⚛️ general relativity

Relativistic MOND Theory from Modified Entropic Gravity

This paper proposes a relativistic extension of Modified Newtonian Dynamics (MOND) derived from temperature-corrected entropic gravity, which successfully reproduces galactic rotation curve data for NGC 3198 with accuracy comparable to dark matter models while offering a unified relativistic framework for MOND phenomenology.

Original authors: A. Rostami, K. Rezazadeh, M. Rostampour

Published 2026-08-18

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CC BY 4.0

Original authors: A. Rostami, K. Rezazadeh, M. Rostampour

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, astronomers have faced a stubborn puzzle: stars at the outer edges of galaxies spin far faster than the visible matter should allow. If you calculate the gravitational pull based only on the stars and gas we can see, those outer stars should be flying off into the void. Instead, they stay in orbit, suggesting either that invisible matter is holding them in place or that our understanding of gravity itself is incomplete. The invisible matter solution, known as dark matter, has become the standard explanation, filling the universe with a ghostly substance that interacts only through gravity. The alternative, proposed by physicist Mordehai Milgrom, suggests that gravity behaves differently when the pull becomes extremely weak, such as in the deep outskirts of a galaxy. This idea, called Modified Newtonian Dynamics, or MOND, fits the observed speeds of stars remarkably well but has struggled to fit into the broader framework of modern physics, which describes gravity as the curvature of spacetime. Without a relativistic version, MOND cannot easily explain other cosmic phenomena, like how light bends around massive objects.

A team of researchers in Iran has now taken a significant step toward solving this gap by weaving MOND into a theory that views gravity not as a fundamental force, but as a consequence of thermodynamics. Their work builds on the idea that gravity emerges from the way information is stored on the boundaries of space, much like a hologram. In this view, the temperature of the universe plays a crucial role. The researchers proposed that the standard rules governing how energy is shared among the microscopic bits of information on these boundaries change when the temperature is very low. By introducing these temperature-dependent corrections, they derived a new set of equations that describe how spacetime curves in the presence of matter. These equations naturally produce a specific threshold of acceleration, below which gravity behaves differently, matching the behavior observed in the outer regions of galaxies without needing to invent invisible matter.

To test their new theory, which they call Relativistic MOND, the team applied it to the real-world data of a spiral galaxy known as NGC 3198. They compared their predictions against two other models: the standard Newtonian view using only visible matter, and the widely accepted dark matter model. The results showed that the standard Newtonian view failed to explain the high speeds of the outer stars, as expected. However, both the dark matter model and the new Relativistic MOND model fit the data very well. The researchers found that while the dark matter model provided the best overall fit, the new theory performed particularly well at the very edges of the galaxy, beyond twenty thousand light-years from the center. In this distant region, the new model matched the observed star speeds even more closely than the dark matter model did.

The study suggests that the strange behavior of stars in the outer galaxy might not be caused by hidden mass, but by a subtle shift in how gravity works when the pull becomes incredibly weak. This shift is linked to the temperature of the vacuum of space, a concept that connects the motion of stars to the fundamental thermodynamic properties of the universe. While the theory successfully explains the rotation of this specific galaxy, the authors acknowledge that it is still a work in progress. It has not yet been tested against other cosmic challenges, such as how galaxy clusters hold together or how light bends around them. Nevertheless, the work offers a compelling, mathematically consistent alternative to dark matter for galactic scales, grounding the mysterious behavior of distant stars in the familiar laws of heat and information. It invites a rethinking of the invisible scaffolding of the universe, proposing that what we perceive as missing mass might simply be a sign that gravity changes its tune in the cold, quiet reaches of space.

Technical Summary: Relativistic MOND Theory from Modified Entropic Gravity

Problem Statement
The paper addresses the persistent discrepancy between Newtonian dynamics and astrophysical observations, specifically the inability of standard Newtonian dynamics (ND) to reproduce galactic rotation curves without invoking non-luminous dark matter (DM). While the Modified Newtonian Dynamics (MOND) paradigm successfully explains these curves phenomenologically, it lacks a consistent relativistic foundation, struggling to account for phenomena like gravitational lensing and cosmological evolution. Conversely, the DM hypothesis, while successful on large scales, introduces a non-luminous component without a fundamental microscopic origin. The authors seek to derive a relativistic extension of MOND (RMOND) from first principles, specifically within the framework of entropic gravity, to provide a thermodynamic underpinning for the observed deviations from Newtonian dynamics.

Methodology
The authors construct their model by modifying Verlinde's entropic gravity framework through the introduction of temperature-dependent corrections to the equipartition law of energy on a holographic screen.

