Need help understanding the twins

[Malorie]

I am having some trouble undestanding why speed would cause time dilation.

Here's why:
Twin A travels at near the speed of light away from twin B for one year from in twin A's frame of reference. When he returns to twin B, he has aged two years but twin B has aged significantly more. Right so far?

What is the difference if it's twin B that does the travelling? Why would twin A now be the one to age faster? From twin A's frame of reference, nothing is any different, other than who experiences the changes in velocity to get up to speed.

This would imply to me that it's not the speed that dilates time, but the changes in velocity used to get up to speed. This would be consistