A kinematical approach to gravitational lensing using new formulae for refractive index and acceleration

Abstract

This paper uses the Schwarzschild metric to derive an effective refractive index and acceleration vector that account for relativistic deflection of light rays in an otherwise classical kinematic framework. The new refractive index and the known path equation are integrated to give accurate results for travel time and deflection angle, respectively. A new formula for coordinate acceleration is derived which describes the path of a massless test particle in the vicinity of a spherically symmetric mass density distribution. A standard ray shooting technique is used to compare the deflection angle and time delay predicted by this new formula with the previously calculated values, and with standard first‐order approximations. Finally, the ray shooting method is used in theoretical examples of strong and weak lensing, reproducing known observer‐plane caustic patterns for multiple masses.

Rotating gravitational lenses: a kinematic approach

“…They showed that their linearized equations were capable of an exact closed-form solution which agreed well with the fully non-linear simulations. In the present paper, the approach of Walters et al (2010) is generalized to include the effects of relativistic frame dragging due to rotation of the lensing object, as described by the Kerr metric. A kinematic description is given in Section 2.…”

“…In order to check this result, we may compare it to the delay (∆t) calculated numerically to high precision using Gaussian quadrature with the formula given by Walters et al (2010). For a ray starting at earth orbit, grazing the sun (r0 = 696000km and rs = 2.95km) and reaching earth-orbit again, the travel time delay is calculated accurately for a range of orbital distances.…”

“…The top picture in Fig. 5 shows the normal diamond caustic without rotation as described by Wambsganss (1997), and calculated here using the numerical procedure described in Walters et al (2010). This was generated using almost 15,000 simulated light rays in a numerical integration of equation (12).…”

“…5 in Section 4, which was described by Wambsganss (1997), and was also plotted previously using 2-dimensional polar co-ordinates in Walters et al (2010) and Walters & Forbes (2011). Interestingly the computations were slightly quicker with this new Cartesian system, as it was not necessary to rotate each ray into the x, y or r, φ plane, and also because the zeroth order terms ofẍ,ÿ andz in equation (12) are now all zero (whereas those ofr andφ are not).…”

“…Recently, Walters, Forbes & Jarvis (2010) undertook a new approach, in which they used the Schwarzschild metric to derive kinematic type laws for the propagation of light rays through a lensing system. They found that the acceleration vectors thus derived gave results in close agreement with those obtained using the simpler model described above.…”