# I Sorry but I'm still going nuts over the twin paradox

#### SamRoss

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Good. That's the answer I was hoping to hear. :)

#### Orodruin

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Good. That's the answer I was hoping to hear. :)
I'm sorry, but I feel that you are simply ignoring what I said after "No", which is the important thing. Your statement is true because it is a tautology.

#### PeterDonis

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The freedom will go away only if I specify the lengths of the lines I want you to draw. If I don't tell you how long I want the lines to be then you can do whatever you want. In the twin paradox, unless you say which twin was under the influence of the force then you won't be able to justify picking one specific twin as the one with the shorter world line.
The lengths of the lines and the force are two different things, and only specifying the former takes all of the freedom away. There is no general rule that says "the one under the influence of the force has the shorter worldline". Both twins could be subjected to a force. And there isn't even a general rule that says "the one who is subjected to more force has the shorter worldline". The only general rule is that specifying the lengths of the lines is what takes the freedom away.

#### SamRoss

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I'm sorry, but I feel that you are simply ignoring what I said after "No", which is the important thing. Your statement is true because it is a tautology.
So your reasoning is that a curved line is the definition of a force, not the result of a force. Therefore, I am being redundant by saying that we must draw a curved line because of the force. Maybe we do have a disagreement here. As I see it, the geometry is only a representation of what's happening in the real world. The force has physical meaning; the curved line is the representation of the trajectory of the object under the influence of that force. The trajectory is not arbitrary - it does not "come first" - but rather follows from physical laws. We could look at the position and velocity of all objects related to the experiment, come up with a Lagrangian, and then use the principle of least action to calculate what the trajectory must be. We are not being redundant if we say that the trajectory follows from the physical constraints in the problem and is not the definition of those constraints. Of course, it would be possible to define the word "force" simply with respect to the curvature of a line, but then we would be doing pure math and not physics. We would lose the connection to the real world.

#### SamRoss

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The only general rule is that specifying the lengths of the lines is what takes the freedom away.
Are we doing pure math or physics? How can we specify the lengths of the lines without looking at what's actually going on in the situation? How can we justify making one line longer or shorter than the other unless there is some physical distinction between the two observers in the real world?

#### PeterDonis

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How can we justify making one line longer or shorter than the other unless there is some physical distinction between the two observers in the real world?
The different lengths of the lines correspond to the different elapsed times on the observers' clocks. That's a physical distinction. And it's the only physical distinction that will always be there in any "twin paradox" situation and will always correspond to the thing that defines the kind of situation you are interested in (different elapsed times for the twins between the same two points in spacetime). Sure, there will be other physical distinctions between the twins, but none of them correspond to a general rule that lets you predict which one will have more elapsed time.

#### SamRoss

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Sure, there will be other physical distinctions between the twins, but none of them correspond to a general rule that lets you predict which one will have more elapsed time.
So if one twin stays on Earth while the other flies off in a rocket and comes back, I can't use those physical distinctions as a general rule to decide which one will be older when they meet? Sometimes it will be the Earth twin and sometimes it will be the rocket twin?

#### Dale

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There is no general rule that says "the one under the influence of the force has the shorter worldline".
But there is a specific rule for flat spacetime that says that the one that is not under the influence of a force has the longer worldline.

#### PeterDonis

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there is a specific rule for flat spacetime that says that the one that is not under the influence of a force has the longer worldline
In the special case where there is one who is not under the influence of a force, yes. But that's a special case.

#### Orodruin

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So your reasoning is that a curved line is the definition of a force
The definition of a curved line is one that has non-zero acceleration. The concept of force is introduced (for a fixed mass object) as mass multiplied by acceleration. This is what a force is, nothing more and nothing less. It turns out that modelling forces in certain ways provides a useful description of some natural phenomena but this is a connection to those phenomena that is utterly irrelevant to what is being regarded here. What is relevant is kinematics, not dynamics.
As I see it, the geometry is only a representation of what's happening in the real world.
This is true for anything yoy describe in physics, including forces.

The force has physical meaning; the curved line is the representation of the trajectory of the object under the influence of that force
The force does not have any more physical meaning than the acceleration. Both are mathematical descriptions of observable phenomena. One is relevant to kinematics, the other not. Both are irrelevant to proper time - only geometry is relevant to proper time. Then you can connect some of that geometry to statements about acceleration in Minkowski space, but the force is still irrelevant to the description. Force only becomes relevant when you ask the question how an object came to follow a particular curved world-line. It is irrelevant to the proper time.

