Exact analytic characteristic initial data for axisymmetric, non-rotating, vacuum spacetimes, with an application to the binary black hole problem

TL;DR

This paper develops an exact analytic method for initial data in axisymmetric, non-rotating vacuum spacetimes, facilitating the study of binary black hole interactions and their gravitational fields.

Contribution

It introduces a new gauge choice and characteristic initial data formulation that yields exact solutions relevant to binary black hole problems.

Findings

Provides a unique expression for the Weyl spinor component $ ext{Ψ}_0$

Derives exact solutions for metric functions on initial null hypersurfaces

Shows Weyl curvature increases as black holes approach each other

Abstract

Bondi's approach to the construction of a coordinate system is used with a different choice of gauge, in accordance with which the radial coordinate r is an affine parameter, to cast the metric tensor into a form suitable for use with the Newman-Penrose null tetrad formalism. The choice of tetrad has the result that the equations and all the functions that appear in them are real-valued. A group classification of the Sachs equations in this gauge leads to a unique expression for the first of the five independent elements $\Psi_0$ of the Weyl spinor, and to the corresponding exact solutions for two of the metric functions on an initial null hypersurface. A proof is presented that the result for $\Psi_0$ constitutes the appropriate characteristic initial value function for all physically realistic axisymmetric, non-rotating vacuum spacetimes. Integration of the field equations on the axis of symmetry when an equatorial symmetry plane is also present, and on the equatorial plane itself when time rates of change can be neglected, shows that these data produce results consistent with Newton's laws and with the Schwarzschild solution in the appropriate limits. The solution on the axis of symmetry indicates that the Weyl curvature increases without limit between black holes as their separation decreases.