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I am not going to rewrite another twin paradox, what i am interested now is that if we assume an observer who moves with acceleration first (during $0<t<T_0$), then he stops his acceleration and moves with constant velocity (during $T_0<t<\infty$), is it possible to use two different metrics (one for his acceleration (like Rindler) and another for his constant speed) to see what he observes in the universe? My first guess was that it should be ok, but actually it's not. And if it's not really ok, how are we going to describe his frame?

My problem: Assume two observers S and S' who are at the rest in $t=t'=0$ but at different places (they can synchronise their clock by exchanging signals, because they are the rest). Now observer S' accelerates with $g=2*10^8 m/s^2$ in its frame for one second, just after that he stops this acceleration and moves with constant speed toward S. We want to show that S will be older than S' according to S' when they meet each other at S's place, by using S' proper time which we can derive from its metric. However though, if our observer accelerate for a mere second we can see that S will become older only for 1.07 second,

https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)

$t=c*sinh(gt'/c)/g=3*sinh(2/3)/2=1.075s$

Where $t$ indicate S elapsed time. This is expected and fine because our accelerated observer sees a faster clock for S. But this is where we encouter a big problem. If we assume that after acceleration (when S' becomes inertia) S' can use Minkowski metric, then due to the gamma factor for the remaining path, he will see a slower clock for S, and if we make their distance large enough, we will see that according to S', S will be younger in their meeting. (gamma factor should be around 1.34, if S' arrive at S location 1y later in S frame, according to S', S' will be 1.34y old while S is 1y old, and let's forget about that mere 0.07 second!). This is twin paradox because S will consider S' younger too. where did i go wrong? Do note that i just said this problem as an example, my question is in the title so don't try to solve paradox without answering the real question.

P.S: I saw around 6 pages of physics.stackexchange about twin paradox. Didn't find something like my question, in all of them S' was considered to be accelerated all of the time (when they wanted to calculate elapsed time directly in frame S') which solves the paradox clearly. However if my question is duplicated, please do enlighten me

Update: Note that we are going to write worldlines in a coordinate system that S' is always at the rest at the origin of coordinate system. We have no problem in the S coordinate system as it was done by @ChiralAnomaly

Paradoxy
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  • None of the discussions of the twin paradox that you refer to assume that "S' was considered to be accelerated all of the time." You are misreading them. – D. Halsey Jun 20 '19 at 20:44
  • @D.Halsey Are you sure in "NONE" of discussion it's done? https://physics.stackexchange.com/questions/242043/what-is-the-proper-way-to-explain-the-twin-paradox or bunch of others? – Paradoxy Jun 20 '19 at 20:51
  • I don't deny the fact that you can draw spacetime diagrams and assume an acceleration in very small time, however, in that case you won't be able to calculate elapsed time in accelerated frame for other observer, that's what i meant above. You had to work with inertia frames in those cases. – Paradoxy Jun 20 '19 at 20:55
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    The flaw in your logic is an incorrect assumption that your initial clock synchronization survives your acceleration. – safesphere Jun 21 '19 at 04:19
  • Case 1: You fly for 1 year, accelerate, and come back younger. Case 2: You fly for 2 years with the same speed, accelerate the same way, and come back younger by twice the age of the first case. The same acceleration cannot account for two different results in aging. – safesphere Jun 21 '19 at 04:34
  • @safesphere from what i am awrare of, you can make a clock which survives from acceleration (like your wristwatch) at least this is what i was being told in university, and it it should be true because we compare an accelerated clock in gravity with a far observer, if synchronization was not possible doing this would be meaningless – Paradoxy Jun 21 '19 at 09:11
  • @safesphere But i assumed there is no come back at all, two observers are at different places, one of them accelerated for a moment and goes towards the other, that's all. Also i am more interested in my question than the twin paradox – Paradoxy Jun 21 '19 at 09:14
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    My example logically shows that a clock synchronization cannot survive acceleration. You have two cases. If the synchronization survives in one, then logically it does not in the other, because the same acceleration cannot compensate for two different time gaps. This example presents a logical proof of the flaw in your reasoning. The logic in your example is the same as in the twin paradox despite your lack of interest in the latter. – safesphere Jun 21 '19 at 15:25
  • @safesphere I see, you are completely right. Then how we are going to calculate the age of S from S' perspective directly? Surely there is no problem in frame S for calculation, however there should be some kind of "metric connector" for S' which let us to calculate whatever we want from its viewpoint. – Paradoxy Jun 21 '19 at 15:45

