The Ryu–Takayanagi Formula

For a static holographic state with a classical Einstein bulk, the leading von Neumann entropy of a boundary region $A$ is the area of the least-area bulk codimension-two surface anchored on $\partial A$ and homologous to $A$, divided by $4G_N$. “Least” means globally least among all admissible candidates, not merely locally stationary. The statement is leading order in $G_N^{-1}$; bulk-field entropy and higher-curvature corrections require different functionals. We use Euclidean and Lorentzian Poincaré $\mathrm{AdS}_3$ of radius $L$, a constant-time reflection-symmetric slice, and a boundary UV cutoff related explicitly to the radial cutoff.

Required background. Regulated subregion entropy defines the cutoff-dependent boundary quantity, and the AdS conformal boundary supplies the radial/UV relation.

Helpful background. Entropy counterterms track scheme dependence; horizon entanglement entropy separates geometric and matter terms; universal geometric terms identify regulator-independent data; and CFT interval entropy supplies the boundary benchmark.

The static Einstein prescription

Let $M$ be a time-reflection-symmetric asymptotically AdS solution and $\Sigma_0$ its fixed spatial slice. The RT surface $\gamma_A\subset\Sigma_0$ obeys

$$\partial\gamma_A=\partial A, \qquad \exists\,R_A\subset\Sigma_0: \quad \partial R_A=A\cup\gamma_A,$$

with orientations understood. The second condition is homology. Among every smooth, piecewise-smooth, connected or disconnected surface satisfying these conditions,

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

For a pure boundary state, the homology condition makes $A$ and its complement share the same admissible surface, so $S_A=S_{\bar A}$. In a thermal geometry, a horizon can enter the homology relation; this is essential for large regions.

The surface is minimal within $\Sigma_0$ and is also extremal in spacetime because reflection symmetry makes both null expansions vanish. In a merely stationary or time-dependent spacetime, the constant-time minimum is not the general prescription.

Interval entropy in vacuum AdS3

On $t=0$, Poincaré $\mathrm{AdS}_3$ has

$$ds^2_{\Sigma_0}=\frac{L^2}{z^2}(dz^2+dx^2).$$

Take the boundary interval $A=[-\ell/2,\ell/2]$, regulated at $z=\epsilon_z$. A curve $z(x)$ has length

$$\mathcal L[z]=L\int_{-\ell/2}^{\ell/2} dx\,\frac{\sqrt{1+z'^2}}{z}.$$

Translation invariance in $x$ gives a first integral,

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

where $z_*$ is the turning point. Integrating gives the semicircle

$$x^2+z^2=z_*^2, \qquad z_*=\frac{\ell}{2}.$$

Parameterize $x=z_*\cos\theta$, $z=z_*\sin\theta$. The regulated length is

$$\begin{aligned} \mathcal L_\gamma &=2L\int_{\theta_\epsilon}^{\pi/2} \frac{d\theta}{\sin\theta} =2L\log\!\left(\frac{2z_*}{\epsilon_z}\right) +O(\epsilon_z^2/\ell^2)\\ &=2L\log\!\left(\frac{\ell}{\epsilon_z}\right) +O(\epsilon_z^2/\ell^2). \end{aligned}$$

The Fefferman–Graham cutoff $z=\epsilon_z$ induces the CFT coordinate cutoff $\epsilon_{\rm CFT}=\epsilon_z$ in this boundary conformal frame. Using the Brown–Henneaux relation

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

as fixed by the asymptotic-symmetry algebra (Brown and Henneaux 1986, pp. 218–222),

RT gives

$$S_A^{(0)}=\frac{\mathcal L_\gamma}{4G_3} =\frac{c}{3}\log\!\left(\frac{\ell}{\epsilon_{\rm CFT}}\right),$$

which exactly matches the vacuum interval result of a two-dimensional CFT at leading holographic order. This is a comparison of regulated quantities in the same conformal frame; replacing $\epsilon_z$ and $\epsilon_{\rm CFT}$ independently would make the match meaningless. The original RT papers established this and many related checks (Ryu and Takayanagi 2006a, pp. 2–3; Ryu and Takayanagi 2006b, §3).

Homology and global competition

Local minimality does not select the entropy when several topologies compete. In a static BTZ geometry, an interval of length $\ell$ on a noncompact thermal line has the connected candidate

$$S_{\rm conn}(\ell)=\frac{c}{3} \log\!\left[ \frac{\beta}{\pi\epsilon} \sinh\!\left(\frac{\pi\ell}{\beta}\right) \right].$$

On a compact boundary circle of circumference $C$, a sufficiently large interval also admits a surface consisting of the geodesic for the complement plus the horizon:

$$S_{\rm disc}(\ell)=S_{\rm conn}(C-\ell)+S_{\rm BH}.$$

Both components are required by homology. The answer is $\min(S_{\rm conn},S_{\rm disc})$. Selecting the locally short connected geodesic beyond the crossing can violate the thermal entropy relation and the $A\leftrightarrow\bar A$ purification structure. For two disjoint intervals, connected and disconnected pairings likewise exchange dominance; a geodesic-by-geodesic choice that allows crossing or ignores joint homology is wrong.

This is the adversarial test: enumerate all noncrossing, anchored, homologous configurations, include horizon pieces, compare their fully regulated areas, and only then remove common divergences. A local extremum that loses this comparison is not the RT surface.

Scope and next prescription

RT applies to a static or time-reflection-symmetric semiclassical Einstein geometry at order $G_N^{-1}$. It does not contain bulk entanglement, surface shifts of order $G_N$, higher-derivative terms, or a Lorentzian rule for time-dependent states. Replica arguments derive it under saddle dominance, replica symmetry, and analytic-continuation assumptions rather than making it an unconditional theorem of every AdS/CFT pair (Lewkowycz and Maldacena 2013, §§2–4).

Use HRT for time dependence, higher-derivative functionals when the action changes, and FLM/QES for semiclassical quantum corrections.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Brown, J. D., and Henneaux, M. (1986). “Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity.” Communications in Mathematical Physics 104, 207–226. DOI.
• Lewkowycz, A., and Maldacena, J. (2013). “Generalized gravitational entropy.” Journal of High Energy Physics 2013(8), 090. DOI.
• Ryu, S., and Takayanagi, T. (2006a). “Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence.” Physical Review Letters 96, 181602. DOI.
• Ryu, S., and Takayanagi, T. (2006b). “Aspects of holographic entanglement entropy.” Journal of High Energy Physics 2006(8), 045. DOI.