Non-perturbative results for the luminosity and area distances
  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.