2 Answers2

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Set $c=1$ to save writing. The Minkowski metric can be expressed as $$ \dot\tau^2 = \dot t^2 - (\dot x^2+\dot y^2+\dot z^2), \tag{1} $$ where the worldline is described by functions $t(\lambda),x(\lambda),y(\lambda),z(\lambda)$, the dot denotes the derivative with respect to the parameter $\lambda$, and $\tau$ is the proper time along the worldline. Using equation (1), we can calculate the proper time along any finite segment of any given worldline, as long as the right-hand side is not negative anywhere (otherwise the proper time is undefined).

The OP describes two worldlines that intersect each other once. Both worldlines can be described using the same $t,x,y,z$ coordinate system, and the whole problem can be solved in that one coordinate system.

The worldline $S$ is $$ (t,x,y,z) = (\lambda_S,X,0,0) \tag{2} $$ where $X$ is a constant and $0\leq\lambda_S\leq\Lambda_S$. The value of $\Lambda_S$ will be determined below.

The worldline $S'$ has two segments: one with a non-zero acceleration, and one with constant velocity. I'll call these segments $A$ and $B$. To describe segment $A$, we can use $$ (t,x,y,z) = \big(a\sinh(\lambda_A),\, a\cosh(\lambda_A),\,0,\,0\big) \tag{3} $$ with $0\leq \lambda_A\leq \Lambda_A$, where $a$ is a constant that determines the acceleration and $\Lambda_A$ controls the duration of this segment. (Relating $a$ to $g$ is a separate exercise.) For segment $B$, we can use $$ (t,x,y,z) = \big(S+\lambda_B,\,C+v\lambda_B,\,0,\,0\big) \tag{4} $$ with $$ S \equiv a\sinh(\Lambda_A) \hskip2cm C \equiv a\cosh(\Lambda_A) \tag{5} $$ and $$ 0\leq\lambda_B\leq\Lambda_B \hskip2cm v = \tanh(\Lambda_A) \tag{6} $$ so that the segments and their first-derivatives both match at the interface. The condition that $S$ and $B$ intersect each other may be used to determine $\Lambda_S$ and $\Lambda_B$: $$ S+\Lambda_B = \Lambda_S \hskip2cm C+v\Lambda_B=X. \tag{7} $$ These can be used to express $\Lambda_S,\Lambda_B$ in terms of the inputs $a$, $X$, and $\Lambda_A$. To compute the elapsed proper times along each worldline:

  • For worldline $S$, use (2) in (1) to get $\dot\tau$, and integrate over $0\leq \lambda_S\leq \Lambda_S$.

  • For segment $A$, use (3) in (1) to get $\dot\tau$, and integrate over $0\leq\lambda_A\leq\Lambda_A$.

  • For segment $B$, use (4) in (1) to get $\dot\tau$, and integrate over $0\leq\lambda_B\leq\Lambda_B$.

This shows how to formulate the problem. The rest is just rote calculation, so I'll stop here.

By the way, by combining this scenario with its time-reflection through $t=0$, we get a scenario in which the worldlines intersect twice, turning it into a more traditional twin-paradox problem. No new calculations are required, because time-reflection symmetry says that we can just double all of the proper-time intervals that were calculated above.

