# Interior solution for the Kerr metric

## TL;DR

This paper extends a method to find interior solutions to Einstein's equations for stationary, rotating bodies, and presents a new anisotropic fluid model matching the Kerr exterior, ensuring physical plausibility and regularity.

## Contribution

It introduces a novel interior solution for the Kerr metric using an ansatz linking interior and exterior metrics, applicable to rotating, anisotropic fluids, and matches smoothly to the Kerr exterior.

## Key findings

- Solution verifies energy conditions for various parameters
- Model converges to known spherical solution in the limit
- Matches smoothly to the Kerr exterior metric

## Abstract

A, recently presented, general procedure to find static and axially symmetric, interior solutions to the Einstein equations, is extended to the stationary case, and applied to find an interior solution for the Kerr metric. The solution, which is generated by an anisotropic fluid, verifies the energy conditions for a wide range of values of the parameters, and matches smoothly to the Kerr solution, thereby representing a globally regular model describing a non spherical and rotating source of gravitational field. In the spherically symmetric limit, our model converges to the well known incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. The physical variables of the energy-momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments.