Gravity from Entropy: Emergent & Modified Gravity

• Gravity from Entropy is a conceptual framework that reinterprets gravitational dynamics as emergent phenomena arising from thermodynamic and entropic principles.
• It employs holographic screens, entropy gradients, and equipartition to reconstruct classical force laws and extend to modified gravity theories.
• The approach connects gravitational dynamics with quantum information, offering novel testable predictions and provoking debates on the fundamental nature of gravity.

Searching arXiv for recent and foundational papers on Gravity from Entropy / entropic gravity. Gravity from Entropy (GfE) denotes a family of programs in which gravitational dynamics are derived, reinterpreted, or constrained by entropy, thermodynamics, holography, or quantum-information principles rather than taken as fundamental from the outset. In the literature surveyed here, GfE ranges from Verlinde-style entropic-force constructions and Jacobson-type thermodynamic equations of state to holographic-entanglement derivations of Einstein’s equation, as well as more recent proposals in which the gravitational action is itself formulated as a relative-entropy functional (Chivukula, 2010, Carroll et al., 2016, Bianconi, 2024). The label does not refer to a single universally accepted theory: some works present gravity as an emergent entropic force, some treat Einstein gravity as an equilibrium condition of entanglement, some generalize the framework to modified gravity, and others argue that the entropic-force interpretation is incomplete or conceptually problematic (Chaichian et al., 2011, Gao, 2010).

1. Conceptual scope and foundational postulates

A central strand of GfE begins from the thermodynamic relation

$$F\,\Delta x = T\,\Delta S \quad\Longrightarrow\quad F = T\,\frac{\Delta S}{\Delta x},$$

interpreting gravitational attraction as an entropic force associated with entropy gradients near a holographic screen (Chivukula, 2010). In this setting, a particle of mass $m$ moving by $\Delta x$ toward the screen is assigned the entropy change

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

while an accelerated observer sees the Unruh temperature

$T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$

Substitution yields $F=ma$, and with additional holographic and equipartition assumptions one recovers Newton’s inverse-square law (Chivukula, 2010, Chaichian et al., 2011).

The standard auxiliary assumptions are the holographic principle, the assignment of screen degrees of freedom proportional to area,

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

and equipartition,

$E=\frac12 N k_B T,\qquad E=M c^2.$

For a spherical screen of area $A=4\pi r^2$, these relations imply a temperature that reproduces

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

In this form, GfE is not merely a claim about force law reconstruction; it ties gravitation to information storage on codimension-one surfaces and to thermal response under acceleration (Chivukula, 2010).

A broader use of the term includes programs where Einstein’s equation is obtained from entropy stationarity or a local Clausius relation rather than from an entropic force in the narrow polymer-like sense. One major distinction is between holographic gravity and thermodynamic gravity. In holographic gravity, Einstein’s equation arises from keeping the entropy of a small causal diamond stationary under variations of geometry and quantum state. In thermodynamic gravity, Einstein’s equation is treated as a local equation of state associated with entropy flux across null lightsheets (Carroll et al., 2016). This division is important because many later debates concern not whether gravity and entropy are related, but which entropy, on which region, and under what variational principle.

A further branch of the literature treats GfE as a genuine modified-gravity framework. In “Gravity from entropy” (Bianconi, 2024), the spacetime metric is related to a quantum operator, matter fields are encoded in a Dirac–Kähler decomposition, and the action is taken to be a quantum relative entropy between the spacetime metric and a matter-induced metric. In this usage, GfE is no longer only an interpretive derivation of Newtonian gravity or Einstein’s equation; it becomes a candidate dynamical theory with its own field content, emergent cosmological term, and second-order modified Einstein equations (Bianconi, 2024).

2. Entropic-force derivations of Newtonian gravity

In the entropic-force formulation, the basic derivation proceeds in two steps. First, the Unruh temperature and the Bekenstein-inspired entropy variation imply Newton’s second law. Second, one combines screen equipartition and bit counting to obtain Newtonian gravitation (Chivukula, 2010).

The derivation can be summarized as follows.

Input	Formula	Role
Entropic force	$m$0	Defines force from entropy change
Entropy change	$m$1	Associates displacement with screen entropy
Unruh temperature	$m$2	Converts acceleration to temperature
Screen bits	$m$3	Implements holography
Equipartition	$m$4, $m$5	Fixes screen temperature

Using these inputs, one obtains $m$6 and then $m$7 for a spherical screen (Chivukula, 2010, Chaichian et al., 2011). In this sense, the original entropic-force program reproduces Newtonian gravity from a small set of thermodynamic and holographic postulates.

