Hydrodynamic and Thermodynamic Gravity

Hydrodynamic and thermodynamic approaches interpret the Einstein equation as an infrared constitutive law rather than a microscopic equation. Their strength is universality: conservation, locality, horizon entropy, and equilibrium can sharply constrain long-wavelength dynamics. Their limitation is the same universality: deriving the equation need not identify the underlying degrees of freedom, quantum state space, or ultraviolet completion.

Required background. Quantum Information supplies entropy, reduced states, and entanglement equilibrium.

Helpful background. Flat-Spacetime Quantum Field Theory supplies stress tensors and vacuum correlations; Einstein Equations supplies the target constitutive relation; Quantum Gravity supplies comparison criteria; and Causality and Global Structure supplies local horizons and causal diamonds.

The local-horizon equation of state

Choose an event $p$ and a small spacelike two-surface through it. Use the past branch of its local Rindler horizon, with future-directed affine generator $k^\mu$, $\lambda\le0$, and $\lambda=0$ at $p$. Choose the flux orientation $d\Sigma^\nu=k^\nu d\lambda\,dA$. Near the horizon, the approximate boost Killing field is $\chi^\mu=-\kappa\lambda k^\mu$, where $\kappa\gt0$ is its acceleration scale. These choices make the displayed heat and area changes positive for positive null energy and focusing curvature on this branch.

The heat flux through the horizon is

$$\delta Q=\int T_{\mu\nu}\chi^\mu\,d\Sigma^\nu =-\kappa\int \lambda T_{\mu\nu}k^\mu k^\nu\,d\lambda\,dA.$$

The Unruh temperature is $T=\kappa/(2\pi)$. If entropy is proportional to area, $\delta S=\eta\,\delta A$, the Raychaudhuri equation supplies the area change. Choose local equilibrium so that expansion and shear vanish at $p$. To leading order,

$$\theta(\lambda)=-\lambda R_{\mu\nu}k^\mu k^\nu+O(\lambda^2), \qquad \delta A=-\int \lambda R_{\mu\nu}k^\mu k^\nu\,d\lambda\,dA.$$

Imposing $\delta Q=T\delta S$ for every local null direction yields

$$\left(R_{\mu\nu}-\frac{2\pi}{\eta}T_{\mu\nu}\right)k^\mu k^\nu=0.$$

Therefore the tensor in parentheses is proportional to $g_{\mu\nu}$. Stress-energy conservation and the contracted Bianchi identity integrate the proportionality to give the Einstein equation with an undetermined cosmological constant. Matching $\eta=1/(4G)$ gives $8\pi G$ Jacobson 1995, pp. 1260–1263.

What is assumed rather than derived

The result assumes a local Lorentzian metric, null congruences, QFT stress energy, the Unruh temperature, an area entropy density, near-equilibrium conditions, and the Clausius relation for all local horizons. It derives the field equation from that package; it does not derive spacetime, the entropy density, or the microscopic carriers of entropy.

This distinction makes the hydrodynamic analogy precise. Fluid equations follow from conservation and constitutive data across many molecular systems. Likewise, Einstein dynamics may be insensitive to microscopic details if diffeomorphism invariance, universal coupling, and local equilibrium dominate the infrared. But viscosity, dispersion, noise, and extra slow variables reveal microphysics in a fluid. Quantum gravity should analogously predict controlled higher-curvature, stochastic, or nonequilibrium corrections.

If the entropy depends on curvature, equilibrium variation generally yields higher-curvature equations or requires entropy production terms. Shear contributes through $\sigma_{\mu\nu}\sigma^{\mu\nu}$ beyond the chosen equilibrium order. Thus the Einstein equation is not the unique outcome of every entropy functional.

Entanglement equilibrium and induced dynamics

Entanglement-equilibrium arguments consider a small causal diamond and demand stationarity of generalized entropy at fixed volume. The matter contribution is related to a modular Hamiltonian; the geometric area variation carries curvature. Under restricted vacuum and small-ball assumptions, the condition yields the linearized Einstein equation Jacobson 2016, pp. 1–4. This route replaces literal heat flow with a variational statement, but still assumes a universal ultraviolet entropy structure.

Induced-gravity ideas integrate out quantum fields and obtain gravitational terms in an effective action. They demonstrate that metric dynamics can arise radiatively. The induced Newton coefficient, cosmological term, and higher-curvature terms depend on field content, regulator, and microscopic completion; without a mechanism controlling those quantities, induction is not a unique prediction.

These mechanisms can complement rather than exclude one another. Microscopic quantum correlations may generate an entropy functional, whose long-wavelength equilibrium variation yields a constitutive gravitational equation. Establishing that chain requires using the same physical algebra and state at every step.

A useful adverse check reverses the usual argument. Start with a modified gravitational action, compute its appropriate horizon-entropy functional, and ask whether the proposed thermodynamic rule recovers the modified field equation rather than forcing the Einstein form. If it fails, the rule was tied to area entropy rather than a universal principle. A second check perturbs away from equilibrium: entropy production and shear terms must appear with signs compatible with the generalized second law. These tests distinguish a derivation with controlled scope from a mnemonic rewriting of an already assumed equation.

Results and open questions through 2026-08-20

This discussion reflects results available through 2026-08-20. There is a robust mathematical result that the stated local-equilibrium premises imply Einstein-like dynamics. The frontier question—whether those premises follow from a specific microscopic quantum-gravity theory—remains unresolved. Competing positions interpret the derivations as evidence that gravity is intrinsically thermodynamic, as a consistency reformulation of general relativity, or as an infrared universality principle compatible with many ontologies.

Argument or mechanism	Evidence or result	Scope and limitation
Local Clausius relation	Derives Einstein’s equation for every local horizon with area entropy	Assumes metric causal structure, equilibrium, temperature, and entropy density
Entanglement equilibrium	Derives linearized dynamics for restricted small-diamond state variations	Modular-energy and ultraviolet-entropy assumptions are substantial
Induced gravitational action	Quantum fluctuations generate $R$, $\Lambda$, and higher-curvature terms	Coefficients and signs depend on field content and regularization
Hydrodynamic universality	Explains why infrared equations can ignore microscopic details	Does not identify microstates or guarantee a quantum completion

The formulation axis includes equilibrium versus nonequilibrium states, area versus generalized entropy, local horizons versus finite causal diamonds, and whether the metric is assumed or reconstructed. Conclusions should not be transferred across those regimes without a derivation.

The conclusions could change if a microscopic model derives the area coefficient, Unruh response, and local equilibrium law in one controlled continuum limit, or if a counterexample satisfies the premises but fails to yield universal coupling; the interpretation could also change with a quantified nonequilibrium correction that distinguishes competing microphysics.

Exercise

Starting from $\eta=1/(4G)$, compute $2\pi/\eta$ and show how the null-contracted relation determines the Einstein equation up to a cosmological constant.

Solution

The coefficient is

$$\frac{2\pi}{\eta}=8\pi G.$$

If $(R_{\mu\nu}-8\pi G T_{\mu\nu})k^\mu k^\nu=0$ for every null $k^\mu$, then $R_{\mu\nu}-8\pi G T_{\mu\nu}=F g_{\mu\nu}$ for some scalar $F$. Taking a divergence and using $\nabla^\mu T_{\mu\nu}=0$ plus $\nabla^\mu R_{\mu\nu}=\tfrac12\nabla_\nu R$ gives $\nabla_\nu(F-\tfrac12R)=0$. Hence $F=\tfrac12R-\Lambda$, where $\Lambda$ is constant, and

$$R_{\mu\nu}-\frac12Rg_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}.$$

The thermodynamic argument fixes the coupling through the entropy density but leaves $\Lambda$ as an integration constant.

References

• Jacobson, Ted (1995), “Thermodynamics of Spacetime: The Einstein Equation of State,” Physical Review Letters 75, 1260–1263, arXiv:gr-qc/9504004.
• Jacobson, Ted (2016), “Entanglement Equilibrium and the Einstein Equation,” Physical Review Letters 116, 201101, arXiv:1505.04753.