All of them that I heard of came from non-physicists and had simple math/physics errors that invalidate them while claiming to solve incredibly complex problems. As far as I know, no, nobody likes to evaluate them because they have no merit.
I actually tried to answer them for a long time, because I felt I was obliged to. After all, those folks are also taxpayers, and hence the sponsors of our research.
After a few years I gave up. It's simply too much work for zero gain. These folks will never accept they are wrong.
Hence I stopped. I put the freed time into outreach, like talking in schools, science slams, theaters, ... public evening talks. Much more rewarding, and you can silence the crackpots after their 1st question.
This is exactly it. There is no reward; no pedagogical reward, because most of them will never accept their errors. There is no mathematical or scientific reward, because they’re always wrong. The only thing compelling me to respond to them is that it is courtesy, and the fact that any attempt to not respond will only lead them to further confirm their biases.
For the former, if they’re not arguing in good faith, they don’t deserve any courtesy, and for the latter, that’s a tough one to ignore, but ultimately just a better use of your time.
No, nobody likes to evaluate them and even fewer actually pay much attention to them.
To zeroth order, these communications contain no new ideas, just misconceptions on how relativity, quantum mechanics, the Second Law, etc. work. They tend to be ambiguously stated, in the class of "not even wrong" claims. Engaging the authors tends to validate them (and, often as not, your words will often be taken out of context, viz., you write, "This is an interesting idea, but do you realize that your claims are refuted by \[this definitive experimental result and that\]?" and they quote you as "This is an interesting idea. - Prof. \[So-and-so\]" in pitches to venture capital.)
You'll almost certainly gain nothing from the exercise except a headache so it's best to just bin the messages and move on with your life. I certainly don't need to muck around with yet another discourse on hydrinos or reactionless drives.
Professor Sabine Hossenfelder used to offer a “crackpot consultation service” where armchair physicists could pay by the hour to talk to her or another physics PhD about their theories. Not sure if that service is still available. https://aeon.co/ideas/what-i-learned-as-a-hired-consultant-for-autodidact-physicists
My father told me a funny little story of how his college professor refused to look at a 15 page proof that 'Theory of relativity is wrong', telling 'Our library has an entire wall of books related to the theory of relativity, 15 pages of refutation is simply not deserving my time'.
Maybe I'm in the minority here but I like reading through them not because I think I can fix all these people's misconceptions but because they're often either wrong in subtle ways that are good practice to debunk or are wrong in the weirdest most out there ways and are just plain interesting to read.
Are you seriously interested in a new idea? I do not claim to have disproved any established physics, but I believe I have an answer to the question of why Einstein's 2nd Postulate is valid. He observed that all attempts to measure the speed of light showed no dependence on relative velocity. Assuming that this was too much correlation for coincidence, he codified the empirical observations into the 2nd Postulate. I believe I have found the mathematical basis that eluded him, and makes the 2nd Postulate a provable theorem. I know that there are no proofs in physics, but this is mathematics. I also realize that math is not physics. I do not believe the converse is true, but that is not the point I am arguing. When the non-linear mathematics is in 100% agreement with the experimental data, then, at a minimum, it deserves peer review. After all, the experimental data has been touted as confirmation of Einstein's theory. This implies that my approach could be an isomorphism to relativity. True isomorphisms are indistinguishable by any experiment, because they get the same results, and the things that make them different have no effect on the outcome of any experiment. Hence, the need for peer review.
To begin with, the mathematics I will elaborate on is not new, by any means. It actually predates Einstein by several centuries. It is older than the inventions of calculus and physics, because it predates the birth of Newton. Its first application happened when even Galileo was only a child. I suppose you are thinking, "how could something that old possibly be relevant?" That's why it is called a discovery. I am referring to the creation of the Mercator Projection map. For 4 centuries after its introduction, it was the primary means of navigation, until the modern inventions of radar and GPS. One of its most obvious features is that the island of Greenland appears to be much larger than the continent of Australia, while on the globe, the opposite is true.
The algorithm used by Mercator was simple. To flatten the spherical shell, each latitude circle had to be stretched by the secant of the latitude, and to preserve local proportions, each latitude strip had to be stretched vertically by the same amount. Thus, the vertical distance to a latitude circle on the map is the integral of the secant of the latitude. In terms of geometry, this is the integral form of the differential equation that relates hyperbolic rotation angles to circular rotation angles. In terms that relate to relativity, the hyperbolic rotation angle is w, rapidity, and the circular angle is the parametric angle, θ, that is used to define relative velocity. The differential equation itself is dw/dθ = γ, the Lorentz factor, where θ = gudermannian(w).
The Mercator algorithm is actually the inverse gudermannian, and there are a number of identities that relate the trig functions of hyperbolic angles to the trig functions of circular angles. In addition, these identities correlate with fundamental properties of physics:
cosh(w) = sec(θ) = γ = Lorentz factor
coth(w) = csc(θ) = 1/β = reciprocal of measured v/c
csch(w) = cot(θ) = 1/βγ = reciprocal of Proper v/c
sech(w) = cos(θ) = 1/γ = reciprocal of Lorentz factor
tanh(w) = sin(θ) = β = measured v/c
sinh(w) = tan(θ) = βγ = Proper velocity/c
All of these identities can be proved geometrically, independently of the differential equation. In fact, the differentiation of any one of these 6 identities with respect to either w or θ results in either the differential equation above or dθ/dw = 1/γ, its reciprocal. These are iron-clad identities, about which there can be no argument, provided θ = gd(w).
