An ADM 3+1 formulation for Smooth Lattice General Relativity

TL;DR

This paper introduces a hybrid numerical relativity scheme combining a 3D lattice with ADM equations, enabling accurate evolution of vacuum spacetimes with flexible gauge choices, demonstrated on Kasner cosmology.

Contribution

It presents a novel ADM 3+1 formulation using a lattice to record the 3-metric, allowing flexible lapse and shift, with zero discretization error shown in a Kasner example.

Findings

Zero discretization error in Kasner cosmology simulation

Flexible gauge choices with arbitrary lapse and shift

Effective combination of lattice and ADM evolution methods

Abstract

A new hybrid scheme for numerical relativity will be presented. The scheme will employ a 3-dimensional spacelike lattice to record the 3-metric while using the standard 3+1 ADM equations to evolve the lattice. Each time step will involve three basic steps. First, the coordinate quantities such as the Riemann and extrinsic curvatures are extracted from the lattice. Second, the 3+1 ADM equations are used to evolve the coordinate data, and finally, the coordinate data is used to update the scalar data on the lattice (such as the leg lengths). The scheme will be presented only for the case of vacuum spacetime though there is no reason why it could not be extended to non-vacuum spacetimes. The scheme allows any choice for the lapse function and shift vectors. An example for the Kasner $T^3$ cosmology will be presented and it will be shown that the method has, for this simple example, zero discretisation error.