Physics:Einstein's Relativity Gravitation from the Theory of Entropicity

Einstein's Relativity and Gravitation from the Theory of Entropicity(ToE)

Theory of Entropicity(ToE) and the Emergence of Gravitational Dynamics

We develop the Theory of Entropicity (ToE), a framework in which entropy is promoted to a dynamical scalar field$S(x)$. In ToE, a single variational principle governing$S(x)$yields both the laws of thermodynamics and the equations of gravitation. In particular, we show that the Clausius relation$\delta Q = T,\delta S$(connecting heat, temperature, and entropy) and Einstein’s field equations (connecting spacetime curvature to stress-energy) emerge from one unified action. This unification treats gravity as an emergent thermodynamic phenomenon: spacetime curvature arises from entropy gradients and their tendency to extremize. By formulating an action for$S(x)$ that includes a canonical kinetic term, a potential, and coupling to matter/geometry, we derive (i) the local second law of thermodynamics as a Noether current, and (ii) the Einstein–Hilbert gravitational dynamics with a specific coupling constant fixed by the entropy–area correspondence. We also incorporate higher-derivative (Fisher information) corrections to the entropy action, indicating how quantum or nonequilibrium effects generate higher-curvature corrections to Einstein gravity. Finally, we discuss observable implications of this theory – showing that it reproduces known tests of general relativity (light bending, perihelion precession, gravitational-wave propagation) while predicting novel, small entropy-production effects that are constrained by gravitational lensing and black hole imaging data.

Introduction

The profound links between gravitation and thermodynamics have been hinted at for decades. Black hole mechanics, for example, closely mirrors the laws of thermodynamics: the area of a black hole horizon behaves as an entropy, and surface gravity as a temperature. Bekenstein and Hawking’s discoveries that black hole area is proportional to entropy and that black holes radiate as black bodies suggested that gravity has deep thermodynamic underpinnings. A milestone in this line of thought was Jacobson’s 1995 result, which showed that one can derive the gravitational field equations by demanding that the Clausius relation $\delta Q = T,dS$ holds for all local Rindler horizons arxiv.org . In Jacobson’s argument, the entropy of a patch of horizon is taken proportional to its area (as in Bekenstein’s area law) and $\delta Q$ is the energy flux crossing the horizon. The requirement $\delta Q = T,\delta S$ – with $T$ identified as the Unruh temperature seen by an accelerating observer – implies the Einstein field equation, $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G,T_{\mu\nu}$, as an equation of state arxiv.org . This remarkable insight suggests that gravity is not a fundamental force but an emergent, macroscopic phenomenon arising from underlying thermodynamic degrees of freedom.

Other developments have reinforced the idea of emergent gravity. Verlinde (2011) proposed that gravity might be an entropic force, arising from the statistical tendency of systems to increase entropy en.wikipedia.org . In his approach, Newton’s law of gravitation was re-derived by assuming that when a test mass moves away from an information-rich holographic screen, the change in entropy induces an attractive force $F = T,\nabla S$ encyclopedia.pub en.wikipedia.org . Similarly, Padmanabhan and others have highlighted that the gravitational field equations have a thermodynamic interpretation: they can be derived by extremizing an entropy functional or by demanding holographic equipartition (i.e. equality between bulk and boundary degrees of freedom) encyclopedia.pub . In all these cases, entropy plays a key role – yet traditionally it has been treated as a book-keeping device (associated with horizons or information screens), rather than as a physical field filling spacetime.

The Theory of Entropicity (ToE) extends these ideas by promoting entropy to a dynamical bulk field $S(x)$ encyclopedia.pub . In ToE, entropy is not confined to horizons; rather, every spacetime point has an associated entropy density $S(x)$ that can vary and propagate. This approach posits that entropy gradients are the fundamental drivers of dynamics – a concept that unifies and generalizes earlier emergent gravity models. The core idea is that a single action principle for the field $S(x)$ is responsible for both (a) the usual laws of thermodynamics (including the second law and familiar entropy formulas), and (b) the laws of gravitation (Einstein’s equations in the appropriate limit). By making entropy an active player in the action, we embed the Clausius relation and related thermodynamic identities into the variational structure of the theory. In short, ToE provides a unified framework in which “gravity is the continuation of thermodynamics by other means.”