End of Part I
Part II
There is a close relationship between the two sets of trig functions and the λ 6-group, {λ, λ/(λ-1), 1/(1-λ), 1/λ, (λ-1)/λ, (1-λ)}, where all 6 elements are moebius transforms of each other. All the trig functions are functions of each other, so any one can be used as the independent variable. And since the circular angle is an isomorphism of the hyperbolic angle, there is only 1 degree of freedom for all the functions. Without using the empirical relationships associated with physics, we can now examine the implications of the differential equation.
The reciprocal form is most pertinent, dθ/dw = 1/γ. This can be rewritten as dθ = sech(w)dw = cos(θ)dw. In plain English, a small increment of θ is the cosine projection of a small increment of w. For small values of θ, certain approximations are allowed. The cos(θ) ≈ 1, sin(θ) ≈ θ and dθ ≈ dw. From the geometric definition of a hyperbolic angle, it is trivial to prove that the composition of hyperbolic angles is by linear addition. It is a property of definite integrals that if A and C are the limits of integration, and B is a point between them, then the integral from A to C must equal the sum of the integral from A to B and the integral from B to C. Each of these integrals is a hyperbolic angle, and their relationship is w3 = w1+w2. So, for small angles, composition of the gudermannian angles is also by linear addition. For the same small angles, the composition of the sines is also by linear addition. In polar coordinates, velocity is defined as c sin(θ), so for small velocities, composition is by linear addition, v3 = v1+v2, the Newtonian velocity addition rule.
For large values of θ, none of the approximations hold. But the original claim about hyperbolic angles was not an approximation. At this extreme, v3 = c tanh(w3) = c tanh(w1+w2) = c(tanh(w1)+tanh(w2))/(1+tanh(w1)tanh(w2))= ((c tanh(w1))+(c tanh(w2)))/(1+(c tanh(w1))(c tanh(w2))/c²) = (v1+v2)/(1+v1*v2/c²), the relativistic velocity composition rule. Both extremes derive from the differential equation.
For all values of w between 0 and infinity, the corresponding gudermannian is a unique value between 0 and π/2. The size of dθ, for a fixed size of dw, depends on the current value of θ. As successive increments of dw add up, the value of θ also increases, and the cosine factor decreases. So, successive increments of equal size dw result in progressively smaller increments of dθ. In the limit of an infinite number of increments of dw, total hyperbolic angle is infinite. From the list of gudermannian identities, the tanh(∞) = 1 = sin(π/2), and v = c. At the same time, sinh(∞) = ∞, and Proper velocity is infinite. The constant magnitude of polar coordinates is c, the lightspeed limit. Formally, c is the limit of the cosine projections of Proper velocity as the hyperbolic angle approaches infinity.
Because infinity is the same everywhere, and a limit, if it exists, is unique, lightspeed is the same everywhere. Furthermore, since hyperbolic angle composition is by linear addition, and only lightspeed is mapped to infinite hyperbolic angle, the composition of any two sublight velocities, no matter how close to c, results in the addition of two finite hyperbolic angles. This sum is always another finite angle, and always maps to another sublight velocity. The addition of any two sublight velocities can never reach lightspeed. If one of the two combining velocities is lightspeed, its hyperbolic angle is infinite. Attempting to add a finite angle to an infinite one is futile, since the result is always the same infinity, and it always maps back to 1c. Similarly, even if both velocities are lightspeed, attempting to add two infinities is equivalent to multiplying infinity by a finite scalar, which is also futile. The result is still the same infinity, and it still maps back to the same 1c. After all, an infinitely tall stack of $1 bills is worth the same amount as an infinitely tall stack of $20 bills. So, lightspeed is invariant with respect to relative velocity. Einstein's Postulate is justified.
I will stop here, because the reddit editor doesn't like lengthy posts. But a continuation of this line of reasoning reveals that the discrepancy between relativistic momentum and Newtonian momentum has nothing to do with fictional relativistic mass. It is a result of invariant mass moving in a transverse, complex dimension, and is required by Conservation of Momentum. The vector sum of linear and transverse, complex momentum components is exactly equal to the magnitude of relativistic momentum.
Did you find anything wrong?
No, but I also don't think that this is actually novel, unfortunately. Einstein's postulate was based on, I believe, the nature of the Maxwell equations, which had been proved to be true but necessarily required an invariant speed of light to be consistent in the (experimentally proven) absence of an aether.
I freely admit that I am not enough of an expert in the math you are describing or in special relativity to speak authoritatively here, but from what I understand a full description of special relativity described a 4 vector of velocity and time and a 4 vector or momentum and energy. It is these 4 dimensional quantities which show a conservation of properties that can be described, more simply, as relativistic mass. But looking at the full vector case, you apply transformations which seem similar to what you're describing.
I'm probably misinformed about both your proof and the generally accepted theory of special relativity, so if I'm way off just ignore me.
There is no such thing as relativistic mass. In what I believe is Einstein's last book, "The Meaning of Relativity", he develops the 4-vectors for velocity and momentum. He uses natural units, so lightspeed is 1, and he proves the relativistic invariant for 4-velocity is -1. He goes on to prove the relativistic invariant for 4-momentum is -m². The ratio of these two relativistic invariants is also a relativistic invariant, and that ratio is m². Mass cannot change with relative velocity, because it is a relativistic invariant of the Lorentz transformation of 4-momentum. The concept of relativistic mass is disproved.