The same logic has been generalized to higher dimensions and higher-curvature theories. In $m$8-dimensional spacetime, the screen area is $m$9, and by replacing the standard area law with the entropy formulas appropriate to Gauss–Bonnet or Lovelock gravity, one derives the corresponding modified Newtonian force law and Friedmann equations (Sheykhi et al., 2011). The paper “Lovelock gravity from entropic force” (Sheykhi et al., 2011) argues that the only theory-specific input is the entropy–area relation, while the postulates $\Delta x$0, $\Delta x$1, and $\Delta x$2 remain unchanged.

Other variants modify the equipartition sector instead of the entropy law. “Modified Entropic Force” (Gao, 2010) replaces classical equipartition with a Debye-model correction at low temperature. In that construction,

$\Delta x$3

leading to a modified relation

$\Delta x$4

The strong-field limit $\Delta x$5 recovers Newtonian gravity, while the weak-field limit yields slower radial falloff and is used there to model late-time cosmic acceleration without dark energy (Gao, 2010).

A different thermodynamic route treats $\Delta x$6 and $\Delta x$7 as both thermodynamic and gravitational sources. “Is gravity entropic force?” (Yang, 2014) shows that gravity can be seen as an entropic force only for systems with constant temperature and zero chemical potential. In that analysis, the Gibbs–Duhem relation

$\Delta x$8

obstructs a pure $\Delta x$9 description whenever $\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$0 varies or $\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$1. This places a strong restriction on the circumstances under which an entropic-force interpretation is viable (Yang, 2014).

3. From entropy variation to Einstein’s equation

A more structurally ambitious class of GfE proposals derives the field equations of gravity from entropy extremization, entropy balance, or modular-energy arguments. These approaches do not necessarily require a force law analogous to polymer elasticity.

In one widely discussed route, Einstein’s equation emerges from a local first-law relation on small causal diamonds. In holographic gravity, one considers a small spacelike ball $\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$2, its causal diamond $\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$3, the reduced density matrix $\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$4, and the entanglement entropy

$\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$5

Under assumptions of entanglement separability, equilibrium at fixed volume, an area–entropy relation,

$\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$6

and a modular-energy relation of conformal-field-theory type, one obtains the semiclassical Einstein equation (Carroll et al., 2016). The entanglement first law,

$\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$7

is the key bridge between quantum information and spacetime geometry. This framework is presented in (Carroll et al., 2016) as the more successful of two entropic derivations because its entropy notion is better controlled.

Thermodynamic gravity adopts instead a local Clausius relation across a null lightsheet. For a small null congruence generated by $\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$8, the heat flux is

$\Delta S = 2\pi k_B\,\frac{m\,c}{\hbar}\,\Delta x,$9

the Unruh temperature is $T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$0, and the area decrement is related to $T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$1 through Raychaudhuri’s equation. Imposing

$T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$2

gives Einstein’s equation with $T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$3 if the entropy assignment is consistent (Carroll et al., 2016). However, the same paper argues that thermodynamic gravity faces a serious entropy-definition problem: natural candidates for local entropy either diverge, fail to vanish in vacuum, or produce the wrong coefficient by an order-unity factor (Carroll et al., 2016). This has become one of the central technical criticisms of local entropic derivations.

A related but distinct line identifies entropy with the gravitational action in the semiclassical saddle-point approximation. “Statistical Origin of Gravity” (Banerjee et al., 2010) begins from

$T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$4

so that

$T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$5

For stationary black holes, the paper derives

$T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$6

with $T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$7 identified as the Komar energy, and shows consistency with the Smarr formula (Banerjee et al., 2010). In that framework, extremizing entropy is equivalent to extremizing the action, thereby reproducing Einstein’s equations.

Entropy-functional methods extend this philosophy beyond Einstein gravity. “Modified gravity from an entropy functional” (Hammad, 2014) generalizes Padmanabhan’s entropy-functional formalism so that extremization with respect to an auxiliary vector field yields, at different orders in curvature, Einstein–Hilbert, Gauss–Bonnet, Lanczos–Lovelock, and more general higher-curvature theories. This suggests that entropy-based variational principles can serve not only as reinterpretations of known dynamics but also as constructive tools for determining admissible higher-curvature corrections (Hammad, 2014).

4. Entanglement, holography, and quantum-information formulations

The informational content of GfE extends beyond thermodynamic analogies. Several formulations make entanglement, modular Hamiltonians, or relative entropy central rather than peripheral.

