Notes: Chapter 15: Digital Physics and Entropic Gravity

Chapter notes for “Chapter 15: Digital Physics and Entropic Gravity”

Notes

¹ John Archibald Wheeler, “Information, Physics, Quantum: The Search for Links,” in Complexity, Entropy, and the Physics of Information (1990). Wheeler’s “It from bit” became a rallying cry for information-theoretic approaches to physics.

² Jacob D. Bekenstein, “Black Holes and Entropy,” Physical Review D 7 (1973): 2333-2346. The Bekenstein bound limits information density in any region of space.

³ Rolf Landauer, “Irreversibility and Heat Generation in the Computing Process,” IBM Journal of Research and Development 5 (1961): 183-191.

^3a Vanchurin, Vitaly, “Towards a theory of quantum gravity from neural networks,” arXiv:2111.00903v3 (2022).

^3b Alexander, Stephon, William J. Cunningham, Jaron Lanier, Lee Smolin, Stefan Stanojevic, Michael W. Toomey, and Dave Wecker, “The Autodidactic Universe,” arXiv:2104.03902 (2021).

⁴ Gerard ’t Hooft, “Dimensional Reduction in Quantum Gravity” (1993); Leonard Susskind, “The World as a Hologram,” Journal of Mathematical Physics 36 (1995): 6377-6396.

⁵ Seth Lloyd, “Computational Capacity of the Universe,” Physical Review Letters 88 (2002): 237901.

⁶ Stephen Wolfram, A New Kind of Science (2002). Wolfram’s computational physics program has been developed further in his Physics Project (wolfram.com/physics).

⁸ Giulio Tononi, “Integrated Information Theory of Consciousness: An Updated Account,” Archives Italiennes de Biologie 150 (2012): 293-329. For a more accessible introduction, see Tononi and Christof Koch, “Consciousness: Here, There and Everywhere?” Philosophical Transactions of the Royal Society B 370 (2015): 20140167.

⁹ Fernando, C. and Sojakka, S., “Pattern Recognition in a Bucket,” in Advances in Artificial Life, ECAL 2003, Lecture Notes in Computer Science, vol. 2801 (2003). Springer. Demonstrated that a bucket of water, stimulated by motors and read by sensors, can perform pattern classification — functioning as a liquid-state machine equivalent to a simple perceptron.

¹⁰ Maldacena, Juan, “The Large N Limit of Superconformal Field Theories and Supergravity,” Advances in Theoretical and Mathematical Physics 2 (1998): 231-252. The foundational paper establishing the Anti-de Sitter/Conformal Field Theory correspondence, showing the exact equivalence between a theory of gravity in a curved higher-dimensional space and a quantum field theory on its boundary — the most rigorously studied realization of the holographic principle.

¹¹ Laplace, Pierre-Simon, A Philosophical Essay on Probabilities (1814). The source of the thought experiment now known as “Laplace’s Demon”: an intelligence that, knowing the position and momentum of every particle, could compute the entire past and future of the universe. Laplace offered it not as a claim about reality but as a statement of deterministic philosophy.

¹² Weberszpil, J. and Sotolongo-Costa, O., “Entropy as a Clock: Foundations and Parametrizations of Emergent Time,” International Journal of Theoretical Physics 64 (2025): 48. Unifies entanglement entropy growth, thermal modular flow, and the Page-Wootters relational time mechanism into a single entropic-time framework.

¹³ Marcucci, G. et al., “Programming Nonlinear Propagation for Efficient Optical Learning Machines,” Science Advances 6, eaay8785 (2020).

¹⁴ Wan, K.Y. and Bhatt, G., “A unicellular walker controlled by a microtubule-based finite state machine,” bioRxiv 2021.02.26.433123 (2021). Demonstrates that Euplotes eurystomus walks using fourteen microtubule-based cirri whose gait transitions decompose into a discrete set of states — embodied computation linking information processing to cell locomotion.

¹⁶ Conway, John and Kochen, Simon, “The Free Will Theorem,” Foundations of Physics 36(10) (2006): 1441–1473. Strengthened in Conway, J. and Kochen, S., “The Strong Free Will Theorem,” Notices of the American Mathematical Society 56(2) (2009): 226–232. Proves that if experimenters’ choices of measurement settings are not determined by prior information (the “free will” assumption), then particles’ responses to those measurements are also not determined by any prior information. The theorem follows from the Kochen-Specker theorem and the violation of Bell inequalities — both experimentally verified.

¹⁷ Jacobson, T. (1995). “Thermodynamics of spacetime: The Einstein equation of state.” Physical Review Letters, 75, 1260–1263. arXiv:gr-qc/9504004. Derives Einstein’s field equations from the proportionality of entropy to the area of local causal horizons, combined with the Clausius relation and the Unruh temperature.