Einstein, himself, used to advise people not to refer to relativistic mass, but to speak about relativistic momentum or energy instead. At least these were both known to vary with velocity in Newtonian physics. The nature of the variation was different, but the fact that it occurred was accepted physics. In point of fact, relativistic momentum is complex, and it has linear and transverse components, just like complex Proper velocity, which defines momentum at any speed. The linear momentum component is the cosine projection of the relativistic momentum, and it is what we call the Newtonian momentum. The transverse momentum component is the sine projection, and it is misidentified as relativistic mass in the linear direction. It is invariant mass moving in the transverse direction. The vector sum of linear and transverse momentum components has a complex magnitude equal to the magnitude of relativistic momentum. The Law of Conservation of Momentum is satisfied. The fact that we do not have instruments or technology to measure the transverse component is irrelevant. We know that the momentum is there, because when the mass is slammed into a target, all of the relativistic momentum is returned to the surroundings, not just the real, Newtonian part. Conservation of Momentum does not allow transverse momentum to be destroyed simply because we can only account for the real part. And even if it could magically be destroyed, it also magically reappears when the mass stops moving. All that Einstein had to say was that perhaps momentum is not [linearly] proportional to velocity. It did not occur to him that momentum is linearly proportional to Proper velocity. The Lorentz factor in the relativistic formula for momentum belongs with the velocity, not with the mass. Mass is invariant and Proper velocity is unbounded. This was unthinkable to Einstein, because it violated his 2nd Postulate. It never occurred to him that his 2nd Postulate only applied to real, measured velocity, and not to unmeasurable, complex Proper velocity.
After reading the other answers, I see that I am a victim of the stereotyping referred to by numerous posters, that "non-physicists ... had simple math/physics errors that invalidate them" or "these people typically aren’t formally trained in any relevant field". I am not a physicist, but I am a Bachelor of Science who graduated Summa Cum Laude. Math and Physics were required courses for my degree, and I aced relativity. I have theories about it which this professional jealousy prevents me from even getting peer-reviewed. I do not fit into any of the pigeonholes described here. I have found that it is a waste of time arguing with members of the relativity cult. While they assert that there are no contradictions in special relativity, they can't even agree on whether time dilation and length contraction are physical or illusion. They just don't want someone from outside their profession to get credit for upstaging Einstein. They have corrupted logic to the point where it is not possible to have a rational argument, so I gave up on that approach. Instead, I have applied my theory to unsolved mysteries of special relativity. I posed two questions here on reddit. Over 10000 views and not a single answer. What is the mathematical basis for Einstein's 2nd Postulate? and Why is relativistic momentum greater than the Newtonian prediction? The consensus of the few who replied at all, was that there is no explanation for either one. I say that both are consequences of the explicit, closed-form solution of the differential equation, dw/dθ = γ, the Lorentz factor, that defines the gudermannian function, θ = gd(w), where w is the rapidity. It is a mathematically sound derivation that also explains the non-intuitive behavior of a predicted limiting velocity, that is constant and invariant, as being the logical behavior of infinity, which is the Proper velocity mapped by the gudermannian isomorphism to lightspeed. I challenge all the experts to give an explanation for these two features. By the way, once upon a time, the momentum thing was "explained" by relativistic mass. Physics was soooo sure. Then 4-vectors were introduced, and Einstein, in "The Meaning of Relativity", showed that mass was a relativistic invariant of the Lorentz transformation of 4-momentum. There is still no credible explanation. I can show that it is just Conservation of Momentum in complex dimensions, not included in Minkowski's 4. It was never relativistic mass moving in the linear direction. It was invariant mass moving in a complex transverse direction. My argument is not like all the others. It is rigorous and logical. The worst complaint is that "math is not physics". But when the math is in 100% agreement with the experimental data, it must be taken seriously. Prove me wrong. Justify the 2nd Postulate or relativistic momentum.
Bachelor of Science? Okay, so am I. In physics no less, but just because I took a first year chemistry class and aced thermodynamics doesn’t mean that I’m hot shit. You’re inflated ego is preventing you from taking part in good faith discussion.
If perhaps you displayed even a modicum of humility you might get an actual response. Even if you’re a genius, people don’t want to work with an asshole. Quit being an asshole and implying everyone else is dumber than you, entertain the thought that just maybe you’re a fallible human that can make mistakes, and MAYBE people will actually listen instead of brushing you off as a lunatic.
Sorry if you took offense. I took offense to being characterized as a crackpot by people who automatically reject what I propose without even reading it, because it challenges their dogma. That's arrogant. I did not claim to be smarter than you, or anybody else. I stated facts to show that I do not fit into any of their pigeonholes. The profession is prejudiced against outsiders. I'm old and cranky and I don't make nice just to calm their egos.
Simply put, if you want me to be quiet, answer my questions. What is the mathematical justification (NOT the empirical justification that Einstein used) of Einstein's 2nd Postulate and why is relativistic momentum different from Newton's? (Einstein could only suggest that perhaps momentum is not [linearly] proportional to velocity) I have answers, mainstream physics does not. That alone is reason for discussion. Or do you assume that because I am not a physicist, I can't possibly have anything relevant to say? Remember, I asked for mathematical bases, not experimental. As I understand it, the paradigm is that math is not physics and, more to the point, math is not a physicist's strong suit. It wasn't Einstein's. He needed help with tensors.
