Ryu-Takayanagi and HRT

The main idea

The Ryu-Takayanagi formula is one of the sharpest forms of the slogan that spacetime geometry is made from quantum degrees of freedom. Given a spatial region $A$ in a boundary CFT, its entanglement entropy is computed, at leading order in the classical bulk limit, by the area of a codimension-two surface in the bulk:

$$S_A^{(0)} = \frac{\mathrm{Area}(\gamma_A)}{4G_N}.$$

This looks almost too simple. It is not. Each symbol hides a condition.

The boundary of the bulk surface must end on the entangling surface,

$$\partial \gamma_A = \partial A,$$

and the surface must be homologous to $A$. In a static state, $\gamma_A$ is the minimal-area surface on a suitable bulk time-reflection slice. In a time-dependent state, the correct surface is the Hubeny-Rangamani-Takayanagi surface $X_A$: a codimension-two extremal surface in Lorentzian spacetime,

$$S_A^{(0)} = \frac{\mathrm{Area}(X_A)}{4G_N}.$$

The superscript $(0)$ is important. This is the leading, classical, order-$N^2$ contribution in an Einstein-like holographic CFT. The next page adds quantum corrections, where the area term is supplemented by bulk entanglement across the surface.

The RT prescription computes $S_A^{(0)}$ from a minimal surface $\gamma_A$ on a static bulk slice. The covariant HRT prescription replaces $\gamma_A$ by a spacelike extremal surface $X_A$ with vanishing null expansions $\theta_+=\theta_-=0$. In both cases the surface is anchored on $\partial A$ and satisfies the homology condition with a bulk region $r_A$.

The formula resembles the Bekenstein-Hawking entropy law, but it is not merely a black-hole formula. For a generic boundary subregion $A$, the surface $\gamma_A$ is usually not a horizon. It is a new geometric object: a surface selected by the quantum entanglement pattern of the boundary state.

A good way to remember the hierarchy is:

Boundary object	Bulk object	Leading holographic formula
Entangling surface $\partial A$	Surface anchor	$\partial\gamma_A=\partial A$
Reduced density matrix $\rho_A$	Entanglement wedge data	$r_A$ bounded by $A$ and $\gamma_A$
Entanglement entropy $S_A$	Area of $\gamma_A$ or $X_A$	$S_A^{(0)}=\mathrm{Area}/(4G_N)$
Entropy inequalities	Geometric inequalities	minimal or extremal surfaces obey constraints
Thermal entropy	Horizon area in special cases	$S_{\mathrm{thermal}}=A_{\mathcal H}/(4G_N)$

The RT/HRT formula is a computational tool, a conceptual bridge, and a consistency condition on any proposed bulk dual. It is also the doorway to entanglement wedges, bulk reconstruction, quantum error correction, and modern black-hole information physics.

Setup and notation

We use the course convention that the boundary theory is a CFT$_d$ and the bulk is asymptotically AdS$_{d+1}$. On a boundary Cauchy slice $\Sigma$, choose a spatial region

$$A\subset \Sigma.$$

The entangling surface is

$$\partial A,$$

which has dimension $d-2$. The RT/HRT surface has dimension $d-1$ in the bulk. Equivalently, it is codimension two in the $(d+1)$-dimensional bulk spacetime.

For a bulk theory that comes from a compactification, such as type IIB string theory on $\mathrm{AdS}_5\times S^5$, one may either compute the area in the full ten-dimensional Einstein-frame geometry or in the consistent lower-dimensional Einstein-frame metric after integrating over the compact space. One should not use the string-frame area unless the prescription has been modified appropriately. Entropy is gravitational, and the classical Bekenstein-Hawking area law uses Einstein frame.

The Newton constant $G_N$ in this page means the Newton constant of the bulk theory in which the area is being computed. For example, for the canonical AdS$_5$/CFT$_4$ duality, the five-dimensional Newton constant satisfies

$$\frac{L^3}{G_5}\sim N^2,$$

so RT areas scale like $N^2$. This is why classical holographic entanglement captures the leading large-$N$ part of the CFT entropy.

