Discussing the energy-momentum tensor in presence of gravitational field, our professor stated that, except some particular cases, it’s not possibile to define a “global” energy and momentum in general relativity. Energy and momentum are only defined locally as densities in the energy-momentum tensor. Why?


2 Answers 2


Well, the situation is even worse.

  1. Globally, energy (momentum) as the integral of an energy (momentum) density over a non-compact space may be divergent/non-integrable, respectively.

  2. Locally, there's no well-defined gravitational stress-energy-momentum (SEM) tensor, cf. e.g. this, this, this, this Phys.SE posts and links therein. Nevertheless, check out the notion of a Landau–Lifshitz pseudotensor, which is not general covariant.

  • $\begingroup$ 1. Just so I don't have nightmares, these topics are out of an introductory course on gr? 2. Our professor just mentioned them, I wanted to know more about it but it's kinda frightening $\endgroup$
    – john
    Feb 5, 2021 at 14:41
  • 1
    $\begingroup$ 1. Probably not. $\endgroup$
    – Qmechanic
    Feb 5, 2021 at 16:15

A crucial problem is that one cannot unambiguously add vectors applied to different events in spacetime in the non-affine spacetime of GR. Hence you cannot add (integrate) the four momenta (densities) located on a spatial slice to construct a global quantity. The problem is even more difficult because one also would like that such a sum or integral be also constant in time (some notion of time associated to a family of spatial slices) in some situations, in order to have a meaningful notion.

Scalars can be instead integrated and this is a way out. Instead of trying to define a global four vector one can define a global component of a fourmomentum with respect to a vector field. Mathematically speaking that object is the integral over a spacelike slice of the spacetime of $${T^a}_b\xi^b n_a,$$ where $n$ is the unit normal to the slice and $\xi$ the said vector field. (That slice is a Cauchy surface if assuming that the spacetime is globally hyperbolic.) If this vector is a Killing vector and if $T$ is symmetric, then the integral of the above quantity is in fact conserved, changing the spatial slice, under mild requirements on the space infinity behaviour of $T$. If $\xi$ is a Killing timelike vector field, this procedure gives rise to a meaningful notion of conserved energy in particular.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.