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Is the covariant conservation of the matter energy momentum tensor T_{μν ; μ }= 0 also valid in a theory of gravity having an action for the gravitational field different from the Einstein Hilbert action ?

I'm asking because in GR the einstein field equations require T_{μν}= G_{μν}

where G_{μν;μ}=0 by construction (Bianchi identities) implying T_{μν;μ}=0 as well

But in an alternative theory of gravity we might have another field equation

T_{μν}= G'_{μν}where the right hand side might not satisfy Bianchi identities...

This troubles me because of the usual argument that says that for any energy momentum tensor conserved in the usual sense in the absence of gravity : ∂_{μ}T_{μν}=0 , we just need to replace by the covariant derivative to get the conservation equation with gravity T_{μν ; μ }= 0, and this argument seems to imply that this covariant conservation equation would have to be satisfied whatever is the geometrical side of the Einstein equation (derived from the Einstein hilbert action or any other action for the gravitational field alone)

My feeling is that in the general case ∂_{μ}T_{μν}=0 can't just be cavariantized into T_{μν ; μ }= 0 because even without gravity the conservation equation we must start with is not ∂_{μ}T_{μν}=0 but ∂_{μ}T_{μν}-∂_{μ}G_{μν}=0 where G_{μν}of course vanishes on flat spacetime yet must not be forgotten in the covariantization process that then leads to T_{μν;μ}-G_{μν;μ}=0 instead of just T_{μν;μ}=0

Then it could be that in a theory different from GR T_{μν;μ}=0 alone is not necessarily valid

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# I Conservation of matter energy momentum tensor beyond GR

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