The static RT prescription

Suppose the boundary state is represented by a static, time-reflection-symmetric bulk geometry. Let $\mathcal M_t$ be the corresponding Riemannian bulk spatial slice. The RT surface $\gamma_A$ is the surface that solves

$$\partial\gamma_A=\partial A,$$

and minimizes the area functional among admissible surfaces satisfying the homology condition. The leading entropy is

$$S_A^{(0)} = \min_{\gamma_A\sim A} \frac{\mathrm{Area}(\gamma_A)}{4G_N}.$$

Here $\gamma_A\sim A$ is shorthand for the homology condition: there exists a bulk spatial region $r_A\subset\mathcal M_t$ such that

$$\partial r_A = A\cup \gamma_A,$$

up to orientations and possible components on horizons or end-of-the-world branes when those are part of the setup.

The three essential conditions are therefore:

Condition	Meaning	Why it matters
Anchoring	$\partial\gamma_A=\partial A$	The bulk surface computes the entropy of that boundary region, not another one.
Minimality	$\gamma_A$ has smallest area among allowed surfaces	Entropy is selected by the dominant saddle.
Homology	$A\cup\gamma_A$ bounds a bulk region $r_A$	Ensures the correct complement property and excludes pathological surfaces.

The word “minimal” means global minimality, not merely a local solution of the Euler-Lagrange equation. In practice one often solves the extremal-surface equation and then compares the areas of all candidate surfaces. This comparison is responsible for many entanglement phase transitions.

The homology condition

The homology condition is not cosmetic. Without it, the formula would fail in simple situations.

Consider a thermal state of a holographic CFT dual to a one-sided AdS black hole. If $A$ is the entire boundary spatial slice, then $\partial A$ is empty. A surface with no boundary could be the empty surface, giving zero entropy. But the thermal density matrix has nonzero entropy. The homology condition forces the relevant surface to include the horizon, giving

$$S_{\mathrm{thermal}} = \frac{\mathrm{Area}(\mathcal H)}{4G_N}.$$

This is exactly the Bekenstein-Hawking entropy of the bulk black hole.

For a pure state on a single connected boundary, the homology condition also helps guarantee

$$S_A=S_{\bar A}.$$

The same surface can be viewed as homologous to $A$ or to $\bar A$, with complementary homology regions. This is the geometric version of the equality of entanglement entropy for complementary regions in a pure state.

There is a useful slogan:

$$\text{RT surface} + A = \text{boundary of the bulk region } r_A.$$

Later, $r_A$ will become the spatial part of the entanglement wedge. The entanglement wedge is not just an auxiliary region for drawing RT pictures; it is the bulk region encoded in the boundary density matrix $\rho_A$.

Example: one interval in AdS$_3$/CFT$_2$

The cleanest RT calculation is the vacuum entropy of an interval in a two-dimensional CFT. Work on a constant-time slice of Poincaré AdS$_3$:

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+dx^2\right).$$

Let the boundary interval be

$$A=[-\ell/2,\ell/2].$$

The RT surface is a bulk geodesic connecting the two endpoints. By symmetry, it is the semicircle

$$x^2+z^2=\left(\frac{\ell}{2}\right)^2.$$

Introduce the cutoff $z=\epsilon$. Parametrize the geodesic by

$$x=\frac{\ell}{2}\cos\theta, \qquad z=\frac{\ell}{2}\sin\theta.$$

The induced line element is

$$ds_{\gamma} = L\frac{d\theta}{\sin\theta}.$$

The cutoff restricts the endpoints to

$$\sin\theta_{\epsilon}=\frac{2\epsilon}{\ell}.$$

Therefore the regulated geodesic length is

$$\mathrm{Length}(\gamma_A) = 2L\int_{\theta_{\epsilon}}^{\pi/2}\frac{d\theta}{\sin\theta} = 2L\log\frac{\ell}{\epsilon}+O(\epsilon^2/\ell^2).$$

The RT formula gives

$$S_A = \frac{\mathrm{Length}(\gamma_A)}{4G_3} = \frac{L}{2G_3}\log\frac{\ell}{\epsilon}+O(1).$$

Using the Brown-Henneaux central charge

$$c=\frac{3L}{2G_3},$$

we obtain

$$S_A = \frac{c}{3}\log\frac{\ell}{\epsilon}+O(1),$$

which is precisely the universal CFT$_2$ vacuum result for a single interval.

