The twin trip, told by the signals each twin receives

Twin B flies out at 0.8 of light speed and comes back; twin A stays home. When they meet, A has aged 10 years and B 6. Both can check this without any theory of simultaneity: each sends a light signal on every birthday, and each counts what arrives. The counts differ in a way that marks B as the one who turned around.

Sandbox Physics · 2026-10-29

The trip and its two clocks

B passes A at $t=0$ moving at $v=0.8$ , coasts for 3 years of B’s own time, turns around at once, and coasts back for another 3 years. With $\gamma=5/3$ each leg takes 5 years of A’s time, and B turns at $t=5,\ x=4$ .

At the reunion A has aged 10 years and B 6, by the proper-time rule of Chapter 4. That rule has been tested on clocks that come back: in the storage-ring measurement of Chapter 4, muons on a closed orbit lived longer by the predicted factor, to about two parts in a thousand. The puzzle is only apparent: during each leg A’s clock runs slow by B’s inertial frame, just as B’s runs slow by A’s. So which one ends up younger? The signals answer that directly.

How fast signals arrive from a moving source

Let a source move away at speed $v$ and send a signal each year of its own time. In the receiver’s frame those emissions are $\gamma$ years apart, and each starts $v\gamma$ light-years further away. The arrivals are therefore spaced by the factor $k$ below. For an approaching source the spacing is $1/k$ .

At $v=0.8$ the factor is 3: a receding twin’s signals arrive once every 3 years, an approaching twin’s three times a year. The same factor applies to both twins, because it depends only on their relative speed.

k=\gamma(1+v)=\sqrt{\frac{1+v}{1-v}}=\sqrt{\frac{1.8}{0.2}}=3

Worked example

Counting the birthdays

Write $N_{\mathrm B}$ for the number of A’s signals that B receives, and $N_{\mathrm A}$ for the number of B’s signals that A receives. B’s count: on the way out B is receding, so for 3 years A’s signals arrive at one per 3 years, which is 1 signal. On the way back B is approaching, so in 3 years 9 more arrive. B receives 10 signals in all, one for each of A’s 10 birthdays.

A’s count: the light from B’s turnaround leaves $x=4$ at $t=5$ and reaches A at $t=9$ . For 9 years A receives the outbound signals at one per 3 years, which is 3. In the final year A receives the 3 inbound signals. A receives 6 signals in all, one for each of B’s birthdays.

Both twins agree on every count. The asymmetry is in when each sees the switch from slow to fast. B sees it at the turnaround, halfway through B’s trip. A sees it only 9 years into a 10-year wait, because the news of the turnaround takes 4 years to arrive. B is the one who changed direction, and the signals record it.

N_{\mathrm B}=3\times\tfrac13+3\times3=10,\qquad N_{\mathrm A}=9\times\tfrac13+1\times3=6

The comoving-frame shortcut jumps

A common shortcut asks what A’s clock reads “now” in B’s momentary rest frame. On the outbound leg B’s simultaneity lines satisfy $t-vx=\text{const}$ , so the home time paired with the turnaround event is $5-0.8\times4=1.8$ years. Just after the turn the lines satisfy $t+vx=\text{const}$ , giving $5+0.8\times4=8.2$ years.

By this bookkeeping, 6.4 years of A’s life pass in no time on B’s clock. Nothing B observes jumps: the signals arriving from A merely change from slow to fast. The jump comes from switching from one inertial frame to another. Dolby and Gull point out that textbook versions of this picture also leave some distant events with several assigned times. Chapter 6 replaces the shortcut with radar time, which gives every event one time and no jump.

Predict, change, explain

Predict how many of A’s yearly signals B receives on the way out and on the way back.

In the twin Lab, choose “Sudden turnaround: 10 years against 6”, then “Gradual turnaround”, and compare the stepped flash counts in the right-hand chart.

Use the moment each twin sees the slow-to-fast switch to explain why their totals differ.

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. Repeat the count for $v=0.6$ with 4 years of B’s time on each leg.

Read an explained answer

Here $k=2$ . B receives 2 signals going out and 8 coming back, 10 in all. B turns at $t=5,\ x=3$ , which A sees at $t=8$ : 4 slow signals in 8 years, then 4 fast ones in the last 2 years, 8 in all. A ages 10 years, B 8.

2. Why can’t B claim that A is the one who travelled?

Read an explained answer

B’s received rate switches halfway through B’s trip, while A’s switches near the end of A’s wait, and the counts agree on this. B also feels the turnaround as an acceleration; A feels nothing.

3. How big is the comoving-frame jump for the trip in the first exercise?

Read an explained answer

The turn is at $t=5,\ x=3$ , so the paired home time jumps from $5-0.6\times3=3.2$ to $5+0.6\times3=6.8$ years.

Sources to read alongside

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.

Bailey et al., Measurements of relativistic time dilatation for positive and negative muons in a circular orbit, Nature 268, 301 (1977)

Abstract only: the storage-ring Lorentz factor 29.33 and the positive-muon lifetime 64.419(58) microseconds. The full paper was not read, so the course compares only these two quoted numbers.

The trip and its two clocks

How fast signals arrive from a moving source

Counting the birthdays

The comoving-frame shortcut jumps

Predict, change, explain

Try explaining it yourself

1. Repeat the count for v=0.6v=0.6v=0.6 with 4 years of B’s time on each leg.

2. Why can’t B claim that A is the one who travelled?

3. How big is the comoving-frame jump for the trip in the first exercise?

Sources to read alongside

1. Repeat the count for $v=0.6$ with 4 years of B’s time on each leg.