Verlinde's Emergent Gravity

• Verlinde's Emergent Gravity is a framework that defines gravity as an emergent entropic force arising from microscopic information on holographic screens.
• It integrates concepts from statistical mechanics, holography, and thermodynamics to reproduce Newtonian gravity and explain galactic rotation curves without dark matter.
• The theory extends to cosmology by introducing a volume-law entropy contribution that accounts for dark energy microstates and observed galactic dynamics.

Verlinde's Emergent Gravity is a theoretical framework positing that gravity and, by extension, spacetime itself are not fundamental entities but rather emergent phenomena arising from the statistical and entropic behavior of underlying microscopic degrees of freedom. This approach synthesizes concepts from holography, statistical mechanics, quantum information, and the thermodynamics of spacetime, offering a fundamentally different paradigm from both general relativity and particle dark matter models for explaining observed galactic and cosmological dynamics.

1. Foundational Principles and Formalism

The core of Verlinde's proposal is that gravity manifests as an entropic force—analogous to entropic forces in polymer physics or thermodynamics—driven by the statistical tendency of a microscopically encoded information ensemble to maximize entropy under external perturbations. Space is conjectured not to exist a priori but to emerge through holographic encoding, with equipotential surfaces ("screens") serving as repositories of microscopic information (Verlinde, 2010, Dieks et al., 2015).

Key Postulates

• Holographic Principle: Each surface of area $A$ carries a finite number of information bits,

$N = \frac{A\,c^3}{G\,\hbar}$

where $G$ is Newton's constant. This is central to the holographic scenario: two-dimensional screens fully encode the information of the three-dimensional bulk they enclose.

• Thermal Equipartition: The energy enclosed by a screen is in equipartition among the $N$ bits,

$E = \frac{1}{2} N k_B T = M c^2$

where $T$ is the screen temperature and $M$ the bulk mass.

• Entropic Force Law: Any displacement $\Delta x$ of a test mass $m$ toward the screen changes the entropy by

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

and exerts an entropic force

$F\,\Delta x = T\,\Delta S$

which in conjunction with thermodynamic and quantum field-theoretic relations (notably the Unruh temperature) recovers both Newton's second law and the inverse square law of gravitation.

• Holographic Emergence: Spatial geometry arises from the data structure implemented on these screens, in a manner fundamentally analogous to coarse-graining steps in renormalization group flows (Dieks et al., 2015).

2. Derivation of Newtonian Gravity, General Relativity, and the Emergence Paradigm

The entropic scenario yields Newton's law of gravitation directly. For a spherical screen of radius $R$ enclosing mass $M$, calculation reveals that

$F = \frac{G M m}{R^2}$

without introducing a fundamental gravitational field as an input (Verlinde, 2010).

The relativistic generalization employs the redshift field associated with a timelike Killing vector $\xi^a$, introduces the Komar mass construction, and (through the Stokes theorem and standard identities) leads directly to the Einstein field equations: $R_{ab} - \frac{1}{2} R g_{ab} = 8\pi G\,T_{ab}$

The novelty and robustness of emergence are exhibited in three respects (Dieks et al., 2015):

• The framework depends critically on the presence and coarse-graining of microphysical degrees of freedom ("bits") on the screen.
• Gravity at the macroscopic level is irreducible to any force present in the underlying micro-dynamics.
• The emergent force law and its scaling are insensitive to the microscopic implementation, mirroring universality in statistical mechanics.

3. Extension to Dark Matter Phenomena and the Cosmological Volume Law

Verlinde's 2016 extension (Verlinde, 2016, Jusufi et al., 2023) introduces a volume-law contribution to the spacetime entropy, relevant in de Sitter backgrounds. The key insight is that dark energy microstates, characterized by long-range entanglement, give rise to a thermal volume contribution,

$S_{\rm vol}(r) = \frac{r}{L} \frac{A(r)}{4G\hbar}$

where $L = c/H_0$ is the cosmological horizon. The displacement of this entropy by baryonic matter gives an effective "apparent dark matter" contribution through an elastic-type response,

$$M_D^2(r) = \frac{a_0 r^2}{6G}\,\frac{d}{dr}[r M_B(r)]$$

with $a_0 = cH_0$, tightly linking the critical acceleration to cosmology.

