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are there spacetime metrics where $nabla_nuT^munu ne 0$ ?

if the divergence of the energy momentum tensor is non-zero, what does that tell us about the spacetime aside from the fact that energy momentum is not conserved?

General relativity does not allow a stress-energy tensor with a nonvanishing divergence. The Einstein field equations set the Einstein curvature tensor proportional to the stress-energy tensor. The Einstein curvature tensor is constructed in such a way that it is automatically divergence-free.

are there spacetime metrics where $nabla_nuT^munu ne 0$ ?

No. If you write down a metric, the way you find the corresponding stress-energy tensor is to use the Einstein field equations, but the field equations will always give a divergence-free stress-energy tensor.

Stress-Energy Tensor is defined, at least in QFT, as the Noehter current associated with infinitesimal space-time translations. Let me explain, consider an infinitesimal space-time translations such as

$x'^mu = x^mu-epsilon^mu$

If we require the field $phi$ to be invariant under the transformation we've just written down, we can then apply Noether's Theorem. Thus, we say there are four conserved currents, one for each space-time direction. Putting this four currents all together in a big $4times4$ matrix, we get the Stress-Energy Tensor $T^munu$

$T^munu = fracpartialmathcalLpartial(partial_mu phi)partial_nu phi-delta^munumathcalL$

It can be easly shown that, since the $T^munu$ is made up of conserved currents, it has zero divergence:

$partial_nu T^munu = 0$

In addition to Ben Crowell's answer, I would state the following:

The Einstein-Hilbert stress-energy-momentum (SEM) tensor is always covariantly conserved, provided one of two conditions are satisfied:

• The gravitational equations of motion hold.




• The matter field equations of motion hold. This however does assume that the matter field action is diffeomorphism-invariant.

It is important and interesting to note that only one of these conditions need to be satisfied.

Let's start with the second. If $psi$ is a matter field, or a collection of matter fields with action $$ S_m[psi,g]=int d^4x mathcal L_m(psi,partialpsi,g,partial g) $$ that is diffeomorphism-invariant, because of diffeomorphism-invariance, the action will be invariant under an infinitesimal diffeomorphism (duh). An infinitesimal diffeo is always generated by a vector field $X$. The variation of the matter action is then $$ delta_X S_m=int d^4xsqrt-g left( fracdelta S_mdeltapsidelta_Xpsi+fracdelta S_mdelta g^munudelta_Xg^munu right)=0. $$ If the matter field EoMs hold, then the first functional derivative vanishes, and we obtain $$ delta_X S_m=int d^4xsqrt-g fracdelta S_mdelta g^munudelta_Xg^munu=0.$$

On the other hand, the fact that the variation is an infinitesimal diffeo, it means that $$ delta_X g^munu=-mathcal L_X g^munu $$ where $mathcal L_X$ is the Lie derivative (the minus sign is conventional - I define transformations via pushforward, and the Lie derivative is change along a pullback). But we have $$ -mathcal L_X g^munu=nabla^mu X^nu + nabla^nu X^mu. $$ If we name $$ T_munu=frac-2sqrt-gfracdelta S_mdelta g^munu, $$ then we have $$ delta_X S_m=int d^4xsqrt-gleft(-frac12right)T_munu(nabla^mu X^nu + nabla^nu X^mu)=-int d^4xsqrt-g T_mununabla^mu X^nu \ =int d^4x sqrt-gnabla_mu T^mu_ nu X^nu=0, $$ and since this is true for all $X$, we obtain $$ nabla_mu T^munu=0.$$ Note that in the last two equalities I have used that $T$ is symmetric and then I have partially integrated (with boundary termsthrown away).

Now, you can also use this procedure but with the gravitational action (without even needing to use the EoMs!) to prove that $$ E_munu=frac1sqrt-gfracdelta S_Gdelta g^munu $$ is divergenceless (if the gravitational action is the usual Einstein-Hilbert action, then this tensor if the $G_munu$ Einstein tensor which is known to be divergenceless, however this procedure work for a general gravitational action).

Then, the gravitational EoMs are $$ E^munu=frac12T^munu, $$ so if the gravitational EoMs hold, then because $E$ is divergenceless, so must be $T$.

The conclusion then is that it is impossible to have the EFE (Einstein's field equations) hold without $nabla_mu T^munu=0$. Of course, sometimes you may consider fields in a "background" spacetime, without any backreaction to gravity, in which case, it might be possible for the SEM tensor to not be conserved, but it must be noted that then the EFE does not hold .