B Development of Special Relativity

[PeroK]

If you assume homogeneity and isotropy (the laws of physics are the same - symmetric - regardless of where you place the origin and which direction you point your axes) then you can derive that the only possibilities are the Galilean transform or the Lorentz transform. There are other similar approaches that emphasize different symmetries.

All roads lead to the Lorentz Transformation!

 

[FactChecker]

If you assume homogeneity and isotropy (the laws of physics are the same - symmetric - regardless of where you place the origin and which direction you point your axes) then you can derive that the only possibilities are the Galilean transform or the Lorentz transform. There are other similar approaches that emphasize different symmetries.

Here is a link to download proof of that in a PDF document. It is not very long.

 

[Wes Tausend]

Although I agree with most of what is said here about learning SR & GR, I think an actual understanding is only complete when one follows the histories.

For books, I most appreciated Asimov's Understanding Physics: Volume II Light, Magnetism and Electricity. Asimov is truly the Great Explainer.

I liked The Mechanical Universe video series so well, I bought their closeout $200 DVD set, but it is now also available on YouTube, when the net is available. It was done by the great Prof. David R. Goodstein at Caltech. I believe Goldstein was once a student of Feynman there. Speaking of the brilliant Feynman, there is an excellent online set of his lectures at http:\\www.feynmanlectures.caltech.edu, the presentation of which Goldstein seems to nearly follow on the DVD series.

The too-short history version of Relativity:
Newton developed laws of motion and pondered light while Faraday carefully drew the geometry of magnetism.
By applying geometry of Faraday, Maxwell was able to mathematically discern that magnetic waves, such as light, traveled at a certain maximum speed, rather contradictory to Newton laws of motion.
Later, Michelson-Morley found their interferometer measured the speed of light as constant no matter how fast it was moving through space, confirming Maxwell.
FitzGerald suggested that the only way this could occur was if the interferometer somehow became shorter in the direction of travel.
Lorentz then to began to calculate what formula would describe how much shorter the distance between mirrors on the interferometer would have to become in the direction of travel for light to travel as it did. This became known as the Lorentz-FitzGerald contraction ratio, aka Lorentz contraction, aka eventually the Lorentz Transformation.
Meanwhile, Einstein, whose family actually built early motors, dynamo's, transformers and generators when he was a child, was greatly interested in solving the Newton-Maxwell electrodynamic conflict. He did, but it is fairly difficult to understand considering the way he had to do it.
And that is the too-short version.

But it is only by understanding the histories of relativities that I suddenly noted a set of insights that, for one thing, I think help me more easily heuristically resolve in my mind, how and why relative distances (lengths) can and would naturally become shorter as relative speeds increase.

 

[GrayGhost]

I think that a derivation of the LTs, either as Einstein did in his 1905 paper, or by Algebra alone (as it's often taught today), is all that's needed. However for many, the Minkowski spacetime diagrams are also needed to reach the full meaning of SR, as a picture can paint a 1000 words. I can certainly say they were necessary for me. It helped me understand the time-desynchronisation of a moving body, and why (and how) that happens. That's the ticket IMO, because one must then consider length-contraction and time-dilation "in collective". The history of SR is fascinating, but I personally did not study the history until after learning and understanding SR first. The history, is not required.GrayGhost

 

[Swamp Thing]

If you are really keen on following how it all played out in terms of the different players' thought processes, you might want to check out "Subtle is the Lord: The Science and Life of Albert Einstein" by Abraham Pais.

There's a DRM'd copy on Internet Archive that you can borrow for 14 days at a time.
(Not that this will help in any way to understand the physics itself, as others have pointed out).

 

[NoahsArk]

I've ordered the Asimov and Pais books. Thank you for the suggestions.

 

[Wes Tausend]

I've ordered the Asimov and Pais books. Thank you for the suggestions.

NoahsArk,

I am pleased that you thought to order these books.

I stand by my high regard for the study of history. History is the composite of our beloved time, after all.

