# Einstein's Equations in Matter

## TL;DR

This paper explores Einstein's equations in matter, deriving their form for relativistic fluids, analyzing modifications to stellar structure equations, and studying matter effects in anti-de Sitter space at various temperatures.

## Contribution

It derives Einstein's equations in matter for relativistic fluids and examines their implications in astrophysics and AdS space, including temperature-dependent susceptibilities and hydrostatic breakdowns.

## Key findings

- Modified Tolman-Oppenheimer-Volkoff equations due to matter response
- Temperature and density dependence of effective Newton's constant
- Breakdown of hydrostatics in AdS at low temperatures

## Abstract

Einstein's equations in matter are gravitational analogues of Maxwell's equations in matter, providing an effective classical description of gravitational fields. We derive Einstein's equations in matter for relativistic fluids, and use them to illustrate how the Tolman-Oppenheimer-Volkoff equations are modified by the matter's response to curvature. For a gas of massive fermions, we evaluate how the effective Newton's constant and other susceptibilities depend on the temperature and density. In anti-de Sitter space, we study the $O(1/(T\ell)^2)$ corrections to the geometries sourced by perfect fluids, and illustrate the breakdown of hydrostatics in AdS at small temperatures.