⚛️ general relativity

Horizon symmetries of exact hydrodynamic actions

This paper demonstrates that horizon symmetries of AdS black holes are realized as gauge symmetries within hydrodynamic actions for the dual stress tensor, explicitly deriving these actions for JT and AdS $_3$ gravity to confirm their connection to effective theories of maximal chaos.

Original authors: Mike Blake, Arpit Das, Richard A. Davison

Published 2026-10-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mike Blake, Arpit Das, Richard A. Davison

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, there is a persistent divide between the very large and the very small. On one side, gravity rules the cosmos, governing the motion of stars and the shape of black holes. On the other, quantum mechanics dictates the behavior of atoms and subatomic particles, where certainty gives way to probability. For decades, physicists have sought a way to unite these two worlds, and one of the most promising bridges is a concept known as holography. This idea suggests that a universe with gravity can be mathematically equivalent to a universe without gravity, living on a lower-dimensional boundary. In this framework, the chaotic, turbulent motion of fluids in the boundary universe is deeply connected to the geometry of black holes in the gravitational universe. Understanding how these two descriptions talk to each other is crucial, especially when trying to understand how information behaves in extreme environments, such as the event horizon of a black hole.

A team of researchers has now taken a significant step in clarifying this connection by deriving exact mathematical descriptions for how fluids behave in these holographic universes. Specifically, they focused on two simplified models of gravity: one that exists in two dimensions and another in three dimensions. In these models, the fluid is not a physical substance like water or air, but rather a representation of energy and momentum in a quantum system. The researchers wanted to understand the rules, or symmetries, that govern this fluid when it is in a state of thermal equilibrium, much like a hot gas sitting still in a container. They were particularly interested in how the surface of a black hole, known as the horizon, influences the behavior of this fluid.

To do this, the scientists constructed a new kind of mathematical action, which is essentially a formula that describes how a system evolves over time. They started with the standard equations that describe gravity in these simplified universes and asked a specific question: what happens if we only solve part of these equations, leaving some details flexible? By doing so, they could isolate the specific movements that correspond to the fluid's behavior. A key part of their method involved making a choice about how to treat the edge of the black hole, the horizon. In physics, the rules you set at a boundary can change the entire solution. The researchers chose a set of rules that did not require adding extra, artificial terms to the equations at the horizon. This choice was motivated by the desire to keep the mathematical description as natural and consistent as possible.

When they applied these rules, they found that the resulting fluid description possessed a very specific set of symmetries. These symmetries are transformations that can be applied to the system without changing its physical outcome. The researchers discovered that these symmetries are not just abstract mathematical curiosities; they are directly linked to the geometry of the black hole horizon itself. In simpler terms, the way the fluid can wiggle and shift without changing its energy is exactly the same as the way the black hole's surface can be deformed without changing its fundamental nature. This confirmed a long-standing hypothesis that the chaotic behavior of black holes is rooted in these hidden symmetries of their horizons.

The team demonstrated this by working through the mathematics for both the two-dimensional and three-dimensional cases. In the two-dimensional case, their result matched a well-known formula called the Schwarzian action, which has been used to describe the chaotic dynamics of certain quantum systems. However, they arrived at this result through a new path, showing that it emerges naturally from the geometry of the black hole horizon. In the three-dimensional case, they derived a new, more complex formula that had not appeared in previous literature. This new formula also described the fluid's behavior correctly and shared the same type of horizon symmetries.

Importantly, the researchers showed that the specific symmetries they found depend entirely on the choice of boundary rules at the horizon. They tested an alternative set of rules, one that is more conventional and involves fixing the shape of the horizon rigidly. Under these different conditions, the fluid description changed. The resulting formula was different, and the symmetries were no longer the same. The exotic, exponentially growing modes that were present in the first case disappeared. This finding is significant because it proves that the connection between black hole horizons and the chaotic behavior of fluids is not a universal, unchangeable law, but rather a consequence of how we choose to define the system at its boundaries.

