The Spacetime Positive Mass Theorem with Multiple Time Dimensions

TL;DR

This paper extends the positive mass theorem to spacetimes with multiple time dimensions, establishing nonnegativity of energy and characterizing the geometry of initial data sets.

Contribution

It introduces a generalized positive mass theorem for multiple time dimensions, linking energy bounds to linear momenta and geometric structures.

Findings

Mass remains nonnegative with multiple time dimensions.

Equality implies a foliation by flat submanifolds.

Under umbilicity, initial data embeds into a generalized pp-wave.

Abstract

We generalize the spacetime positive mass theorem to include multiple time dimensions. In particular, we show that the mass remains nonnegative in the sense that the energy $E$ is bounded from below by the trace norm of the linear momenta $J^1,...,J^m$. Equality in this energy inequality implies a foliation by flat submanifolds of a generalized initial data set. Moreover, under an additional umbilicity assumption, we find that the initial data set isometrically embeds into a generalized pp-wave.