# Special case of the twin paradox in special relativity

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## Main Question or Discussion Point

There is something about the twin paradox in special relativity that has always bothered me. One twin sets out on a journey at a large fraction of the speed of light, turns around and returns. The fact that the returning twin is the one who is younger is explained by the fact that they are the one who had to accelerate, and then slow down, turn around, and accelerate again. Fine. I’ve never had a problem with that. But what I’ve never seen addressed is the scenario in which both twins, starting at the same place, accelerate in opposite directions at a large fraction of light speed, then turn around and head back at the same high speed and meet where their respective journeys began. Whose time was slower in the completely symmetric scenario?

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Orodruin
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They will both experience the same time. It is a symmetric scenario.

Jeff B L and Mihai_B
That's kind of what I was thinking, but I wasn't sure. Relative to each other they are going at a very high speed so at first glance it seems like their time dimension would be experienced differently. But since it is a symmetric scenario, they will, are you point out, experience the same time, but I have to admit I have trouble understanding the details of how it is so.

Orodruin
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The details are the same as in the regular twin paradox. If you blindly apply the time dilation formula without accounting for the relativity of simultaneity you get the wrong result.

You cannot apply a formula without taking into account the implicit assumptions that go into deriving it.

Mihai_B and PeroK
OK. Thanks for explaining.

Ibix
The easiest way to understand is to note that the elapsed time for any object turns out to be the "length" of its path through spacetime. Two objects that start together move apart, and come back together again, are taking different routes through spacetime. They will, in general, experience different elapsed times. But it's always possible to construct a scenario where the different paths have the same length - i.e. the same elapsed time.

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Dale
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But what I’ve never seen addressed is the scenario in which both twins, starting at the same place, accelerate in opposite directions at a large fraction of light speed, then turn around and head back at the same high speed and meet where their respective journeys began.

vanhees71
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I never understood, why there is a puzzle. Suppose you have an observer in an inertial frame of reference with coordinate time ##t=x^0##. Then the proper time of any (poinlike) twin is the age of this twin (measured from the starting point of his journey). It's frame-indendently given by
$$\tau=\int_{t_0}^{t_1} \mathrm{d} t \sqrt{\eta_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}} = \int_{t_0}^{t_1} \mathrm{d} t \sqrt{1-\vec{\beta}^2}.$$
This gives ##\tau## in terms of the coordinate time of the inertial observer. That's it.

You can also compare the proper times of any two non-inertial twins, using the same formula. It tells you unambigously how much each of them aged during their respective journeys.

There's no paradox but just the fact that you can travel from place A to place B in spacetime on different paths, and the travelled path lengths are then usually different. That's very much as in spatial distances in everyday life. You can go from one place to another in many ways, and each choice of your way leads to another distance travelled!

Dale
Dale
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You can also compare the proper times of any two non-inertial twins, using the same formula.
And it generalizes very easily to curved spacetime.

Orodruin
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I never understood, why there is a puzzle.
I do not think there is a puzzle for anyone who understands the argument you followed up with. The problem lies with the laymen who have a hard time getting rid of the concept of absolute simultaneity and therefore do not get the implicit assumptions in the derivation of ##t_0 = t/\gamma##.

stoomart and Ibix
Ibix
I do not think there is a puzzle for anyone who understands the argument you followed up with. The problem lies with the laymen who have a hard time getting rid of the concept of absolute simultaneity and therefore do not get the implicit assumptions in the derivation of ##t_0 = t/\gamma##.
Indeed. A naive reaction to the scenario is to (unthinkingly) adopt an absolute rest frame: the travelling twin comes back younger because he was moving fast. The next step up is to realise that the travelling twin sees the stay-at-home moving fast and (unthinkingly) apply the time dilation formula for both frames and not catch the simultaneity change, which leads to a paradox. The point of the Twin Paradox is to illustrate this particular pitfall, as you say. The final step is to understand relativity in terms of four-dimensional manifolds and then it's almost hard to see why anyone ever thought it paradoxical.

It's quite Zen (or cod Zen, anyway). The novice sees no paradox. The journeyman sees the paradox. The master... sees no paradox.

pervect
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I've tried this explanation before, I'm never sure how well it's received, but I'll try it again.

Suppose you have two cars, with odometers. Both odometers are "synchronized" at the start of the trip, so they both read the same. Then one car takes a straight line path from it's starting location (say Springfield, Ill), to it's ending location (say, Washington DC). The other car doesn't go in a straight line, it goes from Springfield to Atlanta to Washington.

You don't even expect the cars to have the same odometer readings when they both arrive at Washington DC. But you expect that the car that drove in the straight line path has the lowest reading.

The principle behind the odometer reading being a minimum for one car is that "the shortest distance between two points is a straight line". A corollary is that non-straight line paths are longer.

The situation in relativity is somewhat similar, but it turns out that the straight-line path is the longest (proper) time, rather than the shortest. Why this is is interesting, but a little too technical for the point I'm trying to make.