In this chapter, we present a comprehensive formulation of the Theory of Entropicity and demonstrate how gravitational dynamics emerge from entropic dynamics. We begin by formulating the Master Entropic Equation (MEE) – the action functional for the entropy field – and outlining its core postulates. Next, we derive the field equations obtained by varying this action with respect to $S(x)$ and the metric $g_{\mu\nu}$. We show how the variation w.r.t. $S$ yields an equation encoding the local second law of thermodynamics, while variation w.r.t. the metric yields a modified Einstein equation that reduces to the standard one when entropy is in equilibrium. We then focus on local Rindler horizons and near-equilibrium processes, deriving the local Clausius relation $\delta Q = T,\delta S$ from the Noether current of $S(x)$ and the assumed thermality of horizons (Kubo–Martin–Schwinger condition). Using the Raychaudhuri equation to relate area changes to the matter stress-energy flux, we recover the Einstein field equations and identify the emergent Newton’s constant in terms of the fundamental parameters of ToE. We also discuss how beyond-equilibrium situations (where $\delta Q \neq T,\delta S$) correspond to corrections to General Relativity, and how Fisher-information-inspired higher derivative terms in the entropy action lead to higher-curvature gravitational terms (consistent with quantum corrections and extended gravitational theories link.aps.org ). Finally, we consider observational and experimental implications of this theory. A viable ToE must pass all classical tests of GR – we show it does, in the appropriate limit – and we outline how small deviations (such as entropy production in strong fields) could be constrained by precision measurements (gravitational wave propagation speed, lensing anomalies, black hole shadow properties, etc.). An appendix is included with technical details: a derivation of the Unruh temperature via the KMS condition in the entropy field context, and the form of the stress-energy tensor arising from the higher-derivative Fisher term.

\section{Theory of Entropicity: The Master Entropic Equation}

\subsection{Postulates and Action Principle} The Theory of Entropicity is founded on a few key postulates that elevate entropy to a fundamental status on par with space, time, and matter:

\begin{itemize} \item \textbf{Entropy as a Field:} There exists a real scalar field $S(x)$ on spacetime, which represents the local entropy density or content at the event $x$. Classical thermodynamic entropy (Boltzmann $S=k_B \ln \Omega$, Gibbs/Shannon entropy, von Neumann entropy, etc.) is identified with appropriate integrals or limits of this field. In particular, horizon entropy is no longer just an abstract area proportionality – it corresponds to an actual field value or distribution on the horizon.

\item \textbf{Dynamics via Canonical Kinetic Term:} The entropy field $S(x)$ has its own dynamics given by a canonical kinetic term in the action. That is, $S(x)$ can propagate through spacetime as a physical degree of freedom. The simplest choice, and the one adopted in ToE, is a standard quadratic kinetic term: \begin{equation}\label{eq:kinetic} \mathcal{L}{\text{kin}} = -\frac{1}{2},\nabla^{\mu}S,\nabla{\mu}S ,, \end{equation} which ensures that $S$ obeys a wave-like (d’Alembertian) equation of motion and carries finite energy and momentum.

\item \textbf{Entropy Potential:} The field may have a self-interaction potential $V(S)$, introduced in the action as $-V(S)$ (with sign chosen so that $V(S)$ acts like a potential energy density). This potential can encode various entropy-related self-interactions or external constraints. For example, one might choose a quadratic (mass-like) term $V(S) = \frac{1}{2}m_S^2 S^2$ giving $S$ a finite correlation length, or a logarithmic form $V(S) = -2k_B \ln|\psi(S)|$ connecting to a wavefunction’s amplitude $\psi$ in a quantum context encyclopedia.pub . In general, $V(S)$ allows flexibility to incorporate phenomena such as entropy saturation or phase transitions in the entropy field.

