Undergrad Solutions that break the Lorentz invariance...?

[Suekdccia]

TL;DR

Solutions that break the Lorentz invariance...?

I was reading a discussion where some physicists participated* where the topic of Lorentz invariance violations occurring in cosmology is mentioned.

There, they mention that we can imagine a Lorentz-violating solution to the cosmological equations. What do they mean by that? Can anyone specify any example of such solutions (a solution which really breaks the Lorentz symmetry)?

They also said that we don't need a theory which violates the Lorentz invariance to have solutions that are not Lorentz invariant. What do they mean by that? Can you specify any example of such solutions (a solution which really breaks the Lorentz symmetry)?

Thank you

*(https://books.google.com/books?id=W...age&q=inhomogeneities violate lorentz&f=false)

 

[PeterDonis]

You have to distinguish two very different things:

(1) The laws of physics are (locally) Lorentz invariant; but

(2) Particular solutions to the equations, in general, are not.

For example, the particular solution that describes our universe is not Lorentz invariant, because it includes lots of matter and radiation, and the matter and radiation has particular states of motion. The simplest example is the CMB, since it's everywhere; at any event in spacetime, the CMB only looks isotropic (the same temperature in all directions) in one particular Lorentz frame. So the CMB is not Lorentz invariant. But the underlying laws that govern the CMB and everything else are Lorentz invariant.

That is what they are talking about in the reference you give.

 

[vanhees71]

That claim is often made even in scientific papers and textbooks. Of course, a special-relativistic theory is Poincare invariant (and thus also Lorentz invariant). Your example of the cosmic microwave background is of course also Poincare invariant. It's simply electromagnetic radiation in thermal equilibrium at a temperature of about 2.725K. Now sometimes it's claimed that QFT at finite temperature "breaks Lorentz invariance", but that's of course not true. Of course, there is a special (in GR local) inertial frame distinguished by the physical situation, i.e., the rest frame of the CMBR, but this doesn't imply that somehow Poincare invariance is broken. You can formulate it manifestly covariant. You only have to take into account all "elements" you need for its description. We need temperature, which is a Lorentz scalar and the four-velocity $u^{\mu}$ of the observer relative to the (local) rest frame of the CMBR. Then the statistical operator can be written in manifestly covariant form,
$$\hat{R}=\frac{1}{Z} \exp[-u \cdot \hat{p}/(k_{\text{B}} T)], \quad Z=\mathrm{Tr} \exp[-u \cdot \hat{p}/(k_{\text{B}} T)].$$