An Einstein-Bianchi system for Smooth Lattice General Relativity. II. 3+1 vacuum spacetimes

TL;DR

This paper develops a comprehensive Einstein-Bianchi system for evolving smooth lattices in general 3+1 vacuum spacetimes, extending previous work from Schwarzschild to more general cases, ensuring hyperbolic evolution and constraint preservation.

Contribution

It introduces a complete set of evolution equations for all 20 Riemann curvatures alongside lattice leg-lengths in a hyperbolic, constraint-preserving framework for general vacuum spacetimes.

Findings

The system forms a hyperbolic set of equations.

Constraints are shown to be preserved during evolution.

Extension from Schwarzschild to general 3+1 vacuum spacetimes.

Abstract

We will present a complete set of equations, in the form of an Einstein-Bianchi system, that describe the evolution of generic smooth lattices in spacetime. All 20 independent Riemann curvatures will be evolved in parallel with the leg-lengths of the lattice. We will show that the evolution equations for the curvatures forms a hyperbolic system and that the associated constraints are preserved. This work is a generalisation of our previous paper arXiv:1101.3171 on the Einstein-Bianchi system for the Schwarzschild spacetime to general 3+1 vacuum spacetimes.