Dissipative hydrodynamic actions and horizon symmetries in gravity

Technical Summary: Dissipative Hydrodynamic Actions and Horizon Symmetries in Gravity

Problem Statement
The paper addresses the challenge of constructing an effective action principle for dissipative hydrodynamics within the context of the AdS/CFT correspondence. While classical hydrodynamics describes the long-distance, late-time dynamics of conserved quantities, formulating it as an action is non-trivial due to dissipation, necessitating the Schwinger-Keldysh (SK) formalism which doubles the degrees of freedom. Although actions for dissipative hydrodynamics of conserved $U(1)$ charges have been derived from gravity (specifically in [15]), a consistent derivation of a dissipative hydrodynamic action describing the stress tensor dynamics (energy-momentum conservation) for a holographic quantum field theory dual to AdS$_4$ gravity has remained an open problem. Previous attempts [34–36] were limited to one-sided geometries and could not systematically include dissipative effects. Furthermore, there is a conjectured relationship between hydrodynamic actions and many-body quantum chaos, suggesting that such actions should possess specific symmetries related to black hole horizon diffeomorphisms [42–44].

Methodology
The authors employ the Crossley-Glorioso-Liu (CGL) formalism [15], which involves solving bulk gravitational dynamics on a contour where the radial coordinate is analytically continued. This contour possesses two asymptotic boundaries (branches $\sigma=1,2$) and a horizon.

1. Background and Perturbations: The study focuses on the AdS$_4$-Schwarzschild black brane. The authors consider linearized metric perturbations around the thermal equilibrium state, decomposing them into transverse and longitudinal channels based on rotational symmetry in the $(x,y)$ plane.
2. Partially On-Shell Construction: The hydrodynamic action is derived by constructing "partially on-shell" solutions. The authors impose a subset of the Einstein equations (the dynamical equations) but leave the constraint equations (momentum and energy conservation) unimposed. These constraints will later emerge as the equations of motion for the hydrodynamic fields.
3. Hydrodynamic Variables: The dynamical fields of the effective action, denoted $\xi^\sigma_\mu$, are identified as relative diffeomorphisms between the black hole horizon and the asymptotic boundaries of the CGL contour. These are defined as Wilson line-like integrals of the metric perturbations.
4. Horizon Boundary Conditions: A crucial step is imposing specific boundary conditions at the black hole horizon ($r=r_0$) to ensure the variational principle is well-defined and to relate the two branches of the contour. The authors primarily study a set of conditions (Eq. 1.1) that include:
   • Vanishing of time-space components: $\delta g^\sigma_{t\mu}(r_0) = 0$.
   • A specific relation between radial and time derivatives: $2r_0^2 \partial_r \delta g^\sigma_{tt}(r_0) = \partial_t \delta g^\sigma_{xx}(r_0) + \partial_t \delta g^\sigma_{yy}(r_0)$.
   • Continuity of specific metric components ($\delta g_{xy}$ and $\delta g_m = \delta g_{xx} - \delta g_{yy}$) after analytic continuation around the horizon.
5. Derivative Expansion: The calculation is performed to quadratic order in the amplitude of perturbations and to first order in the hydrodynamic derivative expansion.

Key Contributions and Results

• Derivation of the Action: The authors explicitly derive the Schwinger-Keldysh hydrodynamic action $S_{\text{hydro}}[\xi^\sigma_\mu, \delta g^{(s)\sigma}_{\mu\nu}]$ for the stress tensor. The action depends on the hydrodynamic fields $\xi^\sigma_\mu$ and external sources (boundary metric perturbations) $\delta g^{(s)\sigma}_{\mu\nu}$.
• Manifest Diffeomorphism Invariance: The action is expressed in terms of gauge-invariant combinations $B^\sigma_{\mu\nu}$, which represent the perturbations of the induced boundary metric under diffeomorphisms. This formulation ensures the action is invariant under boundary diffeomorphisms.
• Consistency Checks:
  • Equations of Motion: Varying the derived action with respect to $\xi^\sigma_\mu$ yields equations of motion that are exactly equivalent to the unimposed Einstein constraint equations (energy and momentum conservation).
  • Green's Functions: Integrating out the hydrodynamic fields yields the generating functional for real-time two-point functions. The authors explicitly compute the retarded and symmetric Green's functions for the stress tensor and confirm they match known results derived from variational hydrodynamics and the fluctuation-dissipation relation, including the correct shear viscosity $\eta = s/4\pi$.
• Horizon Symmetries and Quantum Chaos:
  • The authors identify the diffeomorphisms that preserve their chosen horizon boundary conditions. These include time-independent shifts and exponentially growing/decaying modes in time.
  • They find that the time-independent shifts are symmetries of the action (verified to the order of their calculation).
  • Regarding the exponentially growing modes (conjectured to be related to quantum chaos [44]), the authors note a distinction: the profile of the growing mode in their setup differs from that proposed in [44].
  • They investigate an alternative set of horizon boundary conditions (Eq. 1.5) which are preserved precisely by the symmetries of [44]. However, they find that these alternative conditions are not manifestly consistent with the variational principle at the horizon for the quadratic Einstein-Hilbert action. Consequently, while the first-order action remains identical, there is a subtlety: the hydrodynamic equations of motion derived from this action are not guaranteed to be equivalent to the full Einstein constraint equations for the exponentially growing modes (specifically at frequencies $\omega = 2\pi T i$).

Significance and Claims
The paper claims to provide the first consistent derivation of a dissipative hydrodynamic action for stress tensor dynamics from gravitational dynamics in the spirit of the CGL prescription [15].

• Technical Breakthrough: It demonstrates that the CGL contour method, previously applied to $U(1)$ charges, can be successfully extended to the more complex stress tensor sector, systematically incorporating dissipation.
• Symmetry Analysis: The work provides a concrete framework to test conjectures linking hydrodynamic symmetries to quantum chaos. It highlights a tension between satisfying the variational principle (which ensures the hydrodynamic EOMs match the bulk Einstein equations) and realizing the specific horizon symmetries proposed in [44]. The authors suggest that the exponentially growing symmetry of the hydrodynamic action may not be identical to the one conjectured in [44] if one strictly adheres to the variational principle, or that different formulations of the action may possess different symmetries.
• Future Directions: The authors note that their calculation is limited to first order in derivatives and quadratic order in perturbations. They suggest that a non-perturbative (in derivatives) action is required to fully verify the symmetries conjectured in [44] and to understand the non-linear generalization of their approach. They also propose that their methodology could be extended to other holographic theories and higher dimensions.

In summary, the paper establishes a robust prescription for deriving dissipative hydrodynamic actions from gravity, validates it against known hydrodynamic correlators, and uses this framework to probe the intricate relationship between horizon symmetries, variational principles, and quantum chaos, revealing subtle distinctions in how these symmetries are realized.