# Energy continuity equation

Is there such an animal as an energy continuity equation, or one involving Pmu or the stress energy tensor?

It suddenly stuck me that if we are to be so inclined by theory as we are by empirical evidence that energy is a conserved quantity, then there should be an equation that describes it in four dimensions.

Rather than bring in general relativity all at once where energy is not well defined (I have reservations), the stress energy tensor is still a tensor in special relativity, and so special relativity might be the better place to start.

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Dale
Mentor
2020 Award
Yes, the continuity equation:
$$\nabla_{\nu}T^{\mu\nu}=0$$

This expresses the conservation of energy and momentum locally in curved spacetime and globally in flat spacetime.

Yes, the continuity equation:
$$\nabla_{\nu}T^{\mu\nu}=0$$

This expresses the conservation of energy and momentum locally in curved spacetime and globally in flat spacetime.
Thanks, I should have done some research sooner. It seems, at this point the validity of \nabla T = 0 in special relativity goes on a case by case basis, until the Einstein equation is invoked where the Bianchi identity is applied to the other side.