Covariant HRT Formula in Holography

Updated 22 May 2026

The covariant HRT formula is a generalization of the Ryu–Takayanagi proposal, prescribing entropy via extremal codimension-two surfaces in Lorentzian geometries.
It employs variational methods like the maximin construction and covariant bit thread formulations to account for time-dependent settings in holographic quantum field theories.
Applications include holographic bulk reconstruction, validation of entropy inequalities, and insights into the emergent geometry and quantum gravity phase-space structure.

The covariant Hubeny-Rangamani-Takayanagi (HRT) formula gives a prescription for computing the von Neumann entropy of a boundary region in holographic quantum field theories with a dynamical, Lorentzian gravity dual. It generalizes the Ryu–Takayanagi (RT) proposal to fully time-dependent settings by extremizing, rather than minimizing, the area functional for codimension-two surfaces in the bulk gravitational spacetime. The HRT surface is used to encode the entanglement structure of quantum field theory states holographically and has become foundational in the modern understanding of quantum gravity, holography, quantum information, and the emergence of bulk spacetime.

1. Covariant Entropy Prescription: Definition and Setup

Given a globally hyperbolic, asymptotically AdS spacetime $\mathcal{M}$ , and a spatial subregion $A$ on a boundary Cauchy slice, the HRT formula determines the entanglement entropy $S_A$ by extremizing over all bulk codimension-2 surfaces $\gamma_A \subset \mathcal{M}$ that are (a) anchored on $\partial A$ and (b) homologous to $A$ . The unique extremal surface $\gamma_A$ satisfies

$\partial \gamma_A = \partial A,\qquad \gamma_A \sim A,\qquad \text{and}\qquad \delta\,\mathrm{Area}[\gamma_A]=0.$

The entropy is then

$S_A = \frac{\mathrm{Area}(\gamma_A)}{4G_N}.$

Here, extremality means vanishing mean curvature in both independent normal directions within the Lorentzian bulk, making the prescription fully covariant (Kaplan et al., 2022, Headrick et al., 2022, Wall, 2012). This construction applies equally to static and dynamical geometries, recovering RT in the time-symmetric limit.

2. Geometric Construction: Extremality, Maximin, and Causal Structure

The HRT surface can be constructed via several equivalent variational or optimization schemes:

Direct extremality: Surface area is stationary under all local deformations—i.e., both null expansions vanish. The extremal surface equation $K^a = 0$ is enforced for both normals (Kaplan et al., 2022).
Maximin construction: For each achronal (Cauchy) bulk slice $A$ 0, find the minimal-area candidate anchored to $A$ 1, then maximize the area over all possible choices of $A$ 2. The resulting maximin surface coincides with the covariant extremal surface when the null curvature condition (NCC) holds (Wall, 2012). This method provides proofs of the existence and stability of HRT surfaces, and links geometric properties such as causal-wedge exclusion and monotonicity (the surface moves outward as $A$ 3 enlarges).

Causal structure plays a critical role: the HRT surface lies outside the causal wedge, reflecting the reach of boundary reconstructions and encoding entanglement wedge reconstruction (Wall, 2012). The region between the HRT surface and the boundary constitutes the entanglement wedge associated to $A$ 4.

3. Functional, Convex, and Flow-Based Reformulations

Several mathematically distinct, yet physically equivalent, reformulations of the HRT prescription have emerged:

Minimax (timelike maximin/minimax) surface: For each piecewise-timelike (or null) hypersurface $A$ 5, maximize the area of a codimension-2 cut homologous to $A$ 6, then minimize over $A$ 7 (Headrick et al., 2022).
Covariant bit thread (V-flow / U-flow) formulations: HRT entropy is expressed as

$A$ 8

where $A$ 9 is a bulk divergenceless 1-form subject to a covariant norm bound. Lagrange duality gives an alternative minimization (U-flow) dual program. Both are convex programs dynamically selecting the HRT surface as the flow bottleneck (Headrick et al., 2022).

Thread distributions: Measurable distributions on the space of bulk curves reproducing the maximization or minimization over flows, providing a measure-theoretic analog of entanglement structure.

These reformulations enable proofs of convexity, duality, and facilitate computational and conceptual analysis of entanglement in covariant settings (Headrick et al., 2022).

4. Entropy Inequalities, Entropy Cone, and Polyhedrality

The RT formula is known to satisfy an infinite set of linear inequalities—subadditivity, strong subadditivity (SSA), monogamy of mutual information (MMI), and generalized multi-party inequalities—which define the "RT entropy cone." The open question was whether the inequalities (and polyhedrality) generalize to covariant HRT entropies.

