Total Mass-Momentum of Arbitrary Initial-Data Sets in General Relativity

TL;DR

This paper generalizes the concept of total mass-momentum in general relativity to arbitrary initial-data sets, extending the mathematical framework to include variable-dimensional spaces of asymptotic spinors.

Contribution

It introduces a new generalized form of mass-momentum applicable to any initial-data set, broadening the understanding of mass-momentum in diverse spacetime configurations.

Findings

Mass-momentum remains a Hermitian quadratic form in the generalized setting.

The space of asymptotic spinors can vary from zero to infinite dimension.

Examples illustrate the properties and applications of the generalized mass-momentum.

Abstract

For an asymptotically flat initial-data set in general relativity, the total mass-momentum may be interpreted as a Hermitian quadratic form on the complex, two-dimensional vector space of ``asymptotic spinors''. We obtain a generalization to an arbitrary initial-data set. The mass-momentum is retained as a Hermitian quadratic form, but the space of ``asymptotic spinors'' on which it is a function is modified. Indeed, the dimension of this space may range from zero to infinity, depending on the initial data. There is given a variety of examples and general properties of this generalized mass-momentum.