# Causal sets

(Redirected from Causal Sets)

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

The programme is based on a theorem[1] by David Malament that states that if there is a bijective map between two past and future distinguishing spacetimes that 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 be 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 "Order + Number = Geometry" to characterise the above argument. The programme provides a theory in which spacetime is fundamentally discrete while retaining local Lorentz invariance.

## Definition

A causal set (or causet) is a set $C$ with a partial order relation $\preceq$ that is

• Reflexive: For all $x \in C$, we have $x \preceq x$.
• Antisymmetric: For all $x, y \in C$, we have $x \preceq y \preceq x \implies x = y$.
• Transitive: For all $x, y, z \in C$, we have $x \preceq y \preceq z$ implies $x \preceq z$.
• Locally finite: For all $x, z \in C$, we have card$(\{y \in C | x \preceq y \preceq z\}) < \infty$.

Here card($A$) denotes the cardinality of a set $A$. We'll write $x \prec y$ if $x \preceq y$ and $x \neq y$.

The set $C$ represents the set of spacetime events and the order relation $\preceq$ represents the causal relationship between events (see causal structure for the analogous idea in a Lorentzian manifold).

Although this definition uses the reflexive convention we could have chosen the irreflexive convention in which the order relation is irreflexive. The causal relation of a Lorentzian manifold (without closed causal curves) satisfies the first three conditions. It is the local finiteness condition that introduces spacetime discreteness.

## Comparison to the continuum

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 criterion is needed however before the embedding is suitable. If, on average, 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 that 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 that are 'manifold-like'. Given a causal set this is a difficult property to determine.

### 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 that (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 $\rho$ 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.

## 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]

### Geodesics

A plot of geodesics between two points in a 180 point causal set made by sprinkling into 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 links 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

1. $x_0 = x\,\!$ and $x_n = y\,\!$
2. 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.

### 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 $k$-length chains 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.

## Dynamics

An ongoing task is to develop the correct dynamics for causal sets. These would provide a set of rules that determine which causal sets correspond to physically realistic spacetimes. The most popular approach to developing causal set dynamics is based on the sum-over-histories version of quantum mechanics. This approach would perform a "sum-over-causal sets" by growing a causal set one element at a time. Elements would be added according to quantum mechanical rules and interference would ensure a large manifold-like spacetime would dominate the contributions. The best model for dynamics at the moment is a classical model in which elements are added according to probabilities. This model, due to David Rideout and Rafael Sorkin, is known as classical sequential growth (CSG) dynamics.[5] The classical sequential growth model is a way to generate causal sets by adding new elements one after another. Rules for how new elements are added are specified and, depending on the parameters in the model, different causal sets result.

## References

1. ^ 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
2. ^ G, Brightwell, R. Gregory, Structure of random discrete spacetime, Phys. Rev. Lett. 66, 260 - 263 (1991)
3. ^ J. Myrheim, CERN preprint TH-2538 (1978)
4. ^ D.D. Reid, Manifold dimension of a causal set: Tests in conformally flat spacetimes, Phys.Rev. D67 (2003) 024034, arXiv:gr-qc/0207103v2
5. ^ D.P. Rideout, R.D. Sorkin; A classical sequential growth dynamics for causal sets, Phys. Rev D, 6, 024002 (2000) arXiv:gr-qc/9904062