The paper also addressed the nature of these symmetries in the context of quantum chaos. In chaotic systems, information about the initial state of a system can spread out and become scrambled very quickly. This scrambling is often measured by a quantity called an out-of-time-ordered correlator, which tracks how much two measurements interfere with each other if taken at different times. The researchers showed that the symmetries they identified are directly responsible for the specific pattern of this scrambling. The mathematical modes that represent these symmetries grow exponentially over time, mirroring the way chaos develops in the system. This provides a concrete, geometric explanation for why black holes are such efficient scramblers of information.

By deriving these exact actions, the researchers have provided a clearer picture of how the microscopic rules of gravity translate into the macroscopic behavior of fluids. They have shown that the symmetries of a black hole's horizon are not just a feature of the black hole itself, but are inherited by the dual fluid description. This work bridges the gap between the geometric description of gravity and the effective theory of hydrodynamics, offering a precise mathematical framework for understanding how chaos arises in these systems. While the models used are simplified, the principles uncovered offer a robust foundation for exploring more complex, realistic scenarios in the future. The study confirms that the choice of boundary conditions is a powerful tool that shapes the physical laws we observe, revealing that the symmetries governing chaos are deeply rooted in the geometry of spacetime.

Technical Summary: Horizon Symmetries of Exact Hydrodynamic Actions

Problem Statement
The macroscopic thermal dynamics of the stress tensor in quantum field theories (QFTs) are governed by hydrodynamics. Recent developments have established that an action principle for this effective theory can be constructed, where the degrees of freedom are diffeomorphism fields mapping between a fluid spacetime and physical spacetimes. A key motivation for this research is the study of holographic QFTs, where hydrodynamic degrees of freedom arise as relative diffeomorphisms between the horizon and asymptotic boundaries of anti-de Sitter (AdS) space.

While generic holographic systems require derivative expansions to construct hydrodynamic actions, special cases exist where exact actions can be derived directly from gravity: specifically, 2-dimensional Jackiw-Teitelboim (JT) gravity and pure gravity in AdS $_3$ . In these theories, the field theory stress tensor is highly constrained by symmetries, rendering the hydrodynamic theory non-dissipative and valid at all scales. A central conjecture posits that the universal form of out-of-time-ordered correlators (OTOCs) in holographic theories, which characterize quantum chaos, is inherited from characteristic symmetries in their hydrodynamic actions. These symmetries were conjectured to correspond to specific geometric symmetries of black hole horizons. However, explicit hydrodynamic actions realizing these horizon symmetries had not been constructed.

Methodology
The authors employ the approach introduced in [14] to derive exact hydrodynamic actions for JT and AdS $_3$ gravity. The methodology involves the following steps:

Partial On-Shell Evaluation: The gravitational action (exterior to the black hole) is evaluated for partially on-shell configurations of the fields, working to quadratic order in perturbations around the thermal state.
Hydrodynamic Degrees of Freedom: The dynamical degrees of freedom are identified as relative diffeomorphisms ( $\xi^\mu$ ) between the asymptotic boundary and the horizon.
Horizon Boundary Conditions: A crucial step is the choice of horizon boundary conditions. The authors primarily utilize a mixed set of boundary conditions (analogous to those in [14]) that ensure the variational principle is satisfied in the absence of a Gibbons-Hawking term at the horizon. These conditions are:
- $\delta g_{tt}|_{r_0} = 0$
- $\delta \kappa|_{r_0} \equiv (\partial_r \delta g_{tt} - 2\partial_t \delta g_{tr} + 2\kappa_0 \delta g_{tr})|_{r_0} = 0$
  (where $\kappa_0$ is the surface gravity).
Symmetry Analysis: The authors identify "horizon symmetries" as diffeomorphisms that preserve these specific horizon boundary conditions while vanishing at the asymptotic boundary. They explicitly demonstrate that these diffeomorphisms act as gauge symmetries (zero modes) on the resulting hydrodynamic actions.
Comparison: To highlight the dependence of symmetries on boundary conditions, the authors also derive actions using Dirichlet boundary conditions on a stretched horizon (with a Gibbons-Hawking term), showing that these yield different actions with different symmetry structures.

