Theorem the hypotenuse would be (30)2 + (1/8)2 = c2 = 900.0156 and therefore c = 30.0003. Using the portion of the law of sines, which we find useful, c/sin C =b/sin B we obtain 30.0003/sin 90° = (1/8)/sin B. Therefore sin B = (1/8)/30.0003 or sin B = .0042. The angle with the sine .0042 is .241° so the degree measure of the entire paper disk at a distance of 30 inches would be .482° which is slightly smaller than the degree measure for either the Sun or the Moon. This is interesting because if you go outside on the night of a full Moon you can make the following observation: close one eye and extend one arm then hold a slip of paper about ¼ of an inch wide in the hand of the arm you have extended. The slip of paper ¼ of an inch wide should cover the moon. This indicates that our approximations are roughly correct.

What use can we make of our approximation that the degree measure of the sun in the daytime sky is about .53°? It seems a typical measure of the degree of radial displacement of a ray of starlight by the gravitational field of the sun for the solar eclipse experiment of May 29, 1919 was .87 arc seconds. In viewing

Thoughts on the solar eclipse experiments continued.

reproductions of the solar eclipse photographs of May 29, 1919, it seems that the sun is about 2 inches in diameter. If we say that 2 inches is approximately .53°, then 1° is about 3.8 inches and 1 minute of arc is approximately .63 inches. One second of arc would be about .001 inches or about one thousandth of an inch. In the solar eclipse photographs from May 29, 1919, the readings for the star's radial displacement of are typically less than one second of arc. It seems that readings with a precision of about .1 arc seconds would be necessary. That would indicate precise measurements must be made on the order of .0001 inches or about one ten thousandth of an inch.

Increasing the Sun's size in the photograph would decrease the precision of the necessary measurements, but the decrease in the called for precision is not that great. For instance, increasing the diameter of the sun to 8 inches leaves us with one tenth of an arc second equal to .00042 inches or about four ten thousandths of an inch. Increasing the size of the Sun in the photograph to 16 inches leaves us with one tenth of an arc second equal to .00084 inches or about eight ten thousandths of an inch.

Einstein gives us an indication that very small measurements are called for in the solar eclipse experiments. He writes in his book Relativity: The Special and the General Theory on page 145 the following: “For a ray of light which passes the sun at a distance of delta sun-radii from its centre, the angle of deflection (a) should amount to a =1.7 seconds of arc/delta. It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical (“curvature”) of the space caused by the sun.” So if a typical prediction of the deflection of a ray of starlight in the eclipse experiments is .87 arc seconds, a typical Newtonian prediction for the deflection of a ray of starlight would be .435 arc seconds. It seems you would need measurements on the order of a tenth of and arc second to distinguish between the Newtonian and Einsteinian predictions.

Another difficulty with the eclipse experiments seems to arise when we try to determine the precise method by which the eclipse photograph and the standard photograph are compared. Einstein seems to indicate an informative but perhaps imprecise method of comparison on page 146 of his book Relativity: The Special and General Theory. He writes, “In practice, the question is tested in the following way. The stars in the neighbourhood of the sun are photographed during a solar eclipse. In addition, a second photograph of the same stars is taken when the sun is situated at another position in the sky, i.e. a few months earlier or later. As compared with the standard photograph, the positions of the stars on the eclipse-photograph ought to appear displaced radially outwards (away from the centre of the sun) by an amount corresponding to the angle a.” Before we continue we should clear up one point. On page 145 of his book Relativity: The Special and General Theory, Einstein writes, “As a result of this theory, we should expect that a ray of light which is passing close to a heavenly body would be deviated towards the latter.” When a ray of starlight is bent toward the sun by the sun’s gravitational field, it appears on photographs to have been displaced radially outwards, away from the center of the sun.

The imprecision in Einstein’s comparison seems to arise because no method is given that allows us to determine the term delta, as it appears in the equation for the angle a. Delta is the distance in sun-radii that a ray of starlight would pass from the position in the sky we denote as the (eclipse-photograph-moment) center of the sun. Since the sun is no longer in that portion of the sky, it is not there to displace the ray of starlight radially outward, away from the center of the sun. It is from delta that we calculate angle a. Do we use the second photograph of the same stars and then somehow place the (eclipse-photograph-moment) center of the sun onto the standard photograph to determine delta?