This computation is worth digesting. The UV divergence of the CFT entropy comes from the infinite length of the bulk geodesic near the asymptotic boundary. The radial cutoff $z=\epsilon$ is the bulk avatar of the boundary UV cutoff. A long-distance boundary entanglement calculation has been replaced by a short geometric minimization problem.

Example: a strip in AdS$_{d+1}$

For higher-dimensional CFTs, exact answers are rarer, but the area-law structure is transparent. Consider a strip region in the vacuum CFT$_d$:

$$A=\left\{ -\frac{\ell}{2}\le x\le \frac{\ell}{2},\quad y_i\in\mathbb R, \quad i=1,\ldots,d-2\right\}.$$

Regulate the transverse volume by

$$V_{d-2}=\int d^{d-2}y.$$

On a constant-time slice of Poincaré AdS$_{d+1}$,

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+dx^2+d\vec y^{\,2}\right),$$

parametrize the surface by $z=z(x)$. The area functional is

$$\mathcal A = L^{d-1}V_{d-2} \int_{-\ell/2}^{\ell/2} dx\, \frac{1}{z^{d-1}}\sqrt{1+(z')^2}.$$

Because the Lagrangian does not depend explicitly on $x$, there is a conserved quantity:

$$\frac{1}{z^{d-1}\sqrt{1+(z')^2}} = \frac{1}{z_*^{d-1}},$$

where $z_*$ is the deepest point of the surface. This gives

$$\frac{\ell}{2} = z_* \int_0^1 du\, \frac{u^{d-1}}{\sqrt{1-u^{2d-2}}} = z_*\sqrt\pi\, \frac{\Gamma\!\left(\frac{d}{2(d-1)}\right)}{\Gamma\!\left(\frac{1}{2(d-1)}\right)}.$$

The regulated area has the form

$$\mathcal A = \frac{2L^{d-1}V_{d-2}}{d-2}\frac{1}{\epsilon^{d-2}} - \kappa_d L^{d-1}\frac{V_{d-2}}{\ell^{d-2}} + \cdots, \qquad d>2,$$

where $\kappa_d>0$ is a dimension-dependent numerical constant. The first term is the expected area-law divergence of a $d$-dimensional QFT:

$$S_A \sim \frac{L^{d-1}}{G_{d+1}}\frac{\mathrm{Area}(\partial A)}{\epsilon^{d-2}}.$$

The finite term is fixed by conformal invariance and scales like $V_{d-2}/\ell^{d-2}$. This is the higher-dimensional analog of the logarithm in CFT$_2$.

Spheres and hyperbolic black holes

For a ball-shaped region in the vacuum of a CFT$_d$,

$$A=\{r\le R\},$$

the RT surface in Poincaré AdS is the hemisphere

$$r^2+z^2=R^2.$$

This looks like the interval calculation, but it has a deeper meaning. The reduced density matrix of a CFT vacuum on a ball is conformally related to a thermal density matrix on hyperbolic space. In the bulk, the corresponding coordinate transformation maps the RT surface to the horizon of a hyperbolic AdS black hole. Thus the RT formula for a ball can be viewed as an ordinary horizon entropy calculation in disguise.

This observation is central in later arguments relating the entanglement first law to the linearized Einstein equations. For a ball in the CFT vacuum, the modular Hamiltonian is local:

$$H_A = 2\pi\int_A d^{d-1}x\, \frac{R^2-r^2}{2R}\,T_{00}(x).$$

The entanglement first law,

$$\delta S_A=\delta\langle H_A\rangle,$$

then becomes a direct bridge between variations of area in the bulk and variations of the CFT stress tensor.

Minimal, extremal, and why HRT is needed

The RT formula assumes a static geometry with a preferred Riemannian spatial slice. Many interesting states are not static: quenches, collapsing shells, shockwaves, evaporating effective geometries, and time-dependent black holes. In a Lorentzian spacetime, “minimal area on a time slice” is not a covariant statement. Different slicings of the same spacetime could give different answers.

