Entropic Gravity: Emergent Force

• Entropic gravity is a theoretical framework where gravity emerges as a macroscopic effect from entropy gradients on holographic screens, linking thermodynamic and information-theoretic principles.
• It builds on Verlinde's entropy formula and holographic principles by relating temperature, equipartition, and entropy change to derive gravitational dynamics akin to Newtonian gravity.
• The model has profound implications for quantum coherence, dark matter phenomenology, and experimental tests, paving the way for novel studies such as matter-wave interferometry and precision orbital measurements.

Entropic gravity is a theoretical framework proposing that gravity is not a fundamental interaction but an emergent, macroscopic effect resulting from statistical, thermodynamic, and information-theoretic considerations. First articulated in its modern holographic and thermodynamic form by Erik Verlinde, entropic gravity hypothesizes that gravitational dynamics arise from entropy gradients associated with information storage on holographic screens, linking spacetime geometry to underlying microscopic degrees of freedom. This view suggests that gravity may be explained as an entropic force, analogous to emergent forces in thermodynamics, and has consequences for the foundations of gravity, quantum mechanics, cosmology, and experimental tests.

1. Foundational Principles and Formulations

Verlinde’s proposal, and subsequent developments, rest on unifying several frameworks: thermodynamics (especially the characterization of entropy and temperature in terms of microscopic microstates), the holographic principle (whereby maximum entropy scales as the area of bounding surfaces, not volume), and observer-dependent notions of information loss.

The key operational postulates include:

• Gravity emerges from the statistical tendency of systems to maximize entropy, encoded in the relation

$F \Delta x = T \Delta S$

with $F$ the effective entropic force, $T$ the local Unruh temperature, and $\Delta S$ the entropy change associated with displacing a test particle toward (or away from) a holographic screen (Verlinde, 2010, Gao, 2010, Chivukula, 2010).

• The entropy variation associated with a particle of mass $m$ moving by $\Delta x$ near a screen is typically taken to be

$\Delta S = 2\pi k_B \frac{mc}{\hbar} \Delta x$

—the so-called Verlinde entropy formula (Verlinde, 2010, Halyo, 2011).

The holographic screen encodes microscopic information about the bulk, with the number of bits $N$ scaling with the area $A$:

$N = \frac{Ac^3}{G\hbar}$

and entropy given by the Bekenstein–Hawking formula,

$S = \frac{k_B A c^3}{4 G \hbar}$

(Lee, 2010, Fursaev, 2010, Chivukula, 2010, Cao, 2014).

Combining these relations with energy equipartition, the entropic derivation yields Newtonian gravity or modifications thereof. Jacobson's earlier derivation of Einstein's equations from local thermodynamics using the Clausius relation $\delta Q = T \delta S$ is also a conceptual precursor (Carroll et al., 2016).

2. Holographic Screens, Information Loss, and Quantum Entanglement

Holographic screens are codimension-1 surfaces that store information about the region they enclose, acting as repositories for the microstates essential to the entropic description. The entropy associated with these screens is not only a function of area but often reflects the entanglement between degrees of freedom separated by the screen (Lee, 2010, Fursaev, 2010, Chivukula, 2010).

A major development is the identification of screens with local Rindler horizons: causally relevant loci for an accelerating observer. When matter (or quantum fields) cross these horizons, the observer loses access to detailed phase space information, and this loss is quantified as Shannon/Von Neumann entropy (Lee, 2010).

Quantum entanglement is now recognized as central: the gravitational entropy of minimal surfaces (e.g., black hole horizons) is interpreted as entanglement entropy of quantum fields across the surface (Fursaev, 2010, Chivukula, 2010, Carroll et al., 2016). Recent advances use modular Hamiltonians and the so-called “Casini entropy” subtracted from the vacuum, yielding consistent definitions in holographic gravity and allowing for a derivation of the Einstein equations from entanglement equilibrium in small causal diamonds (Carroll et al., 2016).

3. Extensions, Modifications, and Model Tests

Various extensions of entropic gravity have been proposed to address both foundational and phenomenological aspects:

• Modified Entropy–Area Laws: Corrections to the standard area law, such as volumetric terms, have been introduced. For instance,

$$S(A)/k_B = \frac{A}{4l_P^2} + \epsilon \left(\frac{A}{2l_P^2}\right)^{3/2}$$

produce deviations in the Newtonian force law, with extra terms in the gravitational potential (Cuéllar et al., 7 Oct 2025). Modifications also appear for $f(R)$ gravity, introducing an effective gravitational constant $G_{\rm eff} = f'(R) G$ in the entropic derivation (Teimouri, 2017).

• Machian and Information-Theoretic Approaches: The conservation of total entropy (geometric plus matter) is postulated, leading to coupled variational principles and “entropic field equations” (Atanasov, 2017, Bianconi, 26 Aug 2024). The geometric entropy density is defined locally (e.g., $S_G = 1/(L^2 R)$), and field equations relate curvature directly to information content.
• Open Quantum System and Decoherence Models: Critics have argued that entropic gravity as a reservoir should produce excessive decoherence for quantum systems in free fall, inconsistent with experiments. However, master-equation treatments (e.g., Lindblad-type dynamics for the density matrix) show, in the strong coupling limit (large dimensionless $\sigma$), that decoherence can be made negligible; in this regime, models reproduce conservative free-fall dynamics, consistent with precision neutron experiments such as qBounce (σ ≳ 250 at 90% C.L.) (Schimmoller et al., 2020, Sung et al., 2023).
• Dirac Fermions and the Relativistic Regime: The DFEG (Decoherence-Free Entropic Gravity) scheme extends to Dirac spinors with Rindler coordinates and proper spin connection, demonstrating preservation of quantum coherence and equivalence principle, even in the presence of zitterbewegung (Sung et al., 2023).

