Causal sets

Causal sets

Discussion

Encyclopedia
The causal sets programme is an approach to quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

. Its founding principle is that spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

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 by David Malament which states that if there is a bijective
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

map between two past and future distinguishing spacetimes which preserves their causal structure
Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.- Introduction :In modern physics spacetime is represented by a Lorentzian manifold...

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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

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 with a partial order relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...

which is
• Reflexive
Reflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...

: For all , we have .
• Antisymmetric
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...

: For all , we have .
• Transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....

: For all , we have implies .
• Locally finite
Locally finite poset
In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements....

: For all , we have card.

Here card() denotes the cardinality of a set . We'll write if and .

The set represents the set of spacetime event
Event (relativity)
In physics, and in particular relativity, an event indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference.Strictly speaking, the...

s and the order relation represents the causal relationship between events (see causal structure
Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.- Introduction :In modern physics spacetime is represented by a Lorentzian manifold...

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 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.

Sprinkling

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
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...

. Thus the probability of sprinkling points into a region of volume is

where 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 .

Geodesics

A link in a causal set is a pair of elements such that but with no such that .

A chain is a sequence of elements such that for . The length of a chain is , the number of relations used.

We can use this to define a geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

between two causal set elements. A geodesic between two elements is a chain consisting only of links such that
1. and
2. The length of the chain, , is maximal over all chains from to .

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

Myrheim 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
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

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 -length chains present in a sprinkling into -dimensional Minkowski spacetime. Counting the number of -length chains in the causal set then allows an estimate for 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 and and counting the number of elements such that (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 -dimensional Minkowski spacetime. Tests in conformally-flat spacetimes 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 spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

s. The most popular approach to developing causal set dynamics is based on the sum-over-histories version of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, is known as classical sequential growth (CSG) dynamics. 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.

Introduction and reviews
• L. Bombelli. Causal Set reference page (Overview)
• L. Bombelli. Causal Sets: Overview and Status, Talk given at Quantum Gravity in the Americas III, August 24–26, 2006; (Introduction, Overview)
• F. Dowker
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, Causal sets and the deep structure of spacetime, arXiv:gr-qc/0508109; (Introduction)
• F. Dowker
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, Causal sets as discrete spacetime, Contemporary Physics, vol. 47, Issue 1, p. 1-9; (Overview, Introduction)
• J. Henson, The causal set approach to quantum gravity, arXiv:gr-qc/0601121; (Introduction, Overview)
• D.D. Reid; Introduction to causal sets: an alternate view of spacetime structure; Canadian Journal of Physics 79, 1-16 (2001); arXiv:gr-qc/9909075; (General);
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; Causal set glossary and bibliography (20 November 2001); (Glossary and bibliography);
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School), In Proceedings of the Valdivia Summer School, edited by A. Gomberoff and D. Marolf; arXiv:gr-qc/0309009; (Introduction, Glossary)

Foundations
• L. Bombelli, J. Lee, D. Meyer, R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, Bombelli et al. Reply, Phys. Rev. Lett. 60, 656 (1988); (Foundations)
• A. Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

, 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
David Finkelstein
David Ritz Finkelstein is currently an emeritus professor of physics at the Georgia Institute of Technology. Finkelstein obtained his Ph.D. in physics at the Massachusetts Institute of Technology in 1953. From 1964 to 1976, he was professor of physics at Yeshiva University.In 1958 Charles W...

, Space-time code, Phys. Rev. 184:1261 (1969); (Foundations)
• D. Finkelstein
David Finkelstein
David Ritz Finkelstein is currently an emeritus professor of physics at the Georgia Institute of Technology. Finkelstein obtained his Ph.D. in physics at the Massachusetts Institute of Technology in 1953. From 1964 to 1976, he was professor of physics at Yeshiva University.In 1958 Charles W...

, "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
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

, Uber die Hypothesen, welche der Geometrie zu Grunde liegen, The Collected Works of B. Riemann (Dover NY 1953); ; (Historical)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; A Finitary Substitute for Continuous Topology, Int. J. Theor. Phys. 30 7: 923-947 (1991); (Foundational)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Gerardus 't Hooft
Gerardus 't Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G...

