Emergent & Entropic Gravity
Gravity not as a fundamental force, but as an emergent macroscopic phenomenon arising from entropy and information
1. Gravity and Thermodynamics
The deep connection between gravity and thermodynamics emerged from Bekenstein and Hawking’s work on black holes. The four laws of black hole mechanics mirror the four laws of thermodynamics exactly, with the identifications:
$$T = \frac{\kappa}{2\pi}, \qquad S = \frac{A}{4\ell_P^2}$$
where $\kappa$ is the surface gravity and $A$ is the horizon area. This is not merely an analogy: the Unruh effect shows that any accelerating observer with acceleration $a$ perceives a thermal bath at temperature:
$$T_U = \frac{\hbar\,a}{2\pi\,c\,k_B}$$
This universality of horizon thermodynamics suggests that gravity itself might be a thermodynamic, or more broadly an entropic, phenomenon rather than a fundamental interaction.
2. Jacobson’s Thermodynamic Derivation
In 1995, Jacobson showed that the Einstein field equations can be derived from the Clausius relation $\delta Q = T\,dS$ applied to local Rindler horizons. Consider a point $p$ in spacetime and a local Rindler horizon generated by an approximate Killing vector $\chi^a$. The heat flux through the horizon is:
$$\delta Q = \int_{\mathcal{H}} T_{ab}\,\chi^a\,d\Sigma^b$$
The entropy change uses the Bekenstein-Hawking relation $dS = \delta A/(4\ell_P^2)$, where the area change of the horizon pencil is computed via the Raychaudhuri equation:
$$\frac{d\theta}{d\lambda} = -\frac{1}{2}\theta^2 - \sigma_{ab}\sigma^{ab} - R_{ab}\,k^a k^b$$
For a pencil of generators with initially vanishing expansion ($\theta = 0$ at $p$), the area change to leading order is $\delta A = -R_{ab}\,k^a k^b\,\delta\lambda\,dA$. Substituting into $\delta Q = T\,dS$:
$$T_{ab}\,k^a k^b = \frac{1}{8\pi G}\,R_{ab}\,k^a k^b$$
Since this must hold for all null vectors $k^a$, one obtains $R_{ab} - \frac{1}{2}g_{ab}R + \Lambda\,g_{ab} = 8\pi G\,T_{ab}$, which is exactly the Einstein equation (with $\Lambda$ arising as an integration constant).
3. Verlinde’s Entropic Gravity
Verlinde (2011) proposed that gravity is an entropic force, analogous to the elastic force of a polymer. Consider a test mass $m$ approaching a holographic screen at distance $r$ from a source mass $M$. The screen has temperature $T$ (the Unruh temperature) and encodes information about $M$.
The entropy change when $m$ moves a distance $\Delta x$ toward the screen is postulated to be:
$$\Delta S = 2\pi\,\frac{mc}{\hbar}\,\Delta x$$
The entropic force is $F = T\,\Delta S / \Delta x$. Using the Unruh temperature $T = \hbar a/(2\pi c k_B)$ where $a = GM/r^2$:
$$F = T\,\frac{\Delta S}{\Delta x} = \frac{\hbar\,a}{2\pi c}\cdot\frac{2\pi mc}{\hbar} = ma = \frac{GMm}{r^2}$$
This reproduces Newton’s law of gravitation from purely thermodynamic reasoning. The holographic screen has $N = A/(4\ell_P^2)$ bits of information, and the equipartition theorem $E = \frac{1}{2}N\,k_B T = Mc^2$ relates the screen data to the enclosed mass.
4. Holographic Entanglement and Gravity
A deeper perspective comes from holographic entanglement entropy. The Ryu-Takayanagi formula relates the entanglement entropy of a boundary region $A$ to the area of the minimal surface $\gamma_A$ in the bulk:
$$S_{\rm EE}(A) = \frac{\text{Area}(\gamma_A)}{4G_N}$$
Van Raamsdonk and others have argued that spacetime connectivity is dual to quantum entanglement: the $\text{ER} = \text{EPR}$ conjecture of Maldacena and Susskind identifies Einstein-Rosen bridges with Einstein-Podolsky-Rosen entanglement. The linearized Einstein equations can be derived from the first law of entanglement entropy:
$$\delta S_{\rm EE} = \delta\langle H_{\rm mod}\rangle \quad \Longrightarrow \quad G_{ab} + \Lambda\,g_{ab} = 8\pi G\,T_{ab}$$
where $H_{\rm mod}$ is the modular Hamiltonian. This “gravity from entanglement” program suggests spacetime geometry is an emergent description of quantum information.
5. Emergent Dark Matter
In 2016, Verlinde extended his framework to argue that the apparent effects attributed to dark matter are actually an emergent gravitational phenomenon. In a de Sitter universe with Hubble parameter $H_0$, the elastic response of the entanglement entropy of the vacuum produces an additional gravitational acceleration at large distances:
$$a_D(r) = \sqrt{\frac{c\,H_0}{6}\,a_N(r)} \quad \text{when } a_N \ll a_0 = c\,H_0$$
This recovers Milgrom’s MOND phenomenology with the acceleration scale $a_0 \sim c\,H_0 \sim 10^{-10}\,\text{m/s}^2$ set by the cosmological constant. For a spherically symmetric mass distribution, the apparent dark matter mass enclosed within radius $r$ is:
$$M_D(r) = \sqrt{\frac{c\,H_0\,M_B(r)\,r^2}{6G}}$$
where $M_B(r)$ is the baryonic mass. This prediction has been tested against galaxy rotation curves with mixed results; it works well for some galaxies but faces challenges with galaxy clusters.
6. Padmanabhan’s Emergent Paradigm
Padmanabhan developed a complementary approach where the expansion of the universe is driven by the discrepancy between surface and bulk degrees of freedom. The cosmic evolution equation takes the form:
$$\frac{dV}{dt} = \ell_P^2\,(N_{\rm surf} - N_{\rm bulk})$$
where $V$ is the Hubble volume, $N_{\rm surf} = A_H/(4\ell_P^2)$ is the number of surface degrees of freedom on the Hubble horizon, and $N_{\rm bulk}$ counts the bulk degrees of freedom. This can be shown to be equivalent to the Friedmann equations. The universe expands because $N_{\rm surf} > N_{\rm bulk}$, and reaches de Sitter equilibrium when $N_{\rm surf} = N_{\rm bulk}$.
This further supports the view that gravitational dynamics and cosmological evolution are emergent thermodynamic phenomena, with the Einstein equations playing the role of an equation of state.
Simulation: Entropic Gravity and Emergent Effects
We numerically verify Verlinde’s derivation of Newton’s force from entropy gradients, visualize the Unruh temperature, demonstrate the Raychaudhuri focusing that underlies Jacobson’s derivation, and show the emergent dark matter effect in the deep MOND regime:
Entropic gravity: from entropy to Newton's force
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