The survey “Gravity as an Entropic Phenomenon” (Chivukula, 2010) explicitly situates entropic gravity within a quantum-information landscape involving quantum microstates, entanglement entropy, AdS/CFT, and string theory. There, the microscopic “bits” on a holographic screen are linked conceptually to underlying quantum microstates, the area law for entanglement entropy, and the reconstruction of bulk geometry from boundary entanglement. This perspective does not by itself provide a unique microscopic completion, but it connects the entropic viewpoint to the broader holographic program (Chivukula, 2010).

In the holographic-gravity approach of (Carroll et al., 2016), the reduced density matrix on a small region,

$T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$8

and the universality of null-limit modular Hamiltonians are the technical foundation for the derivation. The paper emphasizes that recent quantum-field-theoretic results on modular energy partially justify the postulates used. In this version of GfE, the relevant entropy is not a heuristic screen entropy but the entanglement entropy of quantum fields in a localized spacetime region (Carroll et al., 2016).

A more explicit relative-entropy theory appears in “Gravity from entropy” (Bianconi, 2024). There the entropic action is

$T = \frac{\hbar\,a}{2\pi\,c\,k_B}.$9

with $F=ma$0 the topological metric built from the Lorentzian metric and $F=ma$1 a matter-induced metric operator assembled using a Dirac–Kähler decomposition of bosonic matter fields and curvature contributions (Bianconi, 2024). The authors introduce a $F=ma$2-field as a set of Lagrange multipliers, so that the theory reduces to a dressed Einstein–Hilbert action with an emergent small and positive cosmological constant depending only on the $F=ma$3-field. The resulting modified Einstein equations remain second order in the metric and in the $F=ma$4-field (Bianconi, 2024).

This relative-entropy construction has been extended thermodynamically and phenomenologically. “The Thermodynamics of the Gravity from Entropy Theory” (Bianconi, 26 Oct 2025) derives a Hamiltonian for the GfE theory and introduces $F=ma$5-temperature and $F=ma$6-pressure for isotropic FRW metrics, associated with local Geometric Quantum Relative Entropy (GQRE). In the low-energy, small-curvature limit, the resulting modified Friedmann equations reduce to the standard Friedmann equations, while the total GQRE for unit volume is not increasing and the total entropy of Friedmann universes is not decreasing in time (Bianconi, 26 Oct 2025). This suggests a thermodynamic reinterpretation of cosmological evolution internal to the GfE action itself.

A plausible implication is that the term “Gravity from Entropy” now refers to at least two partly disjoint research programs: one rooted in emergent-force and holographic-screen heuristics, and another in action principles built from relative entropy or geometric information measures. The literature does not present these as fully unified.

5. Phenomenology, proposed tests, and cosmological applications

Although much of GfE is conceptual, several papers propose explicit phenomenology.

A notable example is “A testable prediction from entropic gravity” (Süzen, 2016). In that work, Mehmet Süzen combines Verlinde’s entropic-force framework with the Schrödinger–Newton system and the Diósi–Penrose proposal for gravity-induced wave-function collapse. In a one-dimensional toy model, the entropic-force divergence is matched to a Poisson-type equation,

$F=ma$7

and the entropy change is written in terms of a collapsed density matrix,

$F=ma$8

The resulting relation is

$F=ma$9

so that a density-matrix weight depends exponentially on $N = \frac{A\,c^3}{G\,\hbar},$0 (Süzen, 2016). The paper suggests a matter-wave interferometer test based on visibility suppression as the acceleration $N = \frac{A\,c^3}{G\,\hbar},$1 is varied. It also states that current atomic interferometers achieve phase precision of order $N = \frac{A\,c^3}{G\,\hbar},$2 and can sense accelerations down to $N = \frac{A\,c^3}{G\,\hbar},$3, but the Diósi–Penrose criterion requires masses on the order of $N = \frac{A\,c^3}{G\,\hbar},$4 or larger in spatial superposition, which is far beyond present capabilities (Süzen, 2016). The proposal is therefore explicitly futuristic and conceptual.

Quantum phenomenology also appears in critique form. “On gravity as an entropic force” (Chaichian et al., 2011) reexamines claims of conflict between entropic gravity and the GRANIT experiment on gravitationally bound neutron states. It concludes that entropic gravity does not necessarily contradict these states, because in the nonrelativistic limit it is equivalent to Newtonian gravity and yields the same potential $N = \frac{A\,c^3}{G\,\hbar},$5 (Chaichian et al., 2011). However, the same paper argues that spontaneous decays between gravitationally bound states cannot be explained within minimal entropic gravity because there is no mechanism analogous to graviton emission. It cites the estimate

$N = \frac{A\,c^3}{G\,\hbar},$6

for such a spontaneous transition, emphasizing that even conceptually tiny but nonzero rates require a quantized gravitational sector or an equivalent mechanism (Chaichian et al., 2011).