Introduction and reviews
Foundations
• L. Bombelli, J. Lee, D. Meyer, R.D. Sorkin, Spacetime as a causal set, Phys. Rev. Lett. 59:521-524 (1987) ; (Introduction, Foundations)
• C. Moore, Comment on "Space-time as a causal set", Phys. Rev. Lett. 60, 655 (1988); (Foundations)
• L. Bombelli, J. Lee, D. Meyer, R.D. Sorkin, Bombelli et al. Reply, Phys. Rev. Lett. 60, 656 (1988); (Foundations)
• A. Einstein, Letter to H.S. Joachim, August 14, 1954; Item 13-453 cited in J. Stachel,“Einstein and the Quantum: Fifty Years of Struggle”, in From Quarks to Quasars,Philosophical Problems of Modern Physics, edited by R.G. Colodny (U. Pittsburgh Press, 1986), pages 380-381; (Historical)
• D. Finkelstein, Space-time code, Phys. Rev. 184:1261 (1969); (Foundations)
• D. Finkelstein, "Superconducting" Causal Nets, Int. J. Th. Phys 27:473(1988); (Foundations)
• G. Hemion, A quantum theory of space and time; Found. Phys. 10 (1980), p. 819 (Similar proposal)
• J. Myrheim, Statistical geometry, CERN preprint TH-2538 (1978); (Foundations, Historical)
• B. Riemann, Uber die Hypothesen, welche der Geometrie zu Grunde liegen, The Collected Works of B. Riemann (Dover NY 1953); ; (Historical)
• R.D. Sorkin; A Finitary Substitute for Continuous Topology, Int. J. Theor. Phys. 30 7: 923-947 (1991); (Foundational)
• R.D. Sorkin, Does a Discrete Order underly Spacetime and its Metric?, Proceedings of the Third Canadian Conference on General Relativity and Relativistic Astrophysics, (Victoria, Canada, May, 1989), edited by A. Coley, F. Cooperstock, B.Tupper, pp. 82–86, (World Scientific, 1990); (Introduction)
• R.D. Sorkin, First Steps with Causal Sets, General Relativity and Gravitational Physics, (Proceedings of the Ninth Italian Conference of the same name, held Capri, Italy, September, 1990), 68-90, (World Scientific, Singapore), (1991), R. Cianci, R. de Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro (eds.); (Introduction)
• R.D. Sorkin, Spacetime and Causal Sets, Relativity and Gravitation: Classical and Quantum, (Proceedings of the SILARG VII Conference, held Cocoyoc, Mexico, December, 1990), pages 150-173, (World Scientific, Singapore, 1991), J.C. D’Olivo, E. Nahmad-Achar, M.Rosenbaum, M.P. Ryan, L.F. Urrutia and F. Zertuche (eds.); (Introduction)
• R.D. Sorkin, Forks in the Road, on the Way to Quantum Gravity, Talk given at the conference entitled “Directions in General Relativity”, held at College Park, Maryland, May, 1993, Int. J. Th. Phys. 36: 2759–2781 (1997); arXiv:gr-qc/9706002; (Philosophical, Introduction)
• G.'t Hooft, Quantum gravity: a fundamental problem and some radical ideas, Recent Developments in Gravitation (Proceedings of the 1978 Cargese Summer Institute) edited by M. Levy and S. Deser (Plenum, 1979); (Introduction, Foundations, Historical)
• E.C. Zeeman, Causality Implies the Lorentz Group, J. Math. Phys. 5: 490-493; (Historical, Foundations)
PhD theses
Talks
Manifoldness
• L. Bombelli, D.A. Meyer; The origin of Lorentzian geometry; Phys. Lett. A 141:226-228 (1989); (Manifoldness)
• L. Bombelli, R.D. Sorkin, When are Two Lorentzian Metrics close?, General Relativity and Gravitation, proceedings of the 12th International Conference on General Relativity and Gravitation, held July 2–8, 1989, in Boulder, Colorado, USA, under the auspices of the International Society on General Relativity and Gravitation, 1989, p. 220; (Closeness of Lorentzian manifolds)
