Special Relativity Knowledge Challenge (for experts only)

Aer

This is specifically targeted towards those who claim to be experts in the area of Special Relativity.

The following derivation derives the Lorentz equations of special relativity. Follow the derivation and list the assumptions (postulates) used to derive the theory. That is to say, if a postulate must exist for the use of a given equation, name the postulate and the equation number. Finally state whether you think this is a valid or invalid derivation.

Lorentz Equations Derivation

This shouldn't be too difficult for those with extensive experience in Special Relativity. Those without extensive experience are discouraged from participating, but I cannot stop you from doing so.

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Crosson

In my opinion, the first assumption is narrows the field (eliminates the most choices). They assume that the difference between events in one frame depends only on the difference in another. They assume:

$$\Delta x' = f (\Delta x, \Delta t)$$

Rather then the more general case:

$$\Delta x' = f(x_1,x_2,t_1,t_2)$$

The first case is fine for special relativity, in which the universe is the same everywhere in space and time (translational symmetry), but of course only the general case is correct in a universe with gravitation.

Notice that in the next section they use the assumption that x = vt. This is straightfoward, but I will have to read more to see why this does not conflict with the ability to handle accelerations in SR.

In equation (14) they ignore the possibility that E = 0. I had been wondering how they would avoid deriving the galilean transformations, and they did it by implicitly assuming E was not zero (although they do treat this case seperately later).

Overall, this is a good derivation because it includes a lot of generality and is not too difficult to follow.

Mortimer

In Eq. 12, the assumption is made that $$\gamma_{v2}$$ is here still only a function of $v_2$. It may here be a function of both $v_1$ and $v_2$. In that case the equations following Eq. 12 would be considerably more complicated. The condition derived from Eq. 14 that $a$ must be a constant in order to satify it, combined with the earlier assumption, replaces the classical postulate of the universality of lightspeed $c$ as a condition, which brings us back to Einstein. I favor the latter because that condition was empirically found, while the assumption in Eq. 12 is not proven.

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Meir Achuz

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The derivation includes some assumptions that are stated as if obvious.
It should be made more rigorous in this respect.

*In point 1), "We know..." is an assumption that requires justification.

*Point 4) about the group property should be justified.
Using it requires more complicated algebra than using the invariance of c.

*After specializing to 1D, he uses the assumption that an LT cannot depend on the direction of v. But a general LT does depend on the direction of v,
just not in gamma. This can be fixed up in the derivation, but not as it stands.

*"For a general LT... are equal", should be justified.

Actually, the invariance of the speed of light is an unecessary additional postulate. It follows from the equivalence of all inertial frames:

c is not just the speed of light, it is a parameter in most of the equations of EM. This is clearest in Gaussian units. In SI units, this dependence shows up in epsilon0. If the parameter c varied as the earth moved, then most EM phenomena would vary. This is the reason for the title of Einstein's paper. From Maxwell's equations the invariant parameter c is shown to be the speed of light, thus deriving the second postulate from the first.

pervect

Staff Emeritus
Meir Achuz said:
The derivation includes some assumptions that are stated as if obvious.
It should be made more rigorous in this respect.

*In point 1), "We know..." is an assumption that requires justification.
I've just started to look at the paper, but the same point struck me. The paper says

We know that the coordinates y and z perpendicular to the velocity are the same in both reference frames: y = y' and z = z'.
How do we know that in "the most general case"? I do not have a counterexample immediately at hand, but I'm not convinced that this assumption is true, either.

[Edited by Nereid: fixed mis-matched [ QUOTE ] tags]

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robphy

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You can probably apply a symmetry argument. I recall (from Mermin's little book?) an argument explaining why there is no "length contraction" in the perpendicular directions. Suppose each observer had an arm of rest length L along their transverse y-axes, with some marking device (like a pen) at the far ends. When they pass each other (assuming they are properly lined up), they would mark each other inconsistently if the transverse arms were affected by their motion.

From googling, here is a similar argument involving a hole and a bullet, part 5 in
http://cmtw.harvard.edu/Courses/Phys16/l1_latex/l1_latex.html

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I think this paper is actually not as much a formal derivation as a somewhat arm-waving classroom demonstration. As such its aim is not to convince but persuade. Maybe it's also a challenge to the students to come up with just the kind of questions we have created here.

robphy

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This is one of those "Lorentz Transformations without the speed of light" proofs where the value of the speed of light plays its role only in the last step. Of course, some other set of assumptions must be invoked and weaved into the storyline.

Depending on one's tastes on what one regards as fundamental or primitive or even "known by intended audience" (e.g. group structure? causal structure? electromagnetism? projective geometry? experimental facts?), one may or may not like these various approaches.

I have attached a chart from "Spacetime and Electromagnetism" by J.R. Lucas, P.E. Hodgson ( https://www.amazon.com/exec/obidos/tg/detail/-/0198520387/002-5397451-1060834?v=glance ) where they try to diagram the various approaches to obtain the Lorentz Transformations. [Certainly, the chart is not complete... but merely representative.]

Search the American Journal of Physics (http://scitation.aip.org/ajp/ ) for specific papers. Try to search for say, Lorentz, in the keyword field. WOW! You can now search all the way back to the first issue from 1933... and download articles (for free, if your school has access).

My personal favorites (in the terminology of the chart) include pseudoangles, Michelson-Morley, radar, metric, and causality/partial ordering.

PS: If the goal is to move on to GR or beyond, where many of symmetries enjoyed by Minkowski spacetime are not available, some of these approaches might not be so easy to extend.

My \$0.02.

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pmb_phy

robphy said:
My personal favorites (in the terminology of the chart) include pseudoangles, Michelson-Morley, radar, metric, and causality/partial ordering.
There is an article in there about things like extinction in the MMX experiment etc. It seems very complete and a very interesting read when I get around to it.

Pete

Meir Achuz

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"*In point 1), "We know..." is an assumption that requires justification."

I did not mean by that that there aren't several reasonable arguments for that assumption. I just was responding to the "Follow the derivation and list the assumptions (postulates) used to derive the theory." directive in the question.
Even a "somewhat arm-waving classroom demonstration" should not confuse the students or suggest ad hoc assumptions. It certainly shouldn't be put on a website, even if it does lead to discussions about its shortcomings.

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Even a "somewhat arm-waving classroom demonstration" should not confuse the students or suggest ad hoc assumptions. It certainly shouldn't be put on a website, even if it does lead to discussions about its shortcomings.
I sort of agree, although I'm more tolerant of loose derivations ("motivations"). I do like the idea that preserving length-ratios plus "reasonable" extra assumptions like no change in orthogonal directions, gives the LTs, because that would tie in with projective relativity, which I find interesting. If length ratios are preserved then so would be the cross-ratio, no?

Perspicacious

From translational symmetry of space and time, we conclude that the functions f_x(x, t) and f_t(x, t) must be linear functions.
This assertion improperly combines an undefined notion (translational symmetry) with an incorrect statement (implied linearity).

Aer

Sorry, I kind of abandoned this after I posted it. Let me just say that I don't particularly like the derivation upon further analysis. Particularly equations 14 and 26 jump out at me. Feel free to discuss amongst yourselves

neopolitan

zero problem

Having read the derivation, I am struck by what seems to be a problem in equation (5) of the reference. In the lead in paragraph, the author states clearly that x = vt. This is substituted in equation (2) to find that B = -vA which is substituted back into equation (2) (!) and results in the equation x' = A(x - vt).

However, it has only just been stated that x = vt so x' = A(x - vt) = A(0) = 0.

Seems there is some confusion of terms going on here.

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