# Metric-independence of vacuum and force-free electromagnetic fields

## TL;DR

This paper demonstrates that electromagnetic fields, including force-free ones near black holes, are largely insensitive to spacetime geometry, as many solutions remain invariant under a broad class of metric transformations.

## Contribution

It introduces a larger class of metric transformations preserving Maxwell and force-free solutions, revealing the limited geometric information encoded in electromagnetic fields.

## Key findings

- Electromagnetic solutions are invariant under a broad class of metric transformations.
- Many force-free fields near black holes are solutions in flat backgrounds.
- Constructed electromagnetic waves unaffected by certain gravitational waves.

## Abstract

Electromagnetic fields which solve the vacuum Maxwell equations in one spacetime are well-known to also be solutions in all spacetimes with conformally-related metrics. This provides a sense in which electromagnetism alone cannot be used to measure certain aspects of geometry. We show that there is actually much more which cannot be so measured; relatively little of a spacetime's geometry is in fact imprinted in any particular electromagnetic field. This is demonstrated by finding a much larger class of metric transformations---involving five free functions---which preserve Maxwell solutions both in vacuum, without local currents, and also for the force-free electrodynamics associated with a tenuous plasma. One consequence of this is that many of the exact force-free fields which have previously been found around Schwarzschild and Kerr black holes are also solutions in appropriately-identified flat backgrounds. As a more direct application, we use our metric transformations to write down a large class of electromagnetic waves which remain unchanged by a large class of gravitational waves propagating "in the same direction."