Cauchy-characteristic matching for a family of cylindrical vacuum solutions possessing both gravitational degrees of freedom

TL;DR

This paper develops a Cauchy-characteristic matching method for cylindrical vacuum solutions in numerical relativity, enabling global solutions without outer boundary conditions, applied to a specific non-asymptotically flat family with gravitational degrees of freedom.

Contribution

It introduces a modified CCM code for cylindrical solutions that are not asymptotically flat, expanding the applicability of CCM in numerical relativity.

Findings

Successfully applied CCM to a non-asymptotically flat family of solutions.

Modified the cylindrical CCM code to handle regular singular equations.

Generated global solutions for a family with gravitational degrees of freedom.

Abstract

This paper is part of a long term program to Cauchy-characteristic matching (CCM) codes as investigative tools in numerical relativity. The approach has two distinct features: (i) it dispenses with an outer boundary condition and replaces this with matching conditions at an interface between the Cauchy and characteristic regions, and (ii) by employing a compactified coordinate, it proves possible to generate global solutions. In this paper CCM is applied to an exact two-parameter family of cylindrically symmetric vacuum solutions possessing both gravitational degrees of freedom due to Piran, Safier and Katz. This requires a modification of the previously constructed CCM cylindrical code because, even after using Geroch decomposition to factor out the $z$-direction, the family is not asymptotically flat. The key equations in the characteristic regime turn out to be regular singular in nature.