It seems that we cannot use the eclipse photograph itself to determine delta because a ray of starlight that just grazed the edge of the sun would have a delta value of one solar radius and it would be diplaced radially outward from the sun 1.7 arc seconds. If we used the eclipse photagraph of the star to determine delta we would obtain a value for delta that would be approximately 1.0018 solar radii instead of the correct value of one solar radius.

How do we determine delta using the second photograph that was taken when the sun was not present? If the stars in the eclipse photo formed a perfect circle and the sun was at the center of this perfect circle, it would be possible to calculate delta from the second photograph. The stars in the second photograph would also form a perfect circle.

Increasing the Sun's size in the photograph would decrease the precision of the necessary measurements, but the decrease in the called for precision is not that great. For instance, increasing the diameter of the sun to 8 inches leaves us with one tenth of an arc second equal to .00042 inches or about four ten thousandths of an inch. Increasing the size of the Sun in the photograph to 16 inches leaves us with one tenth of an arc second equal to .00084 inches or about eight ten thousandths of an inch.

Einstein gives us an indication that very small measurements are called for in the solar eclipse experiments. He writes in his book Relativity: The Special and the General Theory on page 145 the following: “For a ray of light which passes the sun at a distance of delta sun-radii from its centre, the angle of deflection (a) should amount to a =1.7 seconds of arc/delta. It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical (“curvature”) of the space caused by the sun.” So if a typical prediction of the deflection of a ray of starlight in the eclipse experiments is .87 arc seconds, a typical Newtonian prediction for the deflection of a ray of starlight would be .435 arc seconds. It seems you would need measurements on the order of a tenth of and arc second to distinguish between the Newtonian and Einsteinian predictions.

Another difficulty with the eclipse experiments seems to arise when we try to determine the precise method by which the eclipse photograph and the standard photograph are compared. Einstein seems to indicate an informative but perhaps imprecise method of comparison on page 146 of his book Relativity: The Special and General Theory. He writes, “In practice, the question is tested in the following way. The stars in the neighbourhood of the sun are photographed during a solar eclipse. In addition, a second photograph of the same stars is taken when the sun is situated at another position in the sky, i.e. a few months earlier or later. As compared with the standard photograph, the positions of the stars on the eclipse-photograph ought to appear displaced radially outwards (away from the centre of the sun) by an amount corresponding to the angle a.” Before we continue we should clear up one point. On page 145 of his book Relativity: The Special and General Theory, Einstein writes, “As a result of this theory, we should expect that a ray of light which is passing close to a heavenly body would be deviated towards the latter.” When a ray of starlight is bent toward the sun by the sun’s gravitational field, it appears on photographs to have been displaced radially outwards, away from the center of the sun.

The imprecision in Einstein’s comparison seems to arise because no method is given that allows us to determine the term delta, as it appears in the equation for the angle a. Delta is the distance in sun-radii that a ray of starlight would pass from the position in the sky we denote as the (eclipse-photograph-moment) center of the sun. Since the sun is no longer in that portion of the sky, it is not there to displace the ray of starlight radially outward, away from the center of the sun. It is from delta that we calculate angle a. Do we use the second photograph of the same stars and then somehow place the (eclipse-photograph-moment) center of the sun onto the standard photograph to determine delta?

It seems that we cannot use the eclipse photograph itself to determine delta because a ray of starlight that just grazed the edge of the sun would have a delta value of one solar radius and it would be diplaced radially outward from the sun 1.7 arc seconds. If we used the eclipse photagraph of the star to determine delta we would obtain a value for delta that would be approximately 1.0018 solar radii instead of the correct value of one solar radius.

How do we determine delta using the second photograph that was taken when the sun was not present? If the stars in the eclipse photo formed a perfect circle and the sun was at the center of this perfect circle, it would be possible to calculate delta from the second photograph. The stars in the second photograph would also form a perfect circle.

Final thoughts on the solar eclipse experiments.

It would be possible to determine the center of this circle and thus we could determine delta or how many solar radii each star was from the sun. The stars in the eclipse photographs of May 29, 1919 don’t form a perfect circle with the sun in the center instead they form a quite irregular pattern. So in the second photograph they will form a very irregular pattern, as well. Where do we situate the sun in this irregular pattern of stars in order to determine delta?