Chiral Anomaly
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  • I have upvoted you but let me see if i got it right, 1.I can choose $\lambda$ whatever i want, for $S$ worldline the best choice would be its proper time which gives me $d\tau=dt_s$ obviously because it's at the rest relative to itself. For A and B on the other hand, $\lambda$ should be proper time of S', and in the end i should compare A+B and S proper times right? 2. Why did you use Minkowski metric in the case of A worldline, shouldn't it be Rindler metric now? – Paradoxy Jun 24 '19 at 09:15
  • Our observer is accelerating after all 3.is it necessary to write S' four position from S perspective where $a$ shows up? Or did i misunderstand here, and you meant something else? – Paradoxy Jun 24 '19 at 09:15
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    @Paradoxy You are right that you can parameterize the worldline using whatever $\lambda$ you want. That won't affect the final answer for the proper times, but a good choice can make the calculations easier. In the choice I used for $S$, $\lambda$ happens to be equal to the proper time; but this is not required, and it's usually hard to enforce. – Chiral Anomaly Jun 24 '19 at 13:50
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    @Paradoxy Regarding the Rindler metric: that's not necessary. The Rindler metric and the Minkowski metric are the same metric expressed in different coordinate systems, and the answer doesn't depend on which coordinate system we use (if we do it right). The proper time is something associated with a given worldline, not with a given coordinate system. Proper time is a coordinate-independent concept: it depends only on the worldline and the metric structure, regardless of what coordinate system is used to express those things (as long as we're consistent, of course). – Chiral Anomaly Jun 24 '19 at 13:52
  • Ok, So in other words what you actually have done is that you have described all worldlines in S frame point of view, though what i really wanted was writing all worldlines in the frame of S' (i.e writing equation with respect to $\lambda'$ (S' proper time) not $\lambda$. Ofcourse, there is no problem with what you have done above, because it doesn't really matter to write equation in S' or S. But i am still interested in S' for practice. – Paradoxy Jun 24 '19 at 17:03
  • Actually, i did that myself, but the problem was that we can choose boundary conditions such that effect of A worldline become unnoticeable, as such we can see that $B$ worldline will become $(t',x')=(\lambda',X')$ and $S$ worldline will be $(t',x')=(S'+\lambda'_B,C'-v\lambda_B)$. It will give us a wrong answer because $\tau_B>\tau_S$. (i.e, spaceship will consider his brother younger) – Paradoxy Jun 24 '19 at 17:03
  • @Paradoxy I'm not sure what you mean by the frame of $S'$. Do you mean a coordinate system $T,X,Y,Z$ in which $S'$ is described by $X=Y=Z=0$? Many such coordinate systems can be constructed, but it's not clear to me that any one of them is especially natural or useful. The "frame" wording tends to be associated with certain coordinate systems that make all or part of Poincaré symmetry manifest (like Minkowski or Rindler coordinates), which can be useful when working with worldlines that invariant under one of those symmetry transformations, unlike $S'$. – Chiral Anomaly Jun 25 '19 at 04:05
  • Yes, that's correct, i meant that. Of course, here, we have coordinate system S to make life easier, however we are not lucky in general cases (where for example S accelerate too), that's why i am interested in working with S' frame only, where he is at the rest relative to its coordinate system. I will update my question to emphasize on this matter. – Paradoxy Jun 25 '19 at 10:05
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So S' and S are of equal age when they synchronize. Then there is 1 second of acceleration. At then end, S is now much older according to S', and the time when S synchronized his clock was a long time ago according to S', so it's no problem that S' watches him age slowly as they close in on each other.

Update: So the way this question is phrased isolates the heart of the Twin Paradox. It's not that the traveling twin ages less, it's that he ages less while seeing the at-home twin's clock run slower than his the whole time.

The paradox is resolved when you realize the at home twin's age jumps forward a lot during the turn around (acceleration). But: the acceleration is brief in BOTH reference frames...how can you account for years when each twin experienced only seconds?

That is the Andromeda Paradox: by changing your velocity, the definition of "now" at distance places changes. S' is far away and at rest w.r.t to S, so they can synchronize their clocks. Call those events $s_{\mu}'$ and $s_{\mu}$ respectively. They occur at the same time ($s_0 = s'_0$), with a large space like separation in the initial frames.