Cosmological applications are common. In “Modified Entropic Force” (Gao, 2010), a Debye acceleration $N = \frac{A\,c^3}{G\,\hbar},$7 is introduced, and the modified entropic-force law is used to account for current cosmic speeding up without dark energy. In “Entropy, Gravity and the Mass-Boom” (Alfonso-Faus, 2010), the Hawking–Bekenstein entropy formula

$N = \frac{A\,c^3}{G\,\hbar},$8

is combined with a “Mass-Boom” assumption $N = \frac{A\,c^3}{G\,\hbar},$9, so that increasing mass implies increasing entropy and, via Mach’s principle $E=\frac12 N k_B T,\qquad E=M c^2.$0, cosmic expansion (Alfonso-Faus, 2010). These cosmological constructions are specific to their respective papers and are not presented across the corpus as standard or consensus consequences of GfE.

More recent phenomenology is developed within relative-entropy GfE. “Spherically symmetric black holes in Gravity from Entropy and spontaneous emission” (Thattarampilly et al., 14 Feb 2026) derives modified vacuum field equations for static and dynamical spherically symmetric spacetimes. In the static sector, the Schwarzschild metric receives perturbative corrections scaling as $E=\frac12 N k_B T,\qquad E=M c^2.$1,

$E=\frac12 N k_B T,\qquad E=M c^2.$2

In the dynamical sector, the mass-evolution law near the apparent horizon is

$E=\frac12 N k_B T,\qquad E=M c^2.$3

which includes a large-mass constant leakage term $E=\frac12 N k_B T,\qquad E=M c^2.$4 and an intermediate-scale $E=\frac12 N k_B T,\qquad E=M c^2.$5 term reproducing Hawking-like scaling (Thattarampilly et al., 14 Feb 2026). The paper states that current strong-field observations bound $E=\frac12 N k_B T,\qquad E=M c^2.$6 via the EHT shadow of Sgr A* to

$E=\frac12 N k_B T,\qquad E=M c^2.$7

and argues that the framework remains consistent with present astrophysical tests (Thattarampilly et al., 14 Feb 2026).

6. Critiques, limitations, and open controversies

The most persistent controversy is whether gravity is genuinely an entropic force in the strict thermodynamic sense or whether the entropic formalism is a reformulation that reproduces known classical results without establishing microscopic origin.

A direct critique appears in “Is Gravity an Entropic Force?” (Gao, 2010). That paper argues that Verlinde’s screen-particle construction lacks the defining features of a genuine entropic-force system: no proper heat bath–polymer analogue exists, the screen is not in true thermal equilibrium, the test particle has no independent temperature or entropy, and the direction of energy–entropy flow is reversed relative to standard entropic-force examples (Gao, 2010). In particular, the paper contends that the screen’s entropy increase is caused by gravitational work,

$E=\frac12 N k_B T,\qquad E=M c^2.$8

rather than being the cause of the force. It further argues that one can derive the entropy variation from Newtonian gravity and Unruh temperature, so the entropic derivation may reverse the logical order of implication (Gao, 2010).

A subtler criticism is that minimal entropic gravity is equivalent to Newtonian gravity only in static, nonrelativistic regimes. As emphasized in (Chaichian et al., 2011), once one asks about spontaneous transitions, gravitational radiation, or other genuinely quantum gravitational processes, the minimal entropic picture is underdetermined unless supplemented by essential new ingredients. The paper therefore presents entropic gravity as a possible thermodynamic rewriting of classical gravity rather than a complete dynamical theory (Chaichian et al., 2011).

Even among entropy-based derivations of Einstein’s equation, technical disputes persist over the correct entropy functional. “What is the Entropy in Entropic Gravity?” (Carroll et al., 2016) concludes that holographic gravity is on firmer footing than thermodynamic gravity because the latter lacks a consistent local entropy definition satisfying all required properties. This is not a generic rejection of entropic approaches; rather, it narrows the plausible formulations to those grounded in entanglement entropy and modular Hamiltonians (Carroll et al., 2016).

The status of quantum microphysics also remains unsettled. Some treatments invoke string-theoretic state counting, AdS/CFT, or entanglement area laws as conceptual support (Chivukula, 2010), but these do not amount to a unique derivation of the screen degrees of freedom used in Verlinde-style arguments. A plausible implication is that the entropic-force picture is best interpreted as an effective macroscopic heuristic unless embedded in a more precise microscopic framework.

