Space of spaces as a metric space

Details Space of spaces as a metric space Space of spaces as a metric space

1999-08-31

arxiv.org/abs/gr-qc/9908078v1

Commun.Math.Phys. 209 (2000) 393-405

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

To appear in Communications in Mathematical Physics. 20 pages

Scientific paper

10.1007/s002200050025

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, locally-compactness and the second countability. These facts show that this space of all spaces can be a basic arena for spacetime physics.

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