4. Phenomenological Implications and Constraints

While the entropic framework can reproduce Newtonian gravity at leading order, several concrete predictions and constraints have emerged:

• Solar System and Astrophysical Tests: Modifications to the entropy-area law yield extra perihelion precession in planetary orbits. Empirical data (e.g., Mercury) places extremely stringent bounds on the correction parameter $\epsilon$, with $|\epsilon| \leq 2.8 \times 10^{-58}$—rendering most modifications negligible in the solar system (Cuéllar et al., 7 Oct 2025).
• Dark Matter and Dark Energy: Attempts to explain galactic rotation curves or the cosmological constant via entropic corrections are found to be incompatible with existing solar system and particle-physics constraints on the modification parameters (Cuéllar et al., 7 Oct 2025). Ungravity, another modification driven by scale invariance, also fails to explain observed cosmology when tested against local orbital precession data.
• Quantum Gravity and Collapse: Entropic gravity offers a framework for connecting gravity to quantum state reduction, e.g., via the Verlinde-inspired connection to the Newton–Schrödinger equation and the Diosi–Penrose model. The derived relation between quantum state density and gravitational constant,

$\hat{\rho} \sim \exp(-8\pi^2 G/a),$

links quantum coherence decay rates to the strength of gravity, and can, in principle, be tested using matter-wave interferometry with massive particles (Süzen, 2016).

5. Criticisms, Conceptual Debates, and the Status of Entropic Gravity

Critical analyses dispute the physical necessity of the entropic analogy:

• Thermodynamic Analogy Limitations: Genuine entropic forces, such as those observed in polymer physics, require well-defined heat baths, statistical equipartition, and independence from system-specific parameters. In the Verlinde scenario, neither the holographic screen nor the test particle clearly fulfills these requirements; the entropy increase is a result of external work (gravity) rather than statistical tendency (Gao, 2010).
• Role of Minimum Spacetime Scale: Arguments combining the uncertainty principle and hypothetical spacetime discreteness suggest gravity is a geometric response to energy density, not an entropic force; the gravitational constant can be connected to a fundamental length scale (e.g., the Planck length or beyond), and vanishes in the continuum limit (Gao, 2010).
• Definition of Entropy and Coarse Graining: The identification of the appropriate “entropy” is ambiguous. Approaches such as “holographic gravity” succeed by relating modular (entanglement) entropy variations to area, but “thermodynamic gravity” (relying on Clausius relations on lightsheets) struggles to produce both the correct area–entropy proportionality and Newton’s constant—the Casini entropy on null hypersurfaces gives a mismatch by an order-one factor (Carroll et al., 2016). Observer dependence, as in Unruh or Tolman temperature, is essential for assigning entropy to a screen (Cao, 2014). Coarse graining plays a fundamental role in how gravitational potential encodes entropy information.
• Quantum Mechanical Consistency: Concerns arise regarding the compatibility of entropic gravity with quantum coherence (e.g., in neutron gravitational bound states, the GRANIT experiment), but recent open-system analyses show that the entropic gravity model need not introduce observable decoherence at acceptable parameter values (Chaichian et al., 2011, Schimmoller et al., 2020, Sung et al., 2023).

6. Future Directions, Open Problems, and Connections

Recent theoretical work continues to expand the entropic paradigm:

• Information-theoretic gravity actions using quantum relative entropy between metrics as operators, with topological (Dirac–Kähler) matter field content, yield gravitational equations with emergent cosmological constants and possible phenomenology for dark matter (Bianconi, 26 Aug 2024).
• Explicit links between entanglement, modular energy, and spacetime geometry—supported by recent quantum field theory theorems—offer promising routes to a theory where gravity is a manifestation of underlying quantum entanglement (Carroll et al., 2016).
• Experimental prospects include matter-wave interferometry with massively delocalized objects, optomechanical studies of decoherence at large mass scales, and continued high-precision orbital measurements. Long-term storage experiments may bound or detect the predicted slow energy increase or decoherence rates in entropic models (Schimmoller et al., 2020).

7. Summary Table: Key Theoretical Elements

Principle/Relation	Formula/Interpretation	Source(s)
Entropic Force Law	$F \Delta x = T \Delta S$	(Verlinde, 2010, Lee, 2010)
Verlinde Entropy Variation	$\Delta S = 2 \pi k_B (mc/\hbar) \Delta x$	(Verlinde, 2010, Halyo, 2011)
Bits on Screen (Holography)	$N = (A c^3)/(G\hbar)$	(Lee, 2010, Fursaev, 2010)
Bekenstein–Hawking Entropy	$S = k_B A c^3 / (4G\hbar)$	(Lee, 2010)
Force Correction from Entropy-Mod.	$$F = - (GMm/R^2)[1 + 4l_P^2 \partial \mathcal{S}/\partial A|_{A=4\pi R^2}] \hat{R}$$	(Cuéllar et al., 7 Oct 2025)
Decoherence Rate in DFEG	Suppressed as $\sigma \to \infty$ in Lindblad master equation	(Schimmoller et al., 2020, Sung et al., 2023)
Emergent Gravitational Constant	$G_{\rm eff} = f'(R) G$ in $f(R)$ gravity	(Teimouri, 2017)
Quantum Gravity Link	Relative entropy action $\mathcal{S} = \mathrm{Tr}(g \ln \mathcal{G}^{-1})$	(Bianconi, 26 Aug 2024)

This synthesis brings out central themes and open problems in the field: entropic gravity bridges thermodynamics, quantum information, and gravity, but the correctness and completeness of the analogy remain a subject of significant theoretical and experimental scrutiny. The role of entanglement entropy, observer dependence, operational coarse graining, and the deep connection between geometry and information appear as core topics for future exploration.