On further reflection, the stars in the eclipse photograph would not need to form a perfect circle around the sun in order to determine the proper center of the sun for the standard photograph. We would only need the two following conditions: 1. two stars that were the same distance from the sun and situated due north and due south 2. Two stars that were the same distance from the sun and situated due east and due west. There do not appear to be any pairs of stars that meet this condition in the eclipse photograph. It seems any two pairs of two equidistant stars that formed an orthogonal pattern could be used to determine the proper center for the sun in the standard photograph, but there do not seem to be any stars that meet this condition.

If we knew the precise coordinates of the sun at the precise moment when the eclipse photographs were taken, would this give us the information we needed? We would have the right ascension and declination of the sun, but how would this help us place the sun in the second photograph? The second photograph could have been taken when the stars in the eclipse photograph were at some distance from the precise coordinates of the sun when the eclipse photographs were taken. If we insist that the stars in the second photograph must be very near to the precise coordinates of the sun when the eclipse photographs were taken, we are still left with the question of precisely where should this irregular pattern of stars be in relation to the precise coordinates of the sun as it appeared in the eclipse photographs.

The stars in question with regard to the May 29, 1919 eclipse experiments are a portion of the constellation Taurus, the bull, known as the Hyades. They form a V-shaped cluster that marks the bull’s face. It is interesting that Taurus is the second sign of the zodiac, yet in May the sun is in Taurus. Since the zodiacal calendar begins on March 21 with Aries, the ram, and the second sign of the zodiac, Taurus, runs from April 20 to May 20, you might suspect that by May 29 the sun would be in Gemini, but the signs have slipped westward about one full division, because the position of the earth’s axis has changed in two millennium.

It seems the problem of placing the center of the Sun in its proper place in the standard photograph could be solved this way. The eclipse photographs were taken in the daytime in the spring. The astronomers were looking beyond the sun at what would be seen normally as the fall sky. Perhaps, one could find a fall day the length of whose night (sunset to sunrise) was equal to the length of the daylight (sunrise to sunset) on the spring day of May 29, 1919, the day of the eclipse. One could measure the time interval between sunrise and the taking of the eclipse photograph, and next allow the same time interval to pass after sunset of the fall day whose night is the length of the daylight length of the spring day May 29, 1919. At the appropriate moment take the standard photograph. This method assumes that the constellation Taurus would be rising in the eastern sky at sunset just as it was rising in the eastern sky at sunrise on May 29, 1919. This suggests another method to solve the problem.

Perhaps, another way to solve the problem would be to determine the day when the constellation Taurus rises in the eastern sky at the same time that the sun sets in the west. As Taurus makes its way across the sky wait the same interval of time as between sunrise on May 29, 1919 and the taking of the eclipse photograph, then at the appropriate moment take the standard photograph.

On further reflection, the stars in the eclipse photograph would not need to form a perfect circle around the sun in order to determine the proper center of the sun for the standard photograph. We would only need the two following conditions: 1. two stars that were the same distance from the sun and situated due north and due south 2. Two stars that were the same distance from the sun and situated due east and due west. There do not appear to be any pairs of stars that meet this condition in the eclipse photograph. It seems any two pairs of two equidistant stars that formed an orthogonal pattern could be used to determine the proper center for the sun in the standard photograph, but there do not seem to be any stars that meet this condition.

If we knew the precise coordinates of the sun at the precise moment when the eclipse photographs were taken, would this give us the information we needed? We would have the right ascension and declination of the sun, but how would this help us place the sun in the second photograph? The second photograph could have been taken when the stars in the eclipse photograph were at some distance from the precise coordinates of the sun when the eclipse photographs were taken. If we insist that the stars in the second photograph must be very near to the precise coordinates of the sun when the eclipse photographs were taken, we are still left with the question of precisely where should this irregular pattern of stars be in relation to the precise coordinates of the sun as it appeared in the eclipse photographs.

