# Is ADM energy (momentum) conserved?

Energy is a slippery concept in GR, but in an asymptotically flat spacetime there is a perfectly reasonable concept of energy (indeed, several) which goes by the name the ADM energy: $$E_\mathrm{ADM} := \frac{1}{16 \pi} \lim_{r \to \infty} \int_{S^2_r} \mathrm{d}A \, n_i (\partial_j h_{ij} - \partial_i h_{jj} )$$ Where $h_{ij}$ is the 3-metric on a spatial hypersurface $\Sigma$. My question is simply: is the ADM energy time-independent, for an arbitrary (asymptotically flat) spacetime? Said another way, is the ADM energy independent of the chosen spatial hypersurface? In particular, for a spacetime with no timelike Killing vector field, is the ADM energy conserved? How about the ADM 3-momentum?

• Thanks for the answer. One question: what exactly do you mean by 'conserved at infinity'? Do you just mean that if I removed the limit from my definition, and considered 'the ADM energy at radius $r$', then such a quantity would only be conserved in the limit $r \to \infty$? (The only thing I don't understand is that the ADM energy as I've defined it is a single number, independent of space, and so I don't know what it means to talk about this quantity 'at infinity'). Nov 18, 2016 at 13:19