• L. Bombelli, Causal sets and the closeness of Lorentzian manifolds, Relativity in General: proceedings of the Relativity Meeting "93, held September 7–10, 1993, in Salas, Asturias, Spain. Edited by J. Diaz Alonso, M. Lorente Paramo. ISBN 2-86332-168-4. Published by Editions Frontieres, 91192 Gif-sur-Yvette Cedex, France, 1994, p. 249; (Closeness of Lorentzian manifolds)
• L. Bombelli, Statistical Lorentzian geometry and the closeness of Lorentzian manifolds, J. Math. Phys.41:6944-6958 (2000); arXiv:gr-qc/0002053 (Closeness of Lorentzian manifolds, Manifoldness)
• A.R. Daughton, An investigation of the symmetric case of when causal sets can embed into manifolds, Class. Quant. Grav.15(11):3427-3434 (Nov,1998); (Manifoldness)
• J. Henson, Constructing an interval of Minkowski space from a causal set, Class.Quant.Grav. 23 (2006) L29-L35; arXiv:gr-qc/0601069; (Continuum limit, Sprinkling)
• S. Major, D.P. Rideout, S. Surya, On Recovering Continuum Topology from a Causal Set, J.Math.Phys.48:032501,2007; arXiv:gr-qc/0604124 (Continuum Topology)
• S. Major, D.P. Rideout, S. Surya; Spatial Hypersurfaces in Causal Set Cosmology; Class.Quant.Grav. 23 (2006) 4743-4752; arXiv:gr-qc/0506133v2; (Observables, Continuum topology)
• S. Major, D.P. Rideout, S. Surya, Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory, arXiv:0902.0434 (Continuum topology and homology)
• D.A. Meyer, The Dimension of Causal Sets I: Minkowski dimension, Syracuse University preprint (1988); (Dimension theory)
• D.A. Meyer, The Dimension of Causal Sets II: Hausdorff dimension, Syracuse University preprint (1988); (Dimension theory)
• D.A. Meyer, Spherical containment and the Minkowski dimension of partial orders, Order 10: 227-237 (1993); (Dimension theory)
• J. Noldus, A new topology on the space of Lorentzian metrics on a fixed manifold, Class. Quant. Grav 19: 6075-6107 (2002); (Closeness of Lorentzian manifolds)
• J. Noldus, A Lorentzian Gromov–Hausdorff notion of distance, Class. Quant. Grav. 21, 839-850, (2004); (Closeness of Lorentzian manifolds)
• D.D. Reid, Manifold dimension of a causal set: Tests in conformally flat spacetimes, Phys.Rev. D67 (2003) 024034; arXiv:gr-qc/0207103v2 (Dimension theory)
• S. Surya, Causal Set Topology; arXiv:0712.1648
Geometry
Cosmological constant prediction
• M. Ahmed, S. Dodelson, P.B. Greene, R.D. Sorkin, Everpresent lambda; Phys. Rev. D69, 103523, (2004) arXiv:astro-ph/0209274v1 ; (Cosmological Constant)
• Y. Jack Ng and H. van Dam, A small but nonzero cosmological constant; Int. J. Mod. Phys D. 10 : 49 (2001) arXiv:hep-th/9911102v3; (PreObservation Cosmological Constant)
• Y. Kuznetsov, On cosmological constant in Causal Set theory; arXiv:0706.0041
• R.D. Sorkin, A Modified Sum-Over-Histories for Gravity; reported in Highlights in gravitation and cosmology: Proceedings of the International Conference on Gravitation and Cosmology, Goa, India, 14–19 December 1987, edited by B. R. Iyer, Ajit Kembhavi, Jayant V. Narlikar, and C. V. Vishveshwara, see pages 184-186 in the article by D. Brill and L. Smolin: “Workshop on quantum gravity and new directions”, pp 183–191 (Cambridge University Press, Cambridge, 1988); (PreObservation Cosmological Constant)