¹⁸ Jacobson, T. (2016). “Entanglement equilibrium and the Einstein equation.” Physical Review Letters, 116, 201101. arXiv:1505.04753. Updates the 1995 derivation using entanglement entropy rather than Clausius entropy, obtaining the same result (Einstein’s equations as an equation of state) from quantum-information-theoretic premises.

¹⁹ Verlinde, E. (2011). “On the origin of gravity and the laws of Newton.” Journal of High Energy Physics, 2011:29. arXiv:1001.0785. Derives Newton’s laws of gravity from holographic screens and entropic forces, arguing that gravity is emergent, arising from changes in information associated with material bodies.

²⁰ Verlinde, E. (2017). “Emergent gravity and the dark universe.” SciPost Physics, 2(3), 016. arXiv:1611.02269. Extends entropic gravity to cosmological scales, interpreting the dark matter phenomenon as the elastic response of the entropy associated with dark energy.

²¹ Zurek, W.H. (2009). “Quantum Darwinism.” Nature Physics, 5, 181–188. arXiv:0903.5082. Proposes that the classical world emerges through environmental selection: quantum states that can proliferate redundant copies of themselves in the environment survive decoherence, while fragile states are transformed. Redundancy is the mechanism of objectivity — multiple observers agree because they access independent fragments of the environment that carry the same information.

²² Pikovski, I., Zych, M., Costa, F. & Brukner, Č. (2015). “Universal decoherence due to gravitational time dilation.” Nature Physics, 11, 668–672. arXiv:1311.1095. Demonstrates that gravitational time dilation universally decoheres composite quantum systems by entangling their center-of-mass position with internal degrees of freedom, producing classicality and the arrow of time without any external environmental interaction.

^22b Fields, Chris and Michael Levin, “Metabolic limits on classical information processing by biological cells,” Biosystems 209: 104513 (2021). doi:10.1016/j.biosystems.2021.104513.

²³ Brouwer, M.M. et al. (2017). “First test of Verlinde’s theory of emergent gravity using weak gravitational lensing measurements.” Monthly Notices of the Royal Astronomical Society, 466(3), 2547–2559. arXiv:1612.03034. Measured lensing profiles around 33,613 isolated central galaxies from the KiDS/GAMA survey overlap. Verlinde’s parameter-free prediction showed good agreement across four stellar mass bins. The authors cautioned this was “only a first step.”

²⁴ Yoon, Y. et al. (2023). “Understanding galaxy rotation curves with Verlinde’s emergent gravity.” Classical and Quantum Gravity, 40(2), 02LT01. arXiv:2206.11685. Analysis of 175 disk galaxies from the SPARC database found good agreement between emergent gravity predictions and observed radial accelerations: a mean logarithmic offset μ[log(g_obs) − log(g_Ver)] = −0.060 ± 0.004 dex (scatter 0.137) using Verlinde’s de Sitter acceleration scale a_0 = cH_0, improving to −0.027 dex (scatter 0.129) with the quasi de Sitter value a_0 = 5.41 × 10^-10 m s^-2. The authors note that a_0 remains an approximation in either case, and that a_0/6 computed with the quasi de Sitter value departs from Milgrom’s MOND constant by 30% (against 10% for the de Sitter value).

²⁵ Tamosiunas, A. et al. (2019). “Testing emergent gravity on galaxy cluster scales.” arXiv:1901.05505. Found that emergent gravity fits to X-ray and weak lensing data are significantly worse than GR+CDM for galaxy clusters, with mass predictions exceeding observations by roughly a factor of two at ~1 Mpc scales.

²⁷ Carney, D., Karydas, M., Scharnhorst, T., Singh, R., and Taylor, J., “Entropic gravity models: exploring microscopic mechanisms for emergent gravitational attraction,” arXiv (2025). Two explicit microscopic models (a qubit lattice and a nonlocal qubit bath) in which gravitational-strength 1/r² attraction between massive objects arises from entropy maximization alone. Carney describes the models as ad hoc proofs of principle rather than realistic candidates. For criticism: Van Raamsdonk, M., quoted in Musser, G., “Is Gravity Just Entropy Rising? Long-Shot Idea Gets Another Look,” Quanta Magazine (June 2025), arguing the models lack distinctive gravitational features (equivalence principle, free-fall phenomenology). For Verlinde’s continued development of the program: Verlinde, E., “On the origin of gravity and the laws of Newton,” Journal of High Energy Physics 2011:029 (2011); Verlinde, E., “Emergent Gravity and the Dark Universe,” SciPost Physics 2 (2017): 016.

²⁶ Oppenheim, J., “A postquantum theory of classical gravity?” Physical Review X 13 (2023): 041040. arXiv:2203.17155. Proposes a consistent framework in which gravity remains classical while coupling stochastically to quantum matter. The theory makes a precise testable trade-off: the diffusion (noise) in the gravitational field is bounded below by the decoherence it induces in quantum systems. See also Oppenheim, J., Sparaciari, C., Šoda, B., and Weller-Davies, Z., “Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity,” Nature Communications 14 (2023): 7910, which derives the experimental signatures.

