General Relativistic Entropic Acceleration

• General Relativistic Entropic Acceleration is a theoretical framework where cosmic acceleration emerges from horizon thermodynamics rather than a cosmological constant.
• It employs covariant non-equilibrium formulations that modify Einstein’s equations through entropy gradients and boundary terms.
• The approach extends to weak-field regimes, potentially explaining MOND-like galactic dynamics and altering standard expansion histories.

General Relativistic Entropic Acceleration denotes a class of proposals in which acceleration in gravitating systems is attributed to horizon thermodynamics, entropy production, or entropy gradients formulated within relativistic gravity. In the narrow sense used by the recent GREA program, it is a fully general relativistic, covariant, non-equilibrium mechanism for late-time cosmic acceleration that arises from horizon thermodynamics rather than from a cosmological constant or a dark-energy fluid. In a broader sense, the same theme includes local Rindler-horizon derivations of Einstein gravity, horizon-fluid formulations, cosmological models based on boundary terms or holographic equipartition, and weak-field relativistic constructions in which low-temperature corrections generate MOND-like galactic dynamics (Garcia-Bellido et al., 2021, Espinosa-Portales et al., 2021, Rostami et al., 7 Nov 2025).

1. Horizon thermodynamics as the relativistic starting point

The relativistic entropic picture begins with the statement that an accelerated observer attributes an Unruh temperature and a Bekenstein–Hawking entropy to a local Rindler horizon. In this setting,

$$T_{\rm U}=\frac{\hbar a}{2\pi c k_B},\qquad S_{\rm BH}=\frac{k_B c^3 A}{4G\hbar}.$$

Applying the Clausius relation $\delta Q=T\,dS$ to local causal horizons and using the null Raychaudhuri equation yields the null-null projection of Einstein’s equations; covariance and the Bianchi identity then reconstruct the Einstein field equations with a cosmological constant as an integration constant (Nagle, 2016).

This line of thought was extended by horizon-fluid methods based on the Damour–Navier–Stokes equation. In that formulation, the stretched horizon carries an effective fluid stress tensor with pressure $p=\kappa/(8\pi G)$, shear viscosity $\eta=1/(16\pi G)$, and bulk viscosity $\zeta=-1/(16\pi G)$. The Hajíček 1-form plays the role of surface momentum density, and the tangential-normal projection of Einstein’s equations becomes a Navier–Stokes-like balance law. The resulting picture is not limited to $\delta Q=T\,dS$; it also incorporates the rest of the first law through pressure, momentum transport, and dissipative terms (Nagle, 2016).

A complementary result is that, in Fermi normal coordinates adapted to an accelerated observer, the change in the transverse area of the local Rindler horizon under parametric normal displacements has a thermodynamic structure that explicitly contains the time-time component of the Einstein tensor. In the notation of that analysis,

$$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$

The first term isolates $G(u_{\rm R},u_{\rm R})$, while the second depends on the intrinsic two-curvature of the horizon patch. Demanding that the same identity hold for all accelerated observers elevates the result from a frame-dependent statement to the full Einstein tensor, up to a cosmological-constant term (Kothawala, 2016).

2. Covariant non-equilibrium formulations

A more explicit generally covariant non-equilibrium formulation introduces entropy directly into the gravitational variational principle. In that framework the field equations become

$$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$

where $f_{\mu\nu}$ is a covariant entropic-force tensor. The entropic contribution modifies the dynamical Hamilton equation in ADM form but leaves the Hamiltonian constraint unmodified. In FLRW symmetry the spatial trace of the entropic tensor satisfies

$\delta Q=T\,dS$0

so the acceleration equation becomes

$\delta Q=T\,dS$1

The same formalism identifies an effective bulk viscosity,

$\delta Q=T\,dS$2

and shows that a fluid satisfying the strong energy condition could avoid collapse for a positive and sufficiently large entropic-force contribution (Espinosa-Portales et al., 2021).

This covariant construction distinguishes relativistic entropic acceleration from heuristic force laws of the form $\delta Q=T\,dS$3. The entropic source is not merely appended to Newtonian dynamics; it appears in the relativistic field equations, in the non-conservation law for the matter energy-momentum tensor, and in the Raychaudhuri equation for geodesic congruences. In this sense, acceleration is tied to entropy production rather than to an externally prescribed dark-energy component (Espinosa-Portales et al., 2021).

