# More on acceleration in the Twin “Paradox”

There is a possibly interesting comment thread following my answer to How exactly do Minkowski diagrams prove that acceleration is not needed in resolving the Twin Paradox in Special relativity? – Quora

In it Peter Webb passionately (and sadly sometimes rudely) defends the position that “acceleration has nothing directly to do with the Twin paradox”. This is a position shared by a not insignificant minority of apparently competent physicists (on Quora, Brent Meeker comes to mind as a prominent example), but although I continue to feel that the issue is vastly overblown, I find that the “it’s not acceleration” view is misguided and at least some of its proponents are sufficiently intransigent and aggressive that they demand a further rebuttal.

So, with reference to Peter’s final summary of his position, here goes:

The “paradox”/interest in the TP is that it demonstrates time dilation. The different ages merely demonstrates this.

To be frank, I have no idea what he is talking about here. The usual time dilation of special relativity, which applies in the case of two inertial observers having a constant relative velocity with respect to one another, only exists in the understanding of each observer regarding the clock of the other. There are many ways of demonstrating this effect (most famously with muons from cosmic ray interactions with the upper atmosphere, but also in many other ways), but it is not something that actually happens to either observer in any objective sense – at least not while they both continue in an inertial state of motion.(We may think its obvious that the muon is the one with a dilated lifetime but from the muon’s frame of reference the Earth looks like a very flat pancake and the distance it has to travel can be covered well within its lifetime.)

The Twin “Paradox”, however, is something different in that it does lead to an objective fact of the matter regarding which twin or clock aged more between two specific events in space-time.

Maybe 80% of people with some interest in science think that time dilation is a consequence of acceleration. It isn’t, directly. It is a result of changing reference frames and in the Twin’s paradox that involves acceleration. But time dilation still occurs in the absence of acceleration (eg 3 brothers), and it is easy to show that identical acceleration profiles produce very different amounts of time dilation.

I am not aware of the statistics regarding what “people with some interest in science” think, but I doubt that any significant percentage of them think that the symmetric relative time dilation of unaccelerated motion is a “consequence of acceleration”.

But on the other hand, if they think that the objective observable time difference in the Twins “Paradox” is a consequence of acceleration, then I’m right there with them! (I have no position on how “directly” consequential the acceleration is, but I am happy to see an acknowledgement that indeed having one twin change frames does involve acceleration.)

But now we come to the 3 brothers.

It is true that they provide yet another way of confirming the relative time dilation effect. Indeed, the incoming “brother” brings back to Earth a record of what the outbound one’s clock said when they met  (though of course this is not necessary, and the information could just as easily have been passed by a radio signal). But he (or the radio message) could also inform Earth of the time that the outgoing traveller inferred was on the Earth clock at his idea of when the crossing of paths happened. This would be consistent with the usual unaccelerated time dilation, which is perfectly symmetrical in the sense that while the Earth observer thinks that at the event on the traveller’s path which is concurrent with any time $t_{E}$ on his Earth clock (measured from the outbound traveler’s departure event), the outbound clock reads $t_{O}=\frac{t_{E}}{\gamma}$, and the outbound observer thinks that at the event on Earth which is concurrent with any time $t_{O}$ on his travelling clock (measured again from the shared departure event), the Earth clock reads $t_{E}=\frac{t_{O}}{\gamma}$.

So, at the particular event where the travelers meet, if the Earth brother thinks this occurs at time $t_{M,E}$ then the outbound traveler’s clock reads $t_{M,O}=\frac{t_{M,E}}{\gamma}$ and he thinks that the concurrent time on the Earth clock is just $\frac{t_{M,O}}{\gamma}=\frac{t_{M,E}}{\gamma^{2}}$.

The inference of the inbound traveler on the other hand, after synchronizing his clock with the outbound, is that during the rest of his trip to Earth, the Earth clock only advanced by the same $\frac{t_{M,E}}{\gamma^{2}}$.

But when they synchronize clocks the two travellers can also compare notes on what they think is showing on the Earth clock at that time. And their disagreement on that will show them immediately that everything works out and there is no paradox.

So in the 3 brothers case there is clearly no paradox, as everyone is aware that each traveler considers only a small part of the Earth’s experience as concurrent with their own travel time and there is no reason to expect that those two intervals should add up to the whole time on Earth. So no two observers are ever forced to agree that only one of them is right about the time dilation of the other.

Of course there is not a paradox in the single traveler case either, but it is a more powerful (wrong) intuition that the two intervals judged by the traveller to be concurrent with the legs of his trip should cover the entire Earth time interval (even though we now know from the 3 brothers analysis that they do not).

Note: I am not disputing the validity of the 3brothers scenario as the most appropriate explanation of why there is no paradox. But I will continue to insist that it is not significantly counterintuitive in its own right and that the solution it provides to the single traveler version identifies the “missing” Earth time with the traveller’s “frame jump” and that his “frame jump” (no, not just in his head but between two different frames that he is actually at rest with respect to) is nothing more than an integrated acceleration.

This leads to the question of what we can learn from the sudden jump in the traveler’s inferred Earth time at the turnaround. We learn that when a traveller changes frames  then, as his simultaneity space changes, the times he considers concurrent on clocks that are remote from him in his direction of velocity change are advanced (and those behind retarded).

Of course the sudden turnaround is physically impossible and in any real situation there would be a period of finite acceleration. But it is a simple exercise in calculus to approximate a period of finite acceleration with a number of discrete jumps and conclude that while the traveler is turning around his inferred current time on Earth advances more rapidly than when he is not accelerating (at a rate proportional to both his acceleration and his distance from Earth).

Now none of this made any use of General Relativity. The acceleration effect we have discovered is purely from Special Relativity and I do agree that it’s a monstrous abuse to address the Twin “Paradox” by invoking acceleration=>GR=>”it’s like being in a gravity well which we all know (from the movies!) causes time dilation”.

But the real beauty of all this is that it DOES go the other way!

In SR, acceleration=>time dilation (relative to clocks that are remote in the direction of acceleration), and so, using only the equivalence principle, we learn (without any of the harder GR analysis) that Matt Damon really does outlive his grandchild!

Given Peter’s earlier answer about the Ehrenfest “Paradox” I am totally surprised and disappointed that he doesn’t seem to appreciate this.