Spacelike distance from discrete causal order

Details Spacelike distance from discrete causal order Spacelike distance from discrete causal order

2008-10-09

arxiv.org/abs/0810.1768v3

Class.Quant.Grav.26:155013,2009

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

32 pages, 16 figures, revtex4; journal version

Scientific paper

10.1088/0264-9381/26/15/155013

Any discrete approach to quantum gravity must provide some prescription as to how to deduce continuum properties from the discrete substructure. In the causal set approach it is straightforward to deduce timelike distances, but surprisingly difficult to extract spacelike distances, because of the unique combination of discreteness with local Lorentz invariance in that approach. We propose a number of methods to overcome this difficulty, one of which reproduces the spatial distance between two points in a finite region of Minkowski space. We provide numerical evidence that this definition can be used to define a `spatial nearest neighbor' relation on a causal set, and conjecture that this can be exploited to define the length of `continuous curves' in causal sets which are approximated by curved spacetime. This provides evidence in support of the ``Hauptvermutung'' of causal sets.

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