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

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 Λ. We explain why this does not imply that lensing phenomena are independent of Λ. 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.