This formalism successfully reproduces empirically established galactic scaling laws:

• At large radii in disk galaxies, the emergent gravitational acceleration obeys

$g_{\rm Ver} \approx \sqrt{\frac{a_0}{6} g_B}$

matching the observed flat rotation curves and the Tully–Fisher relation $v^4 \propto M$ (Yoon et al., 2022, Pardo, 2017).

• For a point mass $M$, the emergent gravity acceleration reduces to the MOND formula $\sqrt{a_0 g_B}$, but for extended mass distributions significant deviations from MOND appear in the inner regions (Diez-Tejedor et al., 2016, Lelli et al., 2017).

4. Observational Tests and Phenomenological Constraints

Verlinde's emergent gravity framework has been subjected to multiple empirical probes across astrophysical and cosmological contexts:

Rotation Curves and the Radial Acceleration Relation

Analyses using large galaxy samples (e.g., SPARC) have shown that Verlinde's predictions, while not free from systematics, align well with observed galaxy rotation curves and the empirical radial acceleration relation. The theory, with no free parameters beyond standard cosmological inputs, achieves residuals comparable to or only slightly above empirical fits such as the widely-used McGaugh relation (Yoon et al., 2022). However, EG's detailed predictions for extended baryonic mass distributions exhibit notable departures from pure MOND, especially in the inner regions, and in some galaxies require adjustments to otherwise independently constrained stellar mass-to-light ratios (Lelli et al., 2017).

Dwarf Spheroidal Galaxies and Pressure-Supported Systems

Extended comparisons in isolated dwarf spheroidals reveal that EG with its extra $4\pi G \rho_{\rm bar}(r) r$ term provides significantly better fits to the observed radial acceleration trends in the majority of systems. The preference of EG over MOND reaches $5.2\sigma$ significance across a 23-galaxy sample (Yoon et al., 5 Jan 2026).

Early-Type Galaxies and Stellar Dynamics

Application to elliptical galaxies' central velocity dispersions shows that EG can reproduce the observed stellar mass-to-light ratio trends and implied IMF normalization, assuming maximal elastic strain. The preferred normalization is systematically lower than in standard DM-based or MOND fits but still within observational uncertainty bands (Tortora et al., 2017).

Gravitational Lensing

Weak lensing profiles constructed under the EG hypothesis are consistent with the measured excess surface mass densities around isolated galaxy lenses on Mpc scales, rivaling the performance of ΛCDM+NFW fits. However, a fully covariant, relativistic EG lensing framework remains outstanding (Brouwer et al., 2016).

Black Hole and Strong Gravity Regimes

EG modifications have been incorporated into regular black hole spacetimes, predicting new families of global-monopole-like metrics (characterized by a deficit angle linked to the cosmological acceleration scale) and charged configurations, as well as rotating black holes via Newman–Janis extensions. These solutions manifest novel observational features, including corrections to light bending independent of impact parameter and tiny but nonzero modifications to perihelion advances (Jusufi, 2022, Liu et al., 2016).

5. Theoretical Developments, Covariant Formulations, and Cosmological Implications

Covariant Lagrangian Realizations

Efforts to provide a covariant, field-theoretic foundation for emergent gravity have yielded vector-tensor models in which an "imposter field" couples non-minimally to matter and the background geometry, with non-polynomial kinetic terms reflecting the underlying entropic volume law (Hossenfelder, 2017, Dai et al., 2017). The equations of motion of these models reduce to Verlinde's functional form in appropriate limits but introduce instabilities and require additional matter/radiation couplings for cosmological viability.