, 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
Erik Christopher Zeeman
Sir Erik Christopher Zeeman FRS , is a Japanese-born British mathematician known for his work in geometric topology and singularity theory....

, Causality Implies the Lorentz Group, J. Math. Phys. 5: 490-493; (Historical, Foundations)

PhD theses
• L. Bombelli, Space-time as a Causal Set, PhD thesis (Syracuse University
Syracuse University
Syracuse University is a private research university located in Syracuse, New York, United States. Its roots can be traced back to Genesee Wesleyan Seminary, founded by the Methodist Episcopal Church in 1832, which also later founded Genesee College...

, 1987); (Introduction, Kinematics)
• A.R. Daughton; The Recovery of Locality for Causal Sets and Related Topics; PhD thesis (Syracuse University
Syracuse University
Syracuse University is a private research university located in Syracuse, New York, United States. Its roots can be traced back to Genesee Wesleyan Seminary, founded by the Methodist Episcopal Church in 1832, which also later founded Genesee College...

, 1993); (Locality)
• D. Dou, Causal Sets, a Possible Interpretation for the Black Hole Entropy, and Related Topics; PhD thesis (SISSA
Sissa
Sissa is a comune in the Province of Parma in the Italian region Emilia-Romagna, located about 100 km northwest of Bologna and about 20 km northwest of Parma....

, Trieste, 1999); arXiv:gr-qc/0106024 (Black hole entropy)
• S. Johnston, Quantum Fields on Causal Sets, PhD Thesis (Imperial College London
Imperial College London
Imperial College London is a public research university located in London, United Kingdom, specialising in science, engineering, business and medicine...

, 2010) arXiv:1010.5514 (Quantum Field Theory)
• D.A. Meyer, The Dimension of Causal Sets, PhD thesis (M.I.T., 1988); (Dimension theory)
• L. Philpott, Causal Set Phenomenology, PhD Thesis (Imperial College London
Imperial College London
Imperial College London is a public research university located in London, United Kingdom, specialising in science, engineering, business and medicine...

, 2010); arXiv:1009.1593 (Swerves, Phenomenology)
• D.P. Rideout; Dynamics of Causal Sets; PhD Thesis (Syracuse University
Syracuse University
Syracuse University is a private research university located in Syracuse, New York, United States. Its roots can be traced back to Genesee Wesleyan Seminary, founded by the Methodist Episcopal Church in 1832, which also later founded Genesee College...

2001); arXiv:gr-qc/0212064; (Cosmology, Dynamics)
• R.B. Salgado; Toward a Quantum Dynamics for Causal Sets; PhD Thesis (Syracuse University
Syracuse University
Syracuse University is a private research university located in Syracuse, New York, United States. Its roots can be traced back to Genesee Wesleyan Seminary, founded by the Methodist Episcopal Church in 1832, which also later founded Genesee College...

2008); (Scalar field theory, Quantum Measure)
• R. Sverdlov; Quantum Field Theory and Gravity in Causal Sets; PhD Thesis (University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...

2009); arXiv: 0905.2263 (Quantum Field Theory and Gravity)

Talks
• F. Dowker
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, Causal Set Phenomenology; Talk given at Loops 05, 10–14 October 2005, Potsdam, Max Planck Institute for Gravitational Physics
Max Planck Institute for Gravitational Physics
The Max Planck Institute for Gravitational Physics is a Max Planck Institute whose research is aimed at investigating Einstein’s theory of relativity and beyond: Mathematics, quantum gravity, astrophysical relativity, and gravitational wave astronomy...

(Swerves)
• S. Johnston; Particle Propagators from Discrete Spacetime; Talk given at Perimeter Institute 14/04/2008 (Quantum field theory)
• D.A. Meyer; Talk given at the 1997 Santa Fe workshop: Causal Sets and Feynman diagrams; Presented at "New Directions in Simplicial Quantum Gravity" July 28 - August 8, 1997; (Feynman diagrams, Quantum Dynamics)
• D.P. Rideout; Spatial Hypersurfaces in Causal Set Cosmology; Talk given at Loops 05, 10–14 October 2005, Potsdam, Max Planck Institute for Gravitational Physics
Max Planck Institute for Gravitational Physics
The Max Planck Institute for Gravitational Physics is a Max Planck Institute whose research is aimed at investigating Einstein’s theory of relativity and beyond: Mathematics, quantum gravity, astrophysical relativity, and gravitational wave astronomy...

