Thought Exp: Transform Laws & Feedback Needed

[pervect]

I'd like to get some feedback on the following argument.

Gallilean mechanics, with the Gallilean transformation laws, is a perfectly consistent theory.
Special relativity, with the Lorentz transformation laws, is another perfectly consistent theory.

The question is - can we have some physical quantities that consistently transform according to the Gallilean transform, mixed together with other physical quantites that transform according to the Lorentz transform? My thoughts are that it is logically possible to imagine, as both systems are self consistent. However, such a hybrid approach would not satisfy the principle of relativity, because with two sets of physical quantites transforming differently, there would be a way to detect absolute motion.

It seems simple enough, but it's not something I've seen discussed, and I am wondering if there is some hidden issue with the logic.

 

[anuttarasammyak]

I do not think there exist quantities of such a hybrid transformation.

 

[Helena Wells]

What about QFT which is the unification of SR with QM?

 

[robphy]

There are notions that are the same in both spacetime geometries...
affine notions:
for example, parallelism is preserved, as well as the whole vector space structure (Vector addition)...all without making any reference to a metric.

This is likely the opposite of the hybrid quantities you are probably seeking.

 

[PeterDonis]

can we have some physical quantities that consistently transform according to the Gallilean transform, mixed together with other physical quantites that transform according to the Lorentz transform?

Transformations are not transformations of physical quantities. They are transformations of coordinates, and mathematical objects whose specific components depend on the coordinates.

A mathematical model can define how various mathematical objects transform under various transformations of coordinates however it wants. But those definitions must be consistent with the requirement that any actual physical quantity--anything that can actually be directly measured--is invariant under any transformation allowed by the mathematical model.

For example, in GR, nothing stops you from defining two coordinate charts on the same spacetime that are related by a Galilean transformation. For an example in FRW spacetime, see part 3 of Ned Wright's cosmology tutorial:

http://www.astro.ucla.edu/~wright/cosmo_03.htm

But all physical quantities (for example, the measured redshift of light from one galaxy by observers in another galaxy) must remain invariant under this transformation.

with two sets of physical quantites transforming differently, there would be a way to detect absolute motion

The way you would detect absolute motion is if the physical laws you discovered had a preferred state of motion in them. This is not a property of coordinate transformations; it's a property of the physical laws. In the mathematical model, you would have one particular state of motion in which the physical laws had a certain form, while in every other state of motion they had a different form. How that would relate to coordinates and coordinate transformations would depend on the model, what mathematical objects it had in it, and what coordinate charts and coordinate transformations were allowed.

For example, many physicists in the late 19th century thought that the proper mathematical model for electromagnetism would have Maxwell's Equations only holding exactly in one particular state of inertial motion (the "ether rest frame"), and that to derive the correct laws of electromagnetism for any other state of inertial motion, you would apply the Galilean transformation to Maxwell's Equations, which of course would not leave those equations invariant--extra terms would appear that would predict particular physical effects that would be seen by inertial observers who were not at rest with respect to the ether. One of those predicted effects was a variation in the measured round-trip speed of light, which was what Michelson and Morley originally expected to find when they did their experiment.

But note that none of this contradicts the fact that Maxwell's Equations, mathematically, are invariant under Lorentz transformations, not Galilean transformations. That invariance property suggests, at least in hindsight, that maybe a better physical theory of mechanics could be obtained by assuming that the laws of mechanics should be invariant with respect to Lorentz transformations instead of Galilean transformations--which of course is exactly what led to special relativity. But there is nothing that requires any mathematical model of electromagnetism to have physical laws in it that are Lorentz invariant. We have such a model today because it works, not because it's the only one that's mathematically possible.

 

[pervect]

I do not think there exist quantities of such a hybrid transformation.

I don't believe they exist either, because I think the principle of relativity is valid, and if my logic is correct, the principle of relativity would preclude the existence of such hybrid quantities.

But that conclusion relies on the logic being correct.

 

[anuttarasammyak]

Say B-IFR moves with velocity v in A-IFR, A-IFR moves with velocity -v in B. We can say it for both the transformations, though not quantity.

the principle of relativity would preclude the existence of such hybrid quantities.

Though it is tricky, we may say "mass of particle" conserves in both the transformations. It does not change in Galilei transformation and cannot change in Lorentz transformation because of the convention that all IFRs share mass as energy of particle measured in IFR where it is at rest. The same word but meanings of mass or relation between mass and energy differ.

I think non mechanical quantities, e.g. total number of particles in the system, electric charge both individual and total, temperatures of some defined materials and entropy of the system might have a chance for hybrid transformation, I mean both the transformations give the same result. If there exist a quantity whose transformation does not depend on v in one transformation, it is a good candidate to be so too in another transformation.

 

[Ibix]

The principle of relativity plus finite or infinite invariant speed leads to Galilean or Einsteinian relativity. Combining the two violates the principle of relativity. So unless you've got a derivation of the transforms that doesn't depend on the principle of relativity (and I'm not aware of one) then I'd say you have a problem immediately.