Energy in higher-dimensional Spacetimes

Based on the geometric Hamiltonian formalism we derive new expressions for the total Hamiltonian energy of gravitating systems in higher dimensions in terms of the Riemann tensor, allowing a cosmological constant. In this way, we also obtain the ADM and the Komar expressions of the corresponding spacetimes. We then apply this analysis to a large class of higher dimensional spacetimes with various asymptotics known to us, which satisfy certain conditions. In particular, our analysis covers asymptotically flat spacetimes, Kaluza-Klein asymptotically flat spacetimes, as well as asymptotically (anti-)de Sitter spacetimes. As it turns out, the Komar mass equals the ADM mass in stationary asymptotically flat spacetimes in all dimensions, generalizing the four-dimensional result of Beig. We show that in general, the Hamiltonian mass, the ADM mass and the Komar mass do not coincide with each other in the non-asymptotically flat setting.
This talk is based on my master's thesis under supervision of Prof. P. T. Chrusciel.