Non-perturbative results for the luminosity and area distances

TL;DR

This paper reformulates the concept of luminosity distance in general spacetimes using the van Vleck determinant and null geodesic affine parameter, providing a non-perturbative framework that extends beyond traditional FLRW models.

Contribution

It introduces a non-perturbative formulation of luminosity and area distances using the van Vleck determinant, applicable to almost arbitrary spacetimes.

Findings

Reformulation of luminosity distance in terms of van Vleck determinant

Connection between null geodesic affine parameter and inverse square law

Framework applicable to general, non-FLRW spacetimes

Abstract

The notion of luminosity distance is most often defined in purely FLRW (Friedmann-Lemaitre-Robertson-Walker) cosmological spacetimes, or small perturbations thereof. However, the abstract notion of luminosity distance is actually much more robust than this, and can be defined non-perturbatively in almost arbitrary spacetimes. Some quite general results are already known, in terms of $dA_\mathrm{observer}/d\Omega_\mathrm{source}$, the cross-sectional area per unit solid angle of a null geodesic spray emitted from some source and subsequently detected by some observer. We shall reformulate these results in terms of a suitably normalized null geodesic affine parameter and the van Vleck determinant, $\Delta_{vV}$. The contribution due to the null geodesic affine parameter is effectively the inverse square law for luminosity, and the van Vleck determinant can be viewed as providing a measure of deviations from the inverse square law. This formulation is closely related to the so-called Jacobi determinant, but the van Vleck determinant has somewhat nicer analytic properties and wider and deeper theoretical base in the general relativity, quantum physics, and quantum field theory communities. In the current article we shall concentrate on non-perturbative results, leaving near-FLRW perturbative investigation for future work.