Second of all, every book, every class, every observation, is just that, history... whether the experience tells us the philosophy of ancient thought or expounds on more current thought, the aroma still lingering. History teaches us not only how all the "greats" moved, by leading our mind down their path, but teaching us how to think.

History warns what sort of pitfalls may lurk about, horrid pitfalls in just drawing the quickest, most convenient conclusions. History induces us to ask... what other direction could have been taken, what other direction might end just as well, or better?

Studying a former path well, allows us to follow the same road, rejoicing in the once fresh ideas that came about, came about because someone has taken an old tool and found a new use for it... and knowing history allows us to discover a different path anywhere in time, a different way to employ a tool... just as the best of past humanity has always done.

History helps us invent thee proverbial new tool, the new improved tool almost, but not quite the same as the old tool. I heartily recommend reading Asimov's The Relativity of Wrong, all about the lessons, the shortsighted, half-right tools of history that came first (found free on the net). Do not see only what the giants saw. Stand on their shoulders and see further, even if to the side.

History itself is a tool, the tool-of-tools, so to speak. Tools are really everything we leave behind us, all of it. Some of which we leave behind comes in the form of DNA, in which DNA is basically Nature's wrenches made to fit Nature's bolts. And history otherwise comes in tools left behind as our most valuable possessions, our material and ethereal heirlooms. Besides wrenches, who amongst us does not love ethereal heirlooms such as math and logic, all of which are again, merely human tools we leave behind? It would be so sad were they not fully appreciated.

We live in a Mechanical Universe. That we think we know is so. But as we gradually reverse-engineer it, nobody surely knows how Mother Nature's entire machine really works yet. Don't just use your forefathers wrenches as they are, but go dig up all older tools and then imagine how you might have improved them differently. Be sceptical, as our fine fellows here on PF have so resolutely just admonished me above. Yes, follow the well-worn path most traveled, which is what we promote almost exclusively here on PF, but be equally sceptical of both old and new. Most knowledge is inherently incomplete. Do not fear to consider all other possible paths along the way... whenever permitted.

Most of science normally proceeds by slow, relaxing evolution... tiny, lone insights... but I think, I even hope, we are again ripe for the austere, painful spurt of revolution. A revolution is simply like an upside-down jig-saw puzzle suddenly turned right side up. Not only do all the pieces still fit... the big picture becomes quite evident and, in embarassment, history again speeds ahead for a bit.

Wes

 

[vanhees71]

If we are already at this philosophical level of arguments, I'd not say that we live in a mechanical universe but in a (quantum) field universe.

One should also distinguish between learning the physics of some subject and learning more about the history of this subject. For the former it's better to forget about the history, as many other posters in this thread already emphasized. We have for sure a much better understanding about relativity than the physicists had when they discovered it. That's because it has been applied for nearly 120 years to describe (in parts with utmost precision) real-world phenomena reaching from the micro-cosm resolving length scales below $1 \text{fm}$ to cosmology including length scales of several billion light years.

For example, one of the most important subjects of all is thermodynamics and statistical physics, including kinetic theory and hydrodynamics. Up to the 1960ies there was great confusion about the transformation properties of the thermodynamical quantities like temperature, entropy, various thermodynamical potentials, etc. This has been clarified and brought into a logical scheme since then, and it simply is only confusing for the student when starting this important subject of relativistic many-body theory with old-fashioned ideas.

On the other hand, indeed, a knowledge of the history of physics is very important, because it provides the opportunity to rethink the meaning of the now established theories you learn in the standard physics course, and this indeed leads to a deeper understanding what's behind these theories in physics terms. It also shows how the theories developed in a close interaction between theoretical ideas and quantitative observations and experiments.

 

[David Lewis]

... the most profound part of special relativity is that the passage of time between any two things happening is not the same for everyone, and depends on their relative motion.

Time dilation was proposed by Joseph Larmor in 1897 after George Fitzgerald announced the idea of length contraction in 1889. They and Hendrik Lorentz were trying to explain why luminiferous aether could not be detected, and why the speed of light was frame-independent.

