Fluid/Gravity Correspondence

Fluid/gravity turns a uniform black brane into a long-wavelength solution by promoting its temperature and velocity to slowly varying boundary fields. Einstein’s radial constraint equations then become conservation of the renormalized boundary stress tensor, while the remaining radial equations determine derivative corrections and transport. The construction is controlled when gradients are small compared with the local temperature and nonhydrodynamic transients have decayed enough to be treated separately.

Required background. Gauge Theory and String-Theory Preparation supplies large-N stress tensors and the black-brane setting used below.

Helpful background. Quantum Gravity frames the classical bulk expansion, Quantum Information distinguishes coarse hydrodynamic variables from microscopic states, Black-Hole Perturbation Theory supplies nonhydrodynamic modes, and Gravitational Radiation and Waveforms provides complementary nonlinear-evolution conventions.

The boosted brane seed

In ingoing coordinates, a neutral planar AdS$_{d+1}$ Einstein black brane has the following form, with the AdS radius set to one in this metric:

$$ds^2 =-2u_\mu dx^\mu dr -r^2f(br)u_\mu u_\nu dx^\mu dx^\nu +r^2P_{\mu\nu}dx^\mu dx^\nu,$$

with

$$u^\mu u_\mu=-1, \qquad P_{\mu\nu}=\eta_{\mu\nu}+u_\mu u_\nu, \qquad f(br)=1-(br)^{-d}.$$

For constant inverse-temperature scale $b$ and velocity $u^\mu$, this is exactly a boosted equilibrium solution. The horizon lies at $r=1/b$, and $b=d/(4\pi T)$ in this normalization. Ingoing gauge makes future-horizon regularity transparent.

Now promote $b$ and $u^\mu$ to functions of the boundary coordinates. The promoted metric is no longer an exact solution because derivatives such as $\partial_\mu b$ and $\partial_\mu u_\nu$ source Einstein’s equations. Introduce a bookkeeping parameter $\varepsilon\sim\partial/T$ and correct the metric order by order,

$$g_{MN}=g^{(0)}_{MN}[b(x),u(x)] +\varepsilon g^{(1)}_{MN} +\varepsilon^2g^{(2)}_{MN}+\cdots.$$

Constraints become conservation laws

Decompose Einstein’s equations relative to radial hypersurfaces. Two normal projections give the Hamiltonian constraint; one normal and one tangential projection give momentum constraints. The latter impose, at leading nontrivial order,

$$\nabla_\mu T_{(0)}^{\mu\nu}=0, \qquad T_{(0)}^{\mu\nu} =(\epsilon+p)u^\mu u^\nu+p\,g_{(0)}^{\mu\nu}.$$

These are precisely ideal relativistic hydrodynamic equations. They are not extra boundary assumptions pasted onto gravity: the radial constraints require them for a regular corrected bulk solution. The dynamical radial equations then determine $g^{(1)}_{MN}$ once boundary falloff, gauge, and horizon regularity are fixed.

The constraint statement is independent of the radial slice because the Bianchi identity propagates it once the evolution equations hold. This supplies a useful practical check: evaluate the constraint near the horizon, at an intermediate radius, and in the renormalized boundary limit. The three expressions must encode the same conservation law after the induced fields and counterterms are translated. A radial drift signals an unsolved lower-order equation or numerical error.

At first derivative order in the Landau frame,

$$T^{\mu\nu} =(\epsilon+p)u^\mu u^\nu+p\,g_{(0)}^{\mu\nu} -2\eta\,\sigma^{\mu\nu}+O(\partial^2),$$

where $u_\mu T^{\mu\nu}=-\epsilon u^\nu$ defines the frame. To fix the factor of two explicitly, use

$$\sigma^{\mu\nu}=P^{\mu\alpha}P^{\nu\beta} \left(\nabla_{(\alpha}u_{\beta)} -\frac{P_{\alpha\beta}}{d-1}\nabla_\rho u^\rho\right), \qquad \nabla_{(\alpha}u_{\beta)} =\frac{\nabla_\alpha u_\beta+\nabla_\beta u_\alpha}{2}.$$

For a slow transverse flow $u_y(x)$ this gives $\sigma^{xy}=\partial_xu_y/2$, hence $T^{xy}_{\rm visc}=-\eta\partial_xu_y$, the expected shear stress. Solving the regular radial tensor equation fixes $\eta/s=1/(4\pi)$ for the Einstein seed. The systematic construction is given in Bhattacharyya et al. 2008, §§2–5, pp. 7–35.

Radial equations and regularity

At each order, the source is built from lower-order data and decomposes into scalar, vector, and tensor representations in the local rest frame. Boundary normalizability excludes new sources; a gauge choice fixes residual radial-coordinate freedom; future-horizon regularity removes singular homogeneous solutions. The remaining integration constants correspond to redefinitions of $b$ and $u^\mu$ and are fixed by the hydrodynamic frame.

This logic explains why transport appears as an infrared property yet is determined by a full radial solution. The constraint is radially conserved once lower-order equations hold, so it can be read at the horizon or boundary. The constitutive coefficient, however, is fixed by satisfying both endpoints. A concise derivation and the convention-sensitive tensor basis appear in Rangamani 2009, §§4.1–4.5, pp. 24–34.

Checks and domain of validity

Setting all derivatives to zero must recover the exact boosted brane. Linearizing the first-order constitutive relation gives the shear mode

$$\omega=-i\frac{\eta}{\epsilon+p}k^2+O(k^4),$$

which agrees with the lowest gravitational quasinormal pole in the small-$k$ limit. This is an independent check of the constraint equation, horizon boundary condition, and stress-tensor normalization.

The expansion fails when gradients become comparable to temperature, near caustics of the velocity field, or when another long-lived mode must be included. Hydrodynamization can occur before local isotropy or before nonlocal observables equilibrate, because it means only that the stress tensor is well approximated by a constitutive relation. At any finite truncation one must state the first omitted gradient order and monitor the residual rather than assume convergence.

Curved boundary metrics add covariant derivatives and curvature tensors at higher order, while conserved charges add gauge fields and additional transport. The organizing principle survives, but coefficients and allowed tensor structures change. One must not transplant the neutral conformal constitutive relation into a charged, anomalous, nonconformal, or translation-breaking system without rebuilding its field content and frame choice.

Exercises

Show why promoting the equilibrium temperature while keeping the velocity fixed generally violates a radial constraint.

Solution

The leading boundary stress tensor has $\epsilon$ and $p$ determined by the local temperature. If $T(x)$ varies while $u^\mu$ remains a constant timelike vector, then $\nabla_\mu T_{(0)}^{\mu i}$ contains a pressure gradient transverse to the flow. It does not vanish unless the temperature is spatially constant or acceleration and velocity gradients balance it through the Euler equation. Because the bulk momentum constraint equals this divergence, the naively promoted metric cannot solve Einstein’s equations. Adding the velocity response and derivative correction restores the constraint.

References

• Bhattacharyya, Sayantani, Veronika E. Hubeny, Shiraz Minwalla, and Mukund Rangamani. “Nonlinear Fluid Dynamics from Gravity.” Journal of High Energy Physics 02 (2008): 045. DOI.
• Rangamani, Mukund. “Gravity and Hydrodynamics: Lectures on the Fluid-Gravity Correspondence.” Classical and Quantum Gravity 26 (2009): 224003. DOI.