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A short personal story: While trying to understand special relativity from books, and failing to, I tried doing maths myself based on whatever I had understood

by alok-g 3y ago

A short personal story: While trying to understand special relativity from books, and failing to, I tried doing maths myself based on whatever I had understood till then. I ended up doing what this paper has mentioned. I was surprised. For the next twelve years, I was asking various physicists I knew about what was I doing wrong, however none could give me the time needed. I then saw basically the same proof appear on Wikipedia, so finally had the answer. Einstein's original paper on special relativity, while mentioning the two famous postulates, also later assumes linearity of the transformations (allowing non-linear takes it from special relativity to general relativity). However, the paper did not explicitly call it out as a postulate. Also, Einstein derived a particular equation in the paper from two paths and mentions that there's consistency. This was actually a missed opportunity -- He could have forced consistency as an input assumption, worked backwards, and would then have seen that invariant speed is not needed as an input assumption anymore. With this OP submission, I have now learnt that this has been known for more than a century now, and agree that this should be more popular in books.

Q-Q 3y ago
I find it not believable that - You derived the theory correctly (end result should match textbook result) and thought what you were doing were wrong. - No physicist you asked could explain where you were wrong - No physicist you encountered could see the obvious connection of this and the usual approach It's not really a mysterious or novel fact that you can have these shift of perspectives (at least since Kant who heavily influenced Einstein) e.g. Newton took his three laws as the base, derived relativity as a theorem. Einstein took (modified version) relativity (together with the invariant speed assumption which comes from electromagnetism) as the base and rebuilt the rest of theory of motion that reduces to Newton's theory in the limit. Nothing is amiss here.
phkahler 3y ago
You are probably on the right path IMHO. What wad that quote where Einstein said the mathematicians got hold of relativity, and turned it into something he didn't understand any more?
- Q-Q 3y ago
  Did you even read his post? what you are calling the "right" path is the "mathematician" path, assuming relativity + group structure. What he complained about, the Physics textbook approach was the path taken by Einstein who started from the "invariant speed" assumption to derive relativity: https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/origins_pathway/On-the_electrodynamics/index.html https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/... >"introduce another postulate... namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body." There's also nothing really "wrong" with Einstein's approach either, it's motivated by physical constraint from Maxwell's electromagnetism. Mathematician's approach is motivated by imposing logical constraint of an abstract mathematical structure (group of transformations from one coordinate system to another) The two turn out to be connected (physical constraint selects a math model from alternatives). Nothing really strange.
- bobbylarrybobby 3y ago
  “Since the mathematicians have invaded the theory of relativity I do not understand it myself any more.” https://www.goodreads.com/quotes/78897-since-the-mathematicians-have-invaded-the-theory-of-relativity-i https://www.goodreads.com/quotes/78897-since-the-mathematici...
  - Q-Q 3y ago
    Within historical context: This was said after Minkowski introduced the geometry for special relativity, which can be visualized on the `spacetime diagram`, now recognized as the vastly simpler and more efficient approach than the wall-of-text approach. Einstein initially had some difficulty, that was when he said this, but he caught up quickly. >Strangely enough no personal contacts resulted between his teacher of mathematics, Hermann Minkowski, and Einstein. When, later on, Minkowski built up the special theory of relativity into his 'world-geometry', Einstein said on one occasion: 'Since the mathematicians have invaded the theory of relativity, I do not understand it myself any more'. But soon thereafter, at the time of the conception of the general theory of relativity, he readily acknowledged the indispensability of the four-dimensional scheme of Minkowski Einstein, philosopher-scientist, Schilpp And later Einstein expressed admiration for geometry again and again >This problem remained insoluble to me until 1912, when I suddenly realized that Gauss's theory of surfaces holds the key for unlocking this mystery. I realized that Gauss's surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. Einstein, Kyoto Lecture, 1922 >I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot Hentschel (1998). The Collected Papers of Albert Einstein, Vol. 8 (English): The Berlin Years: Correspondence, 1914-1918
superposeur 3y ago
On the subject of implicit assumptions (like linear trajectories transform to linear trajectories), it has always struck me that there is another assumption: if A sees B at speed v, then B sees A at the same speed v. It’s hard to imagine this symmetry not being true but who knows what’s possible in theory space?
- Q-Q 3y ago
  What are you guys on about >if A sees B at speed v, then B sees A at the same speed v is directly derivable from the Lorentz velocity addition formula which is the result of the usual two postulates. Why would you want that as an extra assumption... https://en.wikipedia.org/wiki/Velocity-addition_formula#Special_relativity https://en.wikipedia.org/wiki/Velocity-addition_formula#Spec... vAA = (vAB + vBA)/(1+(vAB*vBA/c^2) -> 0 = vAB + vBA so velocity of A relative to B and B relative to A both have magnitude v.
  - superposeur 3y ago
    Of course, if you already assume special relativity then this is baked in: taking the inverse of the Lorentz transformation matrix effectively sends v to -v. And velocity addition for sure already assumes you have the Lorentz transformations. The point is that in the derivation of relativity from elementary postulates, the “reflectivity of velocities” assumption is used. For instance the linked paper uses it between equations (12) and (13). All such derivations use it at some stage. Intuitively, “reflectivity of velocities” is why time dilation leads to length contraction with same gamma factor. I don’t think it follows from absence of a preferred frame since one could imagine some complicated group structure relating the velocities of the various observers relative to each other in such a way that all are on equal footing and yet v_AB is not exactly -v_BA. As mentioned, this is certainly a reasonable assumption but who knows what whacky alternatives are out there? After all, in the different context of quantum mechanics, such “obvious” relationships as xp = px no longer hold true.
- LudwigNagasena 3y ago
  Another interesting velocity-related conundrum: It’s impossible to measure the one-way speed of light (from a source to a detector) because we have to synchronize the clocks somehow, which requires an algorithm based on the speed of light to account for time dilation. We can only measure the two-way speed of light (from a source to a reflector and back). So, from the standpoint of pure math, isotropy of the speed of light is purely a convention. The argument can only be made based on, so to speak, philosophical reasoning.