The stars in question with regard to the May 29, 1919 eclipse experiments are a portion of the constellation Taurus, the bull, known as the Hyades. They form a V-shaped cluster that marks the bull’s face. It is interesting that Taurus is the second sign of the zodiac, yet in May the sun is in Taurus. Since the zodiacal calendar begins on March 21 with Aries, the ram, and the second sign of the zodiac, Taurus, runs from April 20 to May 20, you might suspect that by May 29 the sun would be in Gemini, but the signs have slipped westward about one full division, because the position of the earth’s axis has changed in two millennium.

It seems the problem of placing the center of the Sun in its proper place in the standard photograph could be solved this way. The eclipse photographs were taken in the daytime in the spring. The astronomers were looking beyond the sun at what would be seen normally as the fall sky. Perhaps, one could find a fall day the length of whose night (sunset to sunrise) was equal to the length of the daylight (sunrise to sunset) on the spring day of May 29, 1919, the day of the eclipse. One could measure the time interval between sunrise and the taking of the eclipse photograph, and next allow the same time interval to pass after sunset of the fall day whose night is the length of the daylight length of the spring day May 29, 1919. At the appropriate moment take the standard photograph. This method assumes that the constellation Taurus would be rising in the eastern sky at sunset just as it was rising in the eastern sky at sunrise on May 29, 1919. This suggests another method to solve the problem.

Perhaps, another way to solve the problem would be to determine the day when the constellation Taurus rises in the eastern sky at the same time that the sun sets in the west. As Taurus makes its way across the sky wait the same interval of time as between sunrise on May 29, 1919 and the taking of the eclipse photograph, then at the appropriate moment take the standard photograph.

What have you learned about sidewalks recently?

I was walking along some side streets in Elmwood Park that were close to Route 46. Walking west on Martha Avenue, I crossed over Fleisher Brook catching a glimpse of a Baltimore oriole where the brook ran under the street, and then turning north onto Miles Street, I crossed over the Garden State Parkway. In the vicinity of Miles Street there is an elementary school. I noticed that the sidewalks around the school were new, but despite that, some of the ends of the slabs of cement had heaved up and were an inch or so higher than their neighbor. Strangely, the edges of the raised slabs of cement had been ground by some power tool so that they were once again flush with their neighboring slabs. I suspect this was done because the sidewalks were were near the elementary school where many young and trip-prone children walked and ran. The new slabs of cement that made up the sidewalk were white and not profusely dotted with the little, black, angular stones that you see on older sidewalks. I had assumed that new sidewalks were made of cement that did not have little stones added to it, but a close examination of the ground down portions of the new sidewalk revealed that it to was profusely dotted with little, black stones. It seemed remarkable that everyone of the little, black stones sank below the surface of the cement and was only revealed when the surface of the sidewalk had been ground away.

You claim your sarcastically humorous book description alienated many potential readers; please, give an example.

This e-book is free, and it's still overpriced. You should be paid money—a large sum of money—for even considering to read this e-book. The emoluments deserved by the brave readers who actually read this e-book from beginning to end include tax refunds, bearer bonds, precious metals and a balanced portfolio managed by experts. The government should step in and reward you—no better yet it should guarantee your financial security. It’s the fair thing to do. Honestly, "Fathoming Gödel" doesn’t make sense, but then again what does? Will it make sense to readers in the future? I don’t know; my time machine is broken. I would need to get parts for a 1965 Triumph MG to make it work, and I’m afraid if I did so some Phileas Fogg wannabe would sue me for plagiarism. I wish I could use T.S. Eliot’s excuse. When readers opined that his seminal poem "The Waste Land" made no sense, he demurred that his friend, the American Ezra Loomis Pound, had convinced him to remove many explanatory lines. Unfortunately, I don’t have any friends who are crazy, Fascist sympathizers at least not any that could convince me to remove portions of the text so it made less sense. Ezra Pound after writing "Cantos," which some critics claim is the most important long poem in modern literature convinced himself that he was the spider in "Charlotte’s Web."

A frequently asked question is do you need to understand the principles that explain the functioning of the vernier to understand Gödel’s theorems. The answer is no; all that you need to know is that it is an instrument used in measuring lengths and angles. It is named for Pierre Vernier, a French mathematician who invented it in the 1600’s.

Was Kurt Gödel a predecessor of the modern day hacker? His work is certainly as obscure and mysterious as a hacker’s work only Gödel didn’t break into secure computer systems and reveal sensitive data or foul up the workings of the computer systems. Perhaps, instead he burrowed his way into the secure intellectual system of mathematics and wreaked havoc.

