Are moving punctures equivalent to moving black holes?

TL;DR

This paper compares two numerical techniques for simulating binary black-hole systems, demonstrating that the moving-punctures method produces results equivalent to the traditional excision method, thus validating its correctness.

Contribution

The study provides evidence that the moving-punctures technique is equivalent to excision in evolving binary black-hole spacetimes, supporting its validity in numerical relativity.

Findings

Both methods produce identical invariants along the chosen worldline.

Moving-punctures and excision yield the same spacetimes in the causal past.

The results support the correctness of the moving-punctures approach.

Abstract

When simulating the inspiral and coalescence of a binary black-hole system, special care needs to be taken in handling the singularities. Two main techniques are used in numerical-relativity simulations: A first and more traditional one ``excises'' a spatial neighbourhood of the singularity from the numerical grid on each spacelike hypersurface. A second and more recent one, instead, begins with a ``puncture'' solution and then evolves the full 3-metric, including the singular point. In the continuum limit, excision is justified by the light-cone structure of the Einstein equations and, in practice, can give accurate numerical solutions when suitable discretizations are used. However, because the field variables are non-differentiable at the puncture, there is no proof that the moving-punctures technique is correct, particularly in the discrete case. To investigate this question we use both techniques to evolve a binary system of equal-mass non-spinning black holes. We compare the evolution of two curvature 4-scalars with proper time along the invariantly-defined worldline midway between the two black holes, using Richardson extrapolation to reduce the influence of finite-difference truncation errors. We find that the excision and moving-punctures evolutions produce the same invariants along that worldline, and thus the same spacetimes throughout that worldline's causal past. This provides convincing evidence that moving-punctures are indeed equivalent to moving black holes.