Causal Sets

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The causal sets programme is an approach to quantum gravity. Its founding principle is that spacetime is fundamentally discrete and that the spacetime points are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime points.

The programme is based on a theorem^[1] by David Malament which states that if there is a map between two spacetimes which preserves their causal structure then the map is a conformal isomorphism. The conformal factor that is left undetermined is related to the volume of regions in the spacetime. This volume factor can be recovered by specifying a volume element for each spacetime point. The volume of a spacetime region could then by found by counting the number of points in that region.

Causal sets was initiated by Rafael Sorkin who continues to be the main proponent of the programme. He has coined the slogan "Number + Order = Geometry" to characterise the above argument. The programme provides a theory in which spacetime is fundamentally discrete while retaining local Lorentz invariance.

1 Definition
2 Sprinkling
3 Geometry
- 3.1 Geodesics
- 3.2 Dimension Estimators
4 References
5 External links

[edit] Definition

A causal set is a set $C$ with a partial order relation $\prec$ which is

Irreflexive: $x \prec y$ and $y \prec x$ implies $x = y \,\!$
Transitive: $x \prec y \prec z$ implies $x \prec z$
Locally finite: card $(\{y \in C | x \prec y \prec z\}) < \infty$

for all $x, y, z \in C$ . Here card( $A$ ) denotes the cardinality of a set $A$ .

Given a causal set we may ask whether it can be embedded into a Lorentzian manifold. An embedding would be a map taking elements of the causal set into points in the manifold such that the order relation of the causal set matches the causal ordering of the manifold. A further criteria is needed however before the embedding is suitable. If the number of causal set elements mapped into a region of the manifold is proportional to the volume of the region then the embedding is said to be faithful. In this case we can consider the causal set to be 'manifold-like'

A central conjecture to the causal set programme is that the same causal set cannot be faithfully embedded into two spacetimes which are not similar on large scales. This is called the hauptvermutung, meaning 'fundamental conjecture'. It is difficult to define this conjecture precisely because it is difficult to decide when two spacetimes are 'similar on large scales'.

Modelling spacetime as a causal set would require us to restrict attention to those causal sets which are 'manifold-like'. Given a causal set this is a difficult property to determine.

[edit] Sprinkling

A plot of 1000 sprinkled points in 1+1 dimensions

The difficulty of determining whether a causal set can be embedded into a manifold can be approached from the other direction. We can create a causal set by sprinkling points into a Lorentzian manifold. By sprinkling points in proportion to the volume of the spacetime regions and using the causal order relations in the manifold to induce order relations between the sprinkled points, we can produce a causal set which (by construction) can be faithfully embedded into the manifold.

To maintain Lorentz invariance this sprinkling of points must be done randomly using a Poisson process. Thus the probability of sprinkling $n$ points into a region of volume $V$ is

$P(n) = \frac{(\rho V)^n e^{-\rho V}}{n!}$

where $ρ$ is the density of the sprinkling.

Sprinkling points in on a regular lattice would not keep the number of points proportional to the region volume.

[edit] Geometry

Some geometrical constructions in manifolds carry over to causal sets. When defining these we must remember to rely only on the causal set itself, not on any background spacetime into which it might be embedded. For an overview of these constructions, see ^[2].

[edit] Geodesics

The geodesics between two points in a 180 point causal set sprinkled in 1+1 dimensions

A link in a causal set is a pair of elements $x, y \in C\,\!$ such that $x \prec y$ but with no $z \in C\,\!$ such that $x \prec z \prec y$ .

A chain is a sequence of elements $x_0,x_1,\ldots,x_n$ such that $x_i \prec x_{i+1}$ for $i=0,\ldots,n-1$ . The length of a chain is $n$ , the number of relations used.

We can use this to define a geodesic between two causal set elements. A geodesic between two elements $x, y \in C$ is a chain consisting only of links such that

$x_0 = x\,\!$ and $x_n = y\,\!$
The length of the chain, $n$ , is maximal over all chains from $x\,$ to $y\,$ .

In general there will be more than one geodesic between two elements.

Myrheim^[3] first suggested that the length of such a geodesic should be directly proportional to the proper time along a timelike geodesic joining the two spacetime points. Tests of this conjecture have been made using causal sets generated from sprinklings into flat spacetimes. The proportionality has been shown to hold and is conjectured to hold for sprinklings in curved spacetimes too.

[edit] Dimension Estimators

Much work has been done in estimating the manifold dimension of a causal set. This involves algorithms using the causal set aiming to give the dimension of the manifold into which it can be faithfully embedded. The algorithms developed so far are based on finding the dimension of a Minkowski spacetime into which the causal set can be faithfully embedded.

Myrheim-Meyer dimension

This approach relies on estimating the number of chains of $k$ -elements present in a sprinkling into $d$ -dimensional Minkowski spacetime. Counting the number of $k$ -length chains in the causal set then allows an estimate for $d$ to be made.

Midpoint-scaling dimension

This approach relies on the relationship between the proper time between two points in Minkowski spacetime and the volume of the spacetime interval between them. By computing the maximal chain length (to estimate the proper time) between two points $x\,$ and $y\,$ and counting the number of elements $z\,$ such that $x \prec z \prec y$ (to estimate the volume of the spacetime interval) the dimension of the spacetime can be calculated.

These estimators should give the correct dimension for causal sets generated by high-density sprinklings into $d$ -dimensional Minkowski spacetime. Tests in conformally-flat spacetimes^[4] have shown these two methods to be accurate.

[edit] References

^ D. Malament, The class of continuous timelike curves determines the topology of spacetime, Journal of Mathematical Physics, July 1977, Volume 18, Issue 7, pp. 1399-1404
^ G, Brightwell, R. Gregory, Structure of random discrete spacetime, Phys. Rev. Lett. 66, 260 - 263 (1991)
^ J. Myrheim, CERN preprint TH-2538 (1978)
^ D. Reid, Manifold dimension of a causal set: Tests in conformally flat spacetimes, arXiv:gr-qc/0207103v2