Out of Nowhere: Spacetime from causality: causal set theory
2020
Abstract
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wuthrich and under contract with Oxford University Press. (More information at www.beyondspacetime.net.) This chapter introduces causal set theory and identifies and articulates a 'problem of space' in this theory.
References (57)
- Victor Ambarzumian and Dmitri Iwanenko. Zur Frage nach Vermeidung der unendlichen Selbst- rückwirkung des Elektrons. Zeitschrift für Physik, 64(7-8):563-567, 1930.
- Luca Bombelli. Space-time as causal set. PhD thesis, Syracuse University, 1987.
- Luca Bombelli, Joohan Lee, David Meyer, and Rafael Sorkin. Spacetime as a causal set. Physical Review Letters, 59:521-524, 1987.
- Graham Brightwell. Partial orders. In Lowell W Beineke and Robin J Wilson, editors, Graph Connections: Relationships between Graph Theory and other Areas of Mathematics, pages 52-69. Clarendon Press, Oxford, 1997.
- Graham Brightwell and Ruth Gregory. Structure of random discrete spacetime. Physics Review Letters, 66:260-263, 1991.
- Graham Brightwell and Douglas B West. Partially ordered sets. In Kenneth H Rosen, edi- tor, Handbook of Discrete and Combinatorial Mathematics, pages 717-752. CRC Press, Boca Raton, FL, 2000.
- Fay Dowker. Being and becoming on the road to quantum gravity: or, the birth of a baby is not a baby. In Nick Huggett, Keizo Matsubara, and Christian Wüthrich, editors, Beyond Spacetime: The Foundations of Quantum Gravity, pages 133-142. Cambridge University Press, Cambridge, 2020.
- Benjamin F Dribus. On the axioms of causal set theory. Manuscript, 2013. URL http://arxiv. org/abs/1311.2148.
- Benjamin F Dribus. Discrete Causal Set Theory: Emergent Spacetime and the Causal Metric Hypothesis. Springer, Cham, 2017.
- Ben Dushnik and E W Miller. Partially ordered sets. American Journal of Mathematics, 63: 600-610, 1941.
- John Earman. Notes on the causal theory of time. Synthese, 24:74-86, 1972.
- Philipp Ehrlich. Are points (necessarily) unextended? Manuscript, 2015.
- Stefan Felsner, Peter C Fishburn, and William T Trotter. Finite three dimensional partial orders which are not sphere orders. Discrete Mathematics, 201:101-132, 1999.
- Stefan Felsner, Irina Mustatȃ, and Martin Pergel. The complexity of the partial order dimension problem-closing the gap. Manuscript, 2014.
- David Finkelstein. The space-time code. Physical Review, 184:1261-1271, 1969.
- Adolf Grünbaum. Philosophical Problems of Space and Time. Alfred A. Knopf, New York, 1963.
- Adolf Grünbaum. Modern Science and Zeno's Paradoxes. George Allen and Unwin, London, 1967.
- Stephen W Hawking and George F R Ellis. The Large Scale Structure of Space-time. Cambridge University Press, Cambridge, 1979.
- Stephen W Hawking, A R King, and P J McCarthy. A new topology for curved space-time which incorporates the causal, differential, and conformal Journal of Mathematical Physics, 17:174-181, 1976.
- Joe Henson. The causal set approach to quantum gravity. In Daniele Oriti, editor, Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, pages 393-413. Cambridge University Press, Cambridge, 2009.
- David Hilbert and Nick Huggett. Groups in mind. Philosophy of Science, 73:765-777, 2006.
- Toshio Hiraguchi. On the dimension of partially ordered sets. Science Reports of Kanazawa University, 1:77-94, 1951.
- Nick Huggett, Tiziana Vistarini, and Christian Wüthrich. Time in quantum gravity. In Adrian Bardon and Heather Dyke, editors, The Blackwell Companion to the Philosophy of Time, pages 242-261. Blackwell, Chichester, 2013.
- Raluca Ilie, Gregory B Thompson, and David D Reid. A numerical study of the correspondence between paths in a causal set and geodesics in the continuum. Classical and Quantum Gravity, 23:3275-3285, 2006.
- E. H. Kronheimer and Roger Penrose. On the structure of causal spaces. Mathematical Proceed- ings of the Cambridge Philosophical Society, 63:481-501, 1967.
- Koen Lefever. A century of axiomatic systems for ordinal approaches to special relativity theory. Manuscript, 2013.
- Seth Major, David Rideout, and Sumati Surya. Spatial hypersurfaces in causal set cosmology. Classical and Quantum Gravity, 23:4743-4752, 2006.
