General solution for classical sequential growth dynamics of causal sets
Open accessAbstract
A classical precursor to a full quantum dynamics for causal sets has been formulated in terms of a stochastic sequential growth process in which the elements of the causal set arise in a sort of accretion process. The transition probabilities of the Markov growth process satisfy certain physical requirements of causality and general covariance, and the generic solution with all transition probabilities nonzero has been found. Here we remove the assumption of nonzero probabilities, define a reasonable extension of the physical requirements to cover the case of vanishing probabilities, and find the completely general solution to these physical conditions. The resulting family of growth processes has an interesting structure reminiscent of an ``infinite tower of turtles'' cosmology.
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In CSG models, there is nothing corresponding to a three-dimensional space evolving in time; rather the model is four-dimensional from the beginning but incomplete at any stage of growth.
“On the other hand, there is nothing in the model corresponding to a three-dimensional space 'evolving in time'. Rather, one meets with something which is 'four dimensional' from the very beginning, but which at any stage of its growth is still incomplete .”
From Relativity Theory Does Not Imply that the Future Already Exists: A Counterexample (2007) · cites this paper as [2]
The Introduction (p. 2) describes growth by adding elements to a finite causal set at successive stages, supporting the idea that the growing set is incomplete at each stage. Section II (p. 3) likewise describes a path from the empty causal set through finite causal sets. The paper does not establish the claim that the model is four-dimensional from the beginning or explicitly contrast it with a three-dimensional space evolving in time. (LLM, conf=0.83)
Experimental — accuracy not yet measuredA CSG model manifests 'becoming' to the extent that any mathematical model can, via an unceasing cascade of birth-events that build up the causal set.
“On one hand sequential growth seems to me to manifest 'becoming' to the extent that any mathematical model can. It even provides an objective correlate of our subjective perception of 'time passing' in the unceasing cascade of birth-events that build up the causal set, by 'accretion' as it were.”
From Relativity Theory Does Not Imply that the Future Already Exists: A Counterexample (2007) · cites this paper as [2]
The Introduction (p. 2) describes sequential growth as adding a new element at each stage, and Section II (p. 3) describes a path of such transitions from the empty causal set. These passages support the accretion of elements, but the paper does not say that this manifests ‘becoming’ or correlates with the subjective perception of time passing. (LLM, conf=0.83)
Experimental — accuracy not yet measuredThe classical sequential growth (CSG) models of causal set theory are counterexamples to the contention that relativistic spacetime is incompatible with genuine change or becoming.
“I offer the classical sequential growth models of causal set theory as counterexamples.”
From Relativity Theory Does Not Imply that the Future Already Exists: A Counterexample (2007) · cites this paper as [2]
The development of the CSG models is persuasive evidence that the combination of becoming and four-dimensionality can be heuristically fruitful.
“If, on the other hand, we ask not whether becoming is 'logically consistent' with four-dimensionality, but rather whether the combination of the two notions can be heuristically fruitful, then I think that the development of the CSG models is in fact persuasive evidence that it can.”
From Relativity Theory Does Not Imply that the Future Already Exists: A Counterexample (2007) · cites this paper as [2]
A CSG model describes a stochastic process in which elements of the growing causal set are born one by one, each with a definite subset of already born elements as ancestors, defining a Markov process.
“A CSG model describes a stochastic process in which elements e of the growing causal set C are born one by one, each with a definite subset of the already born elements as its ancestors.”
From Relativity Theory Does Not Imply that the Future Already Exists: A Counterexample (2007) · cites this paper as [2]
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- Authors
- Varadarajan, M., & Rideout, D.
- Affiliations
- Raman Research Institute (IN)
- Imperial College London (GB)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2006-05-16
- Volume
- 73
- Issue
- 10
- ISSN
- 0556-2821 (ISSN-L)
- 1550-7998
- 1550-2368
- OpenAlex
- W2058302480
- Cited by
- 40
- References
- 7
- Citations by year
- 2026: 2 · 2024: 5 · 2023: 2 · 2022: 1 · 2021: 1 · 2020: 2 · 2019: 1 · 2016: 2 · 2013: 2 · 2012: 7
- FWCI
- 1.77
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Keywords
- causal sets, stochastic growth processes, Markov processes, transition probabilities, causality, general covariance, quantum gravity, discrete spacetime
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