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At page 9 in Sean Carroll's book Spacetime and Geometry he states:

The proper time between two events measures the time elapsed as seen by an observer moving on a straight path between the events.

He then goes on to explain the "twin paradox". He describes two observers in a coordinate system, one at fixed spatial coordinates and one that moves away with a certain velocity and then comes back to the origin. He marks three events: $A$ as the starting point when the moving observer starts to move away, $B$ as the turning point for the moving observer when he start to turn back to the origin, and $C$ as the point when the moving observer arrives again at the origin (but at different coordinate time $t$).

From the definition that he gave the time measured by the stationary clock is equal to the proper time between events $A$ and $C$ - this is in accord with the definition of proper time between $A$ and $C$ as he defines it above. Then he computes the proper time between $A$ and $C$ as measured by the moving observer. Since the observer is moving, its trajectory is clearly not a straight line between $A$ and $C$ and thus, according to the definition above, we cannot interpret the proper time of the second observer as the time it measures on its journey. But this is exactly what he does to explain that the moving observer has aged less than the static observer.

Is the interpretation he gives for the proper time between two points wrong? Should it be interpreted (as it is on Wikipedia) as:

The proper time along a timelike world line is defined as the time as measured by a clock following that line.

With this meaning for the proper time, the explanation he gives for the apparent "twin paradox" makes sense.

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  • $\begingroup$ If A and C are the same points in space, isn't the only straight line between them a line in time? I mean, the person that stays in A only moves in time and this time is the proper time elapsed between A and C. The proper time for the second twin, moving from A to C via both space and time (so not on a straight line through time only), is the time elapsed on his clock. Which is obviously less because part of the trip is through space. Aa photon going from A to C in space, (say in a circle, by means of mirrors) will not have aged at all. It's kind of confusing what he means, I agree. $\endgroup$ Feb 12 at 10:45

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A more-complete quoting from Carroll's book will clarify your question on his use of terms:

(pg. 9)
The fact that the interval is negative for a timelike line (on which a slower-than-light particle will actually move) is annoying, so we define the proper time $\tau$ to satisfy $$ (\Delta \tau)^2=-(\Delta s)^2= -\eta_{\mu\nu} \Delta x^\mu \Delta x^\nu.\qquad(1.17)$$ A crucial feature of the spacetime interval is that the proper time between two events measures the time elapsed as seen by an observer moving on a straight path between the events.

So, here, he is referring to the "spacetime-interval between a pair of [timelike-related] events"... which could be called the "proper-time interval of this pair of events" (akin to the magnitude of a displacement vector).
In this usage, proper-time (or probably more correctly "proper-time-interval") is a function of a pair of events $\tau(A,C)$.

Then,
in the next paragraph, he uses the proper definition of "proper time" (as you see in Wikipedia) in the parenthetical part ...

(pg.9) A crucial fact is that, for more general trajectories, the proper time and coordinate time are different (although the proper time is always that measured by the clock carried by an observer along the trajectory).

***[I think proper time in this parenthetical section should have been bolded].
In this usage, proper-time (as Minkowski intended) is a function of a worldline segment between events $\tau(\gamma_{AC})$.

Minkowski said (on the page between Figs. 2 and 3 in "Space and Time"):
If we imagine at a world-point $P (x, y, z, t)$ the world-line of a substantial point running through that point, the magnitude corresponding to the time-like vector $dx, dy, dz, dt$ laid off along the line is therefore $$ d\tau = \frac{1}{c}\sqrt{ c^2 dt^2 - dx^2 - dy^2 - dz^2}.$$ The integral $\int d\tau = \tau$ of this amount, taken along the world-line from any fixed starting-point $P_0$ to the variable endpoint $P$, we call the proper time of the substantial point at $P$.

Then, Carroll describes the Clock Effect ( as featured in the Twin Paradox),
with a conclusion starting at the end of p.10

(pg. 10 last word) An
important distinction is that the nonstraight path has a shorter proper time. In space, the shortest distance between two points is a straight line; in spacetime, the longest proper time between two events is a straight trajectory.

So, the proper use of "proper time" is used (but not fully defined) in parenthesis***,
which is unfortunately not-bolded like its use in defining the "interval".


Here is a related discussion where I contributed an answer:

Equivalence of two definitions of proper time in special relativity

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  • $\begingroup$ So the first definiton, the proper-time-interval is just like the distance between two points in Euclidean geometry (the length of a straight line between the two points) and the second definition, the proper-time is like the length of a curve (also in Euclideam geometry) that passes through two points. $\endgroup$
    – ac1643
    Feb 13 at 17:13
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    $\begingroup$ @ac1643 Yes, "interval" is like "magnitude of displacement". "Proper-time" should really be like "arc-length" ("distance along a path") [hence, the use of the integral], where "proper-time-interval" is arc-length along an inertial-path (for small nearby regions). As in ordinary kinematics, magnitude-of-displacement is not the same distance-travelled. $\endgroup$
    – robphy
    Feb 13 at 17:20
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I suppose he means that there are two periods of proper time for the travelling twin (ie the period between A and B and the period between B and C) which, if added, are less than the proper time of the stay at home twin between A and C.

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The definition he give in the book is correct and the same as the one found on Wikipedia.

My mistake in understanding his explanation came in the fact that he said a "an observer moving on a straight path between the events". This straight does not refer to the fact that the path is a straight line on a spacetime diagram, but rather, to the fact that the observer is inertial, i.e. its motion is unaccelerated.

With this in mind we clearly see that the path of the second twin is actually a straight path. And on the next two pages, he goes on to define the proper time on paths that are not necessarily straight (by considering the infinitesimal spacetime interval). This means that the proper time is actually the time between two events as measured by an observer on his path between the two events (not necessarily a straight one this time).

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    $\begingroup$ I think Carroll DOES mean "straight [on a Minkowski spacetime diagram, here in special relativity]" ( that is, inertial) since he describes a "path between events"... and since he uses that phrasing below on p.9 ("one a straight line passing through a halfway point marked B") and in the Fig. 1.6 caption on p.10 ("...straight path through spacetime ABC will age more than someone on the nonstraight path AB' C.") The second twin along "Nonstraight AB'C" is piecewise-inertial. His p.11 introduction of the line-element allows him to extend the result to more than piecewise-inertial worldlines. $\endgroup$
    – robphy
    Feb 12 at 21:13
  • $\begingroup$ But the "nonstraight" path AB'C is acutally a straight path as seen from the inertial frame of the moving twin. So what frame do we use when we specify straighr paths? $\endgroup$
    – ac1643
    Feb 13 at 8:46
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    $\begingroup$ All observers can regard themselves as "at rest" but AB'C is not inertial. (That's the Twin Paradox.) ABC is inertial... on a Minkowski diagram, ABC is a straight segment. AB'C is non-inertial due to the kink from AB' to B'C. (No boost will straighten that kink.) AB'C can try to make its worldline straight on a diagram... but such a diagram is inequivalent to (not a boost of) the spacetime diagram of an inertial observer. See my answer at physics.stackexchange.com/questions/553682/… $\endgroup$
    – robphy
    Feb 13 at 16:17

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