Both proper times are correct. But you should decide which one to use depending on what you want to compute. Proper time is the time which the observer's clock measures. Thus, Earth proper time is what Earth-based clock measures, and spacecraft's proper time is spacecraft clock measurement.
For Earth-based observer you're already given the proper time: it's what the clock on Earth measures. For spacecraft you have another proper time. This proper time is sum of proper times from flying away and going back.
To understand this better, let's make a drawing:
Here the green line is world line of spacecraft, red one is of Earth, and the dashed lines represent the light cone. It's obvious how to find the proper time of the Earth, since it's stationary. Just measure the length of its world line. How do we now compute proper time of spacecraft? We just compute the "length" of its world line. BUT: as space-time is not Euclidean, namely we have $d\tau^2=c^2dt^2-dx^2$, we should take intervals instead of lengths. It becomes important for moving objects. So, we split the world line of spacecraft into two pieces, where the motion is unaccelerated, and get:
$$c\Delta\tau_1=\sqrt{c\Delta t_1^2-\Delta x_1^2},$$
$$c\Delta\tau_2=\sqrt{c\Delta t_2^2-\Delta x_2^2},$$
$$\Delta\tau_\Sigma=\Delta\tau_1+\Delta\tau_2,$$
$\Delta\tau_\Sigma$ is the proper time of spacecraft. So, spacecraft clock will read $\Delta\tau_\Sigma$. You now just have to find the difference between Earth proper time $t$ and spacecraft proper time to get solution of the problem.
From this you can also derive time dilation formula if you wish.
Both of the following statements are true:
(1) Joe observes the lab clocks to run slow
(2) According to the lab clocks, 50 seconds elapse between the creation and decay events of the muon.
Note that in the above, I have used the plural clocks. To determine the elapsed time in the lab, the readings of two spatially separated clocks are recorded, a clock co-located with the creation event of the muon and a clock co-located with the decay event.
Clearly, the lab clocks must by synchronized (in the lab frame of reference) for this to be valid.
But, according to Joe (or any relatively moving reference frame), these two lab clocks are not synchronized (relativity of simultaneity) and this is the resolution to the apparent contradiction.
According to Joe,
- the muon decays in 10 seconds
- the lab clocks run slow compared to his clock
- the two lab clocks show a difference of 50 seconds between the
events since they aren't synchronized.
As always, drawing a spacetime diagram will make the above quite clear.
Best Answer
In my opinion the question is poorly formulated. Its answer depends a great deal on a subjective or context-dependent choice of terminology, not only on physical facts; it leaves some important points unspecified; and it confuses "being undefined" with "being unmeasurable". These concerns are also expressed in joigus's answer and R.W.Bird's comment. Let me address these points in turn.
First, whether the term "proper time" can be used for a lightlike path is a matter of convention, personal choice, and context. If we are dealing with signals that propagate with speed lower than $c$, then the term "proper time" is appropriate and can be calculated or measured (see below) to be larger than zero. If the speed of the signal is increased to have a limit value of $c$, the proper time will have a limit value of zero. In such a circumstance it can be useful to simply keep the term "proper time" and say that its value is zero when the signal becomes lightlike, rather than declaring the term to be suddently inapplicable in the limit case. This is a matter of terminological convention, not of physics.
This kind of limit situations appears often in relativity. Consider for example this quote from Gourgoulhon (2012):
In fact, the radio signal could be propagating through rarefied matter, say, and therefore have a speed lower than $c$. In this case we can speak of a non-zero proper time of the interval between source and target.
Second, being undefined and being unmeasurable are two very different things, and their distinction is very important in physics. For example, "Regarding electromagnetic field quantities, we take the position that they are not measurable except in vacuo" (Hutter & al 2006), and yet we define them also within matter, where they are not measurable. The unmeasurability leads to several different definitions and formulations (the Chu formulation, the Minkowski formulation, and various others). The definitions are inequivalent, but the formulations are nevertheless equivalent (and many of them are indeed used today) because they lead to the same predictions for the quantities that can be measured. For a discussion of these matters see for example Penfield & Haus (1967) or Hutter & al (2006).
In the present case it could be even argued that we don't even have a problem of definition, see below.
Third, it is not clear (at least from the snippet you quote) what the authors mean by "to measure". The proper time of a timelike path represented by a curve $C\colon [a,b] \to \text{spacetime}$ is given by the integral (see eg Misner & al 1973 p. 316) $$ \int_a^b \sqrt{\Bigl\lvert \pmb{g}[\dot{C}(t),\dot{C}(t)]\Bigr\rvert}\ \mathrm{d}t \ , $$ where $\pmb{g}$ is the metric.
This integral, which can be generally called "path length", is also defined for lightlike and spacelike paths. There are several ways to measure the path length. For a timelike path it can be directly measured by one clock moving along that path. For spacelike or mixed paths the procedure is a little more involved but still based on observers carrying clocks; see the illuminating discussion in Frankel (1979) ch. 2, or in Landau & Lifshitz (1996) § 84. It's because of the difference between these possibilities that we use the term "proper time" in the timelike case and "proper length" (see eg Misner & al 1973, p. 324) in the spacelike one.
Yet, in all cases the path length can also be measured by any observer (enough close to the region of the path) equipped with a "radar system". Such an observer is basically just measuring and reconstructing spacetime and its 4D geometry in a local neighbourhood, and can therefore compute the lengths of any paths in that neighbourhood. The path length of the radio signal in the question can be measured by any observer with this same method. I would call this a "measurement" as well. After all, we do measure the universe around us, and we rarely (if ever) do so by really sending observers around with clocks. See for example Dautcourt (1983) on this. Komar (1965) could be quoted here:
To summarize, I would discard all answers provided in the question and would give this answer instead:
If the signal propagates with speed less than $c$, its path length [defined by the integral above] could be directly measured by an observer moving with the signal and carrying a clock. For this reason we call the path length "proper time" in this case. But it could also be measured by any other enough close observer by a system of radar measurements, and all observers would agree on its (non-zero) value. If the signal propagates on a lightlike path, the path length could not be directly measured by the clock-carrying observer, and for this reason we may not want to call it "proper time". But it could still be measured by the radar-system observers, who would agree on a value of zero; in some situations we could agree to still call this "proper time" out of convenience, since its value changes continuously with continuous deformations of signal paths.
I would also counter-propose this (hopefully unrelated) question, taken from Gibson (1964):
References
Dautcourt (1983): The cosmological problem as initial value problem on the observer's past light cone: geometry.
Frankel (1979): Gravitational Curvature: An Introduction to Einstein's Theory (Freeman).
Gibson (1964): Our Heritage from Galileo Galilei.
Gourgoulhon (2012): 3+1 Formalism in General Relativity: Bases of Numerical Relativity (arXiv).
Hutter, van der Ven, Ursescu (2006): Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids.
Komar (1965): Foundations of Special Relativity and the Shape of the Big Dipper.
Landau, Lifshitz (1996): The Classical Theory of Fields.
Misner, Thorne, Wheeler (1973): Gravitation.
Penfield, Haus (1967): Electrodynamics of Moving Media.