What is necessary for a causal set to be manifold-like?

Question

A causal set is a poset which is reflexive, antisymmetric, transitive and locally finite.

As a motivation, there is a programme to model spacetime as fundamentally discrete, with causal sets providing the underlying structure. Typically this is constructed by sprinkling (a poisson process) existing spacetime with elements, endowing these elements with an ordering given by causal cones, and removing spacetime. Volume is then given by a counting metric, which with the causal structure is enough to build a geometry. By the poisson nature of this process, the distribution is Lorentz-invariant.

This only makes sense in nature if the causal set is manifold-like, by which we mean it can be faithfully embedded into a manifold, such that the element count gives volume.

Precisely when is a causal set manifold-like: what are the necessary conditions for the existence of such an embedding? (Are there interesting sufficient conditions?) Do they have natural interpretations?

[This should be tagged quantum-gravity I think.]

Answer 1

In particular the following reference attempts to construct a useful necessary condition:

S. Major, D.P. Rideout, S. Surya, Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory, arXiv:0902.0434 (Continuum topology and homology)

I do not know of any sufficient conditions, beyond giving an explicit embedding. Such a thing would be great!

Answer 2

I noticed that causal sets afford a definition of energy ratios and their quantum, thanks to the relative frequency ratios formed by re-convergent pathways. It is then possible to construct an extendable 4-D lattice in conjunction with higher frequency sequences representing nuclei embedded in the lattice, such that time dilation is proportional to local energy density. This is precisely the local condition that accumulates to General Relativity at the global scale. See [http://vixra.org/pdf/1006.0070v1.pdf][1]. The modeling produced above does not employ random sprinkling in order to make the 4-D lattice conform to Einstein's continuous manifold, but rather substitutes quantized mass-energy values for Einstein's tensor values, reconciling quantum theory with gravity via pure theoretical hypothesis. The coherence of the overall scheme is the only justification for replacing the continuum version of General Relativity with the discrete version. The search for general conditions for faithful embedding of a causal set in a manifold becomes a moot point if the specific modeling above proves to be correct. At the least, an example exists of a causal set with an inherent inflationary metric defined, which shows how non-uniform a discrete causal set manifold can be and still conform to Einstein's well-accepted scheme of variable space-time curvature.

Answer 3

There are some papers that try and construct this embedding, e.g.:

• Manifold dimension of a causal set: Tests in conformally flat spacetimes;
• Manifoldness :: Further Reading (Wikipedia: Causal Sets).

Hope this helps.