Efficient Computation of Null Affine Parameters

Abstract

Finding affine parameters for null geodesics is often of considerable physical importance, especially when studying null geodesics or 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.

Symmetry Transformations in Cosmological and Black Hole Analytical Solutions

“…Here, we have f = f (r) and dΩ 2 = dθ 2 + sin 2 θ dϕ 2 . This encompasses a broad class of spherically symmetric metrics that can be put in a static form [12][13][14]. There is a subclass of FLRW-metrics can be put in the form of (2), such as the de Sitter (dS) and AdS spaces, as well as Milne and Lanczos universes.…”

Black holes, white holes, and near-horizon physics

“…Analyzing the averaged null energy condition (ANEC) would require one to trace the null geodesics through the bounce region, and to unambiguously identify a suitable null affine parameter. Unfortunately, this is one of those situations where (despite recent progress [90]) these issues are still in the "too hard" basket. Overall, we see that key point-wise energy conditions are definitely violated by the black-to-white bounce.…”