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

Julius Serbenta

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.

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.
Paper Structure (20 sections, 65 equations, 3 figures)

This paper contains 20 sections, 65 equations, 3 figures.

Table of Contents

  1. General introduction
  2. Mathematical preliminaries
  3. Introduction
  4. Charts and projections
  5. Geodesic flow
  6. Vertical and horizontal subspaces
  7. Geodesic spray
  8. Lie dragging of vector along $\mathbf G$
  9. Symplectic structure and BGOs
  10. Bilocal geodesic operators in static spherically-symmetric spacetimes
  11. Testing the null energy condition with precise distance measurements
  12. Addendum: observational prospects of distance slip and drift effects
  13. Redshift drift
  14. Parallax and position drift
  15. Distance slip
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Addendum: an illustration of the lift and projection of geodesic motion on the base manifold $\bf M$ and its tangent bundle $\bf {TM}$. The sphere represents the base manifold, the great circle - its geodesic, and the plane tangent to the sphere at a point of the great circle - the tangent space at a point of $\bf M$ containing the tangent vector of the geodesic as well as all the other tangent vectors at that point. The lift of this curve to the tangent bundle is given by the integral curve of the geodesic spray $\bf G$.
  • Figure 2: Addendum: an illustration of the deformation of the geodesic flow generated by the geodesic spray $\bf G$. $\bf Y$ is an infinitesimal vector connecting two integral curves of geodesic sprays, and the span of all such $\bf Y$s define an infinitesimal domain. As we follow the reference integral curve, the deformation tensor $\bf W$ varies as well. This determines the evolution of all $\bf Y$s and, accordingly, the shape of the infinitesimal domain.
  • Figure :