A different covariant entropic theory starts from an entropy functional for geometry plus matter, with a geometric entropy density

$\delta Q=T\,dS$4

and a matter entropy density given by the von Neumann expression $\delta Q=T\,dS$5. Extremizing the total entropy by a Palatini variation yields entropic field equations in which curvature is sourced by $\delta Q=T\,dS$6 and its metric response rather than by the usual stress tensor. In the large-volume limit, the theory gives

$\delta Q=T\,dS$7

so the matter entropy induces an effective cosmological constant. The same paper also presents an “energetic aspect” that restores the repertoire of classical General Relativity up to a different coupling constant (Atanasov, 2017).

3. Cosmological GREA and the causal horizon

In the specific cosmological program called General Relativistic Entropic Acceleration, the decisive input is the Gibbons–Hawking–York boundary term on the relevant cosmological horizon. The background spacetime is FLRW,

$\delta Q=T\,dS$8

and the surface action is written as

$\delta Q=T\,dS$9

In this framework the apparent horizon can be written for completeness as

$p=\kappa/(8\pi G)$0

but its entropy growth is found to be too small to affect the expansion appreciably. The relevant boundary is instead the causal horizon, with physical radius

$p=\kappa/(8\pi G)$1

and the GHY term over that causal horizon becomes

$p=\kappa/(8\pi G)$2

with

$p=\kappa/(8\pi G)$3

The corresponding modified continuity and acceleration equations are

$p=\kappa/(8\pi G)$4

The matter content remains radiation, baryons, and cold dark matter; the entropic effect enters through a boundary term rather than through a bulk dark-energy fluid (Garcia-Bellido et al., 2021).

The explicit causal-horizon background is parameterized in conformal time by

$p=\kappa/(8\pi G)$5

and the Hamiltonian constraint becomes

$p=\kappa/(8\pi G)$6

With parameters consistent with Planck 2018,

$p=\kappa/(8\pi G)$7

acceleration begins around $p=\kappa/(8\pi G)$8, that is $p=\kappa/(8\pi G)$9. Relative to $\eta=1/(16\pi G)$0CDM with the same asymptotic normalization at the CMB, the coasting point occurs at $\eta=1/(16\pi G)$1 rather than $\eta=1/(16\pi G)$2, and the present expansion rate becomes

$\eta=1/(16\pi G)$3

whereas $\eta=1/(16\pi G)$4CDM would give $\eta=1/(16\pi G)$5. The entropic energy fraction today is

$\eta=1/(16\pi G)$6

As the universe expands further, the entropic energy dilutes and the future hypersurface is Minkowski rather than de Sitter (Garcia-Bellido et al., 2021).

A holographic-dual reformulation sharpens the boundary interpretation. In an empty, spatially flat FLRW bulk with a cosmological constant, the bulk $\eta=1/(16\pi G)$7 term can be replaced by a GHY boundary term on the de Sitter horizon,

$\eta=1/(16\pi G)$8

with the proposed correspondence

$\eta=1/(16\pi G)$9

For the de Sitter horizon,

$\zeta=-1/(16\pi G)$0

and one finds

$\zeta=-1/(16\pi G)$1

A bulk observer making long-range electromagnetic and gravitational measurements therefore cannot distinguish acceleration due to a bulk $\zeta=-1/(16\pi G)$2 from acceleration due to the thermodynamic properties of the boundary. When matter is included, the causal horizon evolves, entropy grows, and the theory enters the time-dependent GREA regime; if $\zeta=-1/(16\pi G)$3, the late-time future is Minkowski, whereas $\zeta=-1/(16\pi G)$4 leads asymptotically to de Sitter space (García-Bellido, 24 Nov 2025).

4. Adjacent entropic-cosmology constructions

The broader literature includes several adjacent entropic-cosmology programs that are not identical to covariant GREA. One influential line modifies the Friedmann acceleration equation by horizon-thermodynamic surface terms. In that construction the apparent or Hubble horizon has

$\zeta=-1/(16\pi G)$5

and the entropic pressure is

$\zeta=-1/(16\pi G)$6

The cosmological acceleration law is then written phenomenologically as

$\zeta=-1/(16\pi G)$7

with representative choices $\zeta=-1/(16\pi G)$8 or $\zeta=-1/(16\pi G)$9. In the minimal case,

$\delta Q=T\,dS$0

so a de Sitter-like late attractor appears without inserting a separate dark-energy density by hand. The same program was extended to early-time “entropic inflation” by adding a logarithmic correction to the horizon entropy,

$\delta Q=T\,dS$1

which generates an $\delta Q=T\,dS$2 term in the acceleration law; consistency with the thermal bath at the horizon temperature yields $\delta Q=T\,dS$3 and a quasi–de Sitter inflationary phase that shuts off when the $\delta Q=T\,dS$4 term ceases to dominate (Easson et al., 2010, Easson et al., 2010).