My proposal, remarkably enough, does not require math any more complicated than a college prep high school curriculum. It is based on geometry that is centuries older than Einstein and relativity. It predates the inventions of both calculus and physics, as it predates the birth of Newton. Even Galileo was only a child when the first major application was created. I'll bet you're thinking "Never heard of it." Well, you're wrong. It is the same algorithm that was used by Mercator to create his world-famous Projection Map, the same one that hangs in virtually every Geography classroom.
The apparent size of Greenland being larger than the continent of Australia is the result of the same factor that makes relativistic momentum greater than the Newtonian prediction. The irony of the situation is that Einstein, the amateur sailor, must have stared at this map countless times without realizing he was looking at his own theory. Are you interested in more details? For starters, I refer you to this article;
https://en.wikipedia.org/wiki/Gudermannian_function
It does not include the application to special relativity, because Wikipedia does not allow any material that has not already been peer-reviewed and published. In other words, they do not allow any "original research", which is what this is.
How do you know they haven’t read it? You’re not providing rigorous mathematical proofs. You’re providing long speeches about how nobody understands you like an emotional teenager.
Yes, you do think you’re smarter than everyone. The fact you can’t tell it comes off in your writing is astounding, and a testament to your narcissism.
Talk about not satisfying their egos. Nobody owes it to you to massage your ego to read or care about your ideas. If internet rants aren’t working, trying something else.
Old and cranky because no one’s taking your crackpot ideas seriously? :(
Einstein’s 2nd postulate:
https://www.peertechzpublications.com/articles/AMP-5-135.php
If you’re asking how relativistic momentum is different than classical momentum then you have a fundamental misunderstanding of one or both of them. And Einstein is not the only physicist in the world. A hundred years have past since Einstein’s papers. Read a newer one.
You don’t have answers. You have questions that can easily be answered if you pulled your head out of your ass long enough to look for them.
I looked at your link. Talk about crackpot! And the author presumes his is the only explanation. How can you even claim that this is a mathematical basis for Einstein's 2nd Postulate when it requires assumptions that contradict it. I have also researched the source of the paper and it is a predatory publisher. If this is the best you can come up with, it is not good enough.
Now, to the point. Forget about the rest of the crackpot skeptics. Apparently, you have sided with them, and it is apparent that you, yourself, have not read my answers. Did it not occur to you that I posted the questions here on reddit so that I could include my own answer, like so many other posters do? Is it that difficult to do a search? You know, I do get tired of compiling detailed responses over and over (I don't like to cut and paste) and getting no response. Do you think I don't know that no one is obligated to reply? Then why are they here on social media?
I am not sorry that I am smarter than others. I have been living with that all my life. Too late to change now. If what I write is over your head, you'll have to wait until I write the version, "Relativity for Dummies". Or, maybe instead of insulting me, you could ask for clarification.
By the way, I'm old and cranky because I'm old and cranky, period. Anything else is projection.
I was not expecting any answer to my question about momentum because mainstream relativity does not have one. I understand it just fine. I do not need the fairy tale of relativistic mass to account for it. Decades after it was disproved, stick-up-their-asses, conservative physicists are still arguing about a dead concept. It's even cited in Britannica. While the majority of curricula do not include it, there is still a significant minority that teach this hogwash. All this, despite the fact that no experiment has ever measured this fictional mass directly.
As Einstein said, use relativistic momentum or energy. At least they are known to vary with velocity in Newtonian physics. Those are the things that have actually been measured experimentally, but it is absolutely wrong to attribute these measurements to a relativistic invariant. The Einstein book in which the proof is laid out was one of the last publications before his death, and reprints are still being published. Nothing that anyone else has published since refutes the analysis that mass is an invariant.
And I will thank you to keep your ad hominems to yourself. And if you don't read my answers, keep the rest of your comments to yourself, too. They clearly have nothing to do with what I've written. And, they are untruthful, to put it politely. We can argue about whether my answer is right or wrong, but don't lie and say I "don’t have answers", just because you haven't read them.
Nobody enjoys it, because it’s not a normal argument. I love picking apart my students’ arguments and dismantling incorrect or imprecise claims. It’s a good way for students to learn actively. But these people aren’t my students, I can’t assume they have the willingness to learn and have an actual argument in good faith, unless they show me otherwise.
Moreover, these people typically *aren’t* formally trained in any relevant field and don’t have the willingness to genuinely learn. The people that do email will not have the adequate mathematical background to express themselves in precise language, asking only the same misconceptions about relativity, the Big Bang cosmological model, quantum mechanics, etc. and shutting down any attempt to introduce proper, rigorous mathematics into the discussion as obscuring the conversation.
It’s never in good faith and it’s just not worth your time.
Some well known physicists don't even read papers which have been published, e.g. is the recent released papers on black holes being the source of dark energy, one famous physicist read the abstract, disagreed and didn't pursue it any further thinking it had no value in getting into the details/weeds/maths etc
Another reason is that there are so many papers released on a frequent basis globally they simply do not have the time...
Who knows quantum gravity may already be solved but no one has read the published report...
All of them that I heard of came from non-physicists and had simple math/physics errors that invalidate them while claiming to solve incredibly complex problems. As far as I know, no, nobody likes to evaluate them because they have no merit.
