# Spherically symmetric terrestrial planets with discontinuities are spectrally rigid

## TL;DR

This paper proves spectral rigidity for spherically symmetric planets with interior discontinuities, allowing for different metrics and wave types, which helps in understanding planetary interior structures.

## Contribution

It introduces a new spectral rigidity result for manifolds with interior metric discontinuities, applicable to planetary models with fluid cores and phase transitions.

## Key findings

- Spectral rigidity holds for spherically symmetric manifolds with interior interfaces.
- The length spectrum of the Euclidean ball is shown to be simple.
- The proof utilizes a novel trace formula for these manifolds.

## Abstract

We establish spectral rigidity for spherically symmetric manifolds with boundary and interior interfaces determined by discontinuities in the metric under certain conditions. Rather than a single metric, we allow two distinct metrics in between the interfaces enabling the consideration of two wave types, like P- and S-polarized waves in isotropic elastic solids. Terrestrial planets in our solar system are approximately spherically symmetric and support toroidal and spheroidal modes. Discontinuities typically correspond with phase transitions in their interiors. Our rigidity result applies to such planets as we ensure that our conditions are satisfied in generally accepted models in the presence of a fluid outer core. The proof is based on a novel trace formula. We also prove that the length spectrum of the Euclidean ball is simple.