Optical Geometry and Constructive Gravity

Gravitational lensing provides an important probe of the background geometry.
In this talk, I discuss light propagation on non-Lorentzian backgrounds which
arise in two different contexts. Firstly, we consider optical geometry, which
is a formalism in 3-space for light propagation in Lorentzian spacetimes:
static spacetimes give rise to Riemannian optical geometry and stationary
spacetimes have a Finslerian optical geometry of Randers type. I will
review basic results as well as recent work using the Gauss-Bonnet theorem
and curve-shortening flow. In particular, the first isoperimetric inequality
in this context will be presented. Secondly, we consider a general formalism
for arbitrary tensorial (not necessarily metric) spacetime kinematics and
outline how gravitational dynamics can be derived such that the theory is
causal by construction. This new geometrodynamical approach is called
constructive gravity. I will briefly describe first results for light
propagation on a non-metric (specifically, area metric) background that
allows birefringence to occur and yields deviations from the standard
Etherington distance duality relation which can be probed observationally.