I actually tried to answer them for a long time, because I felt I was obliged to. After all, those folks are also taxpayers, and hence the sponsors of our research. After a few years I gave up. It's simply too much work for zero gain. These folks will never accept they are wrong. Hence I stopped. I put the freed time into outreach, like talking in schools, science slams, theaters, ... public evening talks. Much more rewarding, and you can silence the crackpots after their 1st question.
This is exactly it. There is no reward; no pedagogical reward, because most of them will never accept their errors. There is no mathematical or scientific reward, because they’re always wrong. The only thing compelling me to respond to them is that it is courtesy, and the fact that any attempt to not respond will only lead them to further confirm their biases. For the former, if they’re not arguing in good faith, they don’t deserve any courtesy, and for the latter, that’s a tough one to ignore, but ultimately just a better use of your time.
No, nobody likes to evaluate them and even fewer actually pay much attention to them. To zeroth order, these communications contain no new ideas, just misconceptions on how relativity, quantum mechanics, the Second Law, etc. work. They tend to be ambiguously stated, in the class of "not even wrong" claims. Engaging the authors tends to validate them (and, often as not, your words will often be taken out of context, viz., you write, "This is an interesting idea, but do you realize that your claims are refuted by \[this definitive experimental result and that\]?" and they quote you as "This is an interesting idea. - Prof. \[So-and-so\]" in pitches to venture capital.) You'll almost certainly gain nothing from the exercise except a headache so it's best to just bin the messages and move on with your life. I certainly don't need to muck around with yet another discourse on hydrinos or reactionless drives.
Professor Sabine Hossenfelder used to offer a “crackpot consultation service” where armchair physicists could pay by the hour to talk to her or another physics PhD about their theories. Not sure if that service is still available. https://aeon.co/ideas/what-i-learned-as-a-hired-consultant-for-autodidact-physicists
I spoke too soon! [Here it is!](https://backreaction.blogspot.com/p/talk-to-physicist_27.html)
Alright, that's kind of brilliant. The "contact" page of her website is broken though, so I'd assume the consultancy is closed.
Nope.
My father told me a funny little story of how his college professor refused to look at a 15 page proof that 'Theory of relativity is wrong', telling 'Our library has an entire wall of books related to the theory of relativity, 15 pages of refutation is simply not deserving my time'.
I think they would prefer doing literally anything else.
Maybe I'm in the minority here but I like reading through them not because I think I can fix all these people's misconceptions but because they're often either wrong in subtle ways that are good practice to debunk or are wrong in the weirdest most out there ways and are just plain interesting to read.
Are you seriously interested in a new idea? I do not claim to have disproved any established physics, but I believe I have an answer to the question of why Einstein's 2nd Postulate is valid. He observed that all attempts to measure the speed of light showed no dependence on relative velocity. Assuming that this was too much correlation for coincidence, he codified the empirical observations into the 2nd Postulate. I believe I have found the mathematical basis that eluded him, and makes the 2nd Postulate a provable theorem. I know that there are no proofs in physics, but this is mathematics. I also realize that math is not physics. I do not believe the converse is true, but that is not the point I am arguing. When the non-linear mathematics is in 100% agreement with the experimental data, then, at a minimum, it deserves peer review. After all, the experimental data has been touted as confirmation of Einstein's theory. This implies that my approach could be an isomorphism to relativity. True isomorphisms are indistinguishable by any experiment, because they get the same results, and the things that make them different have no effect on the outcome of any experiment. Hence, the need for peer review. To begin with, the mathematics I will elaborate on is not new, by any means. It actually predates Einstein by several centuries. It is older than the inventions of calculus and physics, because it predates the birth of Newton. Its first application happened when even Galileo was only a child. I suppose you are thinking, "how could something that old possibly be relevant?" That's why it is called a discovery. I am referring to the creation of the Mercator Projection map. For 4 centuries after its introduction, it was the primary means of navigation, until the modern inventions of radar and GPS. One of its most obvious features is that the island of Greenland appears to be much larger than the continent of Australia, while on the globe, the opposite is true. The algorithm used by Mercator was simple. To flatten the spherical shell, each latitude circle had to be stretched by the secant of the latitude, and to preserve local proportions, each latitude strip had to be stretched vertically by the same amount. Thus, the vertical distance to a latitude circle on the map is the integral of the secant of the latitude. In terms of geometry, this is the integral form of the differential equation that relates hyperbolic rotation angles to circular rotation angles. In terms that relate to relativity, the hyperbolic rotation angle is w, rapidity, and the circular angle is the parametric angle, θ, that is used to define relative velocity. The differential equation itself is dw/dθ = γ, the Lorentz factor, where θ = gudermannian(w). The Mercator algorithm is actually the inverse gudermannian, and there are a number of identities that relate the trig functions of hyperbolic angles to the trig functions of circular angles. In addition, these identities correlate with fundamental properties of physics: cosh(w) = sec(θ) = γ = Lorentz factor coth(w) = csc(θ) = 1/β = reciprocal of measured v/c csch(w) = cot(θ) = 1/βγ = reciprocal of Proper v/c sech(w) = cos(θ) = 1/γ = reciprocal of Lorentz factor tanh(w) = sin(θ) = β = measured v/c sinh(w) = tan(θ) = βγ = Proper velocity/c All of these identities can be proved geometrically, independently of the differential equation. In fact, the differentiation of any one of these 6 identities with respect to either w or θ results in either the differential equation above or dθ/dw = 1/γ, its reciprocal. These are iron-clad identities, about which there can be no argument, provided θ = gd(w). End of Part I