\item \textbf{Universal Coupling to Matter and Geometry:} The entropy field couples to other fields universally through the trace of the stress–energy tensor. The Master Entropic Equation introduces a term \begin{equation}\label{eq:coupling} \mathcal{L}{\text{int}} = \eta,S(x),T^{\mu}{}{\mu}(x),, \end{equation} where $T^{\mu}{}{\mu} = g^{\mu\nu}T{\mu\nu}$ is the trace of the energy–momentum tensor of matter (and any other relevant fields) encyclopedia.pub . Here $\eta$ is a new coupling constant (with dimensions to be discussed) that controls how the entropy field back-reacts on matter and spacetime. This term is “universal” in that it couples $S$ to all forms of energy–momentum via their trace – effectively making the entropy field a mediator between matter and geometry. Notably, $T^{\mu}{}{\mu}$ includes the effects of $g{\mu\nu}$ (since $T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \delta S_{SM}/\delta g^{\mu\nu}$), so the $S,T^{\mu}{}_{\mu}$ coupling means that $S$ is linked to spacetime curvature indirectly, even if we do not include an explicit $R$ (Ricci scalar) term in the action. \end{itemize}

Given these postulates, we can write down the master action for the Theory of Entropicity. In addition to the standard action $S_{SM}[g,\Phi]$ for Standard Model fields (and any ordinary matter) which depends on the metric $g_{\mu\nu}$ and matter fields $\Phi$, we add the entropy field action.

The Master Entropic Equation (MEE) is the total action: \begin{equation}\label{eq:master-action} \begin{aligned} S_{\text{ToE}}[,g_{\mu\nu}, S, \Phi,] &= S_{SM}[,g_{\mu\nu}, \Phi,] \ &\quad + \int d^4x,\sqrt{-g};\Big[ -\tfrac{1}{2}(\nabla S)^2 ;-; V(S);+; \eta,S,T^{\mu}{}_{\mu};+; \frac{\xi}{2}F(\nabla S)\Big] ,. \end{aligned} \end{equation}

This action encapsulates the Theory of Entropicity. The first line represents all the usual content (matter, radiation, etc., including possibly any bare gravitational terms we might include – though one of the aims is to show that an explicit Einstein–Hilbert $R/(16\pi G)$ term is not needed, since gravity will emerge). The second line contains the new entropy sector: a kinetic term encyclopedia.pub , potential term, and the universal coupling term encyclopedia.pub . We have also included, at the end, a term $\frac{\xi}{2}F(\nabla S)$ denoting possible Fisher-information corrections – higher-derivative terms suppressed by a small dimensionless parameter $\xi$ (or $\lambda$) encyclopedia.pub . The precise form of $F(\nabla S)$ can vary; a simple choice is $F(\nabla S) = (\nabla S)^2$ (which would just renormalize the kinetic term), but more generally $F$ could involve higher powers or derivatives (e.g. $(\Box S)^2$ or $(\nabla S)^4$) to capture subleading corrections arising from short-distance fluctuations of $S$. We will treat such terms as perturbative (of order $\xi \ll 1$) and focus primarily on the leading terms that drive classical dynamics. The presence of $F(\nabla S)$ highlights that the theory can naturally incorporate information-geometric stiffness or quantum corrections: in information theory language, it corresponds to a Fisher information measure for the field configuration encyclopedia.pub .

Explanation of terms in \eqref{eq:master-action}:

\begin{itemize} \item $g_{\mu\nu}$ is the spacetime metric with signature $(-,+,+,+)$.

\item $\Phi$ denotes collectively all matter/gauge fields. Their stress–energy tensor is $T_{\mu\nu} = -\frac{2}{\sqrt{-g}},\delta S_{SM}/\delta g^{\mu\nu}$, and $T^{\mu}{}_{\mu}$ is its trace encyclopedia.pub .

