# Lorentz transformation without constant speed of light in vacuum reasonable?

So there are many methods to teach the special relativity, such as Bondi's K-caculus, or the Minkowski diagrams from the viewpoint of geometry. But all these methods have two fundamental assumptions, namely

1. All laws of physics are the same in all inertial frames;

2. In empty space, light travels at constant speed $c$ independent of the source or the observer motions.

But we know the constant speed of light in vacuum can appear naturally in Maxwell's wave equation. So can we drop this assumption to derive the Lorentz transformation?

Almost. The approach you speak of is essentially Ignatowski's 1911 Approach. He used the relativity postulate (your assumption 1) but replaced the constant lightspeed assumption with some very basic assumptions about symmetries in spacetime (below). I also cite some more modern references for the approach.

This approach deduces the existence of a speed $c$ that would be observed to be the same in all inertial reference frames. The approach does admit $c=\infty$, i.e. Galilean relativity is admitted by this approach. Moreover, the approach cannot tell you what the invariant speed is if it turns out to be finite. It then becomes a wholly experimental question what this frame invariant speed is, or even whether it is finite. Of course, the Michelson Morley experiment and others showed that the invariant speed is finite, and that the speed of light is either that speed or mighty near to it. Whether the two are indeed the same boils down to the photon mass; they must be if the photon is truly massless and modern measurements give fantastically small upper limits to the photon mass. See the Photon Mass Wikipedia page for further details.

In this way, the universal invariant speed becomes a great deal more basic notion, a geometrical notion that is much deeper than light alone. Experiment then links the notion with the speed of light.

Note that, as in Pentcho Valev's answer, frame invariance of $c$ cannot be deduced from Maxwell's equations without further assumptions. It was simply assumed in the nineteenth century that Maxwell's equations would change their form to reflect the existence of a luminiferous aether until the Michelson Morley experiment suggested that they might not change their form.

The Assumptions

The assumptions you need to derive the Lorentz transformation by Ignatowski's approach are:

1. First, one must postulate that a manifold structure / co-ordinates on spacetime are even meaningful, and then that motion can be described by transformations on these co-ordinates.

2. The relativity postulate then ensures that these transformations depend only on the relative velocity between inertial frames and also completes the group structure for the set of transformations kitted with composition (through enforcing associativity).

3. Assumptions of homogeneity of spacetime together with continuity of the transformations in their dependence on the spacetime co-ordinates then show that the group of transformations is a linear, matrix group;

4. Assumption of continuity of the transformations in their dependence on the relative velocity between frames then shows that this matrix group is a Lie group of $4\times4$ matrices;

5. Isotropy of space then shows that the Lie group is the identity connected component of one of the orthogonal groups $O^+(4)$ or $O^+(1,\,3)$ (or the Galilee group, in the special case where the free parameter $c$ is infinite);

6. Causality then rules out the rotations $O^+(4)$;

7. Experiment shows us that the free parameter $c$ is finite and our group is $O^+(1,\,3)$, not the Galilee group.

References

Palash B. Pal, "Nothing but Relativity," Eur.J.Phys.24:315-319,2003

Jean-Marc Levy-Leblond, "One more derivation of the Lorentz transformation"

• @Jack You are welcome. This is a very interesting topic, and I find a wholly geometric / experimental approach more satisfying than Einstein's original, which leaves one wondering "what's so special about the speed of light?". Answer, aside from an empirical example of the behavior of a massless particle, not much in particular. The notion holds for all massless particles. – Selene Routley Jan 19 '17 at 3:02
• I've been looking for this discussion for some time. Unfortunately search results are always swamped with the usual "c is constant in inertial frames" starting point which is unsatisfying to me. Now that we have excised the speed of light, the next question that follows might be: "Can it be proven that the value of the invariant speed itself must be determined experimentally, or could it possibly be determined theoretically?" – Mr Purple Mar 22 at 7:04

The CONSTANT speed of light did not appear in the Maxwell's equations. That the speed of light is independent of the speed of the light source was a tenet of the ether theory, and that it is independent of the speed of the observer as well was a consequence of this tenet and the principle of relativity.

Lorentz transformation without constant speed of light is unthinkable. Spacetime without constant speed of light is unthinkable as well:

http://community.bowdoin.edu/news/2015/04/professor-baumgarte-describes-100-years-of-gravity/ "Special relativity is based on the observation that the speed of light is always the same, independently of who measures it, or how fast the source of the light is moving with respect to the observer. Einstein demonstrated that as an immediate consequence, space and time can no longer be independent, but should rather be considered a new joint entity called "spacetime."

And yet the speed of light is variable, not constant.

• Your summary of the relationship between Maxwell's Equations and a putative constant $c$ is correct, but I'm not sure the rest addresses quite what the OP is thinking (although it's hard to second guess the exact question sometimes). My reading of it is that the OP is asking whether the constancy of speed of light can come from elsewhere rather than as a postulate. As you show, it can't rigorously come from Maxwell's equations, but there is a whole different approach (Ignatowski's) that deduces an invariant speed from basic symmetries as in my answer. – Selene Routley Jan 18 '17 at 12:09