The HRT prescription fixes this. For a boundary region $A$ lying on a boundary Cauchy slice, the HRT surface $X_A$ is a codimension-two spacelike surface satisfying:

$$\partial X_A=\partial A,$$

with the same homology condition as RT, and with vanishing first variation of area under all local deformations. Equivalently, the two null expansions vanish:

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

This is the natural generalization of a stationary surface to Lorentzian geometry. If there are several extremal candidates, the prescription chooses the one with least area among the admissible extremal surfaces.

In a static spacetime with a time-reflection symmetry, HRT reduces to RT. The extremal surface lies on the symmetric time slice, and extremality in spacetime becomes minimality on that slice.

A useful distinction:

Surface type	Where it lives	Variational condition	Used for
Minimal surface	Riemannian spatial slice	smallest area among candidates	RT in static states
Extremal surface	Lorentzian spacetime	first variation of area vanishes	HRT in time-dependent states
Horizon cross-section	Event/apparent/Killing horizon	causal or trapping condition	black-hole entropy, not generic $S_A$

The HRT surface is generally not a horizon. It is selected by boundary entanglement, not by causal trapping.

The area term as a large-$N$ saddle

The RT/HRT formula is not a statement about an arbitrary CFT. It is a statement about CFTs with a semiclassical gravitational dual. In such theories, one has a large parameter, often denoted $N$, such that

$$\frac{L^{d-1}}{G_N}\sim C_T\sim N^2$$

for matrix-like large-$N$ theories. Consequently,

$$S_A = O(N^2)+O(N^0)+\cdots.$$

The classical area computes the $O(N^2)$ contribution. The $O(N^0)$ term comes from quantum bulk fields entangled across the RT/HRT surface, and from shifts in the surface location. Those corrections are not optional if one asks a fine-grained quantum-gravity question. They are the subject of the next page.

For many classical holographic applications, however, the area term is the leading answer and already contains highly nontrivial physics: entropy inequalities, thermal entropy, confinement/deconfinement transitions, mutual information transitions, and geometric reconstruction regions.

Entanglement phase transitions

The RT prescription involves a minimization. Whenever two candidate surfaces exchange dominance, the leading large-$N$ entropy is continuous but not analytic. These are entanglement phase transitions.

A simple example is the union of two separated intervals in the vacuum of a holographic CFT$_2$:

$$A=[0,\ell], \qquad B=[\ell+s,2\ell+s].$$

There are two natural geodesic pairings for $A\cup B$:

$$S_{A\cup B}^{\mathrm{disc}} = S(\ell)+S(\ell),$$

and

$$S_{A\cup B}^{\mathrm{conn}} = S(s)+S(2\ell+s),$$

where $S(L)=\frac{c}{3}\log\frac{L}{\epsilon}$ at leading order. The RT answer is

$$S_{A\cup B}^{(0)} = \min\left\{S_{A\cup B}^{\mathrm{disc}},S_{A\cup B}^{\mathrm{conn}}\right\}.$$

The connected surface dominates when

$$s(2\ell+s)<\ell^2.$$

In that phase, the leading mutual information is

$$I(A:B)^{(0)} = S_A^{(0)}+S_B^{(0)}-S_{A\cup B}^{(0)} = \frac{c}{3}\log\frac{\ell^2}{s(2\ell+s)}.$$

In the disconnected phase, $I(A:B)^{(0)}=0$. This does not mean the exact mutual information vanishes. It means that it is subleading in $1/N$. Classical geometry sees only the order-$N^2$ part.

This example is a warning against overinterpreting sharp classical transitions. The exact CFT entropy is smooth at finite $N$. The sharp transition appears because the bulk saddle approximation replaces a sum over contributions by the dominant one.

Entropy inequalities from geometry

Entanglement entropy obeys powerful inequalities in quantum theory. The RT formula geometrizes many of them.

The most important is strong subadditivity:

$$S_A+S_B\ge S_{A\cup B}+S_{A\cap B}.$$

For static RT surfaces, this inequality has a beautiful geometric proof. One draws the minimal surfaces for $A$ and $B$, cuts and reglues them into candidate surfaces for $A\cup B$ and $A\cap B$, and uses minimality. Since the RT surfaces for $A\cup B$ and $A\cap B$ have area no larger than the candidate reglue surfaces, the inequality follows.