\item $S(x)$ is the entropy field. In units where $k_B=1$, $S$ is dimensionless; otherwise one may keep explicit factors of $k_B$ in formulas. We often work with $k_B=1$ for convenience.

\item $\eta$ is a coupling constant with dimensions [Length]$^2$ in natural units (as will be seen when relating to Newton’s constant). It governs the strength of feedback between entropy and energy: if $\eta=0$, the $S$ field is decoupled from matter/geometry and would be a test field encoding thermodynamic information without affecting gravity.

\item $\xi$ (or $\lambda$ in some references encyclopedia.pub ) is a small dimensionless parameter controlling the higher-derivative Fisher term. We treat $\xi$ as $\mathcal{O}(\hbar)$, i.e. arising from quantum/informational corrections. \end{itemize}

A few crucial features make this action special:

\begin{enumerate} \item It is the first action principle to identify local entropy with a field value $S(x)$ everywhere in spacetime, rather than only as a boundary or global quantity.

\item It endows entropy with dynamics through the kinetic term, meaning disturbances in entropy propagate (one may interpret this as the ability of “information” or disorder to travel, much like waves or particles).

\item It couples entropy to all forms of energy via the single term $\eta S T^\mu{}_\mu$. This introduces back-reaction: if entropy changes, it can induce stress-energy (and hence curvature), and conversely, the presence of energy/mass curves spacetime which influences $S$’s evolution.

\item From this single action, by applying Euler–Lagrange and Noether’s theorem, one can derive a whole suite of fundamental equations – including the classical entropy formulas (Clausius, Boltzmann, Gibbs, Shannon, von Neumann, etc.) and their associated second-law statements encyclopedia.pub encyclopedia.pub . These appear as special cases or identities once $S(x)$ is identified with the appropriate thermodynamic quantity in various contexts (we will see an example with the Clausius relation shortly). \end{enumerate}

In summary, the MEE unifies previously separate notions: it contains the seeds of the first law of thermodynamics ($\delta Q = T,\delta S$ arises from the $S T^\mu{}\mu$ coupling, as we will show), the second law (a Noether current yields $\nabla\mu J_S^\mu \ge 0$, an entropy production law), and the Einstein field equations (emerging from variation of $g_{\mu\nu}$, once $S$ is related to horizon area). the Table below qualitatively contrasts ToE with prior approaches, illustrating how ToE subsumes and extends them encyclopedia.pub

\clearpage \begin{table}[ht] \centering \scriptsize \setlength{\tabcolsep}{4pt} \renewcommand{\arraystretch}{0.9} \caption{Comparison of prior approaches, their key ideas, and the ToE’s extensions} \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{2.5cm} >{\raggedright\arraybackslash}p{3.5cm} >{\raggedright\arraybackslash}X@{}} \toprule Prior Approach & Key Idea & ToE’s Extension \\ \midrule Jacobson (1995) & $\delta Q = T\,dS$ on Rindler horizons & Promote $S$ to a bulk field $S(x)$ with its own dynamics\cite{dArianoFaggin2020}. \

\\[4pt] Verlinde (2011) & $F = T\,\nabla S$ entropic force on test masses & Embed $\nabla S$ in a Lagrangian with $\tfrac12(\partial S)^2$, so entropy gradients follow from field equations\cite{dArianoFaggin2022}. \

\\[4pt] Padmanabhan (2000s) & Entropy functionals as gravitational surface terms & One bulk \emph{master action} for $S(x)$ unifying gravity and thermodynamics\cite{faggin2024}. \

\\[4pt] Frieden’s EPI (1980s–) & Extremize Shannon entropy + Fisher information & Derive both terms from a single variational principle, making $S$ a physical field\cite{holevo1973}. \\ \bottomrule \end{tabularx} \end{table}

