---
title: "Twin Clocks and Radar Time Interactive Lab | Sandbox Physics"
description: "Which moment at home does a travelling twin call now, and why does the turnaround matter?"
canonical_url: "https://sandboxphysics.com/labs/twin-radar-time/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/twin-radar-time/"
last_modified: "2026-10-29"
---

Special relativity · Accelerated observers

# Twin Clocks and Radar Time

**Which moment at home does a travelling twin call now, and why does the turnaround matter?** The travelling twin comes home younger because the bent path through spacetime accumulates less proper time; neither twin sees the other's clock jump. Which home moment counts as "now" is a convention. Radar time, the average of an echo's send and return readings, gives one smooth answer through the turn, while hopping between momentary rest frames makes home time leap.

Send a twin out and back, then compare how each twin dates events at the other end: radar echoes give a smooth answer, momentary rest frames jump at the turn.

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

Model assumptions

One space dimension, with light speed set to one light-year per year. Observer A rests at the origin. B follows piecewise-constant proper acceleration, integrated in B's own time with fourth-order Runge-Kutta steps of at most 0.02 years; a sudden turn is a jump in rapidity. Radar time follows Dolby and Gull (2001): the average of B's clock readings when an echo leaves and when it returns. Momentary rest-frame simultaneity is drawn only for comparison. The model has no gravity, no rocket fuel budget and no optics of real images.

Green is radar time and purple the momentary rest frame; the right panel counts the flashes each twin receives.

More numbers and the saved record

Proper acceleration while not coasting $\infty$

A's yearly flashes that reach B $10$

B's yearly flashes that reach A $6$

Last home time whose light reaches B $\infty$

The JSON record holds B's sampled worldline, the radar pair for the home event and every flash with its reception time.

[Course: relativity as measurement, in eight chapters](https://sandboxphysics.com/stories/relativity-measurement/)

Physics tutorial

## Two clocks, one reunion

**Background** Twin A stays home. Twin B flies out, turns around and comes back. Both keep sending yearly flashes and radar echoes, and both clocks are compared side by side at the reunion.

**Why it matters** The age difference is not in doubt. The puzzle is how B should date events at home while travelling, and whether the turnaround hides a sudden jump.

Start with the essentials

- **Focus question**: Which moment at home does a travelling twin call now, and why does the turnaround matter?
- **One-sentence intuition**: Proper time along each path decides who is younger. Radar time is a smooth, operational way for B to call a home event simultaneous; momentary rest frames are not.

### Core mathematical model

#### Radar time and distance

$$
\tau=\tfrac12\,(\tau^{+}+\tau^{-}),\qquad \rho=\tfrac12\,c\,(\tau^{+}-\tau^{-})
$$

B sends an echo at its own reading $\tau^{-}$ and receives it back at $\tau^{+}$. Dolby and Gull use this average for any observer, accelerated or not.

#### Proper time along a path

$$
\tau=\int \sqrt{1-v(t)^2/c^2}\;\mathrm{d}t
$$

With a sudden turn at eight tenths of light speed and ten years at home, B's clock gains six years.

#### Momentary rest-frame "now" at home

$$
t_{\mathrm{home}}=t_B-\frac{v\,x_B}{c^2}
$$

Changing v at the turn changes this home time abruptly. In the sudden-turn preset it leaps from 1.8 to 8.2 years.

#### Constant proper acceleration from rest

$$
x(t)=\frac{c^2}{a}\left(\sqrt{1+\frac{a^2t^2}{c^2}}-1\right),\qquad t_{\mathrm{horizon}}=\frac{c}{a}
$$

Light leaving home after the horizon time never catches B. With a equal to half a light-year per year squared, that time is two years.

### Common difficulties

#### The turn is not where the years go

**Typical misconception** B's missing years are lost during the acceleration.

**Better mental model** Shrink the turnaround toward zero: the age gap does not vanish but approaches the value set by the two coasting legs. The difference is the length of each path in proper time.

#### Simultaneity is a convention

**Typical misconception** There is one true home event happening "now" for B.

**Better mental model** Distant simultaneity needs a rule. Radar time is one operational rule that works for accelerated observers without jumps; it agrees with Einstein synchronization for events whose echo leaves and returns during the same coasting leg.

#### What the model leaves out

**Typical misconception** The diagram includes gravity or the rocket's engine.

**Better mental model** One space dimension, with light speed set to one light-year per year. Observer A rests at the origin. B follows piecewise-constant proper acceleration, integrated in B's own time with fourth-order Runge-Kutta steps of at most 0.02 years; a sudden turn is a jump in rapidity. Radar time follows Dolby and Gull (2001): the average of B's clock readings when an echo leaves and when it returns. Momentary rest-frame simultaneity is drawn only for comparison. The model has no gravity, no rocket fuel budget and no optics of real images.

### Run the experiment

1. 01

   #### Count the years at reunion

   Start with the sudden-turn preset and let the trip play.

   **What to observe:** A's clock shows ten years and B's shows six. The flash staircases end at the same pair of numbers.

2. 02

   #### Date a home event by radar

   Drag the round home event up and down A's line.

   **What to observe:** B's radar time changes smoothly; the home event at five years gets radar time three, the middle of B's trip.

3. 03

   #### Compare the two rules for "now"

   Watch the purple and green markers on A's line during the turn, then try the gradual-turn preset.

   **What to observe:** The purple rest-frame marker leaps across years of home time at a sudden turn, while the green radar marker moves on smoothly.

4. 04

   #### Find the horizon

   Choose Endless acceleration hides home and drag the home event past two years.

   **What to observe:** Its light never reaches B, so the radar time and reception time become infinite.

Physical conclusion

Ask two separate questions: whose path holds more proper time, and which rule assigns distant times. The first has one answer; the second needs a convention, and radar time is a smooth one.

## Related experiments and articles

Relativity as measurement · Reading path

### Relativity as measurement: clocks, light signals and four-momentum

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

1. [Article **Proper time is what a moving clock records**](https://sandboxphysics.com/stories/relativity-measurement/proper-time/)
2. [Lab **Light Signals and Moving Clocks** Time one tick three ways](https://sandboxphysics.com/labs/light-signal-clocks/)
3. [Article **The twin trip, told by the signals each twin receives**](https://sandboxphysics.com/stories/relativity-measurement/twins-and-signals/)
4. [Article **Radar time for an observer who accelerates**](https://sandboxphysics.com/stories/relativity-measurement/radar-time-accelerated-observer/)
5. [Article **Reading Dolby and Gull on radar time and the twins**](https://sandboxphysics.com/stories/relativity-measurement/reading-dolby-gull-radar-time/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

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