Setting distant clocks with a light echo

To set a clock 3 light-years away you would add the light’s travel time, but knowing that time already assumes the one-way speed of light, which needs two set clocks to measure. Einstein broke the circle in 1905 with a definition: send light out, catch its echo, and say the reflection happened halfway between. This chapter turns his definition into a measuring tool.

Sandbox Physics · 2026-10-29

Why a distant clock cannot simply be read

Suppose A wants clock B, far away, to agree with A’s own clock. A could send a message saying “it is noon” and let B add the travel time. The travel time is the distance divided by the one-way speed of light. To measure a one-way speed you time the start and the finish with two clocks that already agree, which is what we were trying to arrange.

One clock can measure a round trip. A sends a pulse, a mirror at B reflects it, and A notes how long the pulse was away. Nothing in that number says how the trip divided between going out and coming back.

Einstein’s rule: the reflection happens halfway

Section 1 of Einstein’s 1905 paper defines the missing piece. Light leaves A when A’s clock reads $t_A$ , is reflected at B when B’s clock reads $t_B$ , and arrives back at A at $t'_A$ . The two clocks are called synchronous when the outward and return legs take the same time.

Read the definition as a recipe that A can use alone. If A sends a pulse at $t_1$ and receives its echo at $t_2$ , A gives the reflection the time and distance in the second line below. Dolby and Gull call these the radar time and radar distance of the event. Chapter 6 applies the same recipe to an observer who accelerates.

Einstein assumed the rule is free of contradiction for any number of clocks: if B is synchronous with A, then A is synchronous with B, and two clocks synchronous with A are synchronous with each other. With these assumptions, every clock at rest relative to A shares one time coordinate.

\begin{gathered}t_B-t_A=t'_A-t_B\\ t=\tfrac12(t_1+t_2),\qquad x=\tfrac12(t_2-t_1)\end{gathered}

Worked example

A pulse sent at 2 and received at 8

A sends a pulse at $t_1=2\,\mathrm{yr}$ . It reaches clock B, and the echo returns at $t_2=8\,\mathrm{yr}$ . A dates the reflection at 5 years and places it 3 light-years away. This is the same tick of clock B that Chapter 1 followed.

A receives the echo at the same moment that A sees the flash of the tick itself. Both leave the same event toward A at the speed of light, so they arrive together. The send time is the extra information: it lets A remove the 3-year delay instead of only noticing it.

At the event A dates at 5 years, clock B reads 4 years. That 1-year difference is not a delay, because the delay has already been taken out. It is clock B’s own slower ticking, the subject of Chapter 4.

t=\tfrac12(2+8)=5\,\mathrm{yr},\qquad x=\tfrac12(8-2)=3\,\mathrm{ly}

A definition that still has to work

Is the halfway rule true, or only agreed? A one-way speed of light cannot be measured without clocks that are already synchronized, so the rule cannot be tested on its own. What can be tested is what is built on it. With the halfway rule, light has the same speed in both directions in A’s frame, the clocks of that frame agree with each other consistently, and later predictions, such as the muon lifetime in Chapter 4, match experiment.

The rule defines simultaneity for clocks at rest relative to A. Chapter 3 applies the same rule for an observer moving past A and finds that the two observers sort distant events into “at the same time” differently.

An experiment to think through

Predict, change, explain

With the drawing frame at rest, predict whether A’s radar time for each tick equals the coordinate time in the readouts.
In the Lab, keep the drawing-frame speed at zero and move “Tick of B to inspect” through its range, reading the “sent” and “seen at” labels on A’s worldline.
Show from the two labels that the radar time readout is their average and the radar distance is half their difference.

Time one tick three ways

The clock and twin Labs set light speed to one light-year per year, as these chapters do. The collision Lab uses PDG 2024 masses; the particle collider Lab draws its events from a teaching generator.

Try explaining it yourself

1. A sends a pulse at $t=1\,\mathrm{yr}$ and receives the echo at $t=7\,\mathrm{yr}$ . When and how far away was the reflection?

Read an explained answer

At 4 years, 3 light-years from A. The echo alone does not say which side; A knows that from the direction in which the pulse was sent.

2. Someone proposes that light takes one third of each round trip going out and two thirds coming back. Could A test this with one clock?

Read an explained answer

No. A still measures only $t_2-t_1$ , which is the same under both proposals. The proposal would date every distant reflection at $t_1+\tfrac13(t_2-t_1)$ instead of the midpoint, so it changes the time labels of distant events, not any round-trip measurement.

3. Why does the echo return at the same moment that A sees the tick’s own flash?

Read an explained answer

Both start at the same event, the reflection at the tick, and both travel to A at the speed of light along the same line.

Sources to read alongside

Einstein, On the Electrodynamics of Moving Bodies, Annalen der Physik 17, 891 (1905), English translation by Perrett and Jeffery (1923), Fourmilab edition
Section 1 defines synchronous clocks by a light signal sent from A, reflected at B and returned to A, and states the two consistency assumptions. We read Sections 1 and the opening of Section 2 only.
Dolby & Gull, On Radar Time and the Twin “Paradox”, arXiv:gr-qc/0104077v2 (2002); American Journal of Physics 69, 1257 (2001)
Read in full (nine pages, Figures 1–10). The course reproduces the radar-time formulas of the immediate turn-around, the region formulas of the gradual turn-around and the uniformly accelerating case; the implicit formulas for the four corner regions and the coordinate maps of Figures 8–9 are not reproduced.

Why a distant clock cannot simply be read

Einstein’s rule: the reflection happens halfway

A pulse sent at 2 and received at 8

A definition that still has to work

Predict, change, explain

Try explaining it yourself

1. A sends a pulse at t=1 yrt=1\,\mathrm{yr}t=1yr and receives the echo at t=7 yrt=7\,\mathrm{yr}t=7yr. When and how far away was the reflection?

2. Someone proposes that light takes one third of each round trip going out and two thirds coming back. Could A test this with one clock?

3. Why does the echo return at the same moment that A sees the tick’s own flash?

Sources to read alongside

1. A sends a pulse at $t=1\,\mathrm{yr}$ and receives the echo at $t=7\,\mathrm{yr}$ . When and how far away was the reflection?