 

[HallsofIvy]

To really understand Einstein's "theory or relativity" you should go back to "Gallilean relativity". Gallileo pointed out that, if you were sealed in a carriage that was moving with constant velocity, no bumps, and sealed windows so you cannot see out, no experiment done completely inside the carriage can tell what your speed is. You can only determine your speed "relative" to the outside.

But then Maxwell's equations for electro-magnetic fields (so for light) showed that the force of a magnet field on a charged particle does depend upon the velocity of the particle relative to the magnetic field. That appeared to imply that one could do electro-magnetic experiments in that closed car to find its speed.

Lorenze's equations, based on calculations from experimients, predate Einstein's paper but his theory behind them assumed that, in a moving object, the changed elector-magnetic field of the charged ions in the moving object contracted the object in the direction of motion. It was a pretty theory but further experiment show that even the space between charged objects contracted. That was addressed by Einstein's theory.

 

[NoahsArk]

Time dilation was proposed by Joseph Larmor in 1897 after George Fitzgerald announced the idea of length contraction in 1889. They and Hendrik Lorentz were trying to explain why luminiferous aether could not be detected, and why the speed of light was frame-independent.

My understanding is that although a few others had seen through equations that time was different in different frames, they thought this was just the math, and not reality. One of Einstein's key insights, that he got while walking with his friend Michele Besso, is that time is in fact different in different frames. Is it correct to say that the evidence was there before, but he took the leap of faith before anyone else to come to that conclusion?

I was just reading in an article that his other key insight was that acceleration and gravitation are two ways of describing the same force. It took him 8 more years to prove this insight thought math. While I haven't studied general relativity yet, so don't understand the second insight, is it a fair statement to say that it was more this insight or "aha moment" that was the spark of genius- more so than the working out of the math to prove it (although no doubt that was some feat as well)? Same question regarding his insight that he had on the walk.

 

[Ibix]

Is it correct to say that the evidence was there before, but he took the leap of faith before anyone else to come to that conclusion?

The whole of special relativity is implicit in the structure of Maxwell's equations. But up until Einstein, no-one realized that the incompatibility of Maxwell's equations with Galilean relativity (and hence Newtonian physics) was a problem with Newton and Galileo, not with Maxwell. Einstein seems to have been the first to realize that you could shouldn't regard (what we now call) the Lorentz transforms as a mysterious mathematical patch for Maxwell, but rather that they are a replacement for the Galilean transforms.

I was just reading in an article that his other key insight was that acceleration and gravitation are two ways of describing the same force.

This is a somewhat inaccurate description of the equivalence principle. That principle is, however, a key insight that leads to the possibility of modelling gravity as spacetime curvature.

I think a lot of people were on the trail of general relativity. Einstein got there first, but there were others (such as Hilbert) who were moving in the right direction. My feeling is that Einstein was far ahead of the game with Special Relativity (Poincare actually published all of the maths underlying SR in 1904 - but no-one realized the implications), but with General Relativity I think had Einstein suddenly dropped dead it wouldn't have delayed things very much. (That's unprovable, of course...)

 

[PeterDonis]

One of Einstein's key insights, that he got while walking with his friend Michele Besso, is that time is in fact different in different frames.

"Time is different" is ambiguous.

If it means, for example, that the two twins in the standard twin paradox will in fact agree when they meet up again that one has aged less than the other, then yes, this is a fact (and I think it is basically what Einstein was thinking of in the insight you describe).

But this does not mean that, for example, the two twins while they are traveling will actually "feel" their clocks to be running differently. They won't. Their local experiences of their clocks while they are traveling will be exactly the same. Only by comparing their clocks when they meet up again will they discover that one has aged less than the other.

I was just reading in an article that his other key insight was that acceleration and gravitation are two ways of describing the same force.

That's not quite what this key insight was. His own words (at least, translated from German) were: "when a person falls freely, he will not feel his own weight". This made him realize that "gravity" was not a force, since it couldn't be felt.

 

[NoahsArk]

But up until Einstein, no-one realized that the incompatibility of Maxwell's equations with Galilean relativity (and hence Newtonian physics) was a problem with Newton and Galileo, not with Maxwell.

