Publication: Spacelike distance from discrete causal order

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Title	Spacelike distance from discrete causal order
Authors/Editors*	D. Rideout, P. Wallden
Where published*	arxiv/gr-qc
How published*	Other
Year*	2008
Volume
Number
Pages
Publisher
Keywords	causal sets, quantum gravity, combinatorics, Cactus
Link	http://arxiv.org/abs/0810.1768
Abstract	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 demonstrate numerically 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 important evidence in support of the "Hauptvermutung" of causal sets.