Can space proper-time (SPT) diagrams as used by Lewis Epstein in Relativity Visualized be used to explain the twins paradox of special relativity?

Question

Space proper-time (SPT) diagrams are perhaps easier for the layman to understand than Minkowski diagrams, although I don't much about them.

I have been, so far, unable to understand fully the latter, and I am interested in the twins paradox of special relativity, which I know quite a bit about, although arguably only at a superficial level.

Maybe SPT diagrams could give me a slightly deeper understanding, even though not quite as deep an understanding as a Minkowski diagram could provide hypothetically. Or maybe an SPT diagram could serve as some sort of stepping stone to an ST diagram.

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Edit in light of robphy's comment:

If you Google the title of the book and select title plus "pdf" from the drop down menu, you can see a sort of (nonprintable?) pdf of the book. I did that just now, and verified that Epstein does attempt to explain the twin paradox with an SPT diagram, and even attempts to explain a gravity=caused-motion version of the paradox using a (curved) SPT diagram. I have no idea whether any of the explanations hold water strictly --I couldn't see anything wrong with them.

He also says in a footnote: "A detailed analysis of how this process is seen by the twins is given in the Special Relativity chapter of Conceptual Physics by P. Hewitt, Little Brown, Publishers, 34 Beacon Street, Boston, Massachusetts."

I did not like Epstein's "explanation" for why one twin ages for the other but not vice versa, which is just that "one twin accelerates and the other doesn't" and no mention of the change of frame of reference by the traveling twin, which to me is a more thorough explanation.

I had a look at the P Hewitt book, and it seems Epstein's recommendation was ironic. The book makes no mention of time desynchronization, which is half the story of what happens with twins, and it contains only a signal delay based explanation (more like a proof than an explanation, actually) of time dilation, while Epstein's view is emphatically that signal delay is not the cause of time dilation. He also says that "What you see is not what happens."

Answer 1

Here are portions from Epstein's Relativity Visualized, p.86 and p.88.

Look at the one from p.88.
From what I understand,

• the horizontal coordinate describes the location in space according to the home astronaut

• along each "path in the Epstein SPT diagram" the vertical coordinate describes the proper time elapsed on the wristwatch of its astronaut.
  (I don't want to call the "path in the SPT diagram" a "worldline"... for reasons below)

• the separation event corresponds to point O (where Peter and Danny met with their wristwatches set to zero)

• the reunion event corresponds to point P for Peter and point D for Danny.
  Unlike the usual position-vs-time diagram from PHY 101 and from Minkowski where a meeting event is marked with a single point, on Epstein's SPT-diagram the single reunion event is represented by TWO points. (If there were seven astronauts reuniting at a common reunion event, there would be a single point on a position-vs-time diagram but, in general, seven points on an SPT-diagram.)

What is point Q (which I marked in red)?

• It appears to mark the turn-around event on Danny's trip, which occurs halfway up the PT-axis for Danny (half of OD).

• If we ask Peter when Danny's turn-around event occurred, Peter would presumably say it occurred at the same time as when "Peter's wristwatch read half of what it reads at P" since Danny had the same outgoing and incoming speeds.

• If we then ask Peter to mark the turn around-event as point on the diagram (as if it were a position-vs-time graph), he would mark a point "at 1/2 light year" along the horizontal axis, but higher up than Q along the vertical direction.

• So, point Q only seems to have relevance for Danny.
  OQD is not a worldline that Peter would draw on position-vs-time graph.
  The points on OQD are only meaningful for Danny.
  
  (update3: From the data given, OQ (which represents an outgoing speed of $v=(1/2)c$) has a slope [with respect to the vertical] of $\frac{1/2}{\sqrt{3}/2}=\frac{1}{\sqrt 3}$ on this Epstein SPT-diagram.)

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So, this is how I read Epstein's SPT-diagram for clock-effect/twin-paradox.
Assuming I've interpreted things correctly,
I'll ask you if you think if the above SPT-diagram from Epstein is useful?

(update: I'll admit that I don't quite understand Epstein's statement "Danny's path must also be 2 inches (2 years) long".)

(update2: So, it seems the length of path in a SPT-diagram is equal to the spacetime-interval between the endpoint-events (i.e. the proper-time for the inertial astronaut experiencing the endpoint-events) .)

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By constast, here's my variation of the standard Minkowski spacetime diagram: https://physics.stackexchange.com/a/507592/148184 (go to the "Clock Effect" section halfway in) as my answer to What is the proper way to explain the twin paradox?

Answer 2

The motivation for Epstein's graphs is that the Minkowski metric

$$dτ^2 = dt^2 - dx^2 \mathbin{[-} dy^2-dz^2]$$

can be rearranged to

$$dt^2 = dτ^2 + dx^2 \mathbin{[+} dy^2+dz^2]$$

which looks Euclidean—and human brains have dedicated circuitry for understanding Euclidean geometry (unlike Lorentzian geometry), so you can look at a graph of $τ$ versus $x$ and roughly see $Δt$ as well, since it's the length of the curve.

That is a genuine advantage of Epstein's graphs over standard spacetime diagrams, but it's also the only advantage of them. If you can train your brain to estimate $Δτ=\sqrt{Δt^2-Δx^2}$ instead of Euclidean distance, then you can interpret standard spacetime diagrams in largely the same way as Epstein's. And standard spacetime diagrams have two huge advantages: first, they are true geometric diagrams (Epstein's graphs are merely plots of functions, and, e.g., the intersection points of different graphs plotted on the same axes are meaningless); and second, they are what everyone other than Epstein uses to illustrate things.

I'm actually pretty fond of Epstein's book, because I hate that the pedagogy of special relativity has been practically stagnant for a century, and Epstein at least attempted something very different. But I don't know that his approach is a good one.

I did not like Epstein's "explanation" for why one twin ages for the other but not vice versa, which is just that "one twin accelerates and the other doesn't" and no mention of the change of frame of reference by the traveling twin, which to me is a more thorough explanation.

This may be why you're having trouble with standard spacetime diagrams. There's no such thing as "the change of frame of reference by the traveling twin". Many people have the idea that there is such a thing as "your" reference frame, which is a frame that you must use in order to get the right answer. The truth is just the opposite: you can use any reference frame to solve any problem, because they're all equivalent. Often there is one choice that is more convenient than others, but no choice can ever be wrong.

For the twin paradox, the most convenient reference frame is the one in which the diagram is usually plotted: the inertial frame where the stay-at-home twin is at rest, and the traveling twin has a constant speed on both legs of the trip. You needn't, and shouldn't, worry about any other reference frame. Instead, just compare the length of the two twins' worldlines, since that is the elapsed proper time. You have to remember that the length is not given by the Pythagorean formula, but by a variant of it with a minus sign. But fundamentally, the reason the lengths are different is the same as the reason they would be different in Euclidean geometry. Only the sign of the effect is different (the curved path is shorter instead of longer). When Epstein says that the difference is due to one twin accelerating (one worldline being curved), he's right.

I think Epstein understands that there is no such thing as "your" frame, and citing an explanation of the twin paradox that doesn't use three different inertial frames was probably a deliberate choice. It's not necessary or even convenient to use more than one inertial frame to solve this problem, and trying to solve it that way only obscures the simple geometry of it.

Answer 3

I have written an article that explains Epstein's diagrams in a simple way, using a block universe model of events. See https://it.quora.com/Is-it-possible-to-build-a-model-that-explains-relativity-using-Epstein-diagrams/answer/Renato-Iraldi