# Gravitational lensing beyond geometric optics: II. Metric independence

## TL;DR

This paper demonstrates that high-frequency wave propagation in gravitational lensing is largely insensitive to many metric components, allowing for metric transformations that generate new solutions without altering observable wave behavior.

## Contribution

It reveals the metric independence of wave observables beyond geometric optics, enabling the embedding of configurations into different spacetimes and generating new solutions from existing ones.

## Key findings

- High-frequency wave propagation is insensitive to certain metric transformations.
- Many metric components do not affect scalar or electromagnetic wave observables.
- New wave solutions can be generated via metric transformations.

## Abstract

Typical applications of gravitational lensing use the properties of electromagnetic or gravitational waves to infer the geometry through which those waves propagate. Nevertheless, the optical fields themselves - as opposed to their interactions with material bodies - encode very little of that geometry: It is shown here that any given configuration is compatible with a very large variety of spacetime metrics. For scalar fields in geometric optics, or observables which are not sensitive to the detailed polarization content of electromagnetic or gravitational waves, seven of the ten metric components are essentially irrelevant. With polarization, five components are irrelevant. In the former case, this result together with diffeomorphism invariance allows essentially any geometric-optics configuration associated with a particular spacetime to be embedded into any other spacetime, at least in finite regions. Going beyond the geometric-optics approximation breaks some of this degeneracy, although much remains even then. Overall, high-frequency wave propagation is shown to be insensitive to compositions of certain conformal, Kerr-Schild, and related transformations of the background metric. One application is that new solutions for scalar, electromagnetic, and gravitational waves may be generated from old ones. In one example described here, the high-frequency scattering of a plane wave by a point mass is computed by transforming a plane wave in flat spacetime.