Upload
reuben-delaney
View
35
Download
0
Embed Size (px)
DESCRIPTION
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. Introduction. In the linear approximation, the metric tensor becomes. - PowerPoint PPT Presentation
Citation preview
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