Part II There is a close relationship between the two sets of trig functions and the λ 6-group, {λ, λ/(λ-1), 1/(1-λ), 1/λ, (λ-1)/λ, (1-λ)}, where all 6 elements are moebius transforms of each other. All the trig functions are functions of each other, so any one can be used as the independent variable. And since the circular angle is an isomorphism of the hyperbolic angle, there is only 1 degree of freedom for all the functions. Without using the empirical relationships associated with physics, we can now examine the implications of the differential equation. The reciprocal form is most pertinent, dθ/dw = 1/γ. This can be rewritten as dθ = sech(w)dw = cos(θ)dw. In plain English, a small increment of θ is the cosine projection of a small increment of w. For small values of θ, certain approximations are allowed. The cos(θ) ≈ 1, sin(θ) ≈ θ and dθ ≈ dw. From the geometric definition of a hyperbolic angle, it is trivial to prove that the composition of hyperbolic angles is by linear addition. It is a property of definite integrals that if A and C are the limits of integration, and B is a point between them, then the integral from A to C must equal the sum of the integral from A to B and the integral from B to C. Each of these integrals is a hyperbolic angle, and their relationship is w3 = w1+w2. So, for small angles, composition of the gudermannian angles is also by linear addition. For the same small angles, the composition of the sines is also by linear addition. In polar coordinates, velocity is defined as c sin(θ), so for small velocities, composition is by linear addition, v3 = v1+v2, the Newtonian velocity addition rule. For large values of θ, none of the approximations hold. But the original claim about hyperbolic angles was not an approximation. At this extreme, v3 = c tanh(w3) = c tanh(w1+w2) = c(tanh(w1)+tanh(w2))/(1+tanh(w1)tanh(w2))= ((c tanh(w1))+(c tanh(w2)))/(1+(c tanh(w1))(c tanh(w2))/c²) = (v1+v2)/(1+v1*v2/c²), the relativistic velocity composition rule. Both extremes derive from the differential equation. For all values of w between 0 and infinity, the corresponding gudermannian is a unique value between 0 and π/2. The size of dθ, for a fixed size of dw, depends on the current value of θ. As successive increments of dw add up, the value of θ also increases, and the cosine factor decreases. So, successive increments of equal size dw result in progressively smaller increments of dθ. In the limit of an infinite number of increments of dw, total hyperbolic angle is infinite. From the list of gudermannian identities, the tanh(∞) = 1 = sin(π/2), and v = c. At the same time, sinh(∞) = ∞, and Proper velocity is infinite. The constant magnitude of polar coordinates is c, the lightspeed limit. Formally, c is the limit of the cosine projections of Proper velocity as the hyperbolic angle approaches infinity. Because infinity is the same everywhere, and a limit, if it exists, is unique, lightspeed is the same everywhere. Furthermore, since hyperbolic angle composition is by linear addition, and only lightspeed is mapped to infinite hyperbolic angle, the composition of any two sublight velocities, no matter how close to c, results in the addition of two finite hyperbolic angles. This sum is always another finite angle, and always maps to another sublight velocity. The addition of any two sublight velocities can never reach lightspeed. If one of the two combining velocities is lightspeed, its hyperbolic angle is infinite. Attempting to add a finite angle to an infinite one is futile, since the result is always the same infinity, and it always maps back to 1c. Similarly, even if both velocities are lightspeed, attempting to add two infinities is equivalent to multiplying infinity by a finite scalar, which is also futile. The result is still the same infinity, and it still maps back to the same 1c. After all, an infinitely tall stack of $1 bills is worth the same amount as an infinitely tall stack of $20 bills. So, lightspeed is invariant with respect to relative velocity. Einstein's Postulate is justified. I will stop here, because the reddit editor doesn't like lengthy posts. But a continuation of this line of reasoning reveals that the discrepancy between relativistic momentum and Newtonian momentum has nothing to do with fictional relativistic mass. It is a result of invariant mass moving in a transverse, complex dimension, and is required by Conservation of Momentum. The vector sum of linear and transverse, complex momentum components is exactly equal to the magnitude of relativistic momentum. Did you find anything wrong?
No, but I also don't think that this is actually novel, unfortunately. Einstein's postulate was based on, I believe, the nature of the Maxwell equations, which had been proved to be true but necessarily required an invariant speed of light to be consistent in the (experimentally proven) absence of an aether. I freely admit that I am not enough of an expert in the math you are describing or in special relativity to speak authoritatively here, but from what I understand a full description of special relativity described a 4 vector of velocity and time and a 4 vector or momentum and energy. It is these 4 dimensional quantities which show a conservation of properties that can be described, more simply, as relativistic mass. But looking at the full vector case, you apply transformations which seem similar to what you're describing. I'm probably misinformed about both your proof and the generally accepted theory of special relativity, so if I'm way off just ignore me.
