Time Delay and Light Deflection by a Moving Body Surrounded by a Refractive Medium Adrian Corman and Sergei Kopeikin Department of Physics and Astronomy University of Missouri-Columbia

Introduction

In the linear approximation, the metric tensor becomes Where h αβ is the perturbation to the Minkowski metric. We also impose the harmonic gauge condition

Introduction In the first post-Minkowski approximation, we can use the Retarded Liénard-Wiechert tensor potentials to find h μν. Where And Is the four-velocity of the lens.

Introduction Additionally, the retarded time s in this equation is given by the solution to the null cone equation Giving us

The Optical Metric and Light Geodesics In a medium with constant index of refraction n, the optical metric (as given by Synge) can be written And Where V α is the four-velocity of the medium (equal in this case to u L α, the four-velocity of the lens.) This metric has the usual property

The Optical Metric and Light Geodesics With this metric, the affine connection is given by For the perturbed metric (keeping only terms of linear order in the perturbation) we obtain Where

The Optical Metric and Light Geodesics The null geodesics are given by the usual form

Light Propagation in the Lens Frame We introduce a coordinate system, X α =(cT,X) with the origin at the center of the lens. Using T as a parameter along the light ray trajectory we can write the null geodesic as Where we assumed the unperturbed trajectory of the light ray is a straight line

Light Propagation in the Lens Frame The perturbed trajectory of light is given by the formulas With the boundary conditions

Light Propagation in the Lens Frame Integrating the null geodesic equation along the unperturbed trajectory of the light ray gives the relativistic perturbation to the light’s coordinate velocity Where D = Σ x (X x Σ).

Light Propagation in the Lens Frame Integrating again gives the relativistic perturbation to the light ray trajectory Where we have skipped a constant of integration that can be included in the initial coordinates of the light ray.

Light Propagation in the Observer Frame To determine the form of this equation in the observer’s frame, we must use the Lorentz transformations between the two frames. These are defined in the ordinary way Where

Light Propagation in the Observer Frame In this frame, the perturbed trajectory of the light ray is given by With the boundary conditions

Light Propagation in the Observer Frame The speed of the light ray in the observer frame (c’) can be given in two equivalent forms And

Light Propagation in the Observer Frame The transformation of Σ is given by Where

Light Propagation in the Observer Frame The relationships between the relativistic perturbations of the trajectory and velocity of the light in the two frames are given by

Light Propagation in the Observer Frame The time of propagation between the emitter and observer is given by

Light Propagation in the Observer Frame This becomes With And

Light Propagation in the Observer Frame The angle of light deflection, α i, is given by Where P ij = δ ij – σ i σ j is the projection operator onto the plane orthogonal to the direction of propagation of the light ray in the observer frame. The angle of deflection becomes (where β T i =P ij β j and r T i =P i j r j

Light Propagation in the Observer Frame In retarded time (where r * =x – x L (s), And Giving

Light Propagation in the Observer Frame Additionally