Classical holographic entropies obey additional inequalities that are not true for arbitrary quantum states. A famous example is monogamy of mutual information:

$$I(A:B)+I(A:C)\le I(A:BC),$$

or equivalently

$$S_{AB}+S_{AC}+S_{BC}\ge S_A+S_B+S_C+S_{ABC}.$$

This extra inequality is a signature of the special large-$N$ geometric entropy cone. It should not be imposed on generic quantum states, and it can receive corrections beyond the classical area approximation.

Thermal states and horizon contributions

RT surfaces in black-hole geometries encode both entanglement entropy and ordinary thermal entropy. The distinction depends on the region and the state.

For a finite region $A$ in a thermal CFT state dual to an AdS black brane, the RT surface usually dips toward the horizon. For small regions, it stays mostly near the boundary and the entropy is close to the vacuum result. For large regions, the surface develops a segment that runs near the horizon. The entropy then contains a volume-law piece proportional to the thermal entropy density:

$$S_A \approx s_{\mathrm{thermal}}\,\mathrm{Vol}(A) +\text{boundary terms},$$

where

$$s_{\mathrm{thermal}} = \frac{\mathrm{Area\ density\ of\ horizon}}{4G_N}.$$

For the entire boundary spatial slice in a one-sided thermal state, the entropy is the thermal entropy itself, and the relevant surface is the horizon. For the eternal two-sided black hole dual to the thermofield double state, the entropy of one entire boundary CFT is again the horizon area. The full two-sided state is pure, but either side alone is mixed.

This is the cleanest way to see how RT unifies black-hole entropy and ordinary subregion entanglement entropy.

RT surfaces and the entanglement wedge

Given an RT surface $\gamma_A$, the homology region $r_A$ is the bulk spatial region whose boundary is $A\cup\gamma_A$. The entanglement wedge is the domain of dependence of $r_A$:

$$\mathcal E[A]=D(r_A).$$

For HRT, the analogous construction uses a bulk achronal region bounded by $A\cup X_A$.

The next pages will explain the entanglement wedge in detail. For now, the key point is this:

$$\rho_A \quad\longleftrightarrow\quad \text{bulk physics in }\mathcal E[A].$$

This statement is much stronger than the area formula alone. It says that the reduced density matrix of a boundary subregion knows a particular bulk region. The RT/HRT surface is the boundary of that reconstructable region.

This is also why phase transitions are physically meaningful. When the dominant RT surface changes, the entanglement wedge can jump discontinuously at leading order in $N$. In the exact finite-$N$ theory the transition is smoothed, but in the classical geometry it is a sharp change in the reconstructable bulk region.

The replica idea behind RT

The RT formula was originally proposed as a holographic analog of the Bekenstein-Hawking area law. A later derivation uses the replica trick in the bulk.

In the boundary QFT, Rényi entropies are computed from

$$\mathrm{Tr}\,\rho_A^n.$$

The entanglement entropy is obtained by analytic continuation:

$$S_A = -\left.\partial_n\log\mathrm{Tr}\,\rho_A^n\right|_{n=1}.$$

Holographically, $\mathrm{Tr}\,\rho_A^n$ is computed by a bulk saddle whose boundary is the $n$-fold replicated boundary geometry. In the classical bulk limit, differentiating with respect to $n$ creates a codimension-two conical defect. The regularity condition for this defect selects an extremal surface, and the derivative of the action gives its area divided by $4G_N$.

This is the same mechanism that produces black-hole entropy from a Euclidean conical defect. RT is therefore not a random geometric guess: it is a generalized gravitational entropy formula applied to replicated boundary regions.

For this page, the replica derivation is a consistency check and conceptual anchor. The full quantum-corrected story, including bulk entanglement and quantum extremal surfaces, comes next.

What RT/HRT does not say by itself

RT/HRT is powerful, but one must keep its assumptions visible.

First, the classical formula computes only the leading area term. It does not include bulk-field entanglement across the surface:

$$S_A \ne \frac{\mathrm{Area}(X_A)}{4G_N}$$

as an exact finite-$N$ statement.

Second, the area functional assumes two-derivative Einstein gravity. Higher-derivative bulk theories require a generalized entropy functional. For example, curvature-squared terms do not simply use the area in Planck units.

