Why there can be only one invariant speed

[Rasalhague]

I've just come across the following argument as to why there can be only one invariant speed for massless particles. It's from Applications of Classical Physics by Roger Blandford and Kip Thorne. But I don't understand. Obviously, it's a contradiction to say that the hypothetical speed c_0 is both invariant and not invariant, but why must this contradiction arise if such a speed existed? Or to put it another way, why does this argument for the non-existence of two invariant speeds not apply to the existence of one invariant speed? I thought that there was no rest frame defined for a photon moving at c. Wouldn't it be equally meaningless to assume one for any other hypothetical invariant speed?

Must these [hypothetical massless particles] travel at the same speed as photons? The answer to this question, according to the principle of relativity, is yes. The reason is simple. Suppose there were two such waves (or particles) whose governing laws led to different speeds, c and c_0 < c each the same in all reference frames. If we then move with speed c_0 in the direction of propagation of the second wave, we would bring it to rest, in conflict with our hypothesis. Therefore all signals, whose governing laws require them to travel with a speed that has no governing parameters must travel with a unique speed which we call c".

http://www.pma.caltech.edu/Courses/ph136/yr2004/0401.1.K.pdf

 

[PeterDonis]

why does this argument for the non-existence of two invariant speeds not apply to the existence of one invariant speed? I thought that there was no rest frame defined for a photon moving at c. Wouldn't it be equally meaningless to assume one for any other hypothetical invariant speed?

You are correct that there cannot be a rest frame for an object moving at the invariant speed even if there is only one invariant speed. But if there is only one invariant speed, it is still possible to have objects moving at that speed; no such object has a rest frame, and all of them fail to have a rest frame at the same speed. What the argument you reference points out is that, if there are two or more distinct invariant speeds, the scheme I just described no longer works: you can no longer rely on all objects moving at an invariant speed to fail to have a rest frame at the same invariant speed.

I personally prefer a different way of getting to the result, which is to observe that there is a group of transformations--the Galilean group--that has no invariant speed at all, and there is a group of transformations--the Lorentz group--that has one invariant speed. But there is no group of transformations that has more than one invariant speed. So if there is an invariant speed at all, there can be only one.

 

[robphy]

I would say it this way.

• In 2d euclidean space, they are no invariant directions (no preferred slopes) because there are no eigenvectors of the rotation matrix.
• In a 1+1 Galilean spacetime or 1+1 Minkowski spaetime,
  there are no preferred speeds for timelike directions. [Principle of Relativity]

• In a 1+1 Galilean spacetime, the Galilean boost transformation
  has a spacelike [here, purely spatial] eigenvector [along the plane of simultaneity],
  which corresponds to an infinite speed.
  All observers agree on the same infinite speed.
  (With the timelike-metric, this vector is a null vector.)
• In a 1+1 Minkowski spacetime, the Lorentz boost transformation
  has a pair of lightlike eigenvectors [along the lightcone],
  which both correspond to the speed of light.
  All observers agree on the same finite speed of light. [Speed of Light Principle]
  (Lightlike vectors are null vectors.)

 

[PAllen]

Just for completeness, I note that at least one highly respected physicist has suggested there is a way to accommodate multiple causal cones in a reinterpreted SR:

https://arxiv.org/abs/1005.1614

 

[PeterDonis]

I note that at least one highly respected physicist has suggested there is a way to accommodate multiple causal cones in a reinterpreted SR

Just to be clear, the hypothesis in this paper does not have multiple invariant speeds; it just allows causal influences to propagate faster than the single invariant speed. The causal cones themselves (of which, as you say, there may be multiple ones) are invariant, but the "speed" corresponding to the boundary of any causal cone that is not the future light cone (whether or not the boundary lies inside or outside the future light cone, i.e., whether or not causal propagation faster than light is allowed) is not invariant.

 

[robphy]

Just for completeness, I note that at least one highly respected physicist has suggested there is a way to accommodate multiple causal cones in a reinterpreted SR:

https://arxiv.org/abs/1005.1614

A while back, this Fermilab site had a stream of his colloquium
http://vmsstreamer1.fnal.gov/VMS_Site_03/Lectures/Colloquium/041201Geroch/index.htmHere's a pdf of his slides. (It appears that archive.org has this cached in case this link dies.)
http://vmsstreamer1.fnal.gov/VMS_Site_03/Lectures/Colloquium/presentations/041201Geroch.pdf