Geometric optics in general relativity using bilocal operators

Abstract

We consider the standard problem of observational astronomy, i.e. the observations of light emission from a distant region of spacetime in general relativity. The goal is to describe the changes between the measurements of the light performed by a sample of observers slightly displaced with respect to each other and moving with different 4-velocities and 4-accelerations. In our approach, all results of observations can be expressed as functions of the kinematic variables, describing the motions of the observers and the emitting bodies with respect to their local inertial frames, and four linear bilocal geodesic operators, describing the influence of the spacetime geometry on light propagation. The operators are functionals of the curvature tensor along the line of sight. The results are based on the assumption that the regions of emissions and observations are sufficiently small so that the spacetime curvature effects are negligible within each of them, although they are significant for the light propagation between them. The new formulation provides a uniform approach to optical phenomena in curved spacetimes and, as an application, we discuss the problem of a fully relativistic definition of the parallax and position drifts (or proper motions). We then use the results to construct combinations of observables which are completely insensitive to the motion of both the observer and the emitter. These combinations by construction probe the spacetime geometry between the observation and emission regions and in our formalism we may express them as functionals of the Riemann tensor along the line of sight. For short distances one of these combinations depends only on the matter content along the line of sight. This opens up the possibility to measure the matter content of a spacetime in a tomographylike manner irrespective of the motions of the emitter and the observer.Even if A enters the differential equation linearly, the general solution contains a nonlinear dependence.

Geometric optics in relativistic cosmology: New formulation and a new observable

“…Substituting V XL by W XL is also possible for the other observables discussed below, since they only make use of the transverse components of V XL . It is also noteworthy that the values of D A B do not depend on the choice of the timelike vector, making this way the formalism observer frame-invariant [1,8].…”

“…The main advantage of the BGOs lies in the fact that we can express a number of observables of interest in a unified framework via W XX and V XL (or W XL ) and the kinematical variables describing the momentary motions of the source and the observer in the moments of light emission and observation respectively [1,8]. The observables in question are the angular diameter distance D ang , the luminosity distance D lum , the magnification matrix M A B , the parallax and the position drift (or proper motion) δ O r A .…”

“…[11] for a comparison of four different approaches for the calculation of the luminosity distance in the cosmological context). On the other hand the main result of the new formulation of * Corresponding author: villa@cft.edu.pl [1] emphasizes the advantage of having a unified framework: all the observables -the parallax, the magnification, the position drift, the angular diameter distance etc.…”

“…-registered by a given observer are expressed in terms of one key quantity only, the so-called bilocal geodesic operators (BGOs), and the kinematical variables characterizing the momentary positions and motions of the source and the observer with respect to their local inertial frames. In addition the BGOs can be written as (non-local and non-linear) functionals of the curvature tensor along the line of sight, given by solutions of certain matrix ODEs [1,8]. Therefore, in their turn, the observables can be expressed in terms of the curvature along the line of sight and the momentary 4-velocities and 4-accelerations of both the observer and the source.…”

“…The theoretical justification of this method is based on the flat spacetime analysis of the geometry of light rays and obviously requires a modification if we want to include the curvature effects. The parallax distance in a general, curved spacetime can be defined in many ways [1,17], the differences coming from different methods of averaging over the baseline orientation. In this paper we use the one based on the determinant of the parallax matrix, fully analogous to (26):…”