Covariant HRT Formula in Holography

• The covariant HRT formula is a geometric prescription for computing entanglement entropy in holographic quantum field theories, generalizing the static Ryu–Takayanagi proposal to time-dependent settings.
• It employs an extremal surface condition where both future and past-directed null expansions vanish, ensuring covariance in dynamic bulk spacetimes.
• The formulation underpins bulk reconstruction and maintains key entropy inequalities such as strong subadditivity, crucial for insights into quantum gravity.

The covariant Hubeny-Rangamani-Takayanagi (HRT) formula is the generalization of the static Ryu-Takayanagi minimal surface prescription to arbitrary, possibly time-dependent states in holographic systems. It provides a covariant geometric prescription for computing the fine-grained von Neumann entanglement entropy of a spatial region $A$ in a boundary quantum field theory with a gravitational holographic dual. The HRT formula equates this entropy to the area of a codimension-2 extremal surface $\Gamma_A$ in the bulk asymptotically Anti-de Sitter (AdS) spacetime, with boundary $\partial A$, evaluated as a solution to a Lorentzian extremality condition. The theory is constructed to be manifestly covariant and independent of any choice of time slicing, making it applicable to situations involving nontrivial dynamics, gravitational collapse, or black hole interiors.

1. Foundational Formulation and Covariant Extremal Surface Condition

In a static setting, the Ryu–Takayanagi (RT) proposal computes the entanglement entropy $S_A$ of a (spatial) region $A$ in the CFT as

$$S_A = \frac{\mathrm{Area}(\gamma_A)}{4 G_N^{(d+1)}},$$

where $\gamma_A$ is the minimal-area codimension-2 surface in a $t=\text{const}$ slice of the bulk AdS spacetime, homologous to $A$, and anchored at $\partial A$ on the boundary. However, for Lorentzian bulk spacetimes without a preferred time slice, any such minimization is ambiguous or non-covariant.

The HRT proposal (0705.0016) replaces the minimal-area requirement with a fully covariant extremal surface condition: the desired bulk surface $\Gamma_A$ is the unique codimension-2, spacelike surface anchored at $\partial A$ and homologous to $A$, such that both independent null expansions vanish:

$\theta_+ = 0, \qquad \theta_- = 0$

where $\theta_\pm = h^{\mu\nu} \nabla_{\mu} N_\pm{}_\nu$ are the expansions of the null congruences orthogonal to $\Gamma_A$ (with $N_+^\mu$ and $N_-^\mu$ the two independent future/past-directed null normals, normalized as $N_+^2 = N_-^2=0$, $N_+\cdot N_- = -1$), and $h^{\mu\nu}$ is the induced metric on $\Gamma_A$. The entropy formula then remains:

$$S_A = \frac{\mathrm{Area}(\Gamma_A)}{4 G_N^{(d+1)}}$$

This covariant prescription reduces exactly to the RT formula when a preferred static slicing exists, as any minimal surface in a Riemannian (spacelike) slice automatically satisfies the extremality condition.

2. Geometric and Physical Interpretation

The covariant HRT construction can be equivalently understood via light-sheet methods, paralleling the Bousso entropy bound. Given a region $A$ on the boundary, future- and past-directed bulk light-sheets are launched from $\partial A$. Among all codimension-2 spacelike surfaces anchored at $\partial A$ and lying “between” these light-sheets, the extremal one—where both null expansions vanish—is dynamically selected. This construction is unique, fully Lorentzian, and covariant.

In the regime where $A$ grows to encompass the entire boundary, the extremal surface generically coincides (or asymptotes) to the apparent horizon of a black hole, and its area follows the (generalized) second law of thermodynamics. The time evolution of the HRT surface under, e.g., gravitational collapse, directly encodes entropy growth as the bulk horizon expands.

3. Reductions, Extensions, and Practical Computation

Static Spacetimes

When the bulk metric admits a timelike Killing vector, the HRT surface lies in a static slice, and the extremality condition is equivalent to minimality, reducing the formula to the original RT prescription.

Time-Dependent Spacetimes

For dynamical metrics (e.g., Vaidya–AdS with mass function $m(v)$), the HRT surface can traverse nontrivial regions. In AdS$_3$-Vaidya, where the metric is

$$ds^2 = -[r^2 - m(v)/r^{d-2}]dv^2 + 2dvdr + r^2 dx_i^2,$$

the bulk spacelike geodesics (surfaces of interest in $d=3$) are computed numerically or perturbatively. The length (for $d=3$) or area functionals are extremized subject to the time-dependent bulk boundary, with the appropriate matching conditions at null shells (if present).

An explicit result in Vaidya–AdS$_3$ is:

$$\frac{dL_\mathrm{reg}(v)}{dv} = \frac{h^2}{3} m'(v)$$

where $h$ describes the size of the interval on the boundary. As $m(v)$ increases, the HRT surface's area grows, capturing entropy production consistent with far-from-equilibrium thermalization.

Wormhole Geometries

For Euclidean AdS wormholes with multiple asymptotic boundaries, the minimal-area surface wrapping the bulk “throat” provides a non-zero entanglement entropy, indicating that the noninteracting boundary QFTs are in an entangled state—the geometric connection through the wormhole being its bulk dual.