Modified Equipartition Law: Drawing on the statistical mechanics of condensed matter (specifically the Debye model for heat capacity), the authors propose that the standard equipartition law ( $E = \frac{1}{2}Nk_B T$ ) receives corrections at low temperatures. They model the holographic screen's information bits as interacting fields with vibrational modes, leading to an energy expression involving a two-dimensional Debye function, $D_2(x)$ , where $x$ is a dimensionless variable related to the ratio of the Debye temperature to the system temperature.
Derivation of Modified Einstein Equations: By identifying the Unruh temperature ( $T \propto a$ ) with the acceleration of a static observer, the authors define a thermodynamic mass ( $M_{th}$ ) that includes a temperature-dependent correction function $f(T)$ . Equating this thermodynamic mass to a geometric mass defined via a modified Komar integral, they derive generalized Einstein field equations. In these equations, the geometric sector acquires explicit thermal corrections, resulting in:
$f(T)R_{ab} - e^{-2\phi}\xi_b \nabla_a \xi_c \nabla_c f(T) = 8\pi G \left(T_{ab} - \frac{1}{2}T g_{ab}\right)$
where $f(T)$ acts as a scalar correction function.
Weak-Field Solution: The authors solve these modified equations for a static, spherically symmetric spacetime in the weak-field, low-temperature regime. They approximate the Debye function for low temperatures (low accelerations) and derive analytical forms for the metric coefficients $A(r)$ and $B(r)$ .
Phenomenological Comparison: The resulting metric yields a modified gravitational acceleration. The authors derive the effective MOND interpolation function $\mu_{eff}$ from this metric to demonstrate consistency with standard MOND phenomenology. Finally, they confront the model with observational data from the galaxy NGC 3198, performing a Bayesian parameter inference (using MCMC) to compare the RMOND model against a baryons-only Newtonian model and a standard Dark Matter halo model.

Key Contributions

Relativistic Derivation of MOND: The paper provides a derivation of MOND-like behavior directly from a modified entropic gravity framework, offering a relativistic underpinning for the phenomenological acceleration scale $a_0$ .
Thermodynamic Origin of $a_0$ : The model naturally generates a characteristic acceleration scale ( $a_t$ ) arising from the transition between high-temperature (standard GR) and low-temperature (modified) regimes of the equipartition law.
Metric Formulation: The authors obtain a corrected metric for static, spherically symmetric spacetimes that smoothly approaches Minkowski space at large radii but deviates significantly in the weak-field limit, reproducing flat rotation curves without dark matter.
Conservation Laws: The paper analyzes the Bianchi identities within this framework, showing that the modified equations imply a non-conservation of the matter energy-momentum tensor ( $\nabla_a T^{ab} = J^b$ ), where $J^b$ represents an exchange current between matter and the effective thermal/gravitational sector.

Results

Theoretical Consistency: The derived effective interpolation function $\mu_{eff}$ satisfies the standard MOND conditions: $\mu \to 1$ in the high-acceleration (Newtonian) limit and $\mu \propto |a|/a_0$ in the low-acceleration limit.
Observational Fit (NGC 3198):
- The RMOND model fits the rotation curve data of NGC 3198 significantly better than the baryons-only Newtonian model.
- While the standard Dark Matter (DM) model yields the lowest overall $\chi^2$ value, the RMOND model provides a notably improved fit compared to the DM model at large galactocentric radii ( $r \gtrsim 20$ kpc).
- The transition acceleration $a_t$ derived from the fit is of the order of $10^{-11}$ m/s $^2$ , slightly lower than the canonical Milgrom acceleration ( $a_0 \sim 10^{-10}$ m/s $^2$ ), consistent with the model's definition of the transition to the modified regime.
Parameter Constraints: The MCMC analysis constrained the free parameters, including the disk surface density, disk scale radius, and the RMOND-specific parameters ( $\epsilon$ and transition radius $r_t$ ). The model requires only the baryonic mass of the galaxy disk, avoiding the need for a dark matter halo profile.

Significance and Claims
The authors claim that their work establishes a viable relativistic framework for MOND phenomenology grounded in thermodynamic principles. By embedding MOND within entropic gravity, the model offers a resolution to the dark matter mystery by suggesting that observed galactic dynamics arise from modifications to gravity itself, driven by temperature-dependent statistical corrections to the holographic screen's degrees of freedom.

The paper emphasizes that while the current model successfully explains galactic rotation curves in the weak-field regime without dark matter, it is modest regarding its scope. It explicitly acknowledges that the theory, in its current form, does not yet address cluster-scale dynamics, gravitational lensing, or the Bullet Cluster. The authors position this work as a foundational step, motivating future observational tests (such as lensing and extended galaxy samples) and theoretical extensions to cosmology and quantum gravity. The significance lies in providing a "dark-matter-free" explanation for galactic dynamics that is consistent with general relativity in the strong-field limit and naturally transitions to MOND behavior in the weak-field limit.

Technical Summary: Relativistic MOND Theory from Modified Entropic Gravity

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