(Spatial hyper-surfaces, Dynamics)
• J. Scargle, Testing Quantum Gravity Theories with GLAST; Talk given at Santa Cruz Institute for Particle Physics, April 24, 2007. (Lorentz invariance, Phenomenology)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; Two Talks given at the 1997 Santa Fe workshop: A Review of the Causal Set Approach to Quantum Gravity and A Growth Dynamics for Causal Sets; Presented at ”New Directions in Simplicial Quantum Gravity” July 28 - August 8, 1997; ;;
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; Does quantum gravity give rise to an observable nonlocality?; Talk given at Perimeter Institute 17/01/2007 (d'Alembertian, Locality)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, Some Insights for Quantum Gravity Derived from Work on Causal Sets; Talk given at Loops 05, 10–14 October 2005, Potsdam, Max Planck Institute for Gravitational Physics
Max Planck Institute for Gravitational Physics
The Max Planck Institute for Gravitational Physics is a Max Planck Institute whose research is aimed at investigating Einstein’s theory of relativity and beyond: Mathematics, quantum gravity, astrophysical relativity, and gravitational wave astronomy...

(Overview)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

Is a past-finite causal order the inner basis of spacetime? Talk given at Perimeter Institute 07/09/2005
• S. Surya, Recovering spacetime topology from a causet; Talk given at Loops 05, 10–14 October 2005, Potsdam, Max Planck Institute for Gravitational Physics
Max Planck Institute for Gravitational Physics
The Max Planck Institute for Gravitational Physics is a Max Planck Institute whose research is aimed at investigating Einstein’s theory of relativity and beyond: Mathematics, quantum gravity, astrophysical relativity, and gravitational wave astronomy...

(Topology)
• R. Sverdlov; Introduction of bosonic fields into causal set theory; Talk given at Perimeter Institute 19/02/2008 (Quantum field theory)

Manifoldness
• L. Bombelli, D.A. Meyer; The origin of Lorentzian geometry; Phys. Lett. A 141:226-228 (1989); (Manifoldness)
• L. Bombelli, R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Order (journal)
Order is a quarterly peer-reviewed academic journal on order theory and its applications, published by Springer Science+Business Media. It was founded in 1984 by University of Calgary mathematics professor Ivan Rival; as of 2010, its editor in chief is Dwight Duffus, the Goodrich C...

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
• E. Bachmat; Discrete spacetime and its applications; arXiv:gr-qc/0702140; (Geodesics, Antichains)
• G. Brightwell, R. Gregory; The Structure of Random Discrete Spacetime; Phys. Rev. Lett. 66:260-263 (1991); (Geodesic Length)
• G. W. Gibbons
Gary Gibbons
Gary William Gibbons , FRS, is a British theoretical physicist. Gibbons studied in Cambridge,where in 1969 he became a research student under the supervision of Dennis Sciama. When Sciama moved to Oxford, he became a student of Stephen Hawking, obtaining his PhD from Cambridge in 1973...

, S. N. Solodukhin; The Geometry of Small Causal Diamonds arXiv:hep-th/0703098 (Causal intervals)
• S.W. Hawking
Stephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...

, A.R. King, P.J. McCarthy; A new topology for curved space–time which incorporates the causal, differential, and conformal structures; J. Math. Phys. 17 2:174-181 (1976); (Geometry, Causal Structure
Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.- Introduction :In modern physics spacetime is represented by a Lorentzian manifold...

