Exploiting the hidden symmetry of spinning black holes: conservation laws and numerical tests

TL;DR

This paper explores the hidden symmetry of Kerr black holes, deriving new conservation laws related to the Carter constant, and demonstrates their use in testing the accuracy of numerical simulations of matter near spinning black holes.

Contribution

It introduces novel conservation identities based on the hidden symmetry of Kerr black holes and shows how they can be used to verify numerical simulations.

Findings

Derived new conservation identities involving the Carter constant.

Demonstrated the use of these identities to detect violations in numerical simulations.

Recommended a simplified expression for practical testing purposes.

Abstract

The Kerr black hole is stationary and axisymmetric, which leads to conservation of energy and azimuthal angular momentum along the orbits of free test particles in its vicinity, but also to conservation laws for the evolution of continuum matter fields. However, the Kerr space-time possesses an additional "hidden symmetry" which exhibits itself in an unexpected conserved quantity along geodesics known as the Carter constant. We investigate the possibility of using this hidden symmetry to obtain conservation laws and other identities which could be used to test astrophysical simulations of the evolution of matter fields near spinning black holes. After deriving such identities, we set up a simple numerical toy model on which we demonstrate how they can detect the violations of evolution equations in a numerical simulation. Even though one of the expressions we derive is in the form of a conservation law, we end up recommending an equivalent but simpler expression that is not in the form of a conservation law for practical implementation.