A Is the constancy of c a postulate or a derivable theorem?

[Iskandarani]

Hi everyone, In deriving the Lorentz transformations, the constancy of the speed of light is typically taken as a starting postulate, based on experimental evidence like the Michelson-Morley result.

My question is from a purely theoretical standpoint: Is it possible to construct a self-consistent theory where the Lorentz transformations (and thus all of SR) are derived without postulating the constancy of c? For example, could a theory with a preferred foliation (a universal time coordinate) still reproduce all the observed relativistic effects if the properties of matter and fields (like clocks and rulers) were dynamically altered by their motion relative to this foliation?

In other words, is the modern interpretation of relativity the only logically possible explanation for the experimental results, or are there other, empirically equivalent formulations?

 

[Ibix]

You can add an undetectable preferred foliation - it's called Lorentz ether theory. It's just SR with the claim that one frame is special, but you can't tell which one so the choice of frame is an additional assumption, which is why it's disfavoured by Occamcs razor.

It's off limits for discussion here because it's purely philosophical and adds nothing to SR.

 

[Iskandarani]

Thank you, @Ibix. That is an incredibly helpful and concise answer.

Giving me the precise term "Lorentz Ether Theory" is exactly what I was looking for. I understand why it's disfavored by Occam's razor if it's empirically indistinguishable from SR. I also completely understand and respect that this makes it off-limits for further discussion here.

I appreciate you taking the time to give me such a clear and direct response.

 

[weirdoguy]

Is it possible to construct a self-consistent theory where the Lorentz transformations (and thus all of SR) are derived without postulating the constancy of c?

Yes. Actually that was the way I was taught relativity at my uni. But it's less standard to do so and I guess more suitable for someone who's intersted in theoretical physics. Lecture notes are still online, but they are in polish:
https://www.fuw.edu.pl/~wysmolek/Mechanika-2019-2020
"Wykład 2" is where it starts.

 

[Sagittarius A-Star]

Is it possible to construct a self-consistent theory where the Lorentz transformations (and thus all of SR) are derived without postulating the constancy of c?

You can do this i.e. by demanding, that a velocity composition in the same direction is commutative plus a few other assumptions, as shown in the following link:

I Thread 'Only Minkowski or Galilei from Commutative Velocity Composition'

Jul 31, 2022

The LT can be derived from the first postulate of SR, assuming linearity an that velocity composition is commutative, and that GT can be excluded: $t' \neq t$.

Definition of the constant velocity $v$:

$x' = 0 \Rightarrow x-vt=0\ \ \ \ \ \$(1)

With assumed linearity follows for the only possible transformation, that meets requirement (1), where $A_v$ may be a function of the constant velocity $v$:

$\require{color} x' = \color{red}A_v(x-vt)\color{black}\ \ \ \ \ \$(2)

With SR postulate 1 (the laws of physics are the same in all inertial reference...

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Then the experiment must decide, if the invariant speed is finite (GT vs. LT).

 

[Ibix]

Then the experiment must decide, if the invariant speed is finite (GT vs. LT).

The argument there is whether this counts as a postulate. I've seen people argue that "the universe has a finite/an infinite invariant speed" is a postulate, so "one postulate" approaches just narrow you down to two theories and give you an obvious experimental approach on how to pick the correct one, but still have two postulates.

 

[Sagittarius A-Star]

The argument there is whether this counts as a postulate. I've seen people argue that "the universe has a finite/an infinite invariant speed" is a postulate, so "one postulate" approaches just narrow you down to two theories and give you an obvious experimental approach on how to pick the correct one, but still have two postulates.

Yes. Morin calls it "Relativity without c".

Source (see chapter 11.10 on page XI-38):
https://bpb-us-e1.wpmucdn.com/sites.harvard.edu/dist/0/550/files/2023/11/cmchap11.pdf

via:
https://davidmorin.physics.fas.harvard.edu/books/classical-mechanics/

 

[Sagittarius A-Star]

You can add an undetectable preferred foliation - it's called Lorentz ether theory. It's just SR with the claim that one frame is special, but you can't tell which one so the choice of frame is an additional assumption, which is why it's disfavoured by Occamcs razor.

It's off limits for discussion here because it's purely philosophical and adds nothing to SR.

Yes. A disadvantage of LET is, that it's "absolute time" is not generally equal to what a clock measures.

 

[Dale]

An alternative approach is to just write the most general affine transformation between inertial frames and determine the coefficients by experiment. This was done by Robertson (Robertson, Rev. of Mod. Phys. 21, pg 378, 1949) who showed that you could deduce the Lorentz transform experimentally to within 1% with just the Michelson Morely, Ives Stillwell, and Kennedy Thorndike experiments. Modern tests put much more stringent limits.

 

[ohwilleke]

You can derive the speed of light from Maxwell's equations, and when you do so, you get a result that is independent of the motion of the observer (which seemed like a bug instead of a feature for the roughly quarter of a century until we knew better).

While it takes an ah-ha moment and some serious intellectual heavy lifting to do it this way, this is probably sufficient to derive the Lorentz transform. At a minimum, it provides a strong hint regarding how the Lorentz transform works which can be tested experimentally.

This approach also provides observables from which the speed of light can be determined indirectly that are much easier to measure than a direct measurement of the speed of light, if you are willing to assume that Maxwell's equations are correct.

Maxwell's equations are, of course, correct for this purpose, although Maxwell's equations omit the quantum behavior of light (like the fact that light comes in minimum "packet sizes" at any given frequency, quantum tunneling, quantum entanglement, and the inherent randomness of photon propagation). But these quantum aspects of light are not relevant to this question.

I'm not sure, however, that this derivation can be rigorously generalized from the classical Maxwell's equations to quantum electrodynamics (QED).

 

[Sagittarius A-Star]

You can derive the speed of light from Maxwell's equations, and when you do so, you get a result that is independent of the motion of the observer (which seemed like a bug instead of a feature for the roughly quarter of a century until we knew better).

That's not completely correct. Before Einstein, when the physically relevant coordinate transformation was thought to be the GT, Maxwell's theory was thought to be strictly true in only one inertial frame - that of still ether.

Source: Chapter "6.1 Transformation of the Field Vectors" in W. Rindler "Essential Relativity".
https://www.amazon.com/-/de/dp/3540100903?tag=pfamazon01-20

BTW.: This book contains also a chapter "2.17 Special Relativity without the Second Postulate".

 

[ohwilleke]

Before Einstein, when the physically relevant coordinate transformation was thought to be the GT, Maxwell's theory was thought to be strictly true in only one inertial frame - that of still ether.

Good to know. I hadn't been aware of that.

 

[Nugatory]

That's not completely correct. Before Einstein, when the physically relevant coordinate transformation was thought to be the GT, Maxwell's theory was thought to be strictly true in only one inertial frame - that of still ether.

And this is why the ether was hypothesized in the first place, and the motivation for the MM experiment which would detect it.

At least that's how I understand the history.

 

[PeterDonis]

And this is why the ether was hypothesized in the first place, and the motivation for the MM experiment which would detect it.

At least that's how I understand the history.

That's how I understand it as well.

And the fact that Maxwell's Equations were Lorentz invariant rather than Galilean invariant was also one of the things Einstein seized on in the train of thought that led him eventually to Special Relativity. Everyone else was trying to figure out how to make electrodynamics work in a Galilean invariant framework. Einstein instead asked himself what it would take to make ordinary mechanics work in a Lorentz invariant framework.