by alok-g 3y ago

Q-Q 3y ago

I find it not believable that - You derived the theory correctly (end result should match textbook result) and thought what you were doing were wrong. - No physicist you asked could explain where you were wrong - No physicist you encountered could see the obvious connection of this and the usual approach It's not really a mysterious or novel fact that you can have these shift of perspectives (at least since Kant who heavily influenced Einstein) e.g. Newton took his three laws as the base, derived relativity as a theorem. Einstein took (modified version) relativity (together with the invariant speed assumption which comes from electromagnetism) as the base and rebuilt the rest of theory of motion that reduces to Newton's theory in the limit. Nothing is amiss here.

phkahler 3y ago

You are probably on the right path IMHO. What wad that quote where Einstein said the mathematicians got hold of relativity, and turned it into something he didn't understand any more?

Q-Q 3y ago
Did you even read his post? what you are calling the "right" path is the "mathematician" path, assuming relativity + group structure. What he complained about, the Physics textbook approach was the path taken by Einstein who started from the "invariant speed" assumption to derive relativity: https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/origins_pathway/On-the_electrodynamics/index.html https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/... >"introduce another postulate... namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body." There's also nothing really "wrong" with Einstein's approach either, it's motivated by physical constraint from Maxwell's electromagnetism. Mathematician's approach is motivated by imposing logical constraint of an abstract mathematical structure (group of transformations from one coordinate system to another) The two turn out to be connected (physical constraint selects a math model from alternatives). Nothing really strange.
bobbylarrybobby 3y ago
“Since the mathematicians have invaded the theory of relativity I do not understand it myself any more.” https://www.goodreads.com/quotes/78897-since-the-mathematicians-have-invaded-the-theory-of-relativity-i https://www.goodreads.com/quotes/78897-since-the-mathematici...
- Q-Q 3y ago
  Within historical context: This was said after Minkowski introduced the geometry for special relativity, which can be visualized on the `spacetime diagram`, now recognized as the vastly simpler and more efficient approach than the wall-of-text approach. Einstein initially had some difficulty, that was when he said this, but he caught up quickly. >Strangely enough no personal contacts resulted between his teacher of mathematics, Hermann Minkowski, and Einstein. When, later on, Minkowski built up the special theory of relativity into his 'world-geometry', Einstein said on one occasion: 'Since the mathematicians have invaded the theory of relativity, I do not understand it myself any more'. But soon thereafter, at the time of the conception of the general theory of relativity, he readily acknowledged the indispensability of the four-dimensional scheme of Minkowski Einstein, philosopher-scientist, Schilpp And later Einstein expressed admiration for geometry again and again >This problem remained insoluble to me until 1912, when I suddenly realized that Gauss's theory of surfaces holds the key for unlocking this mystery. I realized that Gauss's surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. Einstein, Kyoto Lecture, 1922 >I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot Hentschel (1998). The Collected Papers of Albert Einstein, Vol. 8 (English): The Berlin Years: Correspondence, 1914-1918

superposeur 3y ago

On the subject of implicit assumptions (like linear trajectories transform to linear trajectories), it has always struck me that there is another assumption: if A sees B at speed v, then B sees A at the same speed v. It’s hard to imagine this symmetry not being true but who knows what’s possible in theory space?

Q-Q 3y ago
What are you guys on about >if A sees B at speed v, then B sees A at the same speed v is directly derivable from the Lorentz velocity addition formula which is the result of the usual two postulates. Why would you want that as an extra assumption... https://en.wikipedia.org/wiki/Velocity-addition_formula#Special_relativity https://en.wikipedia.org/wiki/Velocity-addition_formula#Spec... vAA = (vAB + vBA)/(1+(vAB*vBA/c^2) -> 0 = vAB + vBA so velocity of A relative to B and B relative to A both have magnitude v.
- superposeur 3y ago
  Of course, if you already assume special relativity then this is baked in: taking the inverse of the Lorentz transformation matrix effectively sends v to -v. And velocity addition for sure already assumes you have the Lorentz transformations. The point is that in the derivation of relativity from elementary postulates, the “reflectivity of velocities” assumption is used. For instance the linked paper uses it between equations (12) and (13). All such derivations use it at some stage. Intuitively, “reflectivity of velocities” is why time dilation leads to length contraction with same gamma factor. I don’t think it follows from absence of a preferred frame since one could imagine some complicated group structure relating the velocities of the various observers relative to each other in such a way that all are on equal footing and yet v_AB is not exactly -v_BA. As mentioned, this is certainly a reasonable assumption but who knows what whacky alternatives are out there? After all, in the different context of quantum mechanics, such “obvious” relationships as xp = px no longer hold true.
LudwigNagasena 3y ago
Another interesting velocity-related conundrum: It’s impossible to measure the one-way speed of light (from a source to a detector) because we have to synchronize the clocks somehow, which requires an algorithm based on the speed of light to account for time dilation. We can only measure the two-way speed of light (from a source to a reflector and back). So, from the standpoint of pure math, isotropy of the speed of light is purely a convention. The argument can only be made based on, so to speak, philosophical reasoning.