Conservation of matter energy momentum tensor beyond GR

[fhenryco]

Hello,

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 : ∂_μT_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 ∂_μT_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 ∂_μT_μν=0 but ∂_μT_μν-∂_μG_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

 

[Markus Hanke]

Maybe I am being naive here, but it would seem to me that any theory ( GR or not ) of gravity must reduce to Minkowski space-time everywhere locally, or else it is of little value, physically speaking. It is simply an empirical observation about the universe that small enough patches of space-time seem to be Minkowskian in nature. In Minkowski space-time then, the energy-momentum tensor arises as the conserved Noether current associated with time-translation invariance via Noether's theorem. It is therefore a locally conserved quantity pretty much by definition. If you write down a field theory that locally couples to the energy-momentum tensor as source, and for whatever reason the source ends up not being automatically conserved at each event in that theory, then I think it wouldn't really be much good to us. I think local non-conservation of the energy-momentum tensor is quite simply incompatible with space-time being locally Minkowskian.

Note however that a theory that differs from GR does not necessarily need to couple to the usual energy-momentum tensor as a source - you could modify both sides of the field equation to arrive at something physically reasonable.

 

[fhenryco]

I totally agree with you when you say "I think local non-conservation of the energy-momentum tensor is quite simply incompatible with space-time being locally Minkowskian". But i never said that the energy momentum tensor was not conserved in Minkowski spacetime , at the contrary it is , but it is the complete energy momentum tensor that must be conserved in a theory , not individual pieces that make up the total energy momentum tensor: so it is Tμν-Gμν that is conserved in the usual sense even without gravity so even if this Gμν vanishes on flat space time (this -Gμν is just obtained by replacing the gravitational field by Minkowski in the usual -Gμν and should identify may be to a Noether current associated to the action of the order two tensor field which identifies to Minkowski on flat space-time) allowing to deduce that actually Tμν IS conserved in flat spacetime : the problem is that when we switch on gravity (when we covariantize) we must also switch on gravity in the Gμν term (which is already in a covariant form by construction only its derivative must be replaced by a covariant derivative in the conservation equation) that could have been forgotten because was zero on flat space-time ... so the covariant conservation of energy momentum is the covariant conservation of total energy momentum Tμν-Gμν (of matter , radiation AND gravity) and not of Tμν alone since now Gμν does not vanish. Of course this does not matter in the case of GR where both Tμν and Gμν satisfy separately the Bianchi identities, but not for an alternative theory of gravity with another G'μν ! now do you see better what i meant, may be i did not express myself clearly enough...

thanks for reacting on this, I'm still in trouble, not totally sure what i say makes sense ... but really seems to make sense !

 

[Markus Hanke]

I don't think I am really sure what you are getting at, sorry !

However, we seem to be agreeing on my last sentence - if you modify the field equations, you may need to modify both sides, and couple to a source that is not necessarily just the energy-momentum tensor.

Now, if you talk about the conservation of all forms of energy, then I agree you will need to account for gravity itself as well. In standard GR, you would be doing this by forming a combination of the energy-momentum tensor and the Landau-Lifschitz pseudotensor, and find that the covariant divergence of this complex identically vanishes. With this, you can formulate a global ( as opposed to purely local ) conservation law. Crucially, this pseudotensor can be expressed purely in terms of the connection alone, so this opens up the possibility of modifying the model by choosing different fields as well as a different connection ( other than Levi-Civita ). In fact, this seems to be what happens in models such as Moeller's "tetrad gravity" - though I'm definitely way out of my depths here.

So you are right, in models other than GR you will in general end up with conserved quantities that may be different than just $T_{\mu \nu}$.