Cosmology and Hubble Tension

In the cosmological context, EG can be mapped onto a non-local modification of Newton's law, interpreted as a convolution kernel acting on the baryonic matter distribution (Jusufi et al., 2023). Modifications to the Friedmann equations entail effective shifts in Newton's constant $G \rightarrow G(1+\zeta)$ with $\zeta \sim 10^{-7}$, or rescalings of spatial curvature and the scale factor. Time-evolving "apparent DM" densities can accommodate both Planck- and SH₀ES-inferred values of the Hubble constant by introducing explicit running in the effective DM sector, suggesting partial mitigation of the Hubble tension.

Energy Conditions, Metric Structure, and Universality

Apparent DM in EG modifies the stress-energy tensor but the left side of the Einstein equations remains intact. The induced stress-energy can violate or marginally satisfy standard energy conditions near black hole cores. Generically, apparent DM introduces a solid angle deficit proportional to $\sqrt{a_M M}$, leading to distinctive, scale-invariant geometric signatures (Jusufi, 2022, Liu et al., 2016).

6. Critiques, Open Issues, and Current Status

A number of theoretical and empirical criticisms have been leveled:

• Conservative Force vs. Entropic Interpretation: Standard Newtonian gravity is a conservative, time-reversible force—whereas entropic forces are inherently dissipative. The mapping in EG assumes equilibrium and statically coarse-grained microstates, raising questions about microscopic reversibility (Dai et al., 2017).
• Elasticity Correspondence: Some analyses found inconsistencies in the derivation of the scaling laws for the emergent elasticity tensor and its mapping onto surface densities, but subsequent clarifications established the necessity of properly isolating the dark sector's contribution and utilizing a quadrature sum of accelerations:

$g^2 = g_B^2 + g_D^2$

rather than a linear sum (Yoon, 2020).

• Covariant Closure and Cosmological Stability: The existing field-theoretic covariantizations either lack a stable de Sitter solution or admit instabilities in the absence of substantial matter/radiation; stabilizing the cosmological background requires further ingredients (Dai et al., 2017, Hossenfelder, 2017).

Table: Summary of Observational and Theoretical Status

Domain	Key Results	Status
Galaxy rotation	Matches flat curves, Tully–Fisher, often ~MOND accuracy	Supported (Yoon et al., 2022)
Dwarf spheroidals	Exceeds MOND in fitting detailed radial trends, 5.2σ preference	Supported (Yoon et al., 5 Jan 2026)
Early-type galaxies	Reproduces central dispersions, lower $M_*/L$ normalization	Supported (Tortora et al., 2017)
Weak lensing	Fits ensemble excess density profiles, competitive with ΛCDM	Supported (Brouwer et al., 2016)
Strong gravity	Predicts monopole-like deficit angle, regular black holes	Supported (Jusufi, 2022)
Covariant closure	Vector-tensor models, stability issues unresolved	Open (Hossenfelder, 2017, Dai et al., 2017)
Cosmology	Modifies Friedmann eqs, offers Hubble tension resolution	Supported/Developing (Jusufi et al., 2023)

7. Generalizations, Statistical and Nonlinear Entropic Scenarios

Extensions include generalizations to nonadditive statistical frameworks, such as embedding EG dynamics within Tsallis q-statistics in n dimensions. Here, the emergent entropic force acquires q-dependent tails and modified scaling exponents, resulting in nontrivial short- and long-distance corrections to the gravitational potential and force law, potentially relevant for fundamental and emergent unification schemes (Zamora et al., 2017).

8. Outlook and Prospects

Verlinde's Emergent Gravity provides a microscopically-motivated and conceptually distinctive alternative to both general relativity plus dark matter and modified gravity schemes like MOND. Its successes in predicting rotation curves and dynamical anomalies without free parameters, reliance on cosmological input, and close relation to the thermodynamics of spacetime continue to drive both empirical testing and deeper theoretical development. Critical open questions remain regarding its covariant formulation, the microphysical characterization of the screen information, predictions for early-universe structures, gravitational lensing, and the controlled inclusion of cosmological evolution and departures from equilibrium. The framework's testability in next-generation microlensing, astrometric, and time-domain surveys, as well as its capacity to resolve cosmological tensions, render it an active and consequential area of research.