Darmois matching and $C^3$ matching

TL;DR

This paper compares Darmois and $C^3$ matching conditions in spherically symmetric spacetimes, finding Darmois conditions are satisfied but $C^3$ are not, due to non-physical pressure behavior.

Contribution

It demonstrates the applicability of Darmois matching conditions across different perfect fluid solutions and highlights limitations of $C^3$ matching in these contexts.

Findings

Darmois conditions are satisfied in all tested cases.

$C^3$ conditions are not fulfilled due to pressure issues.

Non-physical pressure behavior affects higher-order matching.

Abstract

We apply the Darmois and the $C^3$ matching conditions to three different spherically symmetric spacetimes. The exterior spacetime is described by the Schwarzschild vacuum solution whereas for the interior counterpart we choose different perfect fluid solutions with the same symmetry. We show that Darmois matching conditions are satisfied in all three cases whereas the $C^3$ conditions are not fulfilled. We argue that this difference is due to a non-physical behavior of the pressure on the matching surface.