# Local Lorentz Invariance and Conformal Metric Transformations

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?

• (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! Dec 6 '16 at 15:00
• 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? Dec 6 '16 at 15:18
• 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. Apr 7 at 16:13
• 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. Apr 10 at 0:12