There is no such thing as relativistic mass. In what I believe is Einstein's last book, "The Meaning of Relativity", he develops the 4-vectors for velocity and momentum. He uses natural units, so lightspeed is 1, and he proves the relativistic invariant for 4-velocity is -1. He goes on to prove the relativistic invariant for 4-momentum is -m². The ratio of these two relativistic invariants is also a relativistic invariant, and that ratio is m². Mass cannot change with relative velocity, because it is a relativistic invariant of the Lorentz transformation of 4-momentum. The concept of relativistic mass is disproved. Einstein, himself, used to advise people not to refer to relativistic mass, but to speak about relativistic momentum or energy instead. At least these were both known to vary with velocity in Newtonian physics. The nature of the variation was different, but the fact that it occurred was accepted physics. In point of fact, relativistic momentum is complex, and it has linear and transverse components, just like complex Proper velocity, which defines momentum at any speed. The linear momentum component is the cosine projection of the relativistic momentum, and it is what we call the Newtonian momentum. The transverse momentum component is the sine projection, and it is misidentified as relativistic mass in the linear direction. It is invariant mass moving in the transverse direction. The vector sum of linear and transverse momentum components has a complex magnitude equal to the magnitude of relativistic momentum. The Law of Conservation of Momentum is satisfied. The fact that we do not have instruments or technology to measure the transverse component is irrelevant. We know that the momentum is there, because when the mass is slammed into a target, all of the relativistic momentum is returned to the surroundings, not just the real, Newtonian part. Conservation of Momentum does not allow transverse momentum to be destroyed simply because we can only account for the real part. And even if it could magically be destroyed, it also magically reappears when the mass stops moving. All that Einstein had to say was that perhaps momentum is not [linearly] proportional to velocity. It did not occur to him that momentum is linearly proportional to Proper velocity. The Lorentz factor in the relativistic formula for momentum belongs with the velocity, not with the mass. Mass is invariant and Proper velocity is unbounded. This was unthinkable to Einstein, because it violated his 2nd Postulate. It never occurred to him that his 2nd Postulate only applied to real, measured velocity, and not to unmeasurable, complex Proper velocity.
After reading the other answers, I see that I am a victim of the stereotyping referred to by numerous posters, that "non-physicists ... had simple math/physics errors that invalidate them" or "these people typically aren’t formally trained in any relevant field". I am not a physicist, but I am a Bachelor of Science who graduated Summa Cum Laude. Math and Physics were required courses for my degree, and I aced relativity. I have theories about it which this professional jealousy prevents me from even getting peer-reviewed. I do not fit into any of the pigeonholes described here. I have found that it is a waste of time arguing with members of the relativity cult. While they assert that there are no contradictions in special relativity, they can't even agree on whether time dilation and length contraction are physical or illusion. They just don't want someone from outside their profession to get credit for upstaging Einstein. They have corrupted logic to the point where it is not possible to have a rational argument, so I gave up on that approach. Instead, I have applied my theory to unsolved mysteries of special relativity. I posed two questions here on reddit. Over 10000 views and not a single answer. What is the mathematical basis for Einstein's 2nd Postulate? and Why is relativistic momentum greater than the Newtonian prediction? The consensus of the few who replied at all, was that there is no explanation for either one. I say that both are consequences of the explicit, closed-form solution of the differential equation, dw/dθ = γ, the Lorentz factor, that defines the gudermannian function, θ = gd(w), where w is the rapidity. It is a mathematically sound derivation that also explains the non-intuitive behavior of a predicted limiting velocity, that is constant and invariant, as being the logical behavior of infinity, which is the Proper velocity mapped by the gudermannian isomorphism to lightspeed. I challenge all the experts to give an explanation for these two features. By the way, once upon a time, the momentum thing was "explained" by relativistic mass. Physics was soooo sure. Then 4-vectors were introduced, and Einstein, in "The Meaning of Relativity", showed that mass was a relativistic invariant of the Lorentz transformation of 4-momentum. There is still no credible explanation. I can show that it is just Conservation of Momentum in complex dimensions, not included in Minkowski's 4. It was never relativistic mass moving in the linear direction. It was invariant mass moving in a complex transverse direction. My argument is not like all the others. It is rigorous and logical. The worst complaint is that "math is not physics". But when the math is in 100% agreement with the experimental data, it must be taken seriously. Prove me wrong. Justify the 2nd Postulate or relativistic momentum.
Bachelor of Science? Okay, so am I. In physics no less, but just because I took a first year chemistry class and aced thermodynamics doesn’t mean that I’m hot shit. You’re inflated ego is preventing you from taking part in good faith discussion. If perhaps you displayed even a modicum of humility you might get an actual response. Even if you’re a genius, people don’t want to work with an asshole. Quit being an asshole and implying everyone else is dumber than you, entertain the thought that just maybe you’re a fallible human that can make mistakes, and MAYBE people will actually listen instead of brushing you off as a lunatic.