- Seth Major, David Rideout, and Sumati Surya. On recovering continuum topology from a causal set. Journal of Mathematical Physics, 48:032501, 2007.
- David B Malament. The class of continuous timelike curves determines the topology of spacetime. Journal of Mathematical Physics, 18:1399-1404, 1977.
- Henryk Mehlberg. Essai sur la théorie causale du temps. Studia Philosophica, 1:119-258, 1935.
- Henryk Mehlberg. Essai sur la théorie causale du temps. Studia Philosophica, 2:111-231, 1937.
- David A Meyer. The Dimension of Causal Sets. PhD thesis, Massachusetts Institute of Tech- nology, 1988.
- David A Meyer. Spherical containment and the minkowski dimension of partial orders. Order, 10:227-237, 1993.
- Jan Myrheim. Statistical geometry. Technical Report TH-2538, CERN preprint, 1978.
- L A Paul and Ned Hall. Causation: A User's Guide. Oxford University Press, Oxford, 2013.
- Henri Poincaré. Science and Hypothesis. Walter Scott Publishing Company, London and Newcastle-on-Tyne, 1905.
- Hans Reichenbach. Axiomatik der relativistischen Raum-Zeit-Lehre. Friedrich Vieweg und Sohn, Braunschweig, 1924.
- Hans Reichenbach. Philosophie der Raum-Zeit-Lehre. Walter de Gruyter, Berlin and Leipzig, 1928.
- Hans Reichenbach. The Direction of Time. University of California Press, Berkeley, 1956.
- David D. Reid. Discrete quantum gravity and causal sets. Canadian Journal of Physics, 79: 1-16, 2001.
- David Rideout and Petros Wallden. Emergence of spatial structure from causal sets. Journal of Physics: Conference Series, 174:012017, 2009.
- Bernhard Riemann. Über die Hypothesen, welche der Geometrie zugrunde liegen. Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Göttingen, pages 133-152, 1868.
- Alfred A Robb. A Theory of Time and Space. Cambridge University Press, Cambridge, 1914.
- Alfred A Robb. The Geometry of Time and Space. Cambridge University Press, Cambridge, 1936.
- Lawrence Sklar. Prospects for a causal theory of space-time. In Richard Swinburne, editor, Space, Time and Causality, pages 45-62. D. Reidel Publishing Company, Dordrecht, 1983.
- Chris Smeenk and Christian Wüthrich. Time travel and time machines. In Craig Callender, editor, The Oxford Handbook of Philosophy of Time, pages 577-630. Oxford University Press, Oxford, 2011.
- Hartland S. Snyder. Quantized space-time. Physical Review, 71:38-41, 1947.
- Rafael Sorkin. A specimen of theory construction from quantum gravity. In Jarrett Leplin, editor, The Creation of Ideas in Physics: Studies for a Methodology of Theory Construction, Univer- sity of Western Ontario Series in Philosophy of Science, pages 167-179. Kluwer Academic Publishers, Dordrecht, 1995.
- Gerard 't Hooft. Quantum gravity: A fundamental problem and some radical ideas. In Maurice Lévy and Stanley Deser, editors, Recent Developments in Gravitation: Cargèse 1978, pages 323-345. Plenum Press, New York and London, 1979.
- Bas C van Fraassen. An Introduction to the Philosophy of Time and Space. Random House, New York, 1970.
- Robert M Wald. General Relativity. The University of Chicago Press, Chicago and London, 1984.
- John A Winnie. The causal theory of space-time. In John Earman, Clark Glymour, and John Stachel, editors, Foundations of Space-Tome Theories, volume 8 of Minnesota Studies in the Philosophy of Science, pages 134-205. University of Minnesota Press, 1977.
- Christian Wüthrich. Challenging the spacetime structuralist. Philosophy of Science, 65:1039- 1051, 2009.
- Christian Wüthrich. The structure of causal sets. Journal for General Philosophy of Science, 43:223-241, 2012.
- Christian Wüthrich. Time travelling in emergent spacetime. In Judit Madarász and Gergely Székely, editors, Hajnal Andréka and István Németi on the Unity of Science: From Computing to Relativity Theory Through Algebraic Logic. Springer, forthcoming.
- Christian Wüthrich and Craig Callender. What becomes of a causal set. British Journal for the Philosophy of Science, 68:907-925, 2017.
- E Christopher Zeeman. Causality implies the Lorentz group. Journal of Mathematical Physics, 5:490-493, 1964.