Once $S'$ is moving (fast) towards $S$, even if he accelerated only for a second in his frame and 1.07 s in $S$, the time coordinate of $s_0$ is way in his past, only because his definition of "now" at $s_i$ jumped into the future, relative to the initial conditions.

Worse: this change is reversible, just by turning his spaceship around.

To quote David Mermin:

"That no inherent meaning can be assigned to the simultaneity of distant events is the single most important lesson to be learned from relativity."

JEB
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  • At the end of the acceleration S is not much older according to S', that's the problem. He will be older like 0.07sec(see my calculation), and this is the core of the problem where S' sees S ages slowly, he will deduce that in their meeting S will be younger. Also note that my question is the title itself, not that example – Paradoxy Jun 24 '19 at 08:57
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    That's why I said last week (and got down voted) that the Andromeda Paradox is at the heart of the Twin Paradox: if you change your velocity, "now" at distance locations can change bigly. The S,S' synchronization events (s, s') are space-like separated, so when S' moves towards S, "s" is now in the distant past. If he then stops, it is only 0.07s in the past, but just by moving, he sends it back to the past. If he turns around, it is then in his future. – JEB Jun 24 '19 at 13:18
  • Can you prove this line "but just by moving, he sends it back to the past" by math? – Paradoxy Jun 24 '19 at 13:41
  • If you draw a Minkowski diagram, the "now" line (also called the hyperplane of simultaneity) tilts (it's always a straight line), but the angle changes depending on velocity. For instance, our definition of "now" at the Andromeda galaxy cycles about 17 hours as we rotate around the globe each day. Events in the morning's past can be in the evening's future. – JEB Jun 24 '19 at 14:00
  • I am not sure how you are going to draw diagram in S' frame (it can be done in S frame easily though) – Paradoxy Jun 24 '19 at 14:14
  • Ok your answer is a lot better now, but can you prove "only because his definition of "now" at $s_i$ jumped into the future, relative to the initial conditions." ?It does make sense, but how can we understand this from formulas? check my other question here: https://physics.stackexchange.com/questions/487692/is-relativity-of-simultaneity-being-violated-in-accelerated-frames it's 100% related! it's as if position of $s_i$ doesn't matter. – Paradoxy Jun 24 '19 at 17:17
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    $t = t' - vx'$. So when $v$ goes from zero to near $c$, events in the forward direction get a new time coordinate that is before where they used to be. All this is happening outside the light cone, so the relative timing really is meaningless. – JEB Jun 25 '19 at 15:22
  • So in other words, just as S' starts its movement, he will consider $t \neq t'$ at $t'=0$. But this arises another question, what he will see from his telescope after a year or so (which light had enough time to cross distance between S and S')? I mean, S' will see S ages a lot suddenly in the middle of way? Isn't it kind of you know, odd? – Paradoxy Jun 25 '19 at 17:02
  • @Paradoxy It's very odd. For any event outside the light cone, say one that happens in 2021 at alpha centauri...there is a reference frame right here on Earth, right now, where it is in the past. – JEB Jun 27 '19 at 05:37
  • I am convinced,your answer also solve my main problem,i should just use proper values for initial conditions,using 2 different metrics doesn't matter.However my last question is, as i said above assume that S' is in the middle of the way where he retrieves a light signal signal from S. He will see S is too old now (because S is sent to the past by factor vx/c^2, where x is the distance between two planets), if S' suddenly stops, what will happen to S age?(He can't be too old by then)You said S' by continuing his motion can send S to the past more,but i don't get it from Lorentz transformation. – Paradoxy Jun 27 '19 at 05:58
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    If you go back to the 2 lightning flashes that are at the front and back to the train car (simultaneous for the guy at the station), if said guy suddenly jumps on the train when the lightning strikes, the front strike moves to his new past, and the rear strike moves to his new future. If he then jumps onto an opposite moving train, the front flash is now in his future and the rear flash is in his past. It's all outside the light cone, so like Mermin said: the simultaneity of distant events has no inherent meaning. Basically you can tilt "now" ($t=0$) as close to light cone as you want. – JEB Jul 01 '19 at 04:00