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The Information Geometry of Space-Time^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Information Geometry of Blurred Space

## 3. Space-Time and the Geometrodynamics of Pure Gravity

## 4. Discussion

**Dimensionless Distance?—**As with any information geometry the distance $d\mathsf{\ell}$ given in Equations (1) and (2) turns out to be dimensionless. The interpretation [23] is that an information distance is measured distances in units of the local uncertainty—the blur. To make this explicit we write the distribution (14) that describes a blurred point in ${\mathrm{RNC}}_{P}$ in the form

**Minimum Length—**To explore the geometry of blurred space it helps to distinguish the abstract “mathematical” points that are sharply defined by the coordinates x from the more “physical” blurred points. We shall call them c-points and b-points respectively. In ${\mathrm{RNC}}_{P}$ the distance between two c-points located at x and at $x+\Delta x$ is given by (27). To find the corresponding distance $\Delta \lambda $ between two b-points located at x and at $x+\Delta x$ we recall that when we say a test particle is at x it is actually located at ${x}^{\prime}=x+y$ so that

**Blur Dilation—**The size of the blur of space is a length but it does not behave as the length of a rod. When referred to a moving frame it does not undergo a Lorentz contraction. It is more analogous to time dilation: just as a clock marks time by ticking along the time axis, so are lengths measured by ticking ${\mathsf{\ell}}_{0}$s along them. By the principle of relativity all inertial observers measure the same blur in their own rest frames — the proper blur ${\mathsf{\ell}}_{0}$. Relative to another inertial frame the blur is dilated to $\gamma {\mathsf{\ell}}_{0}$ where $\gamma $ is the usual relativistic factor. This implies the proper blur ${\mathsf{\ell}}_{0}$ is indeed the minimum attainable.

**The Volume of a Blurred Point: Is Space Continuous or Discrete?—**A b-point is smeared over the whole of space but we can still define a useful measure of its volume by adding all volume elements ${g}^{1/2}\left({x}^{\prime}\right){d}^{3}{x}^{\prime}$ weighed by the scalar density $p\left({x}^{\prime}\right|x)/{g}^{1/2}\left({x}^{\prime}\right)$. Therefore in ${\mathsf{\ell}}_{0}$ units a blurred point has unit volume. This means that we can measure the volume of a finite region of space by counting the number of b-points it contains. It also means that the number of distinguishable b-points within a region of finite volume is finite which is a property one would normally associate to discrete spaces. In this sense blurred space is both continuous and discrete. (See also [26].)

**The Entropy of Space—**The statistical state of blurred space is the joint distribution of all the ${y}_{x}$ variables associated to every b-point x. We assume that the ${y}_{x}$ variables at different xs are independent, and therefore their joint distribution is a product,

**Canonical Quantization of Gravity?—**The picture of space as a smooth blurred statistical manifold stands in sharp contrast to ideas inspired from various models of quantized gravity in which the short distance structure of space is dominated by extreme fluctuations. From our perspective it is not surprising that attempts to quantize gravity by imposing commutation relations on the metric tensor ${g}_{ab}$ have not been successful. The information geometry approach suggests a reason why: quantizing the Lagrange multipliers ${g}_{ab}={\gamma}_{ab}$ would be just as misguided as formulating a quantum theory of fluids by imposing commutation relations on those Lagrange multipliers like temperature, pressure, or chemical potential, that define the thermodynamic macrostate.

**Physical Consequences of a Minimum Length?—**A minimum length will eliminate the short wavelength divergences in QFT. This in turn will most likely illuminate our understanding of the cosmological constant and affect the scale dependence of running coupling constants. One also expects that QFT effects that are mediated by short wavelength excitations should be suppressed. For example, the lifetime of the proton ought to be longer than predicted by grand-unified theories formulated in Minkowski space-time. The nonlocality implicit in a minimum length might lead to possible violations of CPT symmetry with new insights into matter-antimatter asymmetry. Of particular interest would be the early universe cosmology where inflation might amplify minimum-length effects possibly making them observable.