A second line studies corrected entropy–area laws and constrains them with cosmological data. For a monomial infrared correction $\delta Q=T\,dS$5, the effective equation of state in a flat single-fluid regime becomes

$\delta Q=T\,dS$6

In a matter-dominated universe this accelerates for $\delta Q=T\,dS$7. A full MCMC analysis with SNe, BAO, and CMB distance priors gave

$\delta Q=T\,dS$8

with total fit quality comparable to $\delta Q=T\,dS$9CDM. The same study emphasized that ultraviolet entropic corrections based on $$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$0 surface terms or inverse-area entropy corrections face a graceful-exit problem and do not recover the low-curvature GR limit in simple realizations (Koivisto et al., 2010).

Other neighboring constructions use different thermodynamic mechanisms. A Debye-corrected entropic-force model replaces equipartition by

$$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$1

leading to

$$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$2

Its characteristic acceleration is $$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$3, so the model is essentially unchanged from Newtonian gravity in solar-system and galactic environments and instead targets late-time cosmic expansion, with $$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$4 and $$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$5 close to $$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$6CDM for $$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$7 and $$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$8 (Gao, 2010). Padmanabhan-style holographic equipartition,

$$T_{\rm U}\,\delta S = -\frac{1}{3}\left[ \int_{\mathcal H}\frac{1}{8\pi}\,G(u_{\rm R},u_{\rm R})\,\Delta z_0\,d^2x_\perp + \int_{\mathcal H}\frac{1}{16\pi}\,R_{\parallel}\,\Delta z_0\,d^2x_\perp \right].$$9

was shown to be consistent with horizon-entropy maximization and to approach de Sitter equilibrium as $G(u_{\rm R},u_{\rm R})$0 (B et al., 2018). A related FRW analysis based on the unified first law uses the horizon temperature

$G(u_{\rm R},u_{\rm R})$1

and derives modified acceleration and expansion equations, together with a GR branch that reproduces the standard solutions $G(u_{\rm R},u_{\rm R})$2 and the usual acceleration condition $G(u_{\rm R},u_{\rm R})$3 (Tu et al., 2017).

The literature also contains explicitly Machian and Schwarzian-based variants. One Machian proposal interprets the Hubble horizon thermodynamically, with

$G(u_{\rm R},u_{\rm R})$4

and regards cosmic acceleration as a horizon-thermodynamic imprint of the global gravitational potential (Gogberashvili et al., 2010). Another recent construction connects the cosmological Schwarzian derivative to the surface gravity of the apparent horizon and formulates a sum rule

$G(u_{\rm R},u_{\rm R})$5

In standard GR that rule forces an early Milne vacuum, but an entropic source term in the Einstein–Hilbert action modifies the acceleration equation so that the early-time balance condition

$G(u_{\rm R},u_{\rm R})$6

removes the Milne-only restriction while preserving the sum rule (Chakrabarti, 1 Apr 2025).

5. Weak-field realizations and relativistic MOND

General relativistic entropic acceleration also appears in a distinct weak-field program aimed at galaxy dynamics. In that construction the holographic screen carries

$G(u_{\rm R},u_{\rm R})$7

where $G(u_{\rm R},u_{\rm R})$8 is a temperature-dependent correction to equipartition and $G(u_{\rm R},u_{\rm R})$9 is related to acceleration by the Unruh effect. Using

$$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$0

in the low-temperature regime, and identifying

$$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$1

with the MOND scale $$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$2, one obtains modified Einstein equations,

$$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$3

For a static, spherically symmetric vacuum metric,

$$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$4

the weak-field, low-temperature solution is

$$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$5

with $$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$6 and $$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$7. The geodesic acceleration becomes

$$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$8

so the circular velocity scales as $$R_{\mu\nu}-\frac12 R g_{\mu\nu} = \kappa\left(T_{\mu\nu}-f_{\mu\nu}\right), \qquad \nabla^\mu T_{\mu\nu}=\nabla^\mu f_{\mu\nu},$$9, giving nearly flat rotation curves without introducing an explicit MOND interpolating function (Rostami et al., 7 Nov 2025).

The same study confronted the relativistic MOND model (RMOND) with SPARC rotation-curve data for NGC 3198 by MCMC Bayesian inference using Metropolis–Hastings, a multi-chain MPI implementation, GetDist post-processing, and the convergence criterion $f_{\mu\nu}$0. Three models were compared: baryons-only Newtonian dynamics (ND), a dark-matter halo model (DM), and RMOND. In the RMOND fit, $f_{\mu\nu}$1 and $f_{\mu\nu}$2 were fixed because of weak MCMC constraints, while $f_{\mu\nu}$3, $f_{\mu\nu}$4, and $f_{\mu\nu}$5 were inferred (Rostami et al., 7 Nov 2025).

