Why light reaches the AdS boundary in finite coordinate time

Give the map a ruler and a clock

A separation measured in pixels on a map needs a scale before it means kilometers. A spacetime map needs a local rule for both distances and times. The metric supplies that rule. It converts coordinate changes into quantities measured with nearby rulers and clocks.

Use AdS3, with one radial coordinate and one angular coordinate. Set the speed of light to one. The curvature radius $L$ has units of length. Global time $\tau$ , radial angle $\rho$ and azimuth $\phi$ are dimensionless, with $0\leq\rho<\pi/2$ . A central clock measures time $T=L\tau$ . Global time runs over the entire real line. This unwrapping defines the universal cover of AdS.

The negative sign distinguishes the time direction. A clock fixed at a position samples the time term. Distances measured at one global time sample the spatial terms. A light ray has zero spacetime interval. These choices extract different questions from the same metric.

ds^2=\frac{L^2}{\cos^2\rho}\left(-d\tau^2+d\rho^2+\sin^2\rho\,d\phi^2\right)

This is fixed pure global AdS3. Matter does not change its curvature and no black hole is present. A disk depicts spatial coordinates; animation represents time.

Source: João Penedones · Shinsei Ryu and Tadashi Takayanagi

Follow one outgoing pulse

A radial ray keeps $d\phi=0$ . Insert the light condition $ds^2=0$ . The positive overall factor cancels and an outgoing ray obeys $d\tau=d\rho$ . Its coordinate travel time from a source at $\rho_s$ to a cutoff receiver at $\rho_c$ is their difference.

Worked example: place the source at $\rho_s=\pi/6$ and the cutoff at $\rho_c=5\pi/12$ . The pulse takes $\Delta\tau=\pi/4$ , or central-clock time $\Delta T=\pi L/4$ . Moving the cutoff all the way toward the boundary adds a limiting $\pi/12$ in global time. The boundary is a limiting location, not a physical ring of detectors outside the disk.

Multiplying the metric by a positive function leaves its light directions unchanged. Removing the divergent overall factor gives a conformal map that fits infinity on a finite page. Such a map preserves causal reachability. Proper lengths are calculated with the full metric.

\Delta\tau=\rho_c-\rho_s\;\longrightarrow\;\frac{\pi}{2}-\rho_s

Global coordinate time is finite. A light ray has zero proper time. An affine parameter labels position along its null geodesic and diverges at the boundary.

A radial conformal diagram. Equal coordinate steps fit on the page; local rulers grow without bound near the right edge. The gold line is null.

Measure the distance on a simultaneous slice

Now line up stationary observers along a radius at one global time. Add the small lengths that each measures. This procedure lays rulers on a spatial slice instead of following a light ray through spacetime.

The metric gives $d\ell=L\,d\rho/\cos\rho$ . Near the boundary, a small coordinate step covers an increasingly large proper length. Its integral diverges logarithmically. A stationary clock accumulates proper time $d\sigma=L\,d\tau/\cos\rho$ , so clocks farther out also accumulate more proper time per global-time increment.

Dividing a whole spatial distance by the elapsed time of the central clock is therefore not a local speed measurement. Each local observer still measures unit light speed. The next chapter turns to entanglement in two qubits. Learning the geometric and quantum definitions separately gives us something precise to connect later.

\frac{\ell(\rho_s,\rho_c)}{L}=\left[\log(\sec\rho+\tan\rho)\right]_{\rho_s}^{\rho_c}

The proper distance diverges as the cutoff approaches the boundary. The full metric determines AdS curvature; the disk supplies a coordinate map.

Source: João Penedones · Shinsei Ryu and Tadashi Takayanagi

Try the reasoning

Move a source and predict the arrival time

Locate the source, dashed cutoff and conformal boundary. Choose The nearby boundary to send the gold ray radially outward.
Drag the source outward. Predict whether the outgoing pulse arrives earlier or later, then compare the time.
Keep the source fixed. Open Distance and model assumptions and increase Radial cutoff. Explain why proper distance grows rapidly while conformal-boundary arrival time stays fixed.

Exercises and explanations

For a central source, compare global travel time from the center to $\rho=\pi/4$ and from there to the boundary. Are the proper distances equal?

Read the explanation

Each takes $\pi/4$ in global time. The first proper distance is finite; the second extends to the boundary and diverges. Equal coordinate times need not correspond to equal spatial lengths.

If you rescale the plotted radial coordinate by two, does the slope of a light ray change? Does its causal reachability change?

Read the explanation

The plotted slope changes and the metric must transform with the coordinate. Whether the same two events can exchange a signal is unchanged. A forty-five-degree line has meaning only after the axes and metric are specified.

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.

João Penedones, TASI lectures on AdS/CFT, arXiv:1608.04948v1 (2016)

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.

Shinsei Ryu and Tadashi Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, hep-th/0603001v2; PRL 96, 181602 (2006)

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.