Lie Symmetries of Einstein's Vacuum Equations in N Dimensions

TL;DR

This paper systematically determines all Lie symmetries of Einstein's vacuum equations with a cosmological term in N dimensions, revealing the structure of their symmetry algebra using a novel approach.

Contribution

It introduces a new method for computing Lie symmetries of complex PDEs by focusing on derivatives present in the equations, applied here to Einstein's vacuum equations in N dimensions.

Findings

All Lie symmetries of Einstein's vacuum equations with cosmological term are characterized.

The method simplifies symmetry calculations for complex PDEs.

The approach can be adapted to other types of differential equations.

Abstract

We investigate Lie symmetries of Einstein's vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein's equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein's equations), we set to zero the coefficients of derivatives that do not appear in Einstein's equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein's equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations.