Entropic Gravity

Prerequisites: General Relativity, Entropy. Full path (18 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Some forces in physics have no fundamental field behind them. Stretch a rubber band and it pulls back. The pull does not come from a special rubber force between the molecules. A stretched band has its long polymer chains nearly straight, and there are far fewer ways to arrange a straight chain than a crumpled one. Thermal jiggling pushes the chains towards the crumpled arrangements simply because there are so many more of them. The result is a force that we can feel and measure, and it is a statistical effect: it comes from entropy, the count of microscopic arrangements compatible with what we see.

Entropic gravity is the proposal that gravity is a force of this kind. The idea is that spacetime and its curvature are not fundamental. They emerge from some unknown microscopic degrees of freedom, and gravitational attraction is the tendency of these degrees of freedom to move towards states of higher entropy.

There are hints that point in this direction. Black holes have a temperature and an entropy, and the entropy is proportional to the area of the horizon, not to the volume inside. In 1995 Ted Jacobson showed that the Einstein equation of general relativity follows if one assumes that this area law and the ordinary thermodynamic relation between heat, temperature and entropy hold for every local horizon. In 2010 Erik Verlinde gave a heuristic derivation of Newton's law of gravitation as an entropic force, and in 2016 he argued that the same picture, applied to an expanding universe, could produce effects that are usually attributed to dark matter.

Entropic gravity is speculative. The microscopic degrees of freedom have not been identified, and it is debated whether the arguments contain more than a rewriting of known gravity in thermodynamic language.

• The Case Against Dark Matter | Quanta Magazine

Concrete

Entropic forces

For a system whose entropy $S(x)$ depends on a macroscopic coordinate $x$, kept at temperature $T$, the free energy is $F_{\text{free}} = E - TS$. If the energy does not depend on $x$, the force is

$$F = -\frac{\partial F_{\text{free}}}{\partial x} = T \frac{\partial S}{\partial x}.$$

The force points towards larger entropy. For an ideal polymer chain of $N$ links of length $b$ with end-to-end distance $x$, the number of configurations is approximately Gaussian, $S(x) = \text{const} - \frac{3 k_B x^2}{2 N b^2}$, so

$$F = -\frac{3 k_B T}{N b^2}\, x ,$$

a Hooke's law restoring force whose spring constant grows with temperature.

Verlinde's argument for Newton's law

Verlinde's 2010 argument uses three assumptions borrowed from black hole physics and holography.

1. A test particle of mass $m$ near a "holographic screen" changes the entropy of the screen when it moves a distance $\Delta x$ towards it by $\Delta S = 2\pi k_B \frac{mc}{\hbar} \Delta x$. The value is chosen so that $\Delta x$ is measured in units of the Compton wavelength of the particle.
2. The screen is a closed surface of area $A$ that stores $N = \frac{A c^3}{G\hbar}$ bits of information, the number suggested by the Bekenstein-Hawking entropy $S = \frac{k_B A c^3}{4 G \hbar}$ (up to the factor that relates bits to entropy).
3. The bits are in equilibrium at temperature $T$ and obey equipartition, $E = \frac{1}{2} N k_B T$, where $E = Mc^2$ is the energy of the mass $M$ enclosed by the screen.

Take the screen to be a sphere of radius $R$ around $M$, so $A = 4\pi R^2$. Equipartition gives

$$k_B T = \frac{2 M c^2}{N} = \frac{2 M c^2 G \hbar}{4\pi R^2 c^3} = \frac{G M \hbar}{2\pi R^2 c}.$$

The entropic force on the test particle is then

$$F = T\frac{\Delta S}{\Delta x} = \frac{G M \hbar}{2\pi R^2 c}\cdot \frac{2\pi m c}{\hbar} = \frac{G M m}{R^2},$$

which is Newton's law of gravitation. With the same entropy shift, the Unruh temperature $k_B T = \frac{\hbar a}{2\pi c}$ seen by an observer with acceleration $a$ turns $F = T \Delta S/\Delta x$ into $F = ma$.

