# The conundrum of stress energy tensor in GR versus QFT

1. Jul 30, 2010

### jdstokes

Is anyone else concerned by the fact that these quantities have a completely different origin in each theory?!

In Poincare-invariant QFTs, the conserved energy-momentum tensor follows from invariance of the action under translational coordinate transformations.

In GR, the covariantly-conserved energy-momentum follows from invariance of the theory under diffeomorphisms.

My intuition is that covariant conservation of energy and momentum should follow from covariance under general coordinate transformations (after all, translational coordinate transformations are simply a subset of the general coodinate transformations).

The fact that this is not the case is a little worrying. Does anyone have any better intuition to explain this conundrum?

Appendix
Let me clarify a subtle and often overlooked difference between general coordinate transformations and diffeomorphisms. To the best of my knowledge, a general coordinate transformation consists of the following action on coordinates and tensors

$x^\mu \mapsto x'^\mu$
$T_{\mu\nu}(x) \mapsto T'_{\mu\nu}(x')$

where $T'_{\mu\nu}(x')$ is given in terms of $T_{\mu\nu}(x)$ by the well-known tensor transformation rule.

The action of a diffeomorphism, on the other hand, is represented by

$x^\mu \mapsto x^\mu$
$T_{\mu\nu}(x) \mapsto T'_{\mu\nu}(x)$

n.b. the absence of a prime on the argument of T' is intentional.

In the action formulation, it is easy to prove coordinate invariance and (relatively) straightforward to prove that diffeomorphisms are a symmetry of the theory (sometimes regarded as a gauge symmetry).

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