\section{Noether Current and the Second Law of Thermodynamics} Before deriving the field equations, it is illuminating to see how the second law of thermodynamics appears in this field-theoretic context. The entropy field action has a global shift symmetry: the Lagrangian in \eqref{eq:master-action} depends only on derivatives of $S$ and $S$ itself, but is invariant under adding a constant to $S$. Specifically, if we transform $S(x) \to S(x) + S_0$ where $S_0$ is a constant (independent of $x$), the terms $\nabla S$, $V(S)$ and $S T^\mu{}\mu$ either remain unchanged or change by a total derivative. This symmetry reflects the fact that only entropy differences matter physically (the zero of entropy is unobservable). By Noether’s theorem, a continuous global symmetry implies a conserved current. The shift symmetry yields the entropy current $J^\mu_S$: \begin{equation}\label{eq:entropy-current} J^\mu_S ;\equiv; \frac{\partial \mathcal{L}}{\partial(\nabla\mu S)} ;=; -,\sqrt{-g};g^{\mu\nu},\nabla_{\nu}S ,, \end{equation} where we have used $\mathcal{L}{\text{kin}} = -\frac{1}{2}(\nabla S)^2$ so that $\partial \mathcal{L}/\partial(\nabla\mu S) = -\sqrt{-g},g^{\mu\nu}\nabla_\nu S$ (the $\sqrt{-g}$ factor appears because we define currents in generally covariant form) encyclopedia.pub . Conservation of this Noether current (in the absence of explicit breaking of the symmetry) means $\nabla_\mu J^\mu_S = 0$. However, when the entropy field’s equation of motion is used (“on shell”), one finds that $\nabla_\mu J^\mu_S$ is actually non-negative: \begin{equation}\label{eq:second-law} \nabla_\mu J^\mu_S ;=; \Sigma_S(x) ;\ge; 0,. \end{equation} This $\Sigma_S(x)$ represents the local entropy production density. In an ideal reversible process (or in vacuum), $\Sigma_S = 0$ and entropy is conserved locally. In any irreversible process (with dissipation), $\Sigma_S > 0$, consistent with the second law (${dS_{\text{total}}}/{dt} > 0$). The detailed form of $\Sigma_S(x)$ can be derived from the $S$ field’s Euler–Lagrange equation; it arises from terms like $V'(S)$ or from the coupling $\eta S T^\mu{}\mu$ if they cause $S$ changes that are not purely advective. Equation \eqref{eq:second-law} thus realizes the second law in a local, covariant form: as entropy flows and evolves in spacetime, it cannot decrease. The formal structure $\nabla\mu J^\mu_S \ge 0$ is analogous to a continuity equation with a source term, indicating that entropy can be produced (but not destroyed) within any small volume. This is a powerful result of the theory: the tendency of entropy to increase is not just an observational fact, but a mathematical consequence of the underlying symmetry and dynamics of $S(x)$.

Physically, $J^\mu_S$ can be thought of as an entropy flux 4-vector. For a fluid or thermodynamic system, one might identify $J^t_S$ (time component) as the entropy density and $\vec{J}_S$ as the entropy flow vector. In equilibrium, $\vec{J}_S = 0$ and $J^t_S$ is uniform (maximum entropy state). Out of equilibrium, entropy can flow ($\vec{J}S \neq 0$), and \eqref{eq:second-law} implies any entropy leaving one region enters another plus a non-negative increase overall. In later sections, we will apply this to a horizon, interpreting $\int J_S^\mu d\Sigma\mu$ as the entropy passing through a horizon segment.

Finally, it is worth noting that from the master action one can recover standard entropy formulas in appropriate limits encyclopedia.pub encyclopedia.pub . For example, one can show that if $S(x)$ is identified with the logarithm of a probability distribution (via $p(x) = e^{-S(x)/k_B}$), the variational extremum of the action yields the Shannon entropy $H = -\sum p\ln p$ and Fisher information terms encyclopedia.pub encyclopedia.pub . The Clausius relation $dS = \delta Q_{\rm rev}/T$ itself can be obtained by considering quasi-static processes in which the $S T^\mu{}\mu$ coupling yields an energy change $\delta Q{\rm rev}$ for a small entropy increment $\delta S$ encyclopedia.pub . All these consistency checks are described in more detail in foundational papers encyclopedia.pub encyclopedia.pub and confirm that the ToE master action is correctly normalized to the familiar thermodynamic conventions.