Were others thinking that Maxwell was wrong (and also that the results of Michelson and Morley's experiment were wrong) because the speed of light can't be constant without contradicting Newton and Galileo's statements that time is absolute? Interestingly, this article says that when Einstein was on the walk with Besso, he imagined that instead of a passenger on Galileo's ship dropping a ball, a beam of light was sent downward from 186,000 miles high. That's when he realized that because the distance that the light travels is different in the ship frame and dock frame time must also be different since the speed of light is the same in both frames.

https://www.smithsonianmag.com/science-nature/the-year-of-Albert-einstein-75841381/

"Time is different" is ambiguous.

I meant that was his insight that the time elapsed between two events will be measured differently in two different frames?

 

[Dale]

Were others thinking that Maxwell was wrong

Yes. More specifically that Maxwell’s equations were only valid in the rest frame of the aether.

 

[Ibix]

Were others thinking that Maxwell was wrong

Wrong is too strong a word, IMO - the equations clearly worked. The problem was that they were not invariant under the Galilean transforms, which suggested (as Dale observes) that they were missing some terms (small ones, because they worked well enough as was) related to your velocity relative to... something, which got christened "aether". The outcome of Michelson-Morley led to Fitzgerald contraction and eventually Lorentz' transforms. But as far as I understand, the latter were treated as a relationship between Newtonian time and space coordinates and the x and t parameters in Maxwell's equations (which everyone had regarded as position and time).

Einstein realized that he could derive the Lorentz transforms from two postulates, rather than ad hoc from experiments. Then he derived a new version of mechanics, to which Newton was only an approximation - crucially, explaining in the process why no one had noticed before (no one could make anything move fast enough for the difference to be measurable with then-current kit). So he explained that x and t in Maxwell's equations were position and time after all, but the relationship between position and time measured by you in the lab and as measured by me ambling down the corridor weren't quite what everyone thought...

 

[vanhees71]

In view of the lack of Galileo invariance of the Maxwell equations there where 3 possibilities, and all three were considered historically:

(a) Galilei-Newton spacetime is the correct spacetime model and em. phenomena obey the corresponding (proper orthochronous) Galileo symmetry. Then the Maxwell equations are wrong and have to be modified as to obey Galileo symmetry (i.e., the equations of motion should be form-invariant under Galileo transformations).

(b) Galilei-Newton spacetime is the correct spacetime model and the Maxwell equations are correct. Then there must be a preferred inertial frame of reference (IF), where the Maxwell equations hold in their then known form but look different in any non-preferred IF.

(c) Galilei-Newton spacetime is not the correct spacetime model and the Maxwell equations as well as the special principle of relativity are correct, i.e., IFs are a set of preferred reference frames. Then one needs a spacetime model which is compatible with Maxwell's equations, i.e., with a symmetry group that allows for transformations of the electromagnetic quantities (em. field, charge-current densities) under which the Maxwell equations are form-invariant.

ad (a): AFAIK this were the model en vogue before Maxwell's and Faraday's work, but at the latest with the discovery of the em. waves by H. Hertz, which obeyed all properties predicted by Maxwell, these models were ruled out, because they were not compatible with em. waves (at least not with all the properties predicted by Maxwell's equations).

ad (b): This was the common believe of most physicsts at the time, including Maxwell in the beginning: There was a preferred frame of reference, because due to the then mechanistic world view the physicists thought the em. fields, including the em. waves, must be due to the motion of a special "imponderable" medium, called the aether, and then the preferred frame of reference is easily determined as the (local) restframe(s) of the aether. Aether theory delivered predictions about the behavior of the em. phenomena, but it turned out that the aether is a very strange substance in order to fulfill the properties of em. fields as predicted by Maxwell and confirmed by observations. Also aether theory predicted how electromagnetics should look in moving media, where now "moving" means "moving relative to the aether restframe". Famously aether theory turned out to be correct for all phenomena only at the first order in $\beta=v/c$, where $v$ is the speed of the medium and $c$ the vacuum-speed of light but fails for phenomena which depend on $\beta$ at the order $\mathcal{O}(\beta^2)$ and higher. The famous early examples were the negative result of the Michelson-Morley experiment as well as the Trouton-Noble experiment.

