FR09-A · Light finds the boundary

Light Signals in Global AdS

How can light reach the AdS boundary in finite global time? Global AdS time and proper spatial distance measure different things. The metric lets a null ray reach a conformal boundary endpoint at finite global time even as spatial distance and affine parameter diverge.

Drag a light source across a compact map of anti-de Sitter space. Follow the rays and compare when they reach the boundary.

Selected ray arrival $\Delta\tau=1.16$

Earliest boundary arrival $\Delta\tau_{\min}=1.05$

Radial distance to cutoff $D/L=3.13$

Boundary arrival angle $\varphi_{\partial}=0.179\,\mathrm{rad}$

A curved spacetime, a finite map

Send light in another direction

Drag the gold source. The gold ray follows your chosen direction; the blue rays leave at the same time.

Source radius

\rho_s

0.520

Direction from outward radial

\beta

40^\circ

Distance and model assumptions

Radial cutoff

\rho_c

1.52

The dashed circle is a finite radial cutoff. The distance readout follows the radial route from the source to that circle. Moving the cutoff changes this distance; the boundary arrival times refer to the conformal boundary.

Animation speed is illustrative. Keyboard: focus the scene, use left or right to rotate the source, and up or down to change its radius.

Model assumptions

The rays are null geodesics of fixed, empty global AdS in three spacetime dimensions. The disk compresses an infinite spatial distance into a finite radius. Time is global coordinate time in units of the AdS light-crossing scale. Boundary arrivals are conformal endpoints at infinite affine parameter; backreaction, diffraction and a boundary quantum field are omitted.

Fixed-background calculation

Every launch direction has an arrival time

\begin{aligned}\Delta\tau_{\rm outward}&=\frac{\pi}{2}-\rho_s\\\Delta\tau_{\rm inward}&=\frac{\pi}{2}+\rho_s\end{aligned}

Times use the global clock. The circular map preserves the causal paths but compresses proper distances near the boundary.

Start with the essentials

Focus question: How can light reach the AdS boundary in finite global time?
One-sentence intuition: Finite global travel time can coexist with an infinite proper spatial distance because the metric weights space and time differently near the boundary.

Core mathematical model

The fixed global AdS metric

\begin{aligned}ds^2&=\frac{L^2}{\cos^2\rho}\bigl(-d\tau^2+d\rho^2\\&\qquad+\sin^2\rho\,d\varphi^2\bigr)\\0&\le\rho<\pi/2,\qquad \tau=ct/L\end{aligned}

The curvature radius sets the length scale; the azimuth is measured in radians. The universal cover has unbounded global time. Removing the common positive factor preserves null paths, leaving the optical geometry of a hemisphere.

Radial light travel

\begin{aligned}ds^2=0,\ d\varphi=0&\ \Longrightarrow\ |d\rho/d\tau|=1\\\Delta\tau_{\rm near}&=\pi/2-\rho_s\\\Delta\tau_{\rm far}&=\pi/2+\rho_s\end{aligned}

An outward radial ray reaches the nearby boundary first. An inward ray crosses the centre and reaches the opposite boundary later. A central source sends every direction to the boundary after half pi units of global time.

Proper distance on a constant-time slice

\begin{aligned}\frac{D}{L}&=\int_{\rho_s}^{\rho_c}\frac{d\rho}{\cos\rho}\\&=\operatorname{arsinh}(\tan\rho_c)\\&\quad-\operatorname{arsinh}(\tan\rho_s)\end{aligned}

This radial distance diverges as the cutoff approaches the boundary. It measures a path on a constant-time slice. A null ray has zero spacetime interval, and its boundary endpoint lies at infinite affine parameter.

Common difficulties

Different clocks and distances answer different questions

Typical misconceptionIf the boundary is infinitely far away, a finite arrival time must imply faster-than-light travel.

Better mental modelThe local metric gives a null trajectory at every point. Global coordinate time is not the distance divided by a single local clock reading. The divergent affine parameter also prevents treating the endpoint as an ordinary event at a finite distance.

Run the experiment

01
Start at the centre
Choose Every direction arrives together and watch one set of pulses.
What to observe: All directions reach the circular boundary together. Rotational symmetry leaves no preferred launch direction.
02
Compare the near and far side
Compare The nearby boundary with A journey through the centre. Then drag the gold source.
What to observe: Moving outward shortens the near-side travel time and lengthens the far-side time. The arrival chart changes continuously with the source position.
03
Move a finite cutoff
Open Distance and model assumptions and increase Radial cutoff. Compare the distance with the selected arrival time.
What to observe: The radial proper distance grows sharply near the boundary. The conformal boundary arrival time stays fixed because the source and launch direction have not changed.

Light Signals in Global AdS

Every launch direction has an arrival time