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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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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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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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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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