Local Continuity of Angular Momentum and Noether Charge for Matter in General Relativity

TL;DR

This paper extends the concept of local conservation laws in general relativity to include angular momentum and Noether charges, providing new expressions and demonstrating their application in numerical simulations of boson star collisions.

Contribution

It introduces a formalism for local conservation of angular momentum and Noether charges in dynamic spacetimes, with explicit formulas for scalar and Proca fields.

Findings

Derived expressions for Noether charge and flux for complex scalar and Proca fields.

Formulated angular momentum density, flux, and source terms in curved spacetime.

Applied the formalism to simulate boson star collisions in numerical relativity.

Abstract

Conservation laws have many applications in numerical relativity. However, it is not straightforward to define local conservation laws for general dynamic spacetimes due the lack of coordinate translation symmetries. In flat space, the rate of change of energy-momentum within a finite spacelike volume is equivalent to the flux integrated over the surface of this volume; for general spacetimes it is necessary to include a volume integral of a source term arising from spacetime curvature. In this work a study of continuity of matter in general relativity is extended to include angular momentum of matter and Noether currents associated with gauge symmetries. Expressions for the Noether charge and flux of complex scalar fields and complex Proca fields are found using this formalism. Expressions for the angular momentum density, flux and source are also derived which are then applied to a numerical relativity collision of boson stars in 3D with non-zero impact parameter as an illustration of the methods.