# Extensions of Lorentzian spacetime geometry: From Finsler to Cartan and vice versa

April 19, 2013

We briefly review two recently developed extensions of the Lorentzian
geometry of spacetime and prove that they are in fact closely related. The
first is the concept of observer space, which generalizes the space of
Lorentzian observers, i.e., future unit timelike vectors, using Cartan
geometry. The second is the concept of Finsler spacetimes, which generalizes
the Lorentzian metric of general relativity to an observer-dependent Finsler
metric. We show that every Finsler spacetime possesses a well-defined observer
space that can naturally be equipped with a Cartan geometry. Conversely, we
derive conditions under which a Cartan geometry on observer space gives rise to
a Finsler spacetime. We further show that these two constructions complement
each other. We finally apply our constructions to two gravity theories,
MacDowell-Mansouri gravity on observer space and Finsler gravity, and translate
their actions from one geometry to the other.

Keywords:

Finsler geometry, Cartan geometry