On the Riemann Function and Irregular Singular Points for Axisymmetric Black Hole Collisions at the Speed of Light

TL;DR

This paper improves the calculation of the Riemann function related to gravitational radiation in axisymmetric black hole collisions at light speed, enhancing numerical methods for analyzing irregular singular points in hyperbolic equations.

Contribution

It provides an improved evaluation of coefficients in the second-order hyperbolic equation for the Riemann function, facilitating better numerical analysis of gravitational wave emission.

Findings

Enhanced coefficients for the hyperbolic operator improve Riemann function evaluation.

Conditions identified for reducing Green-function calculations to ODE solutions.

Potential for improved numerical simulations of gravitational radiation.

Abstract

The news function providing some relevant information about angular distribution of gravitational radiation in axisymmetric black hole collisions at the speed of light had been evaluated in the literature by perturbation methods, after inverting second-order hyperbolic operators with variable coefficients in two independent variables. More recent work has related the appropriate Green function to the Riemann function for such a class of hyperbolic operators in two variables. The present paper obtains an improvement in the evaluation of the coefficients occurring in the second-order equation obeyed by the Riemann function, which might prove useful for numerical purposes. Eventually, we find under which conditions the original Green-function calculation reduces to finding solutions of an inhomogeneous second-order ordinary differential equation with a non-regular singular point.