The force has physical meaning; the curved line is the representation of the trajectory of the object under the influence of that force. The trajectory is not arbitrary - it does not "come first" - but rather follows from physical laws.
Obviously, but you are still wrong. You do not need to know anything about the force to compute proper time, all you need is the world-line. If you observe an object following a particular world-line it does not matter in the slightest how it came to follow that world-line. The world-line is every bit as physical as the force.

Of course, it would be possible to define the word "force" simply with respect to the curvature of a line, but then we would be doing pure math and not physics. We would lose the connection to the real world.
No, you are simply wrong here. It is how we define force. Then we make a mathematical model of how different forces appear and there is your connection to dynamics. However, again, this is not needed for the physics under consideration. Compare with the situation in classical mechanics where you want to know how long it takes an object to move a distance with constant acceleration. If you are given the acceleration (kinematics), dynamics are irrelevant. Obviously, if you want to know why the object moved that way you need to do dynamics, but this is totally irrelevant to computing the time. If you do not agree with this you essentially have a beef with the entire middle school physics curriculum.

We would lose the connection to the real world.
Again, wrong, we are not trying to explain why an object follows a particular world-line. We are computing the proper time along it. This is a physical observable measured by a clock. You are essentially saying clocks are not connected to the real world.

#### PeterDonis

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if one twin stays on Earth while the other flies off in a rocket and comes back, I can't use those physical distinctions as a general rule to decide which one will be older when they meet? Sometimes it will be the Earth twin and sometimes it will be the rocket twin?
Since the Earth is a gravitating body, spacetime in this scenario is not flat, so you can't apply flat spacetime rules to it. Did you intend to have the Earth there? Or were you really intending to ignore gravity altogether and assume flat spacetime?

To answer the question as you ask it, with the Earth and gravity included, consider two possibilities:

(1) The rocket twin launches upward from Earth at less than escape velocity (we'll pretend there are no other gravitating bodies present anywhere, for simplicity) and then just lets his rocket coast until it reaches maximum height and falls back to Earth, landing at the same spot it took off from. The Earth twin is in a shielded bunker next to the launch point the whole time.

(2) The rocket twin launches upward from Earth at the same velocity as in #1 above, but uses his rocket to turn around long before it would reach maximum height if it were just coasting.

In #1 above, the rocket twin will have aged more. In #2, it depends on the details of when the turnaround happens.

#### Orodruin

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But there is a specific rule for flat spacetime that says that the one that is not under the influence of a force has the longer worldline.
This is a direct result of the fact that finding the stationary world-lines of proper time results in the geodesic equations (when parametrised by proper time).

The OP seems confused in how a mathematical model relates to a physical reality. Just as I do not need to know anything about forces to know how long it takes to travel a distance d with acceleration a in classical mechanics, I do not need to know anything about forces to compute proper time. All I need is geometry and the geometry is physical every bit as much as any force (they are both mathematical descriptions of physical phenomena that we can relate to observations).

#### Dale

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All I need is geometry and the geometry is physical every bit as much as any force (they are both mathematical descriptions of physical phenomena that we can relate to observations)
I agree but I just worry that you are overdoing it here. I think that @SamRoss made some good progress here already and probably doesn’t need the barrage of nitpicking he is getting from you and @PeterDonis.

He now understands the relationship between a bend in a worldline and the acceleration, which he didn’t before. He also rightly points out a previously known relationship between forces and acceleration. He also rightly infers the connection between forces and a bend in a worldline.

That is awesome! He identified and clearly articulated a specific problem, learned something new that addressed his conceptual problem, and correctly integrated the new concept with his existing knowledge. I wish all twin’s paradox threads were that productive.

He chooses to emphasize the force over the geometry, which is somewhat a matter of taste in the standard scenario and the scenarios where it is not a matter of taste are fairly advanced. It is also entirely understandable since the bulk of his physics experience has probably been very focused on forces.

Let him take the win! I think this is the best progress I have ever seen from the OP in a Twin Paradox thread.

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#### SamRoss

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If you are given the acceleration (kinematics), dynamics are irrelevant. Obviously, if you want to know why the object moved that way you need to do dynamics, but this is totally irrelevant to computing the time.
I want to know why the object moved that way!

