Bilocal geodesic operators as a tool of investigating the optical properties of spacetimes

Abstract

In my thesis, I present one particular example of the formalism capable of describing the propagation of a family of light rays in a curved spacetime. It is based on the resolvent operator of the geodesic deviation equation for null geodesics which is known as the bilocal geodesic operator (BGO) formalism. The BGO formalism generalizes the standard treatment of light ray bundles by allowing observations extended in time or performed by a family of neighbouring observers. Furthermore, it provides a more unified picture of relativistic geometrical optics and imposes a number of consistency requirements between the optical observables. The thesis begins with a brief introduction of the transfer matrix and its relativistic versions known as the Jacobi propagators and the bilocal geodesic operators. Then the basics of differential geometry are reviewed, with an emphasis on the geometry of the tangent bundle and the geodesic flow, which later provide the foundation for the BGO formalism. The mathematical introduction is then followed by two articles about the applications of the BGO formalism in the studies of optical distance measures and the conclusion.

arXiv commentUpdated version of PhD thesis reflects referees' comments and include expanded discussion regarding Synge's world function, experimental status of new observables, and other things. Two figures illustrating tangent bundle, geodesic spray, and its deformation tensor were added. The ordering of the bibliography was improved. For original thesis please check the previous version. Comments are welcome!