Recent advances have rigorously established that every entropy vector computed by the HRT prescription satisfies precisely the same set of inequalities as the static RT case (Grimaldi et al., 29 Aug 2025, Zhao, 26 Jan 2026):

No-short-cut theorem: The entropic structure of all HRT surfaces can be encoded in a discrete, causally partitioned graph whose minimal cuts correspond to unions of complete HRT surfaces. Any cut using partial surfaces does not achieve a lower area than a union of full extremal surfaces (Zhao, 26 Jan 2026).
Polyhedrality: The set of all HRT entropy vectors forms a rational polyhedral cone—the covariant entropy cone coincides with the static (RT) cone. A finite basis of inequalities suffices (Zhao, 26 Jan 2026).
Majorization test: Markov-state-based and majorization criteria show that HRT entropies are subject to exactly the same constraints as RT entropies, supporting full equivalence and providing a new combinatorial characterization (Grimaldi et al., 29 Aug 2025).

5. Operator Interpretation and Phase-Space Structure

The area of the HRT surface is promoted to a function(al) on the covariant gravitational phase space. As an operator, the HRT area generates a so-called "boundary-condition-preserving kink transformation" on the canonical data:

Poisson/Peierls bracket: The HRT area operator $S_A$ 0 acts nontrivially only on the transverse extrinsic curvature $S_A$ 1, generating a delta-function jump at the HRT surface—a local Lorentz boost between the two entanglement wedges glued at $S_A$ 2 (Kaplan et al., 2022).
Boundary modular flow: In AdS $S_A$ 3, the induced flow on boundary coordinates corresponds to a one-sided boost (“kink transform”), identifying the geometric origin of the modular Hamiltonian flow in holographic CFTs.
Anomalous cases: In gravitational theories with anomalies (e.g., topologically massive gravity), the geometric entropy operator acquires an additional term, but the generated kink transformation remains, modulo parity-violating commutator structure (Kaplan, 2023).

This phase-space operator structure provides the geometric underpinning for understanding modular properties, relative entropy, and the algebraic structure of the entanglement wedge in bulk gravity.

6. Covariant Derivation from Path Integrals and Quantum Gravity

The HRT formula is derived for generic (non-static) states by mapping the Lorentzian Schwinger-Keldysh path-integral construction for reduced density matrices to the bulk gravitational path integral with replica symmetry:

Replica technique: The analytic continuation of a branched cover, constructed via cyclically sewn Schwinger-Keldysh contours on the boundary, induces a bulk quotient with a conical defect at the locus of $S_A$ 4. Regularity conditions and the equations of motion localize the action variation to this defect, enforcing the extremality condition (Dong et al., 2016).
Quantum generalization: For any UV-complete gravitational theory with standard Euclidean axioms (finiteness, reality, reflection-positivity, etc.), the gravitational path integral defines algebras of observables and associated entropies. In the semiclassical limit, this path integral argument yields the HRT formula (potentially with quantum bulk corrections), clarifying its foundation independent of AdS/CFT (Colafranceschi et al., 2023).

Path integral methods thus connect HRT surfaces to fundamental properties of quantum gravity, including the structure of bulk operator algebras and Hilbert spaces.

7. Holographic Bulk Reconstruction and Applications

The covariant HRT prescription uniquely encodes local bulk geometry in the entanglement data of the boundary theory:

Metric reconstruction: Variations of the areas of families of HRT surfaces anchored on the boundary uniquely determine the bulk metric, via the data of the Jacobi operator governing extremal surface deformations. In dimensions $S_A$ 5, inverse boundary-value theorems guarantee uniqueness wherever a smooth foliation by extremal disks reaches (Bao et al., 2019).
Explicit inversion: In three bulk dimensions, analytic inversion formulas map the Hamilton–Jacobi functional of entanglement entropy as a function of interval length and time separation into the bulk radial metric profile, including frame dragging and causal cone structure. Exemplifying cases include AdS, rotating BTZ, and boosted black branes (Chae, 15 May 2026).
Volume law and thermal states: For large boosted or arbitrarily sliced subregions, the HRT prescription yields an entropy scaling with the projected spatial volume, matching the field-theoretic expectation for thermal states (Guglielmo et al., 2022).
Tensor network models: Covariant random tensor network constructions realize the HRT formula and its Rényi generalizations microscopically. The entanglement spectrum is saturated by discrete "areas" (domain wall lengths), showing how spacetime entanglement structure emerges in fully Lorentzian, non-static, and random settings (Qi et al., 2018).

These results demonstrate that the covariant HRT formula not only governs entanglement in AdS/CFT but provides a central tool for mapping emergent geometry, operator reconstruction, and the structure of quantum information in gravitational theories.

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