Efficient computation of null affine parameters

TL;DR

This paper introduces a method to efficiently compute null affine parameters using conformal transformations and Killing vectors, simplifying the analysis of null geodesics in general relativity and astrophysics.

Contribution

It presents a novel approach to define affine null parameters via conformal Killing vectors without explicitly solving null geodesics.

Findings

Provides a conformal spacetime framework for affine null parameter definition.

Simplifies null geodesic analysis in general relativity.

Applicable to various astrophysical scenarios.

Abstract

Finding affine parameters for null geodesics is often of considerable physical importance, especially when studying the paths swept out by null geodesics or when dealing with conservation laws and/or averaged energy conditions. But explicitly finding null affine parameters is also often quite tedious and can sometimes even be somewhat tricky. Herein we shall demonstrate that the existence of a conformally related spacetime containing a conformal Killing vector, timelike in the domain of outer communication, is quite sufficient to define a preferred set of spatial three-slices -- on which a well-defined "affine" three-metric can be introduced to capture the notion of affine null parameter -- before explicitly finding the null geodesics. The construction depends on the properties of conformal transformations and on the conserved quantity associated with the conformal Killing vector. Having the affine null parameter in hand before attempting to find the actual null geodesics often quite radically simplifies other parts of the analysis. We emphasize that the successful identification of affine null parameters is a general-purpose tool of wide applicability in both general relativistic and astrophysical settings.