Recent thermodynamic extensions underscore the same issue. “Thermodynamic Gravity with Non-Extensive Horizon Entropy and Topological Calibration” (Figliolia et al., 24 Feb 2026) revisits Jacobson’s local-Rindler derivation using generalized entropy

$E=\frac12 N k_B T,\qquad E=M c^2.$9

It finds that the local effective coupling is fixed by the entropy slope,

$A=4\pi r^2$0

and that curvature-dependent entropies with internal entropy production reproduce the field equations of $A=4\pi r^2$1 gravity (Figliolia et al., 24 Feb 2026). However, consistency across scales and topologies yields strong bounds requiring $A=4\pi r^2$2 to be extremely close to unity. This suggests that non-extensive departures from the Bekenstein–Hawking area law are tightly constrained if they are to underpin a viable thermodynamic gravity (Figliolia et al., 24 Feb 2026).

The literature therefore does not support a single verdict. Some works treat GfE as a compelling organizing principle for the gravity–thermodynamics relation; others maintain that the relation is real but insufficient to establish gravity as entropic in origin; still others develop alternative entropic actions whose relation to the original entropic-force paradigm is suggestive rather than direct.

7. Relation to modified gravity and current research directions

A significant current direction treats GfE not as a derivation of GR alone but as a generator of modified-gravity theories.

Entropy-functional methods already showed that higher-curvature theories such as Gauss–Bonnet and Lanczos–Lovelock gravity can emerge from suitable entropy functionals (Hammad, 2014, Sheykhi et al., 2011). More recent work extends this to generalized horizon entropies and $A=4\pi r^2$3 gravity. In (Figliolia et al., 24 Feb 2026), area-type entropies with constant slope reproduce Einstein’s equations, while curvature-dependent entropy densities plus an internal entropy-production term yield the field equations of $A=4\pi r^2$4 gravity. The same paper introduces a Topological Calibration Principle tying the coarse-graining scale to intrinsic geometric data through the Gauss–Bonnet theorem, producing topology-dependent effective coupling $A=4\pi r^2$5 (Figliolia et al., 24 Feb 2026).

The relative-entropy action program pushes further. In (Bianconi, 2024), the action is explicitly information-theoretic, the matter sector is encoded through a topological Dirac–Kähler multiplet, and the dressed Einstein equations contain an emergent small and positive cosmological constant $A=4\pi r^2$6. In (Bianconi, 26 Oct 2025), this theory is given a Hamiltonian formulation and a thermodynamic interpretation for FRW metrics through $A=4\pi r^2$7-temperature, $A=4\pi r^2$8-pressure, and GQRE. In (Thattarampilly et al., 14 Feb 2026), the same framework is applied to black-hole solutions and spontaneous evaporation. These papers collectively define a research line in which GfE is a concrete modified-gravity theory rather than a thermodynamic reinterpretation of GR.

Another informational approach is the “Entropic theory of Gravitation” (Atanasov, 2017), which starts from Landauer’s principle and a total entropy functional

$A=4\pi r^2$9

with a geometric entropy density

$F=\frac{G M m}{r^2}.$0

Using a Palatini variation, the paper derives entropic field equations in which matter entropy, rather than the stress tensor directly, couples to geometry (Atanasov, 2017). In high-entropy regimes it recovers a small negative curvature interpreted there as an effective cosmological constant; in low-entropy condensate regimes it predicts strong local curvature enhancements (Atanasov, 2017). This is another example of how GfE has diversified into multiple entropy-driven modified-gravity frameworks.

Across these developments, a broad pattern emerges. Foundational entropic-force models aim to explain why Newtonian gravity and perhaps Einstein gravity admit thermodynamic reformulations. Entanglement-based models seek a more rigorous derivation of semiclassical geometry from quantum-information principles. Action-based GfE theories reinterpret gravity itself as a relative entropy or information functional and develop independent phenomenology. The resulting field is conceptually unified by the gravity–entropy nexus but methodologically heterogeneous.

This suggests that “Gravity from Entropy” is best understood not as a single doctrine but as an umbrella category for several interlocking research programs. Their common claim is that gravitational dynamics are deeply constrained by entropy, information, and horizon thermodynamics. Their disagreements concern what entropy is relevant, whether gravity is literally an entropic force, how quantum microstates enter, and whether the correct outcome is Einstein gravity, a thermodynamic reformulation of it, or a modified theory with new degrees of freedom and observational signatures (Chivukula, 2010, Carroll et al., 2016, Bianconi, 2024).