^26a Feynman, Richard P., “The Role of Gravitation in Physics,” Chapel Hill Conference (1957); reprinted in Feynman Lectures on Gravitation, ed. Morinigo, F.B., Wagner, W.G., and Hatfield, B. (Addison-Wesley, 1995).

^26b Oppenheim, J., Sparaciari, C., Šoda, B., and Weller-Davies, Z., “Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity,” Nature Communications 14 (2023): 7910.

²⁸ Page, D.N. and Wootters, W.K., “Evolution without evolution: Dynamics described by stationary observables,” Physical Review D 27 (1983): 2885–2892. The foundational paper proposing that time emerges from quantum entanglement between subsystems of a static universe.

²⁹ Foti, C., Coppo, A., Barni, G., Cuccoli, A. and Verrucchi, P., “Time and classical equations of motion from quantum entanglement via the Page and Wootters mechanism with generalized coherent states,” Nature Communications 12 (2021): 1787. Derives both the Schrödinger equation and Hamilton’s classical equations of motion purely from entanglement between a system and a quantum clock.

³⁰ Coppo, A., Cuccoli, A. and Verrucchi, P., “A magnetic clock for a harmonic oscillator,” Physical Review A 109 (2024): 052212. Extends the Page-Wootters framework to show that classical phase-space trajectories emerge naturally when the clock is macroscopic.

³¹ Pearson, A.N. et al., “Measuring the Thermodynamic Cost of Timekeeping,” Physical Review X 11 (2021): 021029. Using a silicon nitride membrane as a mesoscopic clock, demonstrated that the entropy cost of timekeeping scales linearly with clock accuracy.

³² Wadhia, V. et al., “Entropic Costs of Extracting Classical Ticks from a Quantum Clock,” Physical Review Letters 135 (2025): 200407. Double quantum dot experiment showing that reading the clock costs up to a billion times more energy than the clock’s internal ticking mechanism.

³⁴ Julian Barbour, Tim Koslowski, and Flavio Mercati, “Identification of a gravitational arrow of time,” Physical Review Letters 113 (2014): 181101. See also Barbour’s The Janus Point: A New Theory of Time (2020) for the full development of the argument.

³⁵ Wolpert, David, Rovelli, Carlo, and Scharnhorst, Stefan, “Disentangling the Boltzmann Brain Hypothesis and the Second Law,” Entropy 27(12) (December 2025): 1227. Published via MDPI; Wolpert affiliated with the Santa Fe Institute. Demonstrates that standard arguments connecting the past hypothesis, the second law, and Boltzmann brains involve subtle circular reasoning: which moment is treated as “fixed” determines whether entropy increases or decreases. The result does not overturn the second law (which remains observationally ironclad) but exposes the logical gap in standard derivations of time’s arrow from initial conditions.

³⁶ Weinberg, S. (1964). “Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass.” Physical Review 135, B1049–B1056. The foundational bootstrap derivation showing that a massless spin-2 particle must couple universally — yielding the equivalence principle and general relativity from self-consistency alone. See also Weinberg, S. (1965). “Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations.” Physical Review 138, B988–B1002, for the extension to graviton self-interactions.

³⁷ Rodina, L. (2015). “Uniqueness from gauge invariance and the Adler zero.” arXiv:1612.06342. Modernized and generalized Weinberg’s bootstrap proof, showing that locality, unitarity, and Lorentz invariance uniquely fix the graviton’s interactions. Rodina quote from Natalie Wolchover, “Why the Laws of Physics Are Inevitable,” Quanta Magazine, December 9, 2019.

³⁸ Baumann, D., quoted in Wolchover (2019). See also Baumann, D. et al. (2020). “The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds.” Journal of High Energy Physics 2020:204. arXiv:1910.14051. Applies bootstrap methods to constrain the physics of the very early universe, demonstrating that symmetry and consistency conditions strongly limit what could have happened in the first moments after the Big Bang.

³⁹ Gisin, N. (2020). “Mathematical languages shape our understanding of time in physics.” Nature Physics 16, 114–116. Argues that intuitionist mathematics, in which real numbers are finite unfolding processes rather than completed infinite objects, naturally expresses the passage of time and the creation of new information. See also Gisin, N. (2019). “Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?” Erkenntnis 86, 1469–1481. arXiv:1803.06824. Gisin quotes from Natalie Wolchover, “Does Time Really Flow? New Clues Come From a Century-Old Approach to Math,” Quanta Magazine, April 7, 2020.

⁴⁰ Del Santo, F. and Gisin, N. (2019). “Physics without determinism: Alternative interpretations of classical physics.” Physical Review A 100, 062107. arXiv:1909.03697. Reformulates classical mechanics using intuitionist mathematics, obtaining the same predictions as standard equations while casting events as genuinely indeterminate — bridging the conceptual gap between classical determinism and quantum randomness.