• R.D. Sorkin; On the Role of Time in the Sum-over-histories Framework for Gravity, paper presented to the conference on The History of Modern Gauge Theories, held Logan, Utah, July 1987; Int. J. Theor. Phys. 33 : 523-534 (1994); (PreObservation Cosmological Constant)
• R.D. Sorkin, First Steps with Causal Sets, in R. Cianci, R. de Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro (eds.), General Relativity and Gravitational Physics (Proceedings of the Ninth Italian Conference of the same name, held Capri, Italy, September, 1990), pp. 68–90 (World Scientific, Singapore, 1991); (PreObservation Cosmological Constant)
• R.D. Sorkin; Forks in the Road, on the Way to Quantum Gravity, talk given at the conference entitled “Directions in General Relativity”, held at College Park, Maryland, May, 1993; Int. J. Th. Phys. 36 : 2759–2781 (1997) arXiv:gr-qc/9706002 ; (PreObservation Cosmological Constant)
• R.D. Sorkin, Discrete Gravity; a series of lectures to the First Workshop on Mathematical Physics and Gravitation, held Oaxtepec, Mexico, Dec. 1995 (unpublished); (PreObservation Cosmological Constant)
• R.D. Sorkin, Big extra dimensions make Lambda too small; arXiv:gr-qc/0503057v1; (Cosmological Constant)
• R.D. Sorkin, Is the cosmological "constant" a nonlocal quantum residue of discreteness of the causal set type?; Proceedings of the PASCOS-07 Conference, July 2007, Imperial College London; arXiv:0710.1675; (Cosmological Constant)
• J. Zuntz, The CMB in a Causal Set Universe, arXiv:0711.2904 (CMB)
Lorentz and Poincaré invariance, phenomenology
• L. Bombelli, J. Henson, R.D. Sorkin; Discreteness without symmetry breaking: a theorem; arXiv:gr-qc/0605006v1; (Lorentz invariance, Sprinkling)
• F. Dowker, J. Henson, R.D. Sorkin, Quantum gravity phenomenology, Lorentz invariance and discreteness; Mod. Phys. Lett. A19, 1829–1840, (2004) arXiv:gr-qc/0311055v3; (Lorentz invariance, Phenomenology, Swerves)
• F. Dowker, J. Henson, R.D. Sorkin, Discreteness and the transmission of light from distant sources; arXiv:1009.3058 (Coherence of light, Phenomenology)
• J. Henson, Macroscopic observables and Lorentz violation in discrete quantum gravity; arXiv:gr-qc/0604040v1; (Lorentz invariance, Phenomenology)
• N. Kaloper, D. Mattingly, Low energy bounds on Poincaré violation in causal set theory; Phys. Rev. D 74, 106001 (2006) arXiv:astro-ph/0607485 (Poincaré invariance, Phenomenology)
• D. Mattingly, Causal sets and conservation laws in tests of Lorentz symmetry; Phys. Rev. D 77, 125021 (2008) arXiv:0709.0539 (Lorentz invariance, Phenomenology)
• L. Philpott, F. Dowker, R.D. Sorkin, Energy-momentum diffusion from spacetime discreteness; arXiv:0810.5591 (Phenomenology, Swerves)
Black hole entropy in causal set theory
• D. Dou, Black Hole Entropy as Causal Links; Fnd. of Phys, 33 2:279-296(18) (2003); arXiv:gr-qc/0302009v1 (Black hole entropy)
• D.P. Rideout, S. Zohren, Counting entropy in causal set quantum gravity ; arXiv:gr-qc/0612074v1; (Black hole entropy)
• D.P. Rideout, S. Zohren, Evidence for an entropy bound from fundamentally discrete gravity; Class.Quant.Grav. 23 (2006) 6195-6213; arXiv:gr-qc/0606065v2 (Black hole entropy)
Locality and quantum field theory
Causal set dynamics
• M. Ahmed, D. Rideout, Indications of de Sitter Spacetime from Classical Sequential Growth Dynamics of Causal Sets; arXiv:0909.4771
