An exact Jacobi map in the geodesic light-cone gauge

The remarkable properties of the recently proposed geodesic light-cone (GLC) gauge allow to explicitly solve the geodesic-deviation equation, and thus to derive an exact expression for the Jacobi map $J^A_B(s,o)$ connecting a generic source $s$ to a geodesic observer $o$ in a generic space time. In this gauge $J^A_B$ factorizes into the product of a local quantity at $s$ times one at $o$, implying similarly factorized expressions for the area and luminosity distance. In any other coordinate system $J^A_B$ is simply given by expressing the GLC quantities in terms of the corresponding ones in the new coordinates. This is explicitly done, at first and second order, respectively, for the synchronous and Poisson gauge-fixing of a perturbed, spatially-flat cosmological background, and the consistency of the two outcomes is checked. Our results slightly amend previous calculations of the luminosity-redshift relation and suggest a possible non-perturbative way for computing the effects of inhomogeneities on observations based on light-like signals.