The entropy in such cases is

$$S_1 = \frac{\mathrm{Area}(\text{throat})}{4 G_N^{(d+1)}}$$

4. Maximin Construction and Information-Theoretic Properties

The HRT surface can also be obtained by a “maximin” procedure (Wall, 2012): among all achronal (Cauchy) slices $\Sigma$ containing $\partial A$, consider the minimal area surface on $\Sigma$ homologous to $A$, then maximize this area over all possible $\Sigma$. The resulting surface is extremal, and the maximin construction is crucial for proving information-theoretic properties.

A central theorem is that in spacetimes satisfying the null energy condition (NEC), HRT/maximin surfaces obey

• Strong subadditivity (SSA): For any adjacent boundary regions $A$, $B$, $C$, the inequalities

$S(AB) + S(BC) \geq S(ABC) + S(B)$

are satisfied if the underlying bulk metric is NEC-respecting (Callan et al., 2012). Convexity and monotonicity of geodesic lengths (for $d=3$) under NEC are explicitly shown, with NEC violation leading to SSA failure.

• Monogamy of mutual information also holds as a direct geometric consequence of the maximin framework.
• Causal wedge inclusion: The extremal (HRT/maximin) surface always lies outside the causal wedge, allowing bulk reconstruction from the entanglement wedge between the boundary and the extremal surface.

5. Behavior under Bulk Quantum Corrections and Anomalies

Quantum corrections can be incorporated via a quantum extremal surface (QES) prescription, where one extremizes the sum of area and bulk entanglement entropy across the surface. In theories with gravitational anomalies (e.g., topologically massive gravity), the HRT entropy functional acquires additional, explicitly chiral terms dependent on the extrinsic geometry of the surface, precisely reproducing the expected modular properties in the boundary (Kaplan, 2023).

Geometric entropy operators, when promoted to phase space operators, implement “one-sided boosts” that shift the extrinsic curvature of the bulk Cauchy surface along the HRT surface via a delta-function, effecting a "kink transformation" (Kaplan et al., 2022). The action on the phase space commutes with modular flow on constant time slices but can develop UV singularities in general.

6. Extensions, Reformulations, and HRT in Broader Contexts

• Covariant Bit Threads: Reformulations in terms of covariant max-flow/min-cut theorems (bit threads with non-local norm bounds along causal curves) allow convex optimization representations of the HRT entropy, generalizing Riemannian bit thread formulations of RT (Headrick et al., 2022).
• Random Tensor Networks: Space–time random tensor networks with random projections on bulk links realize the covariant HRT formula for Rényi and von Neumann entropies. In these models, covariance is enforced by formulation across the entire spacetime, not any constant-time slice (Qi et al., 2018).
• General Diffeomorphism Invariant Theories: Path integral derivations using the gravitational replica trick, including Lorentzian Schwinger–Keldysh contours, yield a general proof that the HRT prescription (area/4G of the extremal surface) follows for any diffeomorphism-invariant theory in the semiclassical ($\hbar \to 0$) limit, via the variation of the modular (Rényi) entropy under infinitesimal replica index $n$ changes (Dong et al., 2016, Averin, 20 Aug 2025).
• Entropy Cone and HRT Inequalities: Majorization-based tests demonstrate that all static Ryu–Takayanagi entropy inequalities (defining the "RT cone") also hold for entropies computed by the HRT formula ("HRT cone") (Grimaldi et al., 29 Aug 2025). Extensive analytic and numerical evidence establishes that the set of inequalities is unchanged, with the same polyhedral structure.

7. Implications for Metric Reconstruction and Spacetime Emergence

Boundary entanglement entropies, as computed from the HRT areas, uniquely determine the bulk metric (up to diffeomorphism) in any region foliated by the corresponding extremal surfaces (Bao et al., 2019). First and second variations of HRT areas (equivalently, the Jacobi operator's boundary Cauchy data) encode all local curvature information, making the HRT prescription foundational for holographic bulk reconstruction in both static and dynamical spacetimes.

Moreover, in black hole formation scenarios, HRT surfaces can probe regions behind apparent and event horizons at early times, making the entropy growth and eventual saturation (the Page curve) accessible in dynamical AdS/CFT contexts—with moving brane boundary conditions (e.g., METR branes) required for fully time-dependent evaporation models (Chou et al., 2023).

---

Summary Table: Key Features of the Covariant HRT Formula

Property/Aspect	Static (RT) Limit	Covariant (HRT) Extension
Surface Condition	Minimal in time slice	Extremal in Lorentzian spacetime
Null Expansions	$\theta_\pm = 0$	$\theta_\pm = 0$
SSA, Monogamy	Holds (NEC)	Holds (NEC)
Quantum Corrections	Generalized entropy	Quantum extremal surface (QES)
Wormhole Topologies	Non-applicable	Natural entanglement across boundaries
Optimization	Min (area over slices)	Maximin or min-saddle (extremal)
Bit Thread Reform.	Local unit bound	Integrated non-local norm bound

The covariant HRT prescription thus unifies static and dynamical holographic entanglement computations, reproduces all known static inequalities, accommodates fully time-dependent and topologically nontrivial bulk settings, and underlies programmatic efforts at bulk reconstruction and quantum gravity path integral definitions of entropy.