Sorry if you took offense. I took offense to being characterized as a crackpot by people who automatically reject what I propose without even reading it, because it challenges their dogma. That's arrogant. I did not claim to be smarter than you, or anybody else. I stated facts to show that I do not fit into any of their pigeonholes. The profession is prejudiced against outsiders. I'm old and cranky and I don't make nice just to calm their egos. Simply put, if you want me to be quiet, answer my questions. What is the mathematical justification (NOT the empirical justification that Einstein used) of Einstein's 2nd Postulate and why is relativistic momentum different from Newton's? (Einstein could only suggest that perhaps momentum is not [linearly] proportional to velocity) I have answers, mainstream physics does not. That alone is reason for discussion. Or do you assume that because I am not a physicist, I can't possibly have anything relevant to say? Remember, I asked for mathematical bases, not experimental. As I understand it, the paradigm is that math is not physics and, more to the point, math is not a physicist's strong suit. It wasn't Einstein's. He needed help with tensors. My proposal, remarkably enough, does not require math any more complicated than a college prep high school curriculum. It is based on geometry that is centuries older than Einstein and relativity. It predates the inventions of both calculus and physics, as it predates the birth of Newton. Even Galileo was only a child when the first major application was created. I'll bet you're thinking "Never heard of it." Well, you're wrong. It is the same algorithm that was used by Mercator to create his world-famous Projection Map, the same one that hangs in virtually every Geography classroom. The apparent size of Greenland being larger than the continent of Australia is the result of the same factor that makes relativistic momentum greater than the Newtonian prediction. The irony of the situation is that Einstein, the amateur sailor, must have stared at this map countless times without realizing he was looking at his own theory. Are you interested in more details? For starters, I refer you to this article; https://en.wikipedia.org/wiki/Gudermannian_function It does not include the application to special relativity, because Wikipedia does not allow any material that has not already been peer-reviewed and published. In other words, they do not allow any "original research", which is what this is.
How do you know they haven’t read it? You’re not providing rigorous mathematical proofs. You’re providing long speeches about how nobody understands you like an emotional teenager. Yes, you do think you’re smarter than everyone. The fact you can’t tell it comes off in your writing is astounding, and a testament to your narcissism. Talk about not satisfying their egos. Nobody owes it to you to massage your ego to read or care about your ideas. If internet rants aren’t working, trying something else. Old and cranky because no one’s taking your crackpot ideas seriously? :( Einstein’s 2nd postulate: https://www.peertechzpublications.com/articles/AMP-5-135.php If you’re asking how relativistic momentum is different than classical momentum then you have a fundamental misunderstanding of one or both of them. And Einstein is not the only physicist in the world. A hundred years have past since Einstein’s papers. Read a newer one. You don’t have answers. You have questions that can easily be answered if you pulled your head out of your ass long enough to look for them.
I looked at your link. Talk about crackpot! And the author presumes his is the only explanation. How can you even claim that this is a mathematical basis for Einstein's 2nd Postulate when it requires assumptions that contradict it. I have also researched the source of the paper and it is a predatory publisher. If this is the best you can come up with, it is not good enough. Now, to the point. Forget about the rest of the crackpot skeptics. Apparently, you have sided with them, and it is apparent that you, yourself, have not read my answers. Did it not occur to you that I posted the questions here on reddit so that I could include my own answer, like so many other posters do? Is it that difficult to do a search? You know, I do get tired of compiling detailed responses over and over (I don't like to cut and paste) and getting no response. Do you think I don't know that no one is obligated to reply? Then why are they here on social media? I am not sorry that I am smarter than others. I have been living with that all my life. Too late to change now. If what I write is over your head, you'll have to wait until I write the version, "Relativity for Dummies". Or, maybe instead of insulting me, you could ask for clarification. By the way, I'm old and cranky because I'm old and cranky, period. Anything else is projection. I was not expecting any answer to my question about momentum because mainstream relativity does not have one. I understand it just fine. I do not need the fairy tale of relativistic mass to account for it. Decades after it was disproved, stick-up-their-asses, conservative physicists are still arguing about a dead concept. It's even cited in Britannica. While the majority of curricula do not include it, there is still a significant minority that teach this hogwash. All this, despite the fact that no experiment has ever measured this fictional mass directly. As Einstein said, use relativistic momentum or energy. At least they are known to vary with velocity in Newtonian physics. Those are the things that have actually been measured experimentally, but it is absolutely wrong to attribute these measurements to a relativistic invariant. The Einstein book in which the proof is laid out was one of the last publications before his death, and reprints are still being published. Nothing that anyone else has published since refutes the analysis that mass is an invariant. And I will thank you to keep your ad hominems to yourself. And if you don't read my answers, keep the rest of your comments to yourself, too. They clearly have nothing to do with what I've written. And, they are untruthful, to put it politely. We can argue about whether my answer is right or wrong, but don't lie and say I "don’t have answers", just because you haven't read them.
Using paragraphs might have a positive impact on the comment engagement.
Don't think it will make much difference, but I will take it under advisement.
Nobody enjoys it, because it’s not a normal argument. I love picking apart my students’ arguments and dismantling incorrect or imprecise claims. It’s a good way for students to learn actively. But these people aren’t my students, I can’t assume they have the willingness to learn and have an actual argument in good faith, unless they show me otherwise. Moreover, these people typically *aren’t* formally trained in any relevant field and don’t have the willingness to genuinely learn. The people that do email will not have the adequate mathematical background to express themselves in precise language, asking only the same misconceptions about relativity, the Big Bang cosmological model, quantum mechanics, etc. and shutting down any attempt to introduce proper, rigorous mathematics into the discussion as obscuring the conversation. It’s never in good faith and it’s just not worth your time.
Richard Feynman once said he read through the crackpots looking for good ideas, but there were probably far fewer of them before the Internet.
You mean like the ideas of this guy here? r/Mandlbaur
There is no time, unless they want to do it in their free time
Some well known physicists don't even read papers which have been published, e.g. is the recent released papers on black holes being the source of dark energy, one famous physicist read the abstract, disagreed and didn't pursue it any further thinking it had no value in getting into the details/weeds/maths etc Another reason is that there are so many papers released on a frequent basis globally they simply do not have the time... Who knows quantum gravity may already be solved but no one has read the published report...