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Does SR explicitly assume $c$ to be finite?

If so, by what statement in Einstein's original paper is this implied?

If not, what to make of equations containing $c$? (e.g., $E = mc^2$)

Formulating the question in a different way: without knowledge of any physics other than inertial frames, do Einstein's two postulates uniquely identify the transformation between inertial frames as Lorentzian (with a finite limiting speed)? Or would it also allow the Galilean transformation?

I'm trying to figure out what is part of the axiomatic system of special relativity and what is not. I'm assuming the finiteness of the speed of light is part of it, but I'm not sure which statements of the theory make that explicit.

The invariance of the speed of light is explicit in the postulate and conversely it's clear the numerical value of $c$ is left to be determined empirically, as it has no impact on the essence of the theory.

EDIT: Based on the many comments I would like to clarify that my question is not if taking $c \to ∞$ leads to classical mechanics, which is obvious. My question is whether the finiteness of $c$ is part of the core of special relativity or left for empirical determination. In the latter case, the theory (alone) would allow both galilean and lorentzian transformations. In the former case, the postulates select the lorentz transformation uniquely, without the need for further empirical facts. This, I presume, is how special relativity is mostly interpreted, but I'm wondering where the finiteness of the speed of light is stated in Einstein's original paper.

David Bailey
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Real Pattern
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    Special relativity with infinite speed of light is Newtonian mechanics (with Galilean relativity). – Puk Mar 06 '23 at 21:33
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    Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed. – Buzz Mar 08 '23 at 18:33

3 Answers3

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Einstein's paper probably did not make an explicit statement that the speed of light was finite because it never occurred to him that it was necessary:

  • The theory is mathematically valid for any speed of light, but assuming an infinite speed of light was uninteresting except for showing that the theory correctly reduced to Galilean relativity in that limit.
  • Everyone reading the paper would know that the speed of light had been experimentally measured and it was well established that it was finite.
  • Special relativity was inspired by classical electrodynamics, and everyone reading the paper would also know that Maxwell's equations break down for an infinite speed of light.

Stating (and explaining) the obvious can be helpful in pedagogical papers for novices learning a subject, but Einstein's paper was written for experts.

David Bailey
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  • Thanks @DavidBailey. So you don't agree with John Rennie's answer? https://physics.stackexchange.com/a/220821/288306 – Real Pattern Mar 11 '23 at 09:46
  • I'm not sure I agree with your view on stating the obvious when it comes to axiomatic systems. Should Euclid have not stated "All right angles are equal to one another." – Real Pattern Mar 11 '23 at 23:41
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    I am sure that John Rennie and agree that special relativity reduces to galilean relativity in the appropriate limit. Theoretical physics differs from axiomatic mathematics, but even Euclid did not explicitly state every "axiom" he used, e,g, various associative and commutative properties. Much has been written and argued about what are the essential minimal postulates for special relativity. As Dale's answer notes, Einstein did not have the last word. – David Bailey Mar 12 '23 at 15:19
  • Ok, sounds good. So the modern interpretation of SR's postulates would be more in the lines of: 1. RP 2. There is a finite invariant speed (the speed of light). I was hoping that something like "universal constant" or "definite velocity" in Einstein's paper would already mark c as finite. – Real Pattern Mar 13 '23 at 10:21
  • "I am sure that John Rennie and agree that special relativity reduces to galilean relativity in the appropriate limit." I'm sure (almost) everybody would agree on that. I was wondering if SR is the specific flavor of relativity with $c < \infty$ (and not $c \le \infty$) and Newtonian Mechanics the one with $c = \infty$. Note: given the success of NM, SR needs to reduce to NM in the "appropriate limit." – Real Pattern Mar 14 '23 at 10:17
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Does SR explicitly assume c to be finite?

Yes.

If so, by what statement in Einstein's original paper is this implied?

I emphatically reject the premise of this part of the question, which is that only Einstein's original paper defines SR. In science the seminal authors of a concept get the first word on the definition of the theory. They do not have the last word.

All of the tests that are considered to experimentally validate special relativity are tests for things that only occur if the invariant speed is finite. Physics is an experimental science, so the fact that when we test SR we assume a finite c implies that a finite c is assumed as part of SR.

