It is often repeated that Lorentz invariance of Special Relativity (i.e., allowed solutions to Maxwell's Equations) is proven experimentally. This is clearly the case for local experiments and transformations between locally defined inertial frames.

My question is:

To what extent a conformal metric transformation that alters the Minkowski metric but

(a) converts solutions to Maxwell's equations into solutions defined with respect to the transformed space-time),

(b) everywhere preserves local Lorentz invariance (within the limits of experiment), and

(c) leaves the total energy of the field invariant violates Special Relativity or is excluded by experimental evidence?

  • $\begingroup$ (a) is not a special requirement because EM is conformally invariant, I have no idea what (b) is supposed to mean and which field are you talking about in (c). If you are just talking about electromagnetism here, then it's well-known that classical electromagnetism is conformally invariant so why would you expect such a transformation is "excluded by evidence" - we fully expect it to be a symmetry of electromagnetism! $\endgroup$
    – ACuriousMind
    Dec 6 '16 at 15:00
  • $\begingroup$ I agree I had a really hard time seeing the meaning of b). In fact, I kindly ignored it in my answer. Did you mean, by any chance, that your transformation should commute with the action of the Lorentz group? $\endgroup$ Dec 6 '16 at 15:18
  • $\begingroup$ The succinct statement of the question is: Is there any definitive experimental basis for excluding solutions to Maxwell's Equations generated by conformal metric transformations whose spacetime metric M is a function of the interval (R(t)) of signal propagation such that the period/length of each successive wave cycle increases in magnitude as the signal propagates. Technically, the transformations violate Lorentz Invariance (“LI”), but are not locally observable. $\endgroup$
    – EyeSeeTee
    Apr 7 at 16:13
  • $\begingroup$ If I understand your question correctly, you are referring about the scale invariance or the special conformal transformation and, yes, they are symmetries of EM fields in vacuum. That is what is meant by "EM is conformally invariant". This does not really break Lorentz invariance as the notions of time or scale along the propagation loose their meaning in this case. If you couple Maxwell's equation to charged matter, as you need to make any kind of experiment, you break the scale invariance and cannot have the special conformal transformation either. $\endgroup$ Apr 10 at 0:12

Saying the Lorentz invariance is the allowed solutions to Maxwell's equations is odd. Solutions to Maxwell's equations are Lorentz invariant. Electromagnetism (EM) is not the sole evidence for Lorentz invariance. It is also demonstrated experimentally by clocks in motion.

I am not entirely sure of your exact question. Do you mean, to what extent does a conformal transformation as a symmetry that [...] violates Special Relativity?

EM is invariant under the conformal group, unless you add charged particles, in which case scale invariance is broken. So your a) and c) parts are accounted for.

Now, to address your question. First, regarding experimental exclusion, there could never be absolutely real experimental work with conformal invariance. What I mean by that is that the introduction of your measurement apparatus introduces a scale in the experiment which breaks conformal invariance. Conformal theories are somewhat idealized but are used as limit case in the study of real phenomenon, e.g. in condensed matter.

Now, as to the violation of SR, if you posit a conformally invariant theory with a massive spectrum of particles, you will encounter problems with SR. Otherwise, nothing obvious comes to mind. If you consider gravity, then many subtlety arises. For instance, is EM really conformally invariant at high energies when the possibility for geons (bound state of gravity and EM fields) exists. In any case, in extremely curved space, Lorentz invariance becomes more and more meaningless since a local neighbourhood is less and less Minkowskian. What is a translation in curved spacetime?

  • $\begingroup$ The focus of the question is addressed to the issue of the spacetime metric governing the propagation of EMR. Locally, the metric is the Minkowski metric as noted. But, Maxwell's equations are invariant by transformations of the 15 parameter conformal group, not just the orthonormal transformations of the Lorentz Group. The transformations by the Special Conformal Group C4 are understood to alter the spacetime metric of the signal corresponding to wave forms whose magnitude increases during the signal transmission. The only model independent experiment I am aware of is the Sandage Loeb test. $\endgroup$
    – EyeSeeTee
    Apr 7 at 16:40

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