# Constructing Spacetimes

By Steven Carlip and Samuel Loomis

Imagine you are given a bucket of points and asked to assemble them into a spacetime. What kind of “glue” would you need?

In causal set theory, the only added ingredient is the set of causal relations, the knowledge of which points are to the past and future of which. In particular, suppose your points were taken at random from a real spacetime, at some typical length scale ℓ. Then on scales large compared to ℓ, the causal diamonds – the sets formed by intersecting the past of one point with the future of another – determine the topology; the causal relations determine the metric up to a scale factor; and the remaining scale factor is just a local volume, which can be obtained by counting points. As the slogan of Rafael Sorkin, the founder of the field, goes, “Order + Number = Geometry.”

Samuel Loomis and Steven Carlip with their causal set.

Such a causal set – a collection of points and causal relations, with a few technical conditions – can give a good discrete approximation of a spacetime, although there are important open questions of exactly how to define “good”. A causal set taken from at spacetime provides one of very few ways we know to maintain discrete Lorentz invariance. More generally, from the causal relations alone we can determine the dimension of causal set, construct discrete Laplacians, and even write down a discrete version of the curvature and the Einstein-Hilbert action.

Unfortunately, the recovery of observable spacetime from the discrete, disordered causal sets is easier said than done. In fact, the probability of randomly selecting a causal set which is manifold-like is zero – almost all causal sets are of a particular shape, called “Kleitman-Rothschild orders,” which are very much not manifold-like. This is because each KR order has three moments of time; i.e., no event is preceded by more than two events.

A KR order – a causal set with only three moments of time. Points connected by lines
are causally related.

KR orders aren’t the only problem – they are simply the most numerous. Even if those are cast aside, there is an infinite hierarchy of finite-layer sets (after the 3-layers are the 2-layer sets, followed by 4-layers, and so on). Clearly, manifold-likeness is the extreme exception rather than the rule. If causal sets are fundamental, and manifold-like behaviour is emergent, a dynamical process must somehow suppress the vast majority of causal sets, leaving only the very rare manifold-like ones. Finding such a process – especially one that hasn’t been artificially constructed merely to achieve this goal – is not easy.

What we have now shown is that one large collection of “bad” causal sets, the two-level orders, is exponentially suppressed in the gravitational path integral. We start with the ordinary Lorentzian path integral, appropriately discretized, and analytically evaluate the contribution of these sets. The result looks like exp{bN2}, where N is the number of points (which is enormous for any reasonably sized region) and the coefficient b depends on coupling constants. For a large range of couplings, we find that b is negative, and the contribution of the two-level orders is completely negligible. This ideal range of couplings restricts the discreteness scale of the causal set to be larger than 1.5 Planck lengths.

This is far from the end of the story. The two-level orders are particularly simple; the action for other non-manifold-like sets like the KR orders is more complicated, though we have some ideas of how to generalise our results. But if a generalisation is possible, we will have taken a big step toward showing that the causal set path integral is a viable starting point for quantum gravity.