Energy in higher-dimensional spacetimes

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.

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Cited by in corpus (7)

• New perspectives on the TOV equilibrium from a dual null approach
• The hyperbolic positive energy theorem
• On the canonical energy of weak gravitational fields with a cosmological constant $\Lambda \in \mathbb{R}$
• On the energy of the Horowitz-Myers metrics
• Uniqueness and energy bounds for static AdS metrics
• On Asymptotically Locally Hyperbolic Metrics with Negative Mass
• Hyperbolic positive energy theorems