)
• S. He, D.P. Rideout; A Causal Set Black Hole; arXiv:0811.4235 (Causal structure of Schwarzschild spacetime, Sprinklings)
• R. Ilie, G.B. Thompson, D.D. Reid; A numerical study of the correspondence between paths in a causal set and geodesics in the continuum; 2006 Class. Quantum Grav. 23 3275-3285 arXiv:gr-qc/0512073(Geodesic length)
• A.V. Levichev; Prescribing the conformal geometry of a lorentz manifold by means of its causal structure; Soviet Math. Dokl. 35:452-455, (1987); (Geometry, Causal Structure
Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.- Introduction :In modern physics spacetime is represented by a Lorentzian manifold...

)
• D. Malament; The class of continuous timelike curves determines the topology of spacetime; J. Math. Phys. 18 7:1399-1404 (1977); (Geometry, Causal Structure
Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.- Introduction :In modern physics spacetime is represented by a Lorentzian manifold...

)
• D.P. Rideout, P. Wallden; Spacelike distance from discrete causal order; arXiv:0810.1768 (Spatial distances)

Cosmological constant prediction
• M. Ahmed, S. Dodelson, P.B. Greene, R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
C. V. Vishveshwara
C. V. Vishveshwara is an Indian scientist, one of the pioneers in black hole physics. Specializing in Einstein's General Relativity, he has worked extensively on the theory of black holes and made major contributions to this field of research since its very beginning.- Academic Training and...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, Big extra dimensions make Lambda too small; arXiv:gr-qc/0503057v1; (Cosmological Constant)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; Discreteness without symmetry breaking: a theorem; arXiv:gr-qc/0605006v1; (Lorentz invariance, Sprinkling)
• F. Dowker
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, J. Henson, R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, Quantum gravity phenomenology, Lorentz invariance and discreteness; Mod. Phys. Lett. A19, 1829–1840, (2004) arXiv:gr-qc/0311055v3; (Lorentz invariance, Phenomenology, Swerves)
• F. Dowker
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, J. Henson, R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

, 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
• G. Hemion, A discrete geometry: speculations on a new framework for classical electrodynamics; Int. J. Theor. Phys. 27 (1988), p. 1145 (Classical electodynamics)
• S. Johnston; Particle propagators on discrete spacetime; 2008 Class. Quantum Grav. 25 202001; arXiv:0806.3083 (Quantum Field Theory)
• S. Johnston; The Feynman propagator for a Free Scalar Field on a Causal Set; Phys. Rev. Lett. 103, 180401 (2009); arXiv:0909.0944 (Quantum Field Theory)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; Does Locality Fail at Intermediate Length-Scales; Towards Quantum Gravity, Daniele Oriti (ed.) (Cambridge University Press, 2007); arXiv:gr-qc/0703099v1; (d'Alembertian, Locality)
• R. Sverdlov, L. Bombelli; Gravity and Matter in Causal Set Theory; arXiv:0801.0240
• R. Sverdlov; A Geometrical Description of Spinor Fields; arXiv:0802.1914
• R. Sverdlov; Bosonic Fields in Causal Set Theory; arXiv:0807.4709
• R. Sverdlov; Gauge Fields in Causal Set Theory; arXiv:0807.2066
• R. Sverdlov; Spinor fields in Causal Set Theory; arXiv:0808.2956

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
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, 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
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, R.S. Garcia, J. Henson, R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; 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
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, R.S. Garcia, J. Henson, R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; "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
University of Bristol
The University of Bristol is a public research university located in Bristol, United Kingdom. One of the so-called "red brick" universities, it received its Royal Charter in 1909, although its predecessor institution, University College, Bristol, had been in existence since 1876.The University is...

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
Fay Dowker
Helen Fay Dowker is a British Theoretical Physicist at Imperial College London.As a student, she was interested in wormholes and quantum cosmology...

, 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)
• M. Droste, Universal homogeneous causal sets, J. Math. Phys. 46, 122503 (2005); arXiv:gr-qc/0510118; (Past-finite causal sets)
• 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; 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
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; Evidence for a continuum limit in causal set dynamics Phys.Rev.D63:104011,2001; arXiv:gr-qc/0003117(Cosmology, Dynamics)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; Indications of causal set cosmology; Int. J. Theor. Ph. 39(7):1731-1736 (2000); arXiv:gr-qc/0003043; (Cosmology, Dynamics)
• R.D. Sorkin
Rafael Sorkin
Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society...

; 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)