Gravitational Multipoles in General Stationary Spacetimes

TL;DR

This paper extends the Geroch-Hansen formalism for gravitational multipoles to non-vacuum stationary spacetimes, providing a unified approach that applies broadly beyond vacuum solutions.

Contribution

It introduces an improved twist vector construction that generalizes multipole definitions to non-vacuum stationary spacetimes, establishing formal equivalence with Thorne's approach.

Findings

Generalization of multipole formalism to non-vacuum spacetimes

Construction of a well-defined twist vector under mild conditions

Equivalence between Geroch-Hansen and Thorne formalisms established

Abstract

The Geroch-Hansen and Thorne (ACMC) formalisms give rigorous and equivalent definitions for gravitational multipoles in stationary vacuum spacetimes. However, despite their ubiquitous use in gravitational physics, it has not been shown that these formalisms can be generalized to non-vacuum stationary solutions, except in a few special cases. This paper shows how the Geroch-Hansen formalism can be generalized to arbitrary non-vacuum stationary spacetimes for metrics that are sufficiently smooth at infinity. The key is the construction of an improved twist vector, which is well-defined under a mild topological condition on the spacetime (which is automatically satisfied for black holes). Ambiguities in the construction of this improved twist vector are discussed and fixed by imposing natural gauge fixing conditions, which also immediately lead to the equivalence between the Geroch-Hansen and Thorne formalisms for such arbitrary stationary spacetimes.