# $C^3$ matching for asymptotically flat spacetimes

## TL;DR

This paper introduces a $C^3$ matching criterion for connecting interior and exterior solutions in asymptotically flat spacetimes, ensuring smooth curvature properties and physical plausibility at the boundary.

## Contribution

It develops a new $C^3$ differentiability-based matching method using curvature eigenvalues, applicable to perfect fluid solutions in Newtonian and Einstein gravity.

## Key findings

- Matching curvature eigenvalues ensures smooth transition at the boundary.
- Density and pressure vanish at the matching surface for physical consistency.
- The method is validated on various perfect fluid solutions.

## Abstract

We propose a criterion for finding the minimum distance at which an interior solution of Einstein's equations can be matched with an exterior asymptotically flat solution. It is based upon the analysis of the eigenvalues of the Riemann curvature tensor and their first derivatives, implying $C^3$ differentiability conditions. The matching itself is performed by demanding continuity of the curvature eigenvalues across the matching surface. We apply the $C^3$ matching approach to spherically symmetric perfect fluid spacetimes and obtain the physically meaningful condition that density and pressure should vanish on the matching surface. Several perfect fluid solutions in Newton and Einstein gravity are tested.