Energy in higher-dimensional spacetimes

TL;DR

This paper derives Hamiltonian energy expressions for gravitating systems in higher-dimensional spacetimes, analyzing various asymptotic conditions, and clarifies the relationships between different mass definitions, with applications to Kaluza-Klein metrics.

Contribution

It provides new formulas for total energy in higher-dimensional gravity, compares mass definitions across different asymptotics, and explores implications for Kaluza-Klein spacetimes.

Findings

Komar mass equals ADM mass in asymptotically flat spaces across dimensions

Kaluza-Klein asymptotics break the equivalence between Komar and ADM masses

Witten positivity bounds the Hamiltonian mass via electric charge

Abstract

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 $\Lambda \in \mathbb{R}$. 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 ADM mass in asymptotically flat space-times in all dimensions, generalising the four-dimensional result of Beig, and that this is not true anymore with Kaluza-Klein asymptotics. We show that the Hamiltonian mass does not necessarily coincide with the ADM mass in Kaluza-Klein asymptotically flat space-times, 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 Rasheed Kaluza-Klein vacuum metrics, which we study in some detail, pointing out restrictions that arise from the requirement of regularity, seemingly unnoticed so far in the literature.