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I was reading about Unimodular gravity, this is a modified theory of gravity that postulates that the gravity is only invariant under volume preserving diffeomorphism. So it breaks the full diffeomorphism invariance.

And there is a theorem by Lovelock that says that modified theories of gravity can be classified in four different ways , depending on the requirement of the Lovelock theorem that they violate

1) Theories that add extra fields to Einstein field Equations

2) Theories that include higher order derivatives of the metric in the action

3) Theories that add extra dimension.

4) Theories with non-locality or violation of Lorentz-invariance.

However I can't see where is the place for Unimodular gravity in this clasification, I think that Unimodular Gravity is in another classification, maybe a number 5) Theories that break the full diffeomorphism invariance.

I'm correct?

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Unimodular gravity is classically equivalent to standard Einstein's general relativity with a cosmological constant (CC), the only difference is that CC in standard gravity appears directly in the action while in unimodular gravity it is an integration constant. So rather than providing a new way to modify gravity this is an example of the Lovelock's theorem at work: starting with an additional assumption we still arrive at Einstein field equations. For a proof of this equivalence see the following paper:

In quantum theory the path integral would be different for UG and Einstein's GR, so quantum effects would allow one to distinguish between the two theories (at least in principle), see e.g. here:

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  • $\begingroup$ It's hard to me to understand what you say. I think that Unimodular gravity should fit in one of the four categories because, it doesn't preserve the full diffeomorphism group $\endgroup$
    – Nothing
    Commented May 12, 2020 at 20:57
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    $\begingroup$ <UG> should fit in one of the four categories No it shouldn't, because UG leads to the same Einstein equations as GR, this is not a theory of modified gravity. Non-preservation of full diffeomorphism invariance does not lead to any dynamics, classically this is just one of many alternative formulation of GR, now with a specific gauge condition imposed globally. The only change is of cosmological constant now being determined by initial conditions, rather than given in the action. $\endgroup$
    – A.V.S.
    Commented May 13, 2020 at 4:46
  • $\begingroup$ I agree with you, you get the same Field equations, however you need to assume the energy momentum conservation, it does not follow from the field equations, so is not exactly the same. ( I know that the most "natural" thing is to assume the energy momentum conservation) $\endgroup$
    – Nothing
    Commented May 13, 2020 at 5:46
  • $\begingroup$ However is interesting that if you assume the energy momentum conservation then the dynamics of both theories are the exactly the same. $\endgroup$
    – Nothing
    Commented May 13, 2020 at 5:50
  • $\begingroup$ Sorry I was confused, you are right. $\endgroup$
    – Nothing
    Commented May 15, 2020 at 17:08

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