• A.Ash, P. McDonald, Moment Problems and the Causal Set Approach to Quantum Gravity; J.Math.Phys. 44 (2003) 1666-1678; arXiv:gr-qc/0209020
• A.Ash, P. McDonald, Random partial orders, posts, and the causal set approach to discrete quantum gravity; J.Math.Phys. 46 (2005) 062502 (Analysis of number of posts in growth processes)
• D.M.T. Benincasa, F. Dowker, The Scalar Curvature of a Causal Set; arXiv:1001.2725; (Scalar curvature, actions)
• G. Brightwell; M. Luczak; Order-invariant Measures on Causal Sets; arXiv:0901.0240; (Measures on causal sets)
• G. Brightwell; M. Luczak; Order-invariant Measures on Fixed Causal Sets; arXiv:0901.0242; (Measures on causal sets)
• G. Brightwell, H.F. Dowker, R.S. Garcia, J. Henson, R.D. Sorkin; General covariance and the "problem of time" in a discrete cosmology; In ed. K. Bowden, Correlations:Proceedings of the ANPA 23 conference, August 16–21, 2001, Cambridge, England, pp. 1–17. Alternative Natural Philosophy Association, (2002).;arXiv:gr-qc/0202097; (Cosmology, Dynamics, Observables)
• G. Brightwell, H.F. Dowker, R.S. Garcia, J. Henson, R.D. Sorkin; "Observables" in causal set cosmology; Phys. Rev. D67, 084031, (2003); arXiv:gr-qc/0210061; (Cosmology, Dynamics, Observables)
• G. Brightwell, J. Henson, S. Surya; A 2D model of Causal Set Quantum Gravity: The emergence of the continuum; arXiv:0706.0375; (Quantum Dynamics, Toy Model)
• G.Brightwell, N. Georgiou; Continuum limits for classical sequential growth models University of Bristol preprint. (Dynamics)
• A. Criscuolo, H. Waelbroeck; Causal Set Dynamics: A Toy Model; Class. Quant. Grav.16:1817-1832 (1999); arXiv:gr-qc/9811088; (Quantum Dynamics, Toy Model)
• F. Dowker, S. Surya; Observables in extended percolation models of causal set cosmology;Class. Quant. Grav. 23, 1381-1390 (2006); arXiv:gr-qc/0504069v1; (Cosmology, Dynamics, Observables)
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• A.L. Krugly; Causal Set Dynamics and Elementary Particles; Int. J. Theo. Phys 41 1:1-37(2004);; (Quantum Dynamics)
• X. Martin, D. O'Connor, D.P. Rideout, R.D. Sorkin; On the “renormalization” transformations induced by cycles of expansion and contraction in causal set cosmology; Phys. Rev. D 63, 084026 (2001); arXiv:gr-qc/0009063 (Cosmology, Dynamics)
• D.A. Meyer; Spacetime Ising models; (UCSD preprint May 1993); (Quantum Dynamics)
• D.A. Meyer; Why do clocks tick?; General Relativity and Gravitation 25 9:893-900;; (Quantum Dynamics)
• I. Raptis; Quantum Space-Time as a Quantum Causal Set, arXiv:gr-qc/0201004v8
• D.P. Rideout, R.D. Sorkin; A classical sequential growth dynamics for causal sets, Phys. Rev D, 6, 024002 (2000);arXiv:gr-qc/9904062 (Cosmology, Dynamics)
• D.P. Rideout, R.D. Sorkin; Evidence for a continuum limit in causal set dynamics Phys.Rev.D63:104011,2001; arXiv:gr-qc/0003117(Cosmology, Dynamics)
• R.D. Sorkin; Indications of causal set cosmology; Int. J. Theor. Ph. 39(7):1731-1736 (2000); arXiv:gr-qc/0003043; (Cosmology, Dynamics)
• R.D. Sorkin; Relativity theory does not imply that the future already exists: a counterexample; Relativity and the Dimensionality of the World, Vesselin Petkov (ed.) (Springer 2007, in press); arXiv:gr-qc/0703098v1; (Dynamics, Philosophy)
• M. Varadarajan, D.P. Rideout; A general solution for classical sequential growth dynamics of Causal Sets; Phys.Rev. D73 (2006) 104021; arXiv:gr-qc/0504066v3; (Cosmology, Dynamics)