Towards a definition of locality in a manifoldlike causal set

Abstract

It is a common misconception that spacetime discreteness necessarily implies a violation of local Lorentz invariance. In fact, in the causal set approach to quantum gravity, Lorentz invariance follows from the specific implementation of the discreteness hypothesis. However, this comes at the cost of locality. In particular, it is difficult to define a "local" region in a manifoldlike causal set, i.e., one that corresponds to an approximately flat spacetime region. Following up on suggestions from previous work, we bridge this lacuna by proposing a definition of locality based on the abundance of m-element order-intervals as a function of m in a causal set. We obtain analytic expressions for the expectation value of this function for an ensemble of causal set that faithfully embeds into an Alexandrov interval in d-dimensional Minkowski spacetime and use it to define local regions in a manifoldlike causal set. We use this to argue that evidence of local regions is a necessary condition for manifoldlikeness in a causal set. This in addition provides a new continuum dimension estimator. We perform extensive simulations which support our claims.

The causal set approach to quantum gravity

“…Extracting such invariants requires new technical tools and insights sometimes requiring a rethink of familiar aspects of continuum Lorentzian geometry. We will describe some of the progress made in this direction over the years (Myrheim 1978;Brightwell and Gregory 1991;Meyer 1988;Bombelli and Meyer 1989;Bombelli 1987;Reid 2003;Major et al 2007;Rideout and Wallden 2009;Sorkin 2007b;Benincasa and Dowker 2010;Benincasa 2013;Benincasa et al 2011;Glaser and Surya 2013;Roy et al 2013;Buck et al 2015;Cunningham 2018a;Aghili et al 2018;Eichhorn et al 2018). The correlation between order invariants and manifold invariants in the continuum approximation lends support for the Hauptvermutung and simultaneously establishes weaker, observable-dependent versions of the conjecture.…”

“…On the other hand, many of the order invariants we have obtained so far correspond to geometric invariants only in such RNN-type regions. A characterisation of intrinsic localisation was obtained by Glaser and Surya (2013) using the abundance N d m of m element order intervals for C ∈ C(A d , ρ c ). They found the following closed form expression for the associated expectation value…”

“…Moreover, for (1 − q) 1 and n 1 1−q , after a post the universe enters a tree like phase and then a de Sitter-like phase, in which the cardinality of large causal diamonds are de Sitter like functions of the discrete proper time. In Glaser and Surya (2013), it was shown that despite this, the abundance of causal intervals is not de Sitter like, and thus, this is not strictly a manifold-like phase. In Brightwell and Georgiou (2010) and Brightwell and Luczak (2015), moreover, it was shown explicitly that in the asymptotic limit n → ∞ the causal sets limit to "semi-orders" which, though temporally ordered, have no spatial structure at all, and are hence non-manifold-like.…”

Dimensionally restricted causal set quantum gravity: examples in two and three dimensions

“…This distribution is drastically different in the two phases. In the hot phase, it is a reflection of the manifold-like properties of the causal set with, in particular, the interval abundances decreasing monotonically with k, the size of the intervals [28]. In the cold phase, on the other hand, the number of links is very large and the distribution develops "oscillations", with distinct peaks at very large values of k. The abundance of intervals of size k = 54 in d = 2 and k = 89 in d = 3, for example correspond approximately to the size of one of the layers.…”

Spectral dimension on spatial hypersurfaces in causal set quantum gravity

“…A continuum manifold is said to approximate a given causal set if that causal set has a high probability to emerge from the manifold via the sprinkling process. Several methods to check for manifold-likeness have been developed for two-dimensional causal sets in e.g., [34,35] as well as for arbitrary dimension [36][37][38]. Furthermore it was shown in [39] that a causal set obtained through a sprinkling into Minkowski spacetime remains Lorentz invariant in the continuum approximation.…”