Again, wrong, we are not trying to explain why an object follows a particular world-line.
Yes I am! I am trying to explain why an object follows a particular world line! Again, I think we're mainly in agreement, just emphasizing different parts of the problem. For me, what has always been puzzling about the twin paradox is that I would often hear it stated, including in the FAQ, without any reference to the dynamics, leading me to wonder if either observer could arbitrarily be chosen as the one with the curved world line. Obviously they can't. I am not and have never been confused by the idea that one observer has a longer proper time than the other, so long as I can trace the asymmetry in the proper time back to an asymmetry in the physical conditions.

As far as our issue of curvature being the definition of force - perhaps it would have been better for me to stick with the world "potential" but I got talked out of that in earlier posts so I switched to "force". Maybe I should just use "rocket fuel". Anyway, all I'm saying is that the trajectory of the world line follows from something physical. Different physical circumstances between the two observers leads to different world lines. From there, yes, we look to the geometry to calculate the proper times.

#### SamRoss

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At the risk of preempting the end of the back-and-forth, I'd like to thank everyone who participated in this thread. This is exactly the type of rigorous, thoughtful debate I have come to expect and enjoy from PF!

#### PeroK

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I want to know why the object moved that way!

Yes I am! I am trying to explain why an object follows a particular world line! Again, I think we're mainly in agreement, just emphasizing different parts of the problem. For me, what has always been puzzling about the twin paradox is that I would often hear it stated, including in the FAQ, without any reference to the dynamics, leading me to wonder if either observer could arbitrarily be chosen as the one with the curved world line. Obviously they can't. I am not and have never been confused by the idea that one observer has a longer proper time than the other, so long as I can trace the asymmetry in the proper time back to an asymmetry in the physical conditions.

As far as our issue of curvature being the definition of force - perhaps it would have been better for me to stick with the world "potential" but I got talked out of that in earlier posts so I switched to "force". Maybe I should just use "rocket fuel". Anyway, all I'm saying is that the trajectory of the world line follows from something physical. Different physical circumstances between the two observers leads to different world lines. From there, yes, we look to the geometry to calculate the proper times.
Well, this is why the paradox presented without acceleration cuts through all this meaningless soul-searching!

SR fundamentally is a theory of space and time and the four-dimensional geometry thereof. Rocket fuel is about as irrelevant as what the twins ate for their breakfast.

#### PeterDonis

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so long as I can trace the asymmetry in the proper time back to an asymmetry in the physical conditions
And this general form of reasoning is fine: there must always be something physically different about the twins that leads to their different proper times. But "force" will not always be the relevant asymmetry in the physical conditions.

#### Orodruin

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I want to know why the object moved that way!
No you do not. Not if all you are interested in is to compute the time. The dynamics are irrelevant for that. It is pure kinematics.

Yes I am! I am trying to explain why an object follows a particular world line!
Again, no, you are not. It is completely and utterly irrelevant for computing the proper time.

For me, what has always been puzzling about the twin paradox is that I would often hear it stated, including in the FAQ, without any reference to the dynamics, leading me to wonder if either observer could arbitrarily be chosen as the one with the curved world line.
Again, no! The curvature is a geometric property! It has nothing at all to do with dynamics. Dynamics might explain why an object follows a curved world-line, but it has nothing to do with the proper time along that world-line, which is an inherently geometric property.

Anyway, all I'm saying is that the trajectory of the world line follows from something physical.
So what? It is irrelevant for the computation of the proper time.

From there, yes, we look to the geometry to calculate the proper times.
Again. You do not need dynamics for proper times. If you need information about A to compute B, then you need A. If you do not need A to compute B, then it is irrelevant for B. Dynamics are irrelevant for proper time calculation.

so long as I can trace the asymmetry in the proper time back to an asymmetry in the physical conditions.
But the different physical conditions that are relevant to the asymmetry are all geometrical. Your personal desire to explain why something moves along a particular world-line is irrelevant here. Geometry is all there is to it if proper time is all we are asking about - as is the case in differential ageing. The relevant thing is the difference between the world-lines, not why the twins follow the different world-lines.

#### SamRoss

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Let's say there are two observers, A and B. They start out at the same point in space-time. Next, their distance from each other increases at a constant rate. After that, their distance from each other decreases at a constant rate until they are back at the same position. Can you tell me which observer experienced a greater proper time?

