Reading Dolby and Gull on radar time and the twins

Figures 1 and 2: the pictures the paper argues against

Both figures are labelled as typical textbook illustrations. In Figure 1, Barbara’s lines of simultaneity just before and just after her turn meet Alex’s worldline at G and H, far apart. Figure 2 adds a finite turn, during which her lines sweep round from EG to FH.

The introduction says plainly that there is no disagreement about how much each twin ages, nor about when each sees events. The dispute is only about which events happen “now” for Barbara. The common misreading is to treat the paper as a new answer to who is younger. It is an argument about the lines in these two figures, which on Barbara’s left overlap and give one event three times.

Figure 3: earliest and latest

Read the two definitions above Figure 3 slowly. $\tau^+(x)$ is the earliest proper time at which light from event $x$ can reach the observer; $\tau^-(x)$ is the latest proper time at which light can leave the observer and still reach $x$. The superscripts mark reception and emission, not before and after the event.

The paper calls this a rearrangement of Einstein’s $t_B-t_A=t'_A-t_B$ from 1905, popularised by Bondi’s k-calculus. Nothing in it mentions coordinates, so it applies to any observer, which is the point the rest of the paper builds on.

Figure 4 and Eq. (1): rederive one line

Figure 4 sets the turnaround at the origin. Barbara comes in from the right and leaves to the right with speed $s=\tanh\alpha$, and the paper switches to light-cone coordinates $u=x+t$, $v=x-t$. Region I is where both are positive. Alex is not in Figure 4; he first appears in Figure 5, inside region I.

For an event in region I the echo returns on the outgoing leg, so $u_0=e^{\alpha}\tau^+$, and the pulse left on the incoming leg, so $v_0=-e^{\alpha}\tau^-$. Averaging gives Eq. (1). With $s=0.8$, $e^{-\alpha}=1/3$: Barbara’s radar time runs at one third of Alex’s for home events near the turn, as in Chapter 6.

Our Lab is the mirror image: home stays at $x=0$ and B travels towards positive $x$, so the paper’s region I is our home side. The Lab also starts both clocks at departure, so every time differs from the paper’s by a constant.

Figure 5: flat lines on the home side

Figure 5 draws the surfaces for several $\tau_0$. In regions F and P, Barbara sends and receives on the same leg, so she cannot be told apart from an inertial observer at speed $\pm s$ and her surfaces are that observer’s. In regions I and II she sends on one leg and receives on the other, and the surfaces are flat.

Do not read the flat lines in region I as Barbara borrowing Alex’s frame. They are flat because the assignment is symmetric under reversing her proper time, and surfaces one year of hers apart lie $e^{\alpha}$ years apart in Alex’s time, three years when $s=0.8$. Every event gets one surface, so the overlap of Figure 1 is gone.

Figures 6 and 7: a gradual turn, then one that never ends

Figure 6 lets Barbara turn with acceleration $a$ for proper time $\tau_c=\alpha/a$ and divides the plane into nine regions. The text states the result to check: regions I and II are identical to Figure 5, and in F and P each surface only receives a slightly different $\tau_0$, because the turn takes slightly longer. We reproduced these explicit formulas and region U; we did not reproduce the implicit formulas for the four corner regions.

Letting $\tau_c\to\infty$ gives Figure 7, a uniformly accelerating observer. Radar time and radar distance become Rindler coordinates and cover only region U. The paper gives the reason: Barbara cannot send a signal to regions P or II, and no signal reaches her from F or II. This is the horizon of Chapter 6, in the paper’s placement with the observer at $x=1/a$ at time zero.

Figures 8 to 10: new coordinates, same answer

Figure 8 redraws Alex’s surfaces in Barbara’s radar coordinates, where the metric is $ds^2=e^{2a\rho}(d\tau^2-d\rho^2)$. Alex’s path looks curved, yet it is a geodesic of this metric, so he is still the inertial twin and still ages more. Some of his surfaces miss his path in the picture only because these coordinates cover region U alone. Figure 9 does the same for the immediate turn, with a jump in scale along the light lines through the corner.

The paper’s section heading, “Gravity doesn’t matter”, names the misreading to avoid: the twin result does not need general relativity. Barbara’s surfaces depend only on kinematics, and redrawing them in other coordinates changes the picture, not any statement of the form “this happened then”. We did not reproduce the coordinate maps of Figures 8 and 9.

Figure 10, in the conclusion, is the realistic trip in which the twins are together before and after. Inside the causal envelope of the journey, the middle four regions, the surfaces are those of Figure 5. The authors invite the reader to check that the ratio of the two travelling times is $\sqrt{1-s^2}$; for $s=0.8$ that is the 6 years against 10 of Chapter 5.

A checklist for the next paper on simultaneity

Ask first whether the paper disputes ages or only the labelling of distant events; the first is settled by proper time, the second is a choice of definition. Then check whether the time assignment is free of coordinates, whether it gives each event one time, and whether it agrees with proper time on the observer’s own path. Note which events it covers: radar time exists only for events the observer can both signal and hear from. Finally, see what a brief acceleration does to distant events; a good definition changes them only briefly.