Light-bending in Schwarzschild-de-Sitter: projective geometry of the optical metric

TL;DR

This paper explores the projective geometry of optical metrics in Schwarzschild-de Sitter spacetime, revealing how light bending phenomena depend on the cosmological constant despite metric equivalences.

Contribution

It demonstrates the projective equivalence of optical metrics across different cosmological constants and constructs new families of geodesically equivalent metrics with conserved quantities.

Findings

Optical metrics are projectively equivalent regardless of mbda.

Lensing phenomena depend on mbda despite metric equivalence.

New conserved quantities for geodesic flows are identified.

Abstract

We interpret the well known fact that the equations for light rays in the Kottler or Schwarzschild-de Sitter metric are independent of the cosmological constant in terms of the projective equivalence of the optical metric for any value of \Lambda. We explain why this does not imply that lensing phenomena are independent of \Lambda. Motivated by this example, we find a large collection of one-parameter families of projectively equivalent metrics including both the Kottler optical geometry and the constant curvature metrics as special cases. Using standard constructions for geodesically equivalent metrics we find classical and quantum conserved quantities and relate these to known quantities.