Time Delay and Light Deflection by a Moving Body Surrounded by a Refractive

MediumAdrian Corman and Sergei

KopeikinDepartment of Physics and

AstronomyUniversity of Missouri-Columbia

Introduction

IntroductionIn the linear approximation, the metric tensor becomes

Where hαβ is the perturbation to the Minkowski metric.We also impose the harmonic gauge condition

IntroductionIn the first post-Minkowski approximation, we can use theRetarded Liénard-Wiechert tensor potentials to find hμν.

Where

And

Is the four-velocity of the lens.

IntroductionAdditionally, 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 Vα is the four-velocity of the medium (equal in thiscase to uL

α , the four-velocity of the lens.) This metric hasthe 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 linearorder 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, Xα=(cT,X) with the originat 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 unperturbedtrajectory of the light ray gives the relativistic perturbation tothe light’s coordinate velocity

Where D = Σ x (X x Σ).

Light Propagation in the Lens Frame

Integrating again gives the relativistic perturbation to thelight ray trajectory

Where we have skipped a constant of integration that can beincluded 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 twoframes. These are defined in the ordinary way

Where

Light Propagation in the Observer Frame

In this frame, the perturbed trajectory of the light ray isgiven 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 ofthe trajectory and velocity of the light in the two frames aregiven by

Light Propagation in the Observer Frame

The time of propagation between the emitter and observeris given by

Light Propagation in the Observer Frame

This becomes

With

And

Light Propagation in the Observer Frame

The angle of light deflection, αi, is given by

Where Pij = δij – σiσj is the projection operator onto the planeorthogonal to the direction of propagation of the light rayin the observer frame. The angle of deflection becomes (whereβT

i=Pijβj and rTi=Pi

jrj

Light Propagation in the Observer Frame

In retarded time (where r*=x – xL(s),

And

Giving

Light Propagation in the Observer Frame

Additionally