Key Contributions and Results

Derivation of Exact Hydrodynamic Actions:
- JT Gravity: The authors derive a hydrodynamic action for the relative time diffeomorphism $\xi_t$ . With the mixed boundary conditions, this action is precisely the quadratic Schwarzian action coupled to external sources. This provides a new geometric understanding of the Schwarzian's known symmetries in terms of horizon physics.
- AdS $_3$ Gravity: The authors derive a quadratic hydrodynamic action for relative temporal and spatial diffeomorphisms ( $\xi_t, \xi_x$ ). This action differs from previously known geometric Virasoro actions (e.g., [39, 41, 43, 44]) but correctly reproduces the thermal two-point functions of a 2D CFT stress tensor.
Realization of Horizon Symmetries:
The paper explicitly demonstrates that the derived hydrodynamic actions possess gauge symmetries corresponding to the horizon diffeomorphisms that preserve the chosen boundary conditions.
- For JT gravity, the symmetries induce shifts in $\xi_t$ of the form $f(t) = a_1 + a_2 e^{-2\pi T t} + a_3 e^{2\pi T t}$ . These include constant and exponentially growing/decaying modes, which are the zero modes of the Schwarzian action.
- For AdS $_3$ gravity, the symmetries induce shifts in $\xi_t$ and $\xi_x$ with similar exponential time dependence.
- The authors show that these modes are redundancies of the parameterization (carrying no energy density) but are crucial for the chaotic properties of the state.
Dependence on Boundary Conditions:
The paper establishes that the form of the hydrodynamic action and its symmetries are not unique but depend on the choice of horizon boundary conditions.
- Imposing Dirichlet boundary conditions on a stretched horizon (with a Gibbons-Hawking term) yields a different quadratic action. While this alternative action produces the same stress tensor two-point functions (as they are fixed by Ward identities), its horizon symmetries are distinct: they lack the pointwise exponentially growing and decaying modes found in the mixed boundary condition case.
Connection to Chaos:
The hydrodynamic actions derived with mixed boundary conditions contain the specific pointwise exponential zero modes ( $e^{\pm 2\pi T t}$ ) that were proposed in [51] to explain the characteristic profile of OTOCs in maximally chaotic theories. The authors argue that the existence of these modes in the hydrodynamic action provides a concrete realization of the conjecture that horizon symmetries underpin quantum chaos.

Significance
The paper claims significance in three main areas:

Pedagogical Demonstration and Consistency: It serves as a consistency check for the approach of [14], demonstrating that it can be implemented exactly (without derivative expansion) in theories with no bulk gravitons to produce hydrodynamic actions with relative diffeomorphism degrees of freedom.
Explicit Realization of Conjectures: It explicitly constructs hydrodynamic actions that realize the specific gauge symmetries proposed in [55] as black hole horizon symmetries. This bridges the gap between the geometric symmetries of horizons and the effective field theory of hydrodynamics.
Clarification of Symmetry Dependence: It highlights that the "horizon symmetries" relevant for chaos are not universal to all hydrodynamic descriptions of the same physical system but are contingent on the specific boundary conditions chosen to define the effective theory. The mixed boundary conditions used in the main text are shown to be the natural choice for realizing the symmetries associated with maximal chaos, whereas Dirichlet conditions lead to a different symmetry structure.

The authors note that while they focused on linear perturbations, the simplicity of JT and AdS $_3$ gravity suggests that a full non-linear analysis is possible. They also identify technical improvements for future work, such as covariant derivations of the compact actions and a better understanding of $i\epsilon$ regularization directly within the gravitational derivation.

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