The derivation shows that the numbers fit together. It does not identify the microscopic bits, and the assumptions in steps 1 to 3 are inputs. Several of them are motivated by the very gravitational physics they are meant to explain.

• A nice pedagogical introduction to emergent gravity can be found in Chapter 12 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan, especially in Box 12.2

Abstract

The most precise version of the thermodynamic view of gravity is Jacobson's derivation of the Einstein equation as an equation of state. We set $c = k_B = 1$ and keep $\hbar$ and $G$.

Pick a point $p$ in spacetime and a future directed null vector $k^a$ at $p$. Near $p$ consider the local Rindler horizon: the congruence of null geodesics generated by $k^a$ with vanishing expansion $\theta$ and shear at $p$. Let $\lambda$ be an affine parameter with $\lambda = 0$ at $p$, and let $\chi^a = -\kappa \lambda k^a$ be the approximate boost Killing field that is null on the horizon, with surface gravity $\kappa$. Jacobson postulates:

• The horizon carries entropy proportional to its area, $\delta S = \eta\, \delta A$, for some constant $\eta$.
• The temperature is the Unruh temperature of the uniformly accelerated observers just inside the horizon, $T = \frac{\hbar \kappa}{2\pi}$.
• The heat is the boost energy flowing across the horizon, $\delta Q = \int_{\mathcal H} T_{ab}\chi^a d\Sigma^b = -\kappa \int \lambda\, T_{ab} k^a k^b \, d\lambda\, d\mathcal A$, where $d\mathcal A$ is the area element of a cross section. With the orientation in which $\lambda < 0$ before $p$ this is positive for positive energy flux.
• The Clausius relation $\delta Q = T\, \delta S$ holds for all such horizons.

The area change follows from the Raychaudhuri equation. With $\theta = \sigma = 0$ at $p$ and to first order in $\lambda$, $\theta = -\lambda R_{ab}k^a k^b$, so $\delta A = \int \theta\, d\lambda\, d\mathcal A = -\int \lambda R_{ab}k^a k^b\, d\lambda\, d\mathcal A$. Equating $\delta Q$ with $T\eta\,\delta A$ for every null $k^a$ at every point gives

$$T_{ab}k^a k^b = \frac{\hbar \eta}{2\pi} R_{ab}k^a k^b \quad \text{for all null } k^a,$$

which implies $\frac{2\pi}{\hbar\eta} T_{ab} = R_{ab} + f g_{ab}$ for some function $f$. Local conservation $\nabla^a T_{ab} = 0$ and the contracted Bianchi identity $\nabla^a R_{ab} = \frac{1}{2}\nabla_b R$ fix $f = -\frac{1}{2}R + \Lambda$ with a constant $\Lambda$. The result is the Einstein equation

$$R_{ab} - \frac{1}{2} R\, g_{ab} + \Lambda g_{ab} = 8\pi G\, T_{ab}, \qquad G = \frac{1}{4\hbar \eta}.$$

The value $\eta = \frac{1}{4G\hbar}$ is the Bekenstein-Hawking entropy density, so the argument is consistent with black hole thermodynamics. Two features are specific to this route. The cosmological constant appears as an integration constant, not as a parameter of an action. And the field equation holds because the area law and the Clausius relation hold, in the same way that the ideal gas law holds because of the statistics of molecules. Jacobson's derivation uses equilibrium thermodynamics; corrections to the entropy beyond the area law, or departures from equilibrium, are expected to modify the field equations.

Padmanabhan developed a related view in which gravitational field equations, evaluated on horizons, take the form of the thermodynamic identity $T dS = dE + P dV$. Verlinde's 2010 and 2016 proposals start from holographic screens and, in the later work, an additional entropy of de Sitter space that scales with volume instead of area. These constructions are heuristic. A microscopic theory from which the area law and the temperature are derived, and not assumed, is not known.

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Manifold
5. Newton's Second Law
6. Statistics
7. Vector Calculus
8. Curvature
9. Entropy
10. Fiber Bundles
11. Newton's law of Gravity
12. Newtonian Mechanics
13. Parallel Transport
14. Connections
15. Special Relativity
16. Levi-Civita Connection
17. Einstein Equation
18. General Relativity