39 research outputs found
A new topology on the space of Lorentzian metrics on a fixed manifold
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 relative to two space time points and and metrics and . 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.Comment: 33 page
The limit space of a Cauchy sequence of globally hyperbolic spacetimes
In this second paper, I construct a limit space of a Cauchy sequence of
globally hyperbolic spacetimes. In the second section, I work gradually towards
a construction of the limit space. I prove the limit space is unique up to
isometry. I als show that, in general, the limit space has quite complicated
causal behaviour. This work prepares the final paper in which I shall study in
more detail properties of the limit space and the moduli space of (compact)
globally hyperbolic spacetimes (cobordisms). As a fait divers, I give in this
paper a suitable definition of dimension of a Lorentz space in agreement with
the one given by Gromov in the Riemannian case.Comment: 31 pages, 5 figures, submitted to Classical and Quantum gravity,
seriously improved presentatio
A Lorentzian Gromov-Hausdoff notion of distance
This paper is the first of three in which I study the moduli space of
isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I
introduce a notion of Gromov-Hausdorff distance which makes this moduli space
into a metric space. Further properties of this metric space are studied in the
next papers. The importance of the work can be situated in fields such as
cosmology, quantum gravity and - for the mathematicians - global Lorentzian
geometry.Comment: 20 pages, 0 figures, submitted to Classical and quantum gravity,
seriously improved presentatio