There is a Monte Carlo recipe for assembling the blocks into a random spacetime. So you generate thousands and measure what you want and average it over the whole batch. You probably don't have any simple
equation giving probabilities of most of the things you might ask about. You discover probabilities by the Monte Carlo method, numerically, experimentally, so to speak.
I don't recall details of the recipe for generating random CDT spacetimes. But I can try to get you links. I read about this method about 10 years ago, in papers like this:
http://arxiv.org/abs/0711.0273
The Emergence of Spacetime, or, Quantum Gravity on Your Desktop
R. Loll
(Submitted on 2 Nov 2007)
Is there an approach to quantum gravity which is conceptually simple, relies on very few fundamental physical principles and ingredients, emphasizes geometric (as opposed to algebraic) properties, comes with a definite numerical approximation scheme, and produces robust results, which go beyond showing mere internal consistency of the formalism? The answer is a resounding yes: it is the attempt to construct a nonperturbative theory of quantum gravity, valid on all scales, with the technique of so-called Causal Dynamical Triangulations. Despite its conceptual simplicity, the results obtained up to now are far from trivial. Most remarkable at this stage is perhaps the fully dynamical emergence of a classical background (and solution to the Einstein equations) from a nonperturbative sum over geometries, without putting in any preferred geometric background at the outset. In addition, there is concrete evidence for the presence of a fractal spacetime foam on Planckian distance scales. The availability of a computational framework provides built-in reality checks of the approach, whose importance can hardly be overestimated.
22 pages, 11 figures
and earlier papers like these:
http://arxiv.org/abs/hep-th/0604212
Quantum Gravity, or The Art of Building Spacetime
J. Ambjorn,
J. Jurkiewicz,
R. Loll
(Submitted on 28 Apr 2006)
The method of four-dimensional Causal Dynamical Triangulations provides a background-independent definition of the sum over geometries in quantum gravity, in the presence of a positive cosmological constant. We present the evidence accumulated to date that a macroscopic four-dimensional world can emerge from this theory dynamically. Using computer simulations we observe in the Euclidean sector a universe whose scale factor exhibits the same dynamics as that of the simplest mini-superspace models in quantum cosmology, with the distinction that in the case of causal dynamical triangulations the effective action for the scale factor is not put in by hand but obtained by integrating out
in the quantum theory the full set of dynamical degrees of freedom except for the scale factor itself.
22 pages, 6 figures. Contribution to the book "Approaches to Quantum Gravity", ed. D. Oriti, Cambridge University Press
http://arxiv.org/abs/hep-th/0509010
The Universe from Scratch
R. Loll,
J. Ambjorn,
J. Jurkiewicz
(Submitted on 1 Sep 2005)
A fascinating and deep question about nature is what one would see if one could probe space and time at smaller and smaller distances. Already the 19th-century founders of modern geometry contemplated the possibility that a piece of empty space that looks completely smooth and structureless to the naked eye might have an intricate microstructure at a much smaller scale. Our vastly increased understanding of the physical world acquired during the 20th century has made this a certainty. The laws of quantum theory tell us that looking at spacetime at ever smaller scales requires ever larger energies, and, according to Einstein's theory of general relativity, this will alter spacetime itself: it will acquire structure in the form of "curvature". What we still lack is a definitive Theory of Quantum Gravity to give us a detailed and quantitative description of the highly curved and quantum-fluctuating geometry of spacetime at this so-called Planck scale. - This article outlines a particular approach to constructing such a theory, that of Causal Dynamical Triangulations, and its achievements so far in deriving from first principles why spacetime is what it is, from the tiniest realms of the quantum to the large-scale structure of the universe.
31 pages, 5 figures, commissioned review article.
http://arxiv.org/abs/hep-th/0505154
Reconstructing the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht),
J. Jurkiewicz (U. Krakow),
R. Loll (U. Utrecht)
(Submitted on 17 May 2005)
We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time.
52 pages, 20 postscript figures,
Physical Review D.
all the angles of the building blocks are decided at the outset. The tets are REGULAR except they are allowed to be stretched/compressed in the time direction by a percentage chosen in advance.
