SamRoss
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Good. That's the answer I was hoping to hear. :)
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.Good. That's the answer I was hoping to hear. :)
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.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.
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.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.
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?The only general rule is that specifying the lengths of the lines is what takes the freedom away.
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.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?
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?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.
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.There is no general rule that says "the one under the influence of the force has the shorter 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.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
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.So your reasoning is that a curved line is the definition of a force
This is true for anything yoy describe in physics, including forces.As I see it, the geometry is only a representation of what's happening in the real world.
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
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.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.
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.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.
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.We would lose the connection to the real world.
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?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?
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).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.
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.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 want to know why the object moved that way!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.
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.Again, wrong, we are not trying to explain why an object follows a particular world-line.
Well, this is why the paradox presented without acceleration cuts through all this meaningless soul-searching!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.
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.so long as I can trace the asymmetry in the proper time back to an asymmetry in the physical conditions
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.I want to know why the object moved that way!
Again, no, you are not. It is completely and utterly irrelevant for computing the proper time.Yes I am! I am trying to explain why an object follows a particular 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.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.
So what? It is irrelevant for the computation of the proper time.Anyway, all I'm saying is that the trajectory of the world line follows from something physical.
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.From there, yes, we look to the geometry to calculate the proper times.
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.so long as I can trace the asymmetry in the proper time back to an asymmetry in the physical conditions.
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.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
Nope, additional information is required. The accelerometer readings would be sufficient.Can you tell me which observer experienced a greater proper time?
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.Nope, additional information is required. The accelerometer readings would be sufficient.
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: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.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?