With the stage set, we now derive the field equations of the theory by varying the master action \eqref{eq:master-action} with respect to $S(x)$ and $g_{\mu\nu}(x)$. We will see the emergence of both thermodynamic and geometric equations of motion, and prepare for the key results linking entropy dynamics to Einstein gravity.

\section{Field Equations from Variational Principle}

\subsection{Entropic Field Equation (Variation with respect to $S$)}

We vary the action \eqref{eq:master-action} with respect to the entropy field $S(x)$, treating $g_{\mu\nu}$ and matter fields as fixed. The Euler–Lagrange equation for $S$ is: \begin{equation}\label{eq:EL-S} \frac{\partial \mathcal{L}}{\partial S} - \nabla_\mu\Big(\frac{\partial \mathcal{L}}{\partial(\nabla_\mu S)}\Big) = 0,. \end{equation} Using the Lagrangian from \eqref{eq:master-action}, we have:

\begin{itemize} \item $\partial \mathcal{L}/\partial S = -\frac{dV}{dS} + \eta,T^{\mu}{}{\mu}$ (since $\partial(\eta S T^\mu{}\mu)/\partial S = \eta,T^\mu{}_\mu$, and the kinetic term has no explicit $S$ dependence, while the potential gives $-V'(S)$).

\item $\partial \mathcal{L}/\partial(\nabla_\mu S) = -\sqrt{-g},g^{\mu\nu}\nabla_\nu S$ as in \eqref{eq:entropy-current} above. \end{itemize}

Taking the divergence: $\nabla_\mu(\sqrt{-g},g^{\mu\nu}\nabla_\nu S) = \sqrt{-g},\nabla_\mu\nabla^\mu S$ (since $\nabla_\mu(\sqrt{-g} A^\mu) = \sqrt{-g},\nabla_\mu A^\mu$ for any vector $A^\mu$).

Plugging into \eqref{eq:EL-S}, dividing out the common $\sqrt{-g}$ factor, we obtain the entropy field equation:

\begin{equation}\label{eq:S-eom} \nabla^\mu \nabla_\mu S ;-; V'(S) ;+; \eta,T^{\mu}{}_{\mu} ;+; \xi,\frac{\delta F}{\delta S};=;0~, \end{equation} where $\delta F/\delta S$ represents the variation of the Fisher correction term with respect to $S$. In the simplest case where $F(\nabla S) = (\nabla S)^2$, $\delta F/\delta S = 0$ (since $F$ depends only on derivatives of $S$ and not $S$ itself), so the Fisher term does not contribute to the $S$-equation at leading order. If $F$ contains higher derivatives (e.g. $F = \Box S$ or similar), then $\xi,\delta F/\delta S$ would add higher-order terms (e.g. $\xi,\Box^2 S$ for $F=(\Box S)^2$) which we assume to be small.

Equation \eqref{eq:S-eom} is a fundamental equation of ToE. It states that the Laplacian of the entropy field is sourced by two terms: (i) $V'(S)$, which typically drives $S$ towards extrema of the potential (for instance, if $V$ has a minimum, $S$ tends to that value in absence of other effects), and (ii) $\eta,T^\mu{}\mu$, which sources entropy based on the local energy–momentum content. In more intuitive form, we can rewrite it as: \begin{equation}\label{eq:S-eom-int} \Box S ;\equiv; \nabla^\mu\nabla\mu S ;=; V'(S);-;\eta,T^\mu{}\mu ;-; \xi,\frac{\delta F}{\delta S},. \end{equation}

