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Haelfix
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TrickyDicky said:Laws of physics accordint to what specific theory? GR? I thought you said forget about GR, SR? Minkowski spacetimes are strictly empty, the moment you introduce Lorentz invariance breakings is no longer SR.
No not true! Again, the equations must be invariant, but a particular solution or model need not (due to either explicit or spontaneous symmetry breaking).
Another example: Consider Newton and Maxwells laws. They are invariant under time reversal t--> -t, the laws seem to be reversible. However a particular solution that corresponds to the real world need not be. So for instance an icecube that is melting in the sun is irreversible. It never unmelts! This is a consequence of explicit symmetry breaking by initial conditions (in this case, the low entropy configuration preferentially picks out a direction or arrow of time). Note that we don't say that this model violates Newtonian physics. I recommend reading the Feynman lectures volume 1, there is a chapter on symmetries in physics.
TrickyDicky said:I don't think changes of scale (dilations) are usually included in the diffeomorphism group, at least as understood in GR, as the group of coordinate transformations. A dilation is not a diffeomorphism. See this thread post #4 for reference: https://www.physicsforums.com/showthread.php?t=572492
A conformal transformation (for the present purposes sometimes called a conformal isometry) is a diffeomorphism that preserves the metric up to a scale. Eg schematically: F*g = (e^2sigma) g. See Nakahara for the precise definitions.
Now there is a bit of a subtlety b/c whether a conformal transformation is viewed as a symmetry or a diffeomorphism depends on what you keep fixed and what you allow to be transformed, so it's always important to keep that in mind. However the mathematics that comes out are guarenteed to be isomorphic in the classical case of GR, provided you take the appropriate pullbacks when needed.
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