- #1
fhenryco
- 63
- 5
Hello,
Is the covariant conservation of the matter energy momentum tensor Tμν ; 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 Tμν= Gμν
where Gμν;μ=0 by construction (Bianchi identities) implying Tμν;μ=0 as well
But in an alternative theory of gravity we might have another field equation
Tμν= 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 : ∂μTμν=0 , we just need to replace by the covariant derivative to get the conservation equation with gravity Tμν ; 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 ∂μTμν=0 can't just be cavariantized into Tμν ; μ = 0 because even without gravity the conservation equation we must start with is not ∂μTμν=0 but ∂μTμν-∂μGμν=0 where Gμν of course vanishes on flat spacetime yet must not be forgotten in the covariantization process that then leads to Tμν;μ-Gμν;μ=0 instead of just Tμν;μ=0
Then it could be that in a theory different from GR Tμν;μ=0 alone is not necessarily valid
Is the covariant conservation of the matter energy momentum tensor Tμν ; 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 Tμν= Gμν
where Gμν;μ=0 by construction (Bianchi identities) implying Tμν;μ=0 as well
But in an alternative theory of gravity we might have another field equation
Tμν= 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 : ∂μTμν=0 , we just need to replace by the covariant derivative to get the conservation equation with gravity Tμν ; 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 ∂μTμν=0 can't just be cavariantized into Tμν ; μ = 0 because even without gravity the conservation equation we must start with is not ∂μTμν=0 but ∂μTμν-∂μGμν=0 where Gμν of course vanishes on flat spacetime yet must not be forgotten in the covariantization process that then leads to Tμν;μ-Gμν;μ=0 instead of just Tμν;μ=0
Then it could be that in a theory different from GR Tμν;μ=0 alone is not necessarily valid
Last edited: