Interpretation of quantal manifolds

In quantum gravity, one looks for alternative structures to spacetime physics than ordinary real manifolds. Here, we propose an alternative universal construction containing the latter as an equilibrium state under the action of the universal diffeomorphism group. Our theory contains many other previous proposals in the literature as special cases. However, the crucial point we make is that those have to be appreciated in the universal context developed here.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1101.511

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 $A(p,q), \tilde{A} (p,q)$ 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.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