A new topology on the space of Lorentzian metrics on a fixed manifold

Abstract

We give a covariant definition of closeness between (time oriented) Lorentzian metrics on a manifold M, using a family of functions which measure the difference in volume form on one hand and the difference in causal structure relative to a volume scale on the other hand. These functions will distinguish two geometric properties of the Alexandrov sets $A(p,q),\tilde{A}(p,q)$ relative to two space time points $q$ and $p$ and metrics $g$ and $\tilde{g}$. It will be shown that this family generates uniformities and consequently a topology on the space of Lorentzian metrics which is Hausdorff when restricted to strongly causal metrics. This family of functions will depend on parameters for a volume scale, a length scale (relative to the volume scale) and an index which labels a submanifold with compact closure of the given manifold M.

33 pages