Use ilja-schmelzer.de/gravity/LorentzInvarianceArgument.php instead.

In the blog Shores of the Dirac Sea, Moshe proposes an argument in favour of Lorentz invariance as being fundamental titled

The argument is the following:

February 1, 2009 by Moshe:

...Nature is relativistic, and this fact is crucially important!

In fact, Lorentz invariance is so important, and relativistic systems are so entirely different from non-relativistic ones, that there is whole discipline (theoretical high energy physics) which is devoted almost exclusively to the study of Lorentz invariance and its consequences. Along the way we stumbled upon a few things that may possibly be interesting and relevant in our search for that underlying structure.

The message has two parts really, both need some unpacking so I'll only summarize them here. First, fundamental violations of Lorentz invariance, meaning the idea that Lorentz symmetry is not an ingredient of that holy grail, the fundamental theory of our universe, are strongly excluded by experimental constraints. It is not known how to break Lorentz symmetry in a way that induces only small effects on observable physics (unlike for example symmetries like baryon number that can be violated by a small amount; technically: there are many new relevant and marginal operators once you allow for Lorentz violations at high energy, their coefficients are constrained to a ridiculous degree by experiment). Therefore, all the evidence we have indicates that the fundamental theory is Lorentz invariant.

In the following discussion, he argues:

on February 3, 2009 at 12:28 am Moshe:

At low energies we know that any violation of 4dim LI, if it exists at all, should be small. This means that any EFT at much higher energies should not have any (4dim) Lorentz violating relevant operators.

With his well-known modesty, Luboš Motl supports this argument too:

on February 3, 2009 at 8:34 am Luboš Motl:

How many dozens of decimal points of agreement do you have to see to graciously begin to consider the possibility that Lorentz invariant might be real and not just an illusion that should be a permanent subject of silly trash-talking? How many centuries of perfectly successful tests of relativity do we have to live through before we will be allowed to say that those who ignore relativity or expect it to go away are simply not doing physics?

.... Even if I insist on the most experimentally verified input only, a good theory must be able to reproduce relativity to vastly higher boosts than a generic fundamentally non-relativistic theory ever could.

I disagree. And for a counterexample we don't have to look far away – our environment contains a lot of examples of particular forms of Lorentz invariance in effective field theories, where the fundamental theory breaks this Lorentz invariance, having a discrete structure in space but not in time. These examples are known as waves in condensed matter theories, in particular as sound waves. Indeed, if we have some wave equation in condensed matter theory, we can consider it's linear approximation for small disturbances ψ(x). This is already a linear wave equation. We can, as well, ignore in some limit derivatives of order higher than two of ψ(x). Then, the general form of such a self-adjoint wave equation is

∂_{μ} f^{μν}(x,t) ∂_{μ}ψ(x,t) = 0

for some symmetric fields f^{μν}(x,t) of coefficients. This field depends on the particular type of condensed matter as well as some local properties, like temperature, pressure, density and so on. But, whatever these coefficients f^{μν}(x,t), we can define an (in general curved) Lorentz metric g_{μν}(x) so that

f^{μν}(x,t) = g^{μν}(x,t) (-g)^{1/2},

and, therefore, the wave equation becomes the harmonic equation for the Lorentz metric g_{μν}(x,t), thus, appears to be a locally Lorentz-invariant equation. The formula how to compute the g_{μν}(x) for a given f^{μν}(x) is elementary, it is sufficient to take the determinant of both sides to obtain f^{-1} = g, (f = det(f_{μν})), and, therefore,

g^{μν}(x) = f^{μν}(x) (-f)^{1/2}.

In arXiv:0711.4416, arXiv:gr-qc/0505065, and the references therein one can find lot's of examples of such effective Lorentz metrics in condensed matter theory. Now, given that we have identified the condensed matter wave equation with the harmonic equation for some Lorentz metric, we have found some part of reality – namely all the waves governed by this particular wave equation – which has the corresponding local Lorentz symmetry. This local Lorentz symmetry has been obtained from a fundamental theory – some atomic model – which obviously does not have it: It's fundamental Lorentz invariance, with the speed of light c, has nothing to do with the sound Lorentz metric we have found, the atomic model itself is discrete in space but continuous in time, thus, also obviously breaks this sound Lorentz symmetry.

I recognize very well that this Lorentz invariance is only a very particular one, usually restricted to only one type of waves, moreover, that the speed of sound in general also depends on the frequency. But, however particular, it is a local Lorentz symmetry in all its beauty, it appears in an effective field theory, and the corresponding fundamental theory does not have it. Thus, it defines a counterexample to the thesis proposed by Moshe that Lorentz symmetry has to be fundamental. We also see that a generic fundamentally non-relativistic theory can give local Lorentz invariance it it's large distance limit: Wave equations appear certainly in generic condensed matter theories.

At least for the argument, which is about a discrete structure in general, as well as for the point which interests me (ether theories), condensed matter theory is generic enough.

Let's try to use sound waves to clarify where the argument goes wrong. Note that the speed of sound of our local Lorentz metric depends on x. Therefore, we cannot obtain it starting with some fixed Minkowski metric η_{μν}. Or, more accurate, whatever the Minkowski metric we start with, we will find that it is somewhere violated, that the physical Lorentz metric g_{μν}(x) differs from η_{μν}. But that means that for the description of the physical metric g_{μν}(x) we need some η_{μν}-symmetry-violating terms which are relevant at low distances. Thus, there is nothing wrong with the observation that there are lot's of relevant and marginal operators breaking Lorentz invariance. But this doesn't prove that one cannot obtain Lorentz invariance in the low energy limit. Instead, they are a necessary prerequisite to obtain local Lorentz invariance, with different coordinate speeds of sound at different points. Moreover, we cannot say about these operators that "their coefficients are constrained to a ridiculous degree by experiment". Observation can support only the local sound-Lorentz invariance g_{μν}(x), not a global Lorentz invariance η_{μν}. Thus, operators violating η_{μν}-symmetry are not constrained at all. As well, operators violating g_{μν}(x,t) at one point may be necessary to describe the g_{μν}(x,t') even at the same place but at another time.

One could argue that the fundamental Lorentz symmetry has been established with much higher accuracy. This is true, but not the point. Last but not least, the argument is that that we do not have to reach the Planck energy scales to establish Lorentz symmetry for distances much below the Planck scale, because of the way effective field theory works, increasing any violations of Lorentz symmetry in the long distance limit:

on February 2, 2009 at 11:10 am Luboš Motl:

...Quite generally, symmetries are restored at short distances and broken at long distances. The Lorentz case is really no different and you’re still getting it upside down.

It is this thesis which seems to be, in my opinion, in conflict with the sound waves, which follow a Lorentz-symmetric wave equation at large distances with some (however low) accuracy, but are described by discrete atomic models, which have no trace of any sound Lorentz invariance at all. Thus, Lorentz symmetry is something which can appear in the large distance limit, and even regularly appears there in quite generic situations.