## Acknowledgments

## References and Notes

- For an introduction to the extensive literature on canonical quantization of gravity, loop quantum gravity, string theory, and causal sets see e.g., [2,3]
- Kiefer, C. Quantum Gravity; Oxford U.P.: Oxford, UK, 2007. [Google Scholar]
- Ashtekar, A.; Berger, B.; Isenberg, J.; MacCallum, M. (Eds.) General Relativity and Gravitation; Cambridge U.P.: Cambridge, UK, 2015. [Google Scholar]
- Caticha, A. The Entropic Dynamics approach to Quantum Mechanics. Entropy
**2019**, 21, 943, doi:10.3390/e21100943, arXiv**2019**arXiv:1908.04693. [Google Scholar] [CrossRef] - Ipek, S.; Abedi, M.; Caticha, A. Entropic Dynamics: Reconstructing Quantum Field Theory in Curved Spacetime. Class. Quantum Grav.
**2019**, 36, 205013, arXiv**2018**, arXiv:1803.07493. [Google Scholar] [CrossRef] - Ipek, S.; Caticha, A. An Entropic Dynamics approach to Geometrodynamics. arXiv
**2019**, arXiv:1910.01188. [Google Scholar] - Caticha, A. Entropic Inference and the Foundations of Physics; International Society for Bayesian Analysis-ISBrA: Sao Paulo, Brazil, 2012; Available online: http://www.albany.edu/physics/ACaticha-EIFP-book.pdf (accessed on 20 September 2019).
- The subject of information geometry was introduced in statistics by Fisher [9] and Rao [10] with important later contributions by other authors [11,12,13,14]. Important aspects were also independently discovered in thermodynamics [15,16]
- Fisher, R.A. Theory of statistical estimation. Math. Proc. Camb. Philos. Soc.
**1925**, 22, 700–725. [Google Scholar] [CrossRef] - Rao, C.R. Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc.
**1945**, 37, 81. [Google Scholar] - Amari, S. Differential-Geometrical Methods in Statistics; Springer: Berlin, Germany, 1985. [Google Scholar]
- Čencov, N.N. Statistical Decision Rules and Optimal Inference; American Mathematical Soc.: Providence, RI, USA, 1981; Volume 53. [Google Scholar]
- Rodríguez, C.C. The metrics generated by the Kullback number. In Maximum Entropy and Bayesian Methods; Skilling, J., Ed.; Kluwer: Dordrecht, The Netherlands, 1989. [Google Scholar]
- Ay, N.; Jost, J.; Vân Lê, H.; Schwanchhöfer, L. Information Geometry; Springer: Berlin, Germany, 2017. [Google Scholar]
- Weinhold, F. Metric geometry of equilibrium thermodynamics. J. Chem. Phys.
**1975**, 63, 2479. [Google Scholar] [CrossRef] - Ruppeiner, G. Thermodynamics: A Riemannian geometric model. Phys. Rev. A
**1979**, 20, 1608. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black holes and entropy. Phys. Rev. D
**1973**, 7, 2333. [Google Scholar] [CrossRef] - Hawking, S. Black Holes and Thermodynamics. Phys. Rev. D
**1976**, 13, 191. [Google Scholar] [CrossRef] - Jacobson, T. Thermodynamics of space-time: the Einstein equation of state. Phys. Rev. Lett.
**1995**, 75, 1260. [Google Scholar] [CrossRef] [PubMed] - Padmanabhan, T. Thermodynamical aspects of gravity: New insights. Rep. Prog. Phys.
**2010**, 73, 046901. [Google Scholar] [CrossRef] - Verlinde, E.P. On the origin of gravity and the laws of Newton. J. High Energy Phys.
**2011**, 2011, 29. [Google Scholar] [CrossRef] - Jacobson, T. Entanglement equilibrium and the Einstein equation. Phys. Rev. Lett.
**2016**, 116, 201101. [Google Scholar] [CrossRef] [PubMed] - Caticha, A. Geometry from Information Geometry. In Bayesian Inference and Maximum Entropy Methods in Science and Engineering; Giffin, A., Knuth, K., Eds.; AIP American Institute of Physics: College Park, MD, USA, 2016; Volume 1757, p. 030001, arXiv
**2015**, arXiv:1512.09076. [Google Scholar] - Ingraham, R.L. Stochastic Space-time. Nuovo Cimento
**1964**, 34, 182. [Google Scholar] [CrossRef] - It is possible that there is some connection with ideas proposed by Kempf [26] expressed in the language of spectral geometry. This is a topic for future research
- Kempf, A. Information-theoretic natural ultraviolet cutoff for spacetime. Phys. Rev. Lett.
**2009**, 103, 231301. [Google Scholar] [CrossRef] [PubMed] - Hojman, S.A.; Kuchar̆, K.; Teitelboim, C. Geometrodynamics Regained. Ann. Phys.
**1976**, 96, 88. [Google Scholar] [CrossRef] - Brodie, D.J.; Hughston, L.P. Statistical Geometry in Quantum Mechanics. Proc. R. Soc. Lond. Ser. A
**1998**, 454, 2445–2475. [Google Scholar] [CrossRef] - Teitelboim, C. How Commutators of Constraints Reflect the Spacetime Structure. Ann. Phys.
**1973**, 79, 542. [Google Scholar] [CrossRef] - Kuchař, K. Canonical Quantization of Gravity. In Relativity, Astrophysics, and Cosmology; Israel, W., Ed.; Reidel: Dordrecht, The Netherlands, 1973; pp. 237–288. [Google Scholar]
- The quotes in “group” and “algebra” are a reminder that the set of deformations do not form a group. The composition of two successive deformations is itself a deformation but it depends on the surface to which the first deformation is applied

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Caticha, A.
The Information Geometry of Space-Time. *Proceedings* **2019**, *33*, 15.
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Caticha A.
The Information Geometry of Space-Time. *Proceedings*. 2019; 33(1):15.
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Caticha, Ariel.
2019. "The Information Geometry of Space-Time" *Proceedings* 33, no. 1: 15.
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