Frankenstein's Glue: Transition functions for approximate solutions

TL;DR

This paper formalizes a method for smoothly joining approximate solutions to Einstein's equations using transition functions, ensuring the combined solution remains valid within the approximation order, demonstrated with black hole binary systems.

Contribution

It introduces sufficient conditions on transition functions that guarantee the joined approximate solutions satisfy Einstein's equations to the same order, with an application to black hole binaries.

Findings

Transition functions satisfying the conditions preserve Einstein equation validity.

Violating conditions introduces matter shells affecting spacetime.

Method applicable to post-Newtonian and Schwarzschild approximations.

Abstract

Approximations are commonly employed to find approximate solutions to the Einstein equations. These solutions, however, are usually only valid in some specific spacetime region. A global solution can be constructed by gluing approximate solutions together, but this procedure is difficult because discontinuities can arise, leading to large violations of the Einstein equations. In this paper, we provide an attempt to formalize this gluing scheme by studying transition functions that join approximate solutions together. In particular, we propose certain sufficient conditions on these functions and proof that these conditions guarantee that the joined solution still satisfies the Einstein equations to the same order as the approximate ones. An example is also provided for a binary system of non-spinning black holes, where the approximate solutions are taken to be given by a post-Newtonian expansion and a perturbed Schwarzschild solution. For this specific case, we show that if the transition functions satisfy the proposed conditions, the joined solution does not contain any violations to the Einstein equations larger than those already inherent in the approximations. We further show that if these functions violate the proposed conditions, then the matter content of the spacetime is modified by the introduction of a matter-shell, whose stress-energy tensor depends on derivatives of these functions.