Third, the surface must be computed in the appropriate Einstein-frame geometry. Using the wrong frame or forgetting internal compact directions can change the answer.

Fourth, minimal and extremal are not interchangeable words. In Lorentzian signature, one extremizes; in a static Riemannian slice, one minimizes.

Fifth, the RT surface is not generally the same as a causal horizon. Causal wedges and entanglement wedges are related but distinct. In many important examples, the entanglement wedge extends deeper than the causal wedge.

Common mistakes

Mistake	Correction
“The RT surface is always a horizon.”	Only in special cases, such as the entropy of an entire thermal boundary or a ball after a conformal map.
“Any extremal surface is the answer.”	One must impose anchoring, homology, and choose the least-area admissible extremal surface.
“The disconnected mutual information phase means no correlations.”	It means no order-$N^2$ mutual information. Subleading correlations can remain.
“The area law divergence is mysterious in the bulk.”	It comes from the infinite area near the asymptotic boundary, regulated by $z=\epsilon$.
“RT is exact.”	Classical RT is the leading term. Quantum corrections produce generalized entropy and quantum extremal surfaces.
“HRT is minimal on some chosen time slice.”	HRT is covariantly extremal in Lorentzian spacetime; the maximin construction is a theorem, not the definition used in calculations.

Exercises

Exercise 1: The interval entropy in AdS$_3$

Consider the constant-time slice of Poincaré AdS$_3$,

$$ds^2=\frac{L^2}{z^2}(dz^2+dx^2),$$

and the interval $A=[-\ell/2,\ell/2]$. Show that the geodesic $x^2+z^2=(\ell/2)^2$ has regulated length

$$\mathrm{Length}=2L\log\frac{\ell}{\epsilon}+O(1),$$

and hence

$$S_A=\frac{c}{3}\log\frac{\ell}{\epsilon}+O(1).$$

Solution

Parametrize the geodesic by

$$x=\frac{\ell}{2}\cos\theta, \qquad z=\frac{\ell}{2}\sin\theta.$$

Then

$$dx^2+dz^2=\left(\frac{\ell}{2}\right)^2d\theta^2,$$

so

$$ds=\frac{L}{z}\frac{\ell}{2}d\theta =L\frac{d\theta}{\sin\theta}.$$

The cutoff $z=\epsilon$ gives

$$\sin\theta_\epsilon=\frac{2\epsilon}{\ell}.$$

The full geodesic has two equal halves, so

$$\mathrm{Length} =2L\int_{\theta_\epsilon}^{\pi/2}\frac{d\theta}{\sin\theta} =2L\log\cot\frac{\theta_\epsilon}{2} =2L\log\frac{\ell}{\epsilon}+O(\epsilon^2/\ell^2).$$

Using RT,

$$S_A=\frac{\mathrm{Length}}{4G_3} =\frac{L}{2G_3}\log\frac{\ell}{\epsilon}.$$

The Brown-Henneaux relation is

$$c=\frac{3L}{2G_3},$$

hence

$$S_A=\frac{c}{3}\log\frac{\ell}{\epsilon}.$$

Exercise 2: The strip first integral

For the strip in AdS$_{d+1}$, derive the conserved quantity

$$\frac{1}{z^{d-1}\sqrt{1+(z')^2}} = \frac{1}{z_*^{d-1}}.$$

Solution

The area density is

$$\mathcal L(z,z')=z^{-(d-1)}\sqrt{1+(z')^2}.$$

Since $\mathcal L$ has no explicit $x$ dependence, the Hamiltonian-like quantity

$$\mathcal H=z'\frac{\partial\mathcal L}{\partial z'}-\mathcal L$$

is conserved. We compute

$$\frac{\partial\mathcal L}{\partial z'} =z^{-(d-1)}\frac{z'}{\sqrt{1+(z')^2}},$$

and therefore

$$\mathcal H =z^{-(d-1)}\frac{(z')^2}{\sqrt{1+(z')^2}} -z^{-(d-1)}\sqrt{1+(z')^2} =-\frac{1}{z^{d-1}\sqrt{1+(z')^2}}.$$

At the turning point $z=z_*$, the profile has $z'=0$. Thus

$$\mathcal H=-\frac{1}{z_*^{d-1}},$$

which gives

$$\frac{1}{z^{d-1}\sqrt{1+(z')^2}} = \frac{1}{z_*^{d-1}}.$$

Exercise 3: Homology and the thermal entropy

In a one-sided thermal state dual to an AdS black hole, take $A$ to be the entire boundary spatial slice. Explain why the homology condition selects the horizon and gives the thermal entropy.

Solution

For the entire boundary spatial slice, $\partial A=\varnothing$. Without homology, the empty surface would be an allowed surface with zero area. That would incorrectly give $S_A=0$ for a thermal density matrix.

The homology condition requires a bulk region $r_A$ whose boundary is $A\cup\gamma_A$. In the exterior region of a black-hole geometry, the boundary slice by itself is not the full boundary of a compact bulk region; the horizon cross-section must be included. Thus the admissible surface is the horizon cross-section:

$$\gamma_A=\mathcal H.$$

The RT formula gives

$$S_A=\frac{\mathrm{Area}(\mathcal H)}{4G_N},$$

which is the Bekenstein-Hawking entropy of the black hole and the thermal entropy of the boundary state.

Exercise 4: A mutual-information phase transition

For two equal intervals in a holographic CFT$_2$ vacuum,

$$A=[0,\ell], \qquad B=[\ell+s,2\ell+s],$$

use $S(L)=\frac{c}{3}\log\frac{L}{\epsilon}$ to show that the connected RT pairing dominates when

$$s(2\ell+s)<\ell^2.$$

Find the critical value of $s/\ell$.

Solution

The disconnected candidate is

$$S^{\mathrm{disc}}_{A\cup B} =2S(\ell) =\frac{2c}{3}\log\frac{\ell}{\epsilon}.$$

The connected candidate is

$$S^{\mathrm{conn}}_{A\cup B} =S(s)+S(2\ell+s) =\frac{c}{3}\log\frac{s(2\ell+s)}{\epsilon^2}.$$

The connected candidate dominates when

$$S^{\mathrm{conn}}_{A\cup B}<S^{\mathrm{disc}}_{A\cup B}.$$

This is equivalent to

$$\log\frac{s(2\ell+s)}{\epsilon^2} < \log\frac{\ell^2}{\epsilon^2},$$

so

$$s(2\ell+s)<\ell^2.$$

Let $u=s/\ell$. Then

$$u(2+u)<1,$$

or

$$u^2+2u-1<0.$$

The positive root is

$$u_*=-1+\sqrt2.$$

Thus the connected phase dominates for

$$\frac{s}{\ell}<\sqrt2-1.$$

Exercise 5: Why HRT is not “minimal on a time slice”

Explain why the phrase “choose the minimal surface on the boundary time slice” is not a covariant prescription in a time-dependent Lorentzian geometry.

Solution

A time-dependent Lorentzian spacetime does not come with a unique preferred bulk time slice anchored to a given boundary time. Different bulk foliations can agree at the boundary but differ in the interior. A prescription that minimizes area on a chosen slice would therefore depend on a gauge-like choice of slicing rather than on the spacetime geometry itself.

The HRT prescription avoids this by selecting a spacelike codimension-two surface $X_A$ that is extremal in the full Lorentzian spacetime. Its first variation of area vanishes under local deformations in both independent normal directions. Equivalently, its two null expansions vanish:

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

This condition is covariant. In static geometries with time-reflection symmetry, the HRT surface lies on the symmetric slice and reduces to the RT minimal surface.

Further reading

The original proposal is Ryu and Takayanagi, “Holographic Derivation of Entanglement Entropy from AdS/CFT”. The covariant extension is Hubeny, Rangamani, and Takayanagi, “A Covariant Holographic Entanglement Entropy Proposal”. The replica derivation is Lewkowycz and Maldacena, “Generalized Gravitational Entropy”. For geometric entropy inequalities, see Headrick and Takayanagi, “A Holographic Proof of the Strong Subadditivity of Entanglement Entropy”. A broad review is Nishioka, Ryu, and Takayanagi, “Holographic Entanglement Entropy: An Overview”.