A Matrix Formulation of Einstein's Vacuum Field Equations

TL;DR

This paper introduces a matrix-based formulation of Einstein's vacuum field equations using tensor-to-matrix correspondence and matrix calculus, aiming to improve numerical relativity computations.

Contribution

It presents a novel matrix formulation of Einstein's equations, enabling potentially more efficient numerical solutions in relativistic modeling.

Findings

Derived Einstein's vacuum equations as a differential-matrix equation

Established tensor-matrix correspondence using Kronecker products

Facilitated potential numerical implementation improvements

Abstract

We develop a correspondence between arbitrary tensors and matrices based on the use of Kronecker products and associated identities. Utilizing the rules of matrix differentiation we derive the vacuum Einstein field equations as a differential-matrix equation. This formulation may facilitate their efficient use in numerical relativistic models.