---
title: "Light Signals in Global AdS Interactive Lab | Sandbox Physics"
description: "How can light reach the AdS boundary in finite global time? Global AdS time and proper spatial distance measure different things."
canonical_url: "https://sandboxphysics.com/labs/ads-boundary-signals/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/ads-boundary-signals/"
last_modified: "2026-10-28"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Light Signals in Global AdS

**Background** A map can fit an infinite space on one page. Anti-de Sitter space has a natural compact map whose edge marks its conformal boundary. Light follows the causal structure of that map.

**Why it matters** Boundary conditions matter in AdS because light can connect an interior event to a boundary endpoint at a finite global time. Establishing that causal relation comes before discussing a boundary quantum theory.

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 misconception** If the boundary is infinitely far away, a finite arrival time must imply faster-than-light travel.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Read the metric before interpreting a compact map: causal arrival, proper distance and affine parameter are separate quantities.

## Related experiments and articles

Holography and quantum gravity · Reading path

### From AdS signals to entanglement geometry and quantum recovery

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why light reaches the AdS boundary in finite coordinate time**](https://sandboxphysics.com/stories/holography/clocks-and-curvature/)
2. [Article **Reading boundary correlations from a bulk field**](https://sandboxphysics.com/stories/holography/fields-at-the-boundary/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)