Space of Spaces as a Metric Space
Abstract
In spacetime physics, we frequently need to consider a set of all spaces (`universes') as a whole. In particular, the concept of `closeness' between spaces is essential. However, there has been no established mathematical theory so far which deals with a space of spaces in a suitable manner for spacetime physics. Based on the scheme of the spectral representation of geometry, we construct a space of all compact Riemannian manifolds equipped with the spectral measure of closeness. We show that this space of all spaces can be regarded as a metric space. We also show other desirable properties of this space, such as the partition of unity, locallycompactness and the second countability. These facts show that this space of all spaces can be a basic arena for spacetime physics.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2000
 DOI:
 10.1007/s002200050025
 arXiv:
 arXiv:grqc/9908078
 Bibcode:
 2000CMaPh.209..393S
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 To appear in Communications in Mathematical Physics. 20 pages