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This is related to the thread on the meaning of diffeomorphism invariance but is adressing a distinct point (at least I think so, but I may be proven wrong).
As Rovelli discusses in his book, the action of the Standard Model coupled to gravity has three types of invariance: under the gauge group of the SM, under local Lorentz transformations (which may be seen as a gauge group too) and under diffeomorphisms.
Presented this way, the issue of Local Lorentz invariance and of diffeomorphism invariance seem distincts. Is that really the case? Could we build a theory with Local Lorentz invariance but without diffeomorphism invariance (and vice versa)?
I thought (for some unclear reason) that the two issues were intimately linked but mathematically they are not, as far as I can tell. Let's say that Einstein had never thought of special relativity, but still had had his brillant insight that in free fall gravity is unobservable (up to tidal forces effect, as usual). What kind of theory would he have come up with in that case? He would still have been led to using tensors, right? Would this approach have necessarily forced him to special relativity or is it possible to make the whole theory diffeomorphism invariant without ever using Lorentz invariance?
It's probably a stupid question with a very obvious answer. If it is, feel free to tell me
As Rovelli discusses in his book, the action of the Standard Model coupled to gravity has three types of invariance: under the gauge group of the SM, under local Lorentz transformations (which may be seen as a gauge group too) and under diffeomorphisms.
Presented this way, the issue of Local Lorentz invariance and of diffeomorphism invariance seem distincts. Is that really the case? Could we build a theory with Local Lorentz invariance but without diffeomorphism invariance (and vice versa)?
I thought (for some unclear reason) that the two issues were intimately linked but mathematically they are not, as far as I can tell. Let's say that Einstein had never thought of special relativity, but still had had his brillant insight that in free fall gravity is unobservable (up to tidal forces effect, as usual). What kind of theory would he have come up with in that case? He would still have been led to using tensors, right? Would this approach have necessarily forced him to special relativity or is it possible to make the whole theory diffeomorphism invariant without ever using Lorentz invariance?
It's probably a stupid question with a very obvious answer. If it is, feel free to tell me