Model	Key setting or result	$f_{\mu\nu}$6
ND	baryons-only Newtonian dynamics	277.26
DM	Begeman halo with $f_{\mu\nu}$7 fixed	57.12
RMOND	$f_{\mu\nu}$8, $f_{\mu\nu}$9 fixed; $\delta Q=T\,dS$00	61.37

For NGC 3198 the RMOND fit also gave

$\delta Q=T\,dS$01

The global $\delta Q=T\,dS$02 is slightly higher than the DM fit, but RMOND was reported to provide notably improved agreement at $\delta Q=T\,dS$03, where it closely tracks the observed flattening. Specific lensing predictions were not developed, although the relativistic nature of the metric was presented as setting the stage for such tests (Rostami et al., 7 Nov 2025).

This galaxy-scale relativistic MOND construction should be distinguished from the earlier Debye-corrected entropic-force model of Gao. There the central relation is

$\delta Q=T\,dS$04

and in the deep weak-field limit

$\delta Q=T\,dS$05

so $\delta Q=T\,dS$06, not $\delta Q=T\,dS$07. Because $\delta Q=T\,dS$08, that model predicts negligible deviations in galaxies and does not reproduce MOND-like phenomenology (Gao, 2010).

6. Conceptual distinctions, tests, and unresolved issues

A recurrent source of confusion is that not all “entropic acceleration” models operate at the same theoretical level. Covariant GREA derives the additional acceleration from a generally covariant non-equilibrium variational principle and a causal-horizon boundary term (Espinosa-Portales et al., 2021, Garcia-Bellido et al., 2021). By contrast, the surface-term and corrected-entropy cosmologies of the earlier literature are typically effective FRW constructions, and Gao’s Debye model explicitly states that a fully covariant set of field equations is not presented, giving instead a modified Poisson equation and a modified Raychaudhuri/Friedmann relation (Gao, 2010). This difference is technical rather than terminological: it determines whether the theory modifies the geometric sector of Einstein’s equations, the background expansion alone, or only the weak-field force law.

The observational programs are correspondingly different. The cosmological GREA framework predicts a background expansion history nearly indistinguishable from $\delta Q=T\,dS$09CDM at high redshift, but with a slightly earlier coasting point, a time-dependent $\delta Q=T\,dS$10, and a present Hubble rate near $\delta Q=T\,dS$11 when normalized to the same CMB asymptote that gives $\delta Q=T\,dS$12 in $\delta Q=T\,dS$13CDM; the authors identify $\delta Q=T\,dS$14, $\delta Q=T\,dS$15, $\delta Q=T\,dS$16, $\delta Q=T\,dS$17, growth, and ISW diagnostics as future discriminants (Garcia-Bellido et al., 2021). The holographic-dual extension adds the possibility of testing whether cosmic acceleration is boundary-induced and whether the future is Minkowski or de Sitter (García-Bellido, 24 Nov 2025). The relativistic MOND program emphasizes rotation curves at large radii and motivates future lensing analyses and extended galaxy samples (Rostami et al., 7 Nov 2025).

Several open problems remain explicit in the literature. The microphysical nature of the boundary quantum degrees of freedom responsible for $\delta Q=T\,dS$18 is not known in the holographic-dual account (García-Bellido, 24 Nov 2025). A full perturbation analysis is left for future work in the causal-horizon GREA model, even though the paper states that early-universe observables such as BBN, $\delta Q=T\,dS$19, and the sound horizon $\delta Q=T\,dS$20 are unaffected because the background matches the standard expansion at high redshift (Garcia-Bellido et al., 2021). In the relativistic MOND construction, energy conditions, stability, and causality are not analyzed in detail, and lensing predictions are deferred (Rostami et al., 7 Nov 2025). In the entropy-corrected cosmology literature, ultraviolet $\delta Q=T\,dS$21 or inverse-area corrections can drive inflationary behavior but face a graceful-exit problem and fail to recover the low-curvature GR limit in simple forms (Koivisto et al., 2010).

This suggests that the central unresolved issue is not whether entropic terms can be written in a relativistic setting, but whether a single covariant and observationally complete framework can simultaneously account for background expansion, perturbations, lensing, and galaxy dynamics. The present literature demonstrates several technically distinct ways in which entropy, horizon thermodynamics, and relativistic geometry can be coupled. What remains unsettled is which, if any, of those couplings survives precision tests across all scales.