Transfer matrix approach to general relativistic geometrical optics




Center for Theoretical Physics PAS

January 11, 2023 12:30 PM

The theory of transfer matrices, propagators, and evolution operators is a fairly common tool in theoretical physics and mathematics. For many physical systems, the evolution of the system can be described using appropriate operators mapping the initial state of the physical system to its state at some later time. This approach has been used in many branches of physics ranging from the theory of elasticity to the quantum field theory.

In astrophysics and cosmology, most observations require a good understanding of the propagation of electromagnetic waves through a curved spacetime. In the high-frequency limit, the propagation of waves, which is often applicable in astrophysical situations, can be well approximated by propagation along null geodesics. Still, for sources of a finite size, a whole family of null geodesics has to be considered. While the general problem of light ray propagation is non-linear, it can be simplified for geodesics remaining close to a given one. The behaviour of neighbouring null geodesics can be described reasonably well by the first order geodesic deviation equation (GDE). As a system of linear ordinary differential equations, it admits the transfer matrix formulation known as the bilocal geodesic operator (BGO) formalism.

In my talk, I will introduce BGO formalism and highlight its main features. Then I will show how BGOs can help us study the optical properties of spacetime models with or without symmetries, with particular attention given to the angular diameter and parallax distances and the influence of the motions of the emitter and the observer.