The point I'm making is that the very idea that the two clocks, or cars, should have the same reading on their instruments, their times, or their odometers, is not a logical necessity. The idea that clocks "should" have the same reading comes from the notion of absolute time, usually. This would be the only thing I needed to say, if everyone who read this post knew what "absolute time" actually meant. But they most likely don't (or they would have figured this all out for themselves, ages ago), so I'm going to try the above example to explain.

It's possible to go into a whole lot more detail about the geometry, and at some point it even becomes a good idea. What I'm advocating is to start out with the idea that two clocks don't have to read the same time if they take different paths through space-time. And that there is something special about straight-line motion, which in the case of cars gives the shortest reading on their odometers when they re-unite, and in the case of clocks, gives the longest reading on the clock.

To try and explain to the people who "don't get" where the confusion is, I'll suggest the following model. The people who are confused believe in absolute time, and never thought about time in any other way. They then try to treat acceleration as a physical effect that modifies an underlying absolute time. And they note that this approach just doesn't work. But the idea of throwing out "absolute time" seems too radical to them, it's even hard to talk about it because they'd have to realize an alternative to absolute time even existed before they could think about it. And the idea that it might not apply hasn't occured to them, and the words pointing out that it doesn't apply don't make any sense, because they don't have the conceptual background to appreciate them.

Now, I can't guarantee that my general model of "why people don't get it" works, but I think it's a good working hypothesis in 90+ percent of the cases.

stoomart, Ibix and Dale
vanhees71
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I do not think there is a puzzle for anyone who understands the argument you followed up with. The problem lies with the laymen who have a hard time getting rid of the concept of absolute simultaneity and therefore do not get the implicit assumptions in the derivation of ##t_0 = t/\gamma##.
Of course, this is usually the greatest obstacle to get used to the relativistic spacetime structure when it comes to discussuion of the (usually much overpronounced) kinematical "paradoxes". Math (formulae not so much spacetime/Minkowski diagrams in my case) usually resolves such troubles quite easily. Thus, one should keep Pauli's advice in mind: Don't make so many words. On the other hand, there's this investigation on the citations of physics papers containing many formulae ;-)) (I'm pretty sure one cannot take it too seriously, at least not in the theoretical-physics community):

I do not think there is a puzzle for anyone who understands the argument you followed up with. The problem lies with the laymen who have a hard time getting rid of the concept of absolute simultaneity and therefore do not get the implicit assumptions in the derivation of ##t_0 = t/\gamma##.
Total science layman here...is the simultaneity change you and Ibix are talking about related to the concept that a person at the top of the Empire State Building for a day ages slower than someone in the lobby during the same time?

Ibix
No. That's gravitational time dilation, which is an unrelated phenomenon (at least for the puroses of this discussion).

The situation in relativity is somewhat similar, but it turns out that the straight-line path is the longest (proper) time, rather than the shortest. Why this is is interesting, but a little too technical for the point I'm trying to make.

The point I'm making is that the very idea that the two clocks, or cars, should have the same reading on their instruments, their times, or their odometers, is not a logical necessity. The idea that clocks "should" have the same reading comes from the notion of absolute time, usually.
I was having a hard time understanding your example until I read this part a few times and saw the following visual on Wikipedia, thanks for the explanation.

To verify I'm understanding this correctly, would it be accurate to say: If I shine a laser at a mirror so that it reflects onto my shirt, the age of the light on my shirt is shorter than the time I experienced for it to reach me.

Ibix
To verify I'm understanding this correctly, would it be accurate to say: If I shine a laser at a mirror so that it reflects onto my shirt, the age of the light on my shirt is shorter than the time I experienced for it to reach me.
Age isn't defined for things moving at the speed of light. But if you replace your light pulse with a ball bounced off a wall, then yes.

stoomart
Indeed. A naive reaction to the scenario is to (unthinkingly) adopt an absolute rest frame: the travelling twin comes back younger because he was moving fast. The next step up is to realise that the travelling twin sees the stay-at-home moving fast and (unthinkingly) apply the time dilation formula for both frames and not catch the simultaneity change, which leads to a paradox. The point of the Twin Paradox is to illustrate this particular pitfall, as you say. The final step is to understand relativity in terms of four-dimensional manifolds and then it's almost hard to see why anyone ever thought it paradoxical.

It's quite Zen (or cod Zen, anyway). The novice sees no paradox. The journeyman sees the paradox. The master... sees no paradox.
I'm still not sure I completely understand on an intuitive level the relativity of simultaneity (other than working out the train "paradox"), but I do know that if you actually work out the time measured by each twin up and back, taking into account length contraction and the time for a light signal to bounce in between them, it seems pretty clear that the traveling twin has to age less. I think we had a discussion here on it a few months ago where it was pointed out that what I was working out was related to the Doppler effect, but in any event, the math very clearly showed the traveling twin ages less.

I guess space time diagrams are easier, but it works out really well for a novice like myself just to calculate the times for each portion of the trip for each twin, in my humble opinion.

As far as the symmetrical example the OP describes, for me that one is much easier to understand intuitively than the standard twin paradox example. No asymmetries to worry about. I think the phrase "moving clocks run slowly" might confuse people some, however.

Ibix