If we ignore $V$ and $\xi$ for the moment, this reduces to $\Box S = -\eta,T^\mu{}\mu$. In a simple scenario, consider the trace of stress-energy for a (non-relativistic) fluid or particle rest mass: $T^\mu{}_\mu = -\rho c^2 + 3p \approx -\rho c^2$ (rest energy dominates). Then $\Box S \approx \eta \rho c^2$, implying that mass-energy generates convex (positive Laplacian) entropy – loosely speaking, matter tends to increase $S$ in its vicinity. This is consistent with the idea that gravitating matter induces entropy (for example, matter accreting onto a black hole increases horizon entropy).

A crucial consequence of \eqref{eq:S-eom} is the local second-law mentioned earlier. Taking the divergence $\nabla_\mu$ of \eqref{eq:S-eom} and using the fact that $\nabla_\mu\nabla^\mu\nabla_\nu S$ commutes up to the Riemann tensor (which introduces terms $\sim R_{\nu}{}^{\alpha}\nabla_\alpha S$) and that $\nabla_\mu T^{\mu}{}{\mu}=0$ (from energy–momentum conservation), one can derive $\nabla\mu S$ times a curvature term plus other pieces equals something like $-\eta \nabla_\mu T^\mu{}\mu = 0$. Without delving into those algebraic details, the upshot is that $\nabla\mu S$ is driven by $\nabla_\mu V'(S)$ and higher corrections, which when substituted back show $\nabla_\mu J_S^\mu = V'(S),(\dots) + \xi (\dots)^2 \ge 0$. In short, \eqref{eq:S-eom} implies $\nabla_\mu J^\mu_S \ge 0$, with equality for reversible/quasi-static evolutions of $S$.

Eq. \eqref{eq:S-eom} can be regarded as a generalized heat equation or diffusion equation for entropy, with additional source terms. In the limit of near-equilibrium, $S$ will adjust until $\Box S$ balances $\eta T^\mu{}\mu$, which can be viewed as establishing a relation between entropy and geometry. Indeed, in a steady state ($\partial_t S = 0$, spatial gradients balanced), one might solve $-\nabla^2 S = \eta T^\mu{}\mu$, suggesting $S$ increases in regions of high energy density until an equilibrium is reached. We will see later that a constant $S$ solution (maximum entropy) corresponds to a state of no gravitational acceleration, whereas gradients in $S$ correspond to the presence of gravitational fields (in line with Verlinde’s entropic force picture $g \sim \nabla S$).

Before moving on, let’s remark on a special case: if $T^\mu{}_\mu = 0$ (e.g. radiation or conformal matter in 4D) and if we set $V'(S)=0$ (either $S$ is at an extremum of $V$ or $V$ is negligible), then $\Box S \approx 0$. This means $S$ obeys a free wave equation. One solution is $S = ,$constant, which corresponds to a state of uniform entropy (no entropy gradients, no forces). Another is a propagating entropy wave or fluctuation. The existence of wave-like solutions for $S$ underscores that information/entropy can propagate dynamically in this theory, rather than being constrained to increase monotonically everywhere. Only when sources or potentials act does entropy production ($\Sigma_S$) occur.

\section {Gravitational Field Equation (Variation with respect to $g_{\mu\nu}$)} Next, we vary the action with respect to the metric $g_{\mu\nu}(x)$ to obtain the gravitational field equations. We treat $S(x)$ and matter fields as fixed, and consider an arbitrary variation $\delta g_{\mu\nu}$. It is convenient to write the total action as:

\[ S_{\mathrm{ToE}} \;=\; \int d^4x \;\sqrt{-g}\;\mathcal{L}_{\mathrm{tot}}\bigl(g_{\mu\nu},S,\Phi\bigr)\,. \] where $\mathcal{L}_{\text{tot}} = \mathcal{L}_{SM}(g,\Phi) + \mathcal{L}_{S}(g,S)$. Then: