--- title: "Calculate interval entanglement from a geodesic" description: "Derive an AdS3 spatial geodesic, integrate its finite-cutoff length and remove the shared divergence to obtain interval entropy differences." canonical_url: "https://sandboxphysics.com/stories/holography/regulated-geodesics/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/holography/regulated-geodesics/" last_modified: "2026-10-28" --- [Holography and quantum gravity](https://sandboxphysics.com/stories/holography/) L3 · Chapter 4 # Calculate interval entanglement from a geodesic Sandbox Physics · 2026-10-28 For a two-dimensional CFT vacuum with a classical gravitational description, the leading interval entropy is the length of a minimal curve in AdS3 divided by a gravitational constant. The length diverges at the boundary. Comparing two intervals with the same cutoff removes their common divergence. We will follow the geometry until its logarithm appears. ## Before you begin Read the Poincaré metric in Chapter 3 and reduced states in Chapter 2. Derivatives and definite integrals are needed. The length functional, first integral and substitution are explained below. Endpoint and cutoff conventions are fixed explicitly. - Derive the geodesic circle from the variational problem. - Distinguish exact finite-cutoff geometry, the continuum approximation and regulator-independent length differences. ## Write the quantity that the shortest curve minimizes Set $t=0$ in Poincaré coordinates. The boundary direction is $x$ and depth $z>0$ increases into the bulk. Use the same length units for $x,z,L$. Put endpoints at $(-\ell/2,\epsilon)$ and $(\ell/2,\epsilon)$. Here $\ell$ means their horizontal separation on the cutoff surface. Hold $\epsilon>0$ fixed while finding the shortest bulk curve. For a curve $z(x)$, a small proper length is $L\sqrt{dx^2+dz^2}/z$. The factor $1/z$ makes motion near the boundary expensive. A curve can travel inward to reduce the cost of horizontal motion. The competition between descent and horizontal distance determines its depth. $$ \mathcal L[z]=L\int_{-\ell/2}^{\ell/2}\frac{\sqrt{1+(z')^2}}{z}\,dx $$ This is a positive spatial length functional. The curve is spacelike, rather than the trajectory of a light pulse through spacetime. Source: João Penedones · Shinsei Ryu and Tadashi Takayanagi ## Translation symmetry determines a circle The integrand $F(z,z^{\prime})=\sqrt{1+(z^{\prime})^2}/z$ has no explicit dependence on $x$. Its Euler–Lagrange equation therefore has first integral $F-z^{\prime}\,\partial F/\partial z^{\prime}=\mathrm{constant}$. At maximum depth the slope vanishes. Call that depth $R$, so the constant is $1/R$. Rearrange to obtain $z\sqrt{1+(z^{\prime})^2}=R$. Separating variables and using left-right symmetry gives $x^2+z^2=R^2$. Requiring the circle to pass through our cutoff anchors fixes $R=\sqrt{(\ell/2)^2+\epsilon^2}$. Starting instead with a circle of radius $\ell/2$ anchored at the coordinate boundary and then clipping it moves the cutoff endpoints inward. That is a different regulator convention. $$ \frac{1}{z\sqrt{1+(z')^2}}=\frac1R,\qquad z(x)=\sqrt{R^2-x^2} $$ The first integral and endpoint conditions determine the curve. Its circle center lies on the coordinate boundary; the displayed endpoints lie at finite cutoff depth. An authored coordinate schematic. Both gold anchors lie on the same cutoff. The arc follows the spatial hyperbolic metric; it is not a light ray. ## Integrate along the arc Substitute $x=R\cos\theta$ and $z=R\sin\theta$. The Euclidean line element becomes $R\,d\theta$. Dividing by depth gives $L\,d\theta/\sin\theta$. The endpoints have angles $\theta_0$ and $\pi-\theta_0$, with $\sin\theta_0=\epsilon/R$. An antiderivative is $\log\tan(\theta/2)$. Evaluating both endpoints gives the exact expression below. The inverse hyperbolic sine is $\operatorname{arsinh}u=\log(u+\sqrt{1+u^2})$. Expanding for small $\epsilon/\ell$ produces the logarithmic leading term. The first correction is quadratic in the cutoff ratio. Worked example: choose $L=1$, $\ell=2$ and $\epsilon=0.1$. The exact length is $2\operatorname{arsinh}10\simeq5.99645$; the leading approximation is $2\log20\simeq5.99146$. Their difference is about $0.00498$. Halving the cutoff reduces this error by roughly four, although the length itself grows by nearly $2\log2$. $$ \mathcal L_\epsilon(\ell)=2L\log\frac{R+\ell/2}{\epsilon}=2L\operatorname{arsinh}\frac{\ell}{2\epsilon} $$ This exact common-height endpoint distance is the Lab’s geometric readout and the target of an independent numerical line integral. $$ \mathcal L_\epsilon(\ell)=2L\log\frac\ell\epsilon+2L\frac{\epsilon^2}{\ell^2}+O\!