Dale
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  • Thanks @Dale. The assumption still seems implicit. – Real Pattern Mar 11 '23 at 09:47
  • I was hoping that something like "universal constant" or "definite velocity" in Einstein's paper would already mark $c$ as finite. But ok. I guess the modern interpretation of SR's postulates would be more in line with 1. RP 2. There is a finite invariant speed (speed of light). – Real Pattern Mar 13 '23 at 10:18
  • I guess this makes your point, Wikipedia's mathematical formulation of the second postulate: There exists an absolute constant $0 < c < \infty$ with the following property... – Real Pattern Mar 14 '23 at 10:20
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    @RealPattern well, you can argue about explicit or implicit. I always feel that a person's or group's actions speak louder than their words. So that is why the experimental fact makes it so clear to me. But I can see your position that it remains implicit in that way. As you say, many more modern texts (e.g. Wikipedia) state it plainly. – Dale Mar 14 '23 at 15:15
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In Wikipedia's modern terms:

  1. First postulate (principle of relativity)

The laws of physics take the same form in all inertial frames of reference.

  1. Second postulate (invariance of $c$)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity $c$ that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.

or in Einstein (1905 Electrodynamics of moving bodies) words:

In the following we make these assumptions (which we shall subsequently call the Principle of Relativity) and introduce the further assumption, —an assumption which is at the first sight quite irreconcilable with the former one— that light is propagated in vacant space, with a velocity $c$ which is independent of the nature of motion of the emitting body.

If $c<\infty$ these postulates lead to Lorentz transformations as shown on Einstein's paper (clearly he uses as a finite quantity as he manipulates $c$ as a finite variable). If $c\to\infty$ you recover Galilean transformations.

The value of $c$ in your unit system is to be determined empirically.

Mauricio
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  • Thanks @Maurico. I know that if $c < \infty$ this leads to the Lorentz transformation. I'm asking if the theory alone uniquely leads to the Lorentz transformation. – Real Pattern Mar 11 '23 at 12:23
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    @RealPattern what does that mean? Clearly if you assume the two postulates you get to Lorentz transformations. – Mauricio Mar 11 '23 at 13:06
  • Thanks for your comment. I mean, it seems that we need the 2 postulates and additionally an empirical determination of the finiteness of $c$. Without this extra proposition we don't exclude the galilean transformation. Maybe I'm getting too strict here. – Real Pattern Mar 11 '23 at 13:25
  • "If $c < \infty$ these postulates lead to Lorentz transformations." This means that the postulates only conditionally lead to the Lorentz transformation. – Real Pattern Mar 11 '23 at 20:03
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    @RealPattern sure, the validity of Lorentz transformations has to be validated by empirical data, which is the case. – Mauricio Mar 11 '23 at 21:26
  • "Clearly if you assume the two postulates you get to Lorentz transformations." Well, from your answer I got the feeling that $c < \infty$ was not part of the postulates, so in order to get the Lorentz transformation you need the two postulates plus the assumption that $c < \infty$. Is that correct? – Real Pattern Mar 11 '23 at 22:06
  • @RealPattern why are you so bothered with that specific detail? Anyway no, special relativity as postulated by Einstein does not EXPLICITLY state that light has a finite c. You either have to derive that from empirical results or modify postulate 2 to make it say something like "As measured in any inertial frame of reference, light is always propagated in empty space with finite velocity that is independent of the state of motion of the emitting body" – Mauricio Mar 11 '23 at 22:16
  • Ok thanks. I'm trying to figure out if SR leaves the door open for galilean relativity. – Real Pattern Mar 12 '23 at 11:19
  • @RealPattern special relativity was invented because Galilean relativity was not compatible with experiments – Mauricio Mar 12 '23 at 11:52
  • yes, I understand. I was hoping that something like "universal constant" or "definite velocity" in Einstein's paper would already mark c as finite. But ok. Wikipedia states as mathematical formulation of the second postulate: There exists an absolute constant $0<c<\infty$ with the following property... – Real Pattern Mar 13 '23 at 10:27