Energy in higher-dimensional spacetimes

We derive expressions for the total Hamiltonian energy of gravitating systems in higher-dimensional theories in terms of the Riemann tensor, allowing a cosmological constant Λ R. Our analysis covers asymptotically anti-de Sitter spacetimes, asymptotically flat spacetimes, as well as Kaluza-Klein asymptotically flat spacetimes. We show that the Komar mass equals the Arnowitt-Deser-Misner (ADM) mass in stationary asymptotically flat spacetimes in all dimensions, generalizing the four-dimensional result of Beig, and that this is no longer true with Kaluza-Klein asymptotics. We show that the Hamiltonian mass does not necessarily coincide with the ADM mass in Kaluza-Klein asymptotically flat spacetimes, and that the Witten positivity argument provides a lower bound for the Hamiltonian mass - and not for the ADM mass - in terms of the electric charge. We illustrate our results on the five-dimensional Rasheed metrics, which we study in some detail, pointing out restrictions that arise from the requirement of regularity, which have gone seemingly unnoticed so far in the literature.