ad (c): That was the famous solution of the problem by Einstein, using of course previous work by "the Maxwellians", FitzGerald, Heaviside as well as Lorentz and Poincare. Thus there was no aether necessary and no preferred IF at all but the spacetime structure had to be described by Minkowski spacetime and with the corresponding symmetry group, which is the proper orthochronous Poincare group rather than the proper orthochronous Galileo group. That's the status today (modulo the refinements when it comes to the description of gravity in terms of the General Relativity Theory) and confirmed by observation at a very high level of accuracy.

 

[Dale]

@vanhees71 can you explain the distinction between your cases (a) and (b)? I don’t see the difference. Aren’t the “look different in any non-preferred IF” terms in (b) the same as the “have to be modified as to obey Galileo symmetry” terms in (a)? If not, then what is the distinction?

 

[vanhees71]

The difference is in the conclusions you draw. In case (a) the idea is that Maxwell's equations are wrong and you need to modify them such that the theory is Galilei invariant. In case (b) you claim Maxwell's equations are correct but that there's a preferred IF, which is interpreted as the restframe of a mechanical fluid called aether. Then Maxwell's equations do not need to be Galilei invariant though Galilei-Newton spacetime is the correct spacetime model.

That's like in acoustics, where the air is the medium, which defines a preferred reference frame by its rest frame, and thus the Doppler effect doesn't only depend on the relative velocity of the observer and the source but on the velocities of the source and the observer relative to the medium restframe, and that's true also in the relativistic theory: As soon as you have a medium, there's a preferred reference frame. Of course all the physical laws are compatible with Poincare symmetry, you only have the four-velocity field of the fluid as additional building block, and you define all intrinsic quantities of the medium in the (local) restframe of the fluid (temperature, chemical potential, various densities of thermodynamical potentials etc).

 

[Tendex]

(c) is the non-degenerate metric mathematical formalization of the Galilean degenerate spacetime(b)

 

[Herman Trivilino]

Were others thinking that Maxwell was wrong (and also that the results of Michelson and Morley's experiment were wrong) because the speed of light can't be constant without contradicting Newton and Galileo's statements that time is absolute?

I think that ever since it was established in the early 1800's that light is a wave, the concept of an aether was inferred because all other experiences with waves involved a medium that's doing the waving. Therefore it was assumed that Maxwell's equations were valid in the rest frame of the aether. Michaelson-Morley attempted to detect the speed of Earth through the aether.

The notion of the speed of light being independent of the speed of the source was something that Einstein came up with, as far as I know.

 

[Tendex]

The notion of the speed of light being independent of the speed of the source was something that Einstein came up with, as far as I know.

That's just a consequence of taking the finite constancy of light seriously as he did together with the relativity postulate that rejects unique privileged rest frames like the aether. Once it was empirically clear that the speed of light was not infinite it could either depend on the sources privileging some frame or else follow a relativity principle like the galilean but with a finite speed of light instead(Einstein way).

But in purity a galilean infinite speed of light didn't depend on the source either as it was always infinite, it just was obvious from Galileo and earlier that it wasn't infinite and the only satisfactory mathematical explanation available from the time of Maxwell and until Poincaré-Einstein-Minkowski time of this finiteness was a Newtonian preferred frame or aether.

 

[Herman Trivilino]

That's just a consequence of taking the finite constancy of light seriously as he did together with the relativity postulate that rejects unique privileged rest frames like the aether.

Einstein was careful to state his 2nd Postulate as an assertion that the speed of light is independent of the speed of the source. It's not a consequence in his theory, it's a postulate.

 

[Tendex]

Einstein was careful to state his 2nd Postulate as an assertion that the speed of light is independent of the speed of the source. It's not a consequence in his theory, it's a postulate.

What did you think "taking the finite constancy of light seriously " mean here?