# On the link between the Maxwell and linearised Einstein equations on Schwarzschild

## TL;DR

This paper establishes a mathematical connection between solutions of the linearised Einstein equations and Maxwell equations on Schwarzschild spacetime, linking spin 2 and spin 1 field equations through gauge-invariant constructs.

## Contribution

It demonstrates how to derive Maxwell solutions from linearised Einstein solutions on Schwarzschild, revealing a novel link between spin 2 and spin 1 equations.

## Key findings

- Constructed Maxwell solutions from Einstein solutions using gauge-invariant quantities.
- Linked Regge--Wheeler and Zerilli equations to Maxwell currents.
- Provided a new perspective on the relation between gravitational and electromagnetic perturbations.

## Abstract

In this short note we shall demonstrate that given a smooth solution $\gamma$ to the linearised Einstein equations on Schwarzschild which is supported on the $l\geq 2$ spherical harmonics and expressed relative to a transverse and traceless gauge then one can construct from it a smooth solution to the sourced Maxwell equations expressed relative to a generalised Lorentz gauge. Here the Maxwell current is constructed from those gauge-invariant combinations of the components of $\gamma$ which are determined by solutions to the Regge--Wheeler and Zerilli equations. The result thus provides an elegant link between the spin 1 and spin 2 equations on Schwarzschild.