Spacelike distance from discrete causal order

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 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 some evidence in support of the "Hauptvermutung" of causal sets.Contents 2 Note that this 'small scale' may be after some coarse graining of the causal set, to get to the macroscopic, continuum-like scales. It is of course quite open what the smallest scales of the causal set will look like.

The causal set approach to quantum gravity

“…The former underestimates the spatial distance compared to the continuum, and the latter overestimates it. The spatial distance functions of both Brightwell and Gregory (1991) and Rideout and Wallden (2009) are however strictly "predistance" functions since they do not satisfy the triangle inequality.…”

Dimensional reduction in causal set gravity

“…In the continuum, asymptotic silence is an "anti-Newtonian" limit, in which the speed of light goes to zero and nearby spacelike separated points become (nearly) causally disconnected. In the causal set context, defining "nearby" is nontrivial, but not impossible [34], and one can measure the minimum number of links N required for two nearby points to share a common point in the future. The short distance asymptotic silence conjecture is that while N should behave classically for pairs of points with large spatial separations, it should become much larger than its classical value as the spatial distance shrinks.…”

Spacetime is as spacetime does

“…Major et al (2006Major et al ( , 2007 construct a spatial topology by 'thickening' the 'spatial' antichains of a causal set, i.e., subsets of events which are pairwise incomparable, and exploiting the arising causal structure;Rideout and Wallden (2009) offer the most penetrating analysis for the case of spacelike distances.…”