\left(L\frac{\epsilon^4}{\ell^4}\right) $$ The cutoff must be much smaller than the interval. An exact cutoff distance is not automatically the exact entropy of an arbitrary regulated field theory. ## Subtract before removing the cutoff Give intervals of widths $\ell_1$ and $\ell_2$ the same cutoff. Their common term $-2L\log\epsilon$ cancels. The difference has a finite limit. At finite cutoff the exact difference still contains power corrections, so the pure logarithmic difference is only the limiting result. Increasing the width from two to four gives a limiting length difference $2L\log2$. At $\epsilon=0.1$, the exact difference is about $1.38256L$, slightly below $1.38629L$. Numerical quadrature should reproduce the finite value; a cutoff scan tests convergence toward the limit. $$ \lim_{\epsilon\to0}\left[\mathcal L_\epsilon(\ell_2)-\mathcal L_\epsilon(\ell_1)\right]=2L\log\frac{\ell_2}{\ell_1} $$ Both intervals must use the same regulator prescription. Scheme-dependent finite constants in individual entropies require separate care. ## When a length becomes an entropy The static Ryu–Takayanagi prescription divides the length of a minimal curve homologous to the boundary interval by $4G_3$. Set $\hbar=1$. The three-dimensional Newton constant $G_3$ has length units, so the entropy is dimensionless. In an AdS3/CFT2 description governed by Einstein gravity, the central charge $c=3L/(2G_3)$ measures the scale of boundary degrees of freedom. The leading vacuum interval entropy is then $(c/3)\log(\ell/\epsilon)$. Doubling the width changes it by $(c/3)\log2$. This interpretation assumes a semiclassical bulk, large central charge and a controlled gravitational approximation. Chapter 7 introduces bulk quantum corrections, which matter when interpreting a vanishing leading mutual information. $$ S_A^{(0)}=\frac{\mathcal L(\gamma_A)}{4G_3},\qquad \Delta S_A^{(0)}=\frac c3\log\frac{\ell_2}{\ell_1} $$ The superscript denotes the classical leading term. Exact cutoff lengths and continuum leading entropies are labeled separately in the experiment. Source: Shinsei Ryu and Tadashi Takayanagi ## Try the reasoning ### [Test a logarithm by moving the cutoff](https://sandboxphysics.com/labs/ads-interval-entanglement/) 1. Choose One interval reaches deeper and set width to one. Predict how increasing width to two changes maximum depth and total length, then drag an endpoint to check. 2. Fix the interval, open cutoff settings and decrease the radial cutoff. Calculate the leading logarithm from this chapter’s expression and compare with the single-curve regulated total; test roughly quadratic error reduction. 3. Read the left plot’s length difference between the current interval and a unit interval at the same cutoff. Explain why the difference converges even though both individual lengths diverge. ## Exercises and explanations 1. Two interval widths have ratio three and $c=6$. What is their leading continuum entropy difference? Read the explanation Cancel the common cutoff, then multiply by $c/3=2$. The result is $2\log3\simeq2.197$ nats. This difference does not fix either interval’s absolute entropy. 2. A straight horizontal line looks shorter on the drawing than an arc that descends into the bulk. Why is it not the shortest path? Read the explanation The drawing uses Euclidean coordinate lengths. Hyperbolic proper length divides each segment by its depth. A horizontal segment near the boundary pays a large scale factor throughout. The variational problem includes that cost. [Previous: Reading boundary correlations from a bulk field](https://sandboxphysics.com/stories/holography/fields-at-the-boundary/) [Next: How two intervals can share a bulk region](https://sandboxphysics.com/stories/holography/intervals-and-wedges/) ## Sources and recalculated scope The course derives its worked examples and draws its own schematics. The numerical references recalculate specified analytic geometry and a finite five-qubit encoding. The source locations below distinguish those calculations from theoretical reading topics. 1. [João Penedones, TASI lectures on AdS/CFT, arXiv:1608.04948v1 (2016)](https://arxiv.org/html/1608.04948v1) Sections 3.1–3.3: global and Poincaré coordinates, fixed-background fields, source boundary conditions and dynamical gravity; section 4.1: conditions for a CFT with a weakly coupled bulk description. The course derives its radial optical paths and scalar near-boundary exponents from these metrics and field equations. 2. [Shinsei Ryu and Tadashi Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, hep-th/0603001v2; PRL 96, 181602 (2006)](https://arxiv.org/html/hep-th/0603001v2) Equation (5), static area prescription; equations (6)–(10) and Figure 1, vacuum AdS3 interval geometry and the circle result. We independently integrate the Poincaré geodesic and compare regulator limits. Our equal-height finite-cutoff anchors are a separately defined geometric diagnostic. Multiple-interval minima use Headrick, not equation (12) of this early paper.