Can a combination of the conformal thin-sandwich and puncture methods yield binary black hole solutions in quasi-equilibrium?

TL;DR

This paper investigates whether combining the conformal thin-sandwich formalism and puncture method can produce binary black hole initial data in quasi-equilibrium, revealing fundamental conflicts in their boundary condition requirements.

Contribution

The study identifies inherent conflicts between the boundary conditions of the two methods, showing that a fully compatible combination for quasi-stationary slices is not feasible.

Findings

Conflicting boundary conditions prevent simultaneous quasi-stationarity and singularity avoidance.

A positive lapse function cannot be maintained across slices when combining methods.

Relaxing some conditions may allow solutions but reduces the benefits of the combined approach.

Abstract

We consider combining two important methods for constructing quasi-equilibrium initial data for binary black holes: the conformal thin-sandwich formalism and the puncture method. The former seeks to enforce stationarity in the conformal three-metric and the latter attempts to avoid internal boundaries, like minimal surfaces or apparent horizons. We show that these two methods make partially conflicting requirements on the boundary conditions that determine the time slices. In particular, it does not seem possible to construct slices that are quasi-stationary and avoid physical singularities and simultaneously are connected by an everywhere positive lapse function, a condition which must obtain if internal boundaries are to be avoided. Some relaxation of these conflicting requirements may yield a soluble system, but some of the advantages that were sought in combining these approaches will be lost.