Who is the predecessor of Gödel? Is it possible to speculate that brooms were once the equivalent of smart phones? Were brooms devices of advanced technology that were designed to make life easier? Smooth wooden floors would be the equivalent of the internet. Would that make witches the predecessors of hackers? Individuals that used advanced technology to disrupt the social order.

Or, is it all more prosaic, a kind of fashion. Walking down an old railroad track, you can see that to securely fasten a nut a hole was drilled in the bolt above the nut and a cotter pin was inserted into the hole. Walking under new high tension towers, you can see the modern method of securely fastening a nut involves special kinds of lock washers.

A frequently asked question is do you need to understand the principles that explain the functioning of the vernier to understand Gödel’s theorems. The answer is no; all that you need to know is that it is an instrument used in measuring lengths and angles. It is named for Pierre Vernier, a French mathematician who invented it in the 1600’s.

Was Kurt Gödel a predecessor of the modern day hacker? His work is certainly as obscure and mysterious as a hacker’s work only Gödel didn’t break into secure computer systems and reveal sensitive data or foul up the workings of the computer systems. Perhaps, instead he burrowed his way into the secure intellectual system of mathematics and wreaked havoc.

Who is the predecessor of Gödel? Is it possible to speculate that brooms were once the equivalent of smart phones? Were brooms devices of advanced technology that were designed to make life easier? Smooth wooden floors would be the equivalent of the internet. Would that make witches the predecessors of hackers? Individuals that used advanced technology to disrupt the social order.

Or, is it all more prosaic, a kind of fashion. Walking down an old railroad track, you can see that to securely fasten a nut a hole was drilled in the bolt above the nut and a cotter pin was inserted into the hole. Walking under new high tension towers, you can see the modern method of securely fastening a nut involves special kinds of lock washers.

Reflections on the Michelson-Morley Experiment and the Ineluctable Self-Interview
by Jim Spinosa

Price:
Free!
Words: 56,260.
Language:
English.
Published: July 21, 2017.
Categories:
Nonfiction » Science and Nature » Physics, Essay » Author profile

This free e-book consists of two books: the first is the very short “Reflections on the Michelson-Morley Experiment” and the second is “The Ineluctable Self-Interview,” which is longer but brightened by patches of humor. The high points of the interview are serious attempts to falsify Albert Einstein’s theory of general relativity. The interview ends scrutinizing covariant differentiation.

Fathoming Gödel
by Jim Spinosa

Price:
Free!
Words: 20,010.
Language:
English.
Published: October 22, 2015.
Categories:
Nonfiction » Science and Nature » Nature, Nonfiction » Science and Nature » Mathematics

(1.00)
The conclusion reached in "Fathoming Gödel" is that Gödel's 1931 paper is a shell game. It is based on several errors that are well camouflaged. Some shortcomings in the paper are openly admitted although they are downplayed, and errors are also produced in an effort to force a particular conclusion. This critique is limited to Gödel's first incompleteness theorem as translated by Martin Hirzel.

Bell's Inequality Untwisted
by Jim Spinosa

Price:
Free!
Words: 35,750.
Language:
American English.
Published: September 24, 2014.
Categories:
Nonfiction » Science and Nature » Physics

“Bell’s Inequality Untwisted” is a unique book. The author’s aim is to explain in detail all the equations and statements in John S. Bell’s ground-breaking paper “On the Einstein Podolsky Rosen Paradox.” He attempts an in depth explanation of Bell’s paper that is understandable to a wide audience. As the explanation proceeds, it becomes clear that Bell’s paper is a series of incoherent equations.

Nuts And Bolts: Taking Apart Special Relativity
by Jim Spinosa

Price:
Free!
Words: 46,210.
Language:
English.
Published: October 6, 2010.
Categories:
Nonfiction » Science and Nature » Physics

(1.00)
Nuts and Bolts: Taking Apart Special Relativity is an attempt to disprove Einstein's theory of special relativity. It is written to appeal to a wide audience. Nuts and Bolts explains the formidable equations of special relativity in unprecedented detail. Soon everyone will conclude that special relativity is invalid, and I mean soon in the geological sense time, may science make us immortal.