Hyperbolic Hamiltonian equations for general relativity

TL;DR

This paper introduces a hyperbolic Hamiltonian formulation of general relativity under a specific gauge, enabling improved numerical analysis of black hole spacetimes through conformal and complex coordinate techniques.

Contribution

It develops a new hyperbolic Hamiltonian system for general relativity with a specific gauge choice, facilitating numerical simulations of black hole spacetimes using conformal compactification.

Findings

The system is applicable to asymptotically quiescent black hole spacetimes.

Conformal flat initial data are invariant and relate to a pendulum analogy.

Newton's law emerges in the relaxation of specific surface deformations.

Abstract

The 3+1 Hamiltonian formulation in the gauge $D_tN=-K$ on the lapse function fixes the direction of time associated with the trace $K$ of the extrinsic curvature tensor. The Hamiltonian equations hereby become hyperbolic. We study this new system for black hole spacetimes that are asymptotically quiescent, which introduces analyticity properties that can be exploited for numerical calculations by compactification in spherical coordinates with complex radius following a M\"obius transformation. Conformal flat initial data of two black holes are hereby invariant, and correspond to a turn point in a pendulum, up for a pair of separated black holes and down for a single black hole. Here, Newton's law appears in the relaxation of $l=2$ deformations of semi-infinite poloidal surface elements, defined by the moment of inertia of the binary.