Geodesics for the Painlevé–Gullstrand form of Lense–Thirring Spacetime

Recently, the current authors have formulated and extensively explored a rather novel Painlevé–Gullstrand variant of the slow-rotation Lense–Thirring spacetime, a variant which has particularly elegant features—including unit lapse, intrinsically flat spatial 3-slices, and a separable Klein–Gordon equation (wave operator). This spacetime also possesses a non-trivial Killing tensor, implying separability of the Hamilton–Jacobi equation, the existence of a Carter constant, and complete formal integrability of the geodesic equations. Herein, we investigate the geodesics in some detail, in the general situation demonstrating the occurrence of “ultra-elliptic” integrals. Only in certain special cases can the complete geodesic integrability be explicitly cast in terms of elementary functions. The model is potentially of astrophysical interest both in the asymptotic large-distance limit and as an example of a “black hole mimic”, a controlled deformation of the Kerr spacetime that can be contrasted with ongoing astronomical observations.

Role of lambda in the cosmological lens equation

The cosmological constant Lambda affects cosmological gravitational lensing. Effects due to Lambda can be studied in the framework of the Schwarzschild-de Sitter spacetime. Two novel contributions, which cannot be accounted for by a proper use of angular diameter distances, are derived. First, a term deltaalpha[over ;]_(Lambda)=2mbLambda/3 has to be added to the bending angle, where m is the lens mass and b the impact parameter. Second, Lambda brings about a difference in the redshifts of multiple images. Both effects are quite small for real astrophysical systems, deltaalpha[over ;]_(Lambda) less, similar1muarcsec and Deltaz_(s) less, similar10(-7).