#### Mister T

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Summary: I read through "A Geometrical View of Time Dilation and the Twin Paradox" from the FAQ but I still have questions.

Why is the green guy always the bent one
Because he has the distinction of being the traveling twin. Somebody has to be the one to turn around, otherwise there would be no reunion and hence no way to compare ages.

#### Dale

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Can you tell me which observer experienced a greater proper time?

#### SamRoss

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This post was actually in response to a post from Orodruin. I probably should have included a quote. It makes more sense in that context.

#### PeroK

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Let's say there are two observers, A and B. They start out at the same point in space-time. Next, their distance from each other increases at a constant rate. After that, their distance from each other decreases at a constant rate until they are back at the same position. Can you tell me which observer experienced a greater proper time?
Take any inertial reference frame (IRF). Let $v_A(t)$ and $v_B(t)$ be the velocities of A and B in that IRF, with $t$ the coordinate time in that IRF. In general, the proper time experienced by A and B during the period $t = 0$ to $t = T$ is:
$$\tau_A = \int_0^T \sqrt{1- \frac{v_A^2}{c^2}} dt, \ \ \tau_B = \int_0^T \sqrt{1- \frac{v_B^2}{c^2}} dt$$
There are two important points:
a) the proper time depends only on the velocity profile of A and B.
b) If A and B meet twice (i.e. their paths cross twice), then $\tau_A, \tau_B$ (their proper time between those two events) is invariant across all IRF's.

There's another general point I'd like to make. It is very difficult or almost impossible to learn SR from the twin paradox. But, if you learn SR, then you can understand the twin paradox.

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#### Orodruin

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Let's say there are two observers, A and B. They start out at the same point in space-time. Next, their distance from each other increases at a constant rate. After that, their distance from each other decreases at a constant rate until they are back at the same position. Can you tell me which observer experienced a greater proper time?
No, because you have insufficiently defined the geometry. It is exactly the same in regular geometry: You draw two curves. First the distance between them increases, then it decreases. Which curve is longer? You do not know because your specification is incomplete. You need to specify which curve has a kink in it because that kink is part of the geometry.

Your information needs to be sufficient to identify both the spacetime geometry (in SR assumed to be Minkowski space) and the world-lines as geometrical objects in that spacetime.

#### Federation 2005

Acceleration is everything -- and it's the only thing. That comes straight out of the action principle for the law of inertia. For the action principle, you use the negative of proper time as the action. So "least action" -- which is what gives you inertial motion -- means "greatest proper time". So, as a consequence, that means that for any two trajectories that cross paths at the start and end of an interval, the one which has more inertial motion and has spent more tine in 0G will have less action -- meaning more proper time; while the one which has less inertiality in its motion (and more acceleration) will have more action (and less proper time)

It should actually be possible to take the function a = A(s), that maps one's proper time s to one's acceleration a in one's rest frame acceleration and derive, from it, the trajectory as a function of coordinate position and time. That, is, one should be able to prove a theorem like this:

Theorem: Given the initial position r(0) = 0, and initial velocity v(0) = V, with proper time s set to 0 at t = 0, and given the acceleration (in one's instantaneous rest frame) a = A(s) as a function of proper time, then there is a unique trajectory r(t) for which each of these are true. Assume that A is continuous.

Let T > 0 be the end time, S the proper time and (without any real loss of generality) assume r(T) = 0. Then T - S can be expressed entirely as a functional of A such that (i) T - S > 0 if A is non-zero for any proper time between 0 and S (inclusive); (ii) T - S = 0 if A is 0 between proper times 0 and S.

Proof:
Exercise and your next published paper. The details are very hairy; lots of coupled differential equations that need to have explicit solutions by quadrature found, before you can get an actual (integral) expression for T - S in terms of the function A.

For instantaneous changes in velocity (which are technically illegal since they are unphysical) you have to use delta functions for A and that WILL produce a non-zero contribution (as you can see by treating the delta function as a limit of smooth functions and examining the limiting value of T - S as you allow the smooth functions to approach the delta function).

Reference:
The twin paradox: the role of acceleration
J. Gamboa, F. Mendex, M. B. Paranjape, Benoit Sirois

This sets the story straight and gets rid of all the folklore myths that are STILL being perpetuated even by professional and mainstream physicists on this issue; and the faulty analyses usually seen for this problem; that contribute to the misunderstanding of the issue.

"Sorry but I'm still going nuts over the twin paradox"

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