Rotating gravitational lenses: a kinematic approach

Abstract

This paper uses the Kerr geodesic equations for massless particles to derive an acceleration vector in both Boyer-Lindquist and Cartesian coordinates. As a special case, the Schwarzschild acceleration due to a non-rotating mass has a particularly simple and elegant form in Cartesian coordinates. Using forward integration, these equations are used to plot the caustic pattern due to a system consisting of a rotating point mass with a smaller non-rotating planet. Additionally, first and second order approximations to the paths are identified, which allows for fast approximations of paths, deflection angles and travel-time delays.

A Simple and Practical Algorithm for Accurate Gravitational Magnification Maps

“…In order to apply a fully consistent first-order approximation to a ray passing by several lenses, we begin with the first-order path equations derived from the acceleration vector for a massless test particle in the gravitational field of a massive lens located at the origin [see Walters & Forbes (2014)]. These are where r s is the Schwarzschild radius of the lens, K is the square of the impact parameter, and r is the normal position vector, and r = | r | is the distance from the origin.…”

“…By also specifying that the ray leaves the origin at some trajectory (ϕ, θ), where ϕ is the azimuthal angle and θ is the inclination angle above the plane, along with the speed constraint equation, C 1 C 11 + C 3 C 31 + C 5 C 51 = 0 [see Walters & Forbes (2014)], we can solve for the other constants as follows: …”

“…In order to apply a fully consistent first order approximation to a ray passing by several lenses, we begin with the first order path equations derived from the acceleration vector for a massless test particle in the gravitational field of a massive lens located at the origin (see Walters & Forbes (2014)). These are:…”

A Note on the Overall Magnification of a Gravitational Point-Source—Point-Lens System

“…In order also to account for the ‘backward’ rays, we will now apply a fully consistent first-order approximation to the question of overall flux due to gravitational lensing. Following the approach given in Walters & Forbes (2014), we can identify the path equations for a ray leaving the origin at τ = 0 in a system containing a single massive lensing object with Schwarzschild radius r s located at ( x m , y m , z m ). The ray has initial trajectory with azimuthal angle ϕ, and angle of inclination θ above the x – y plane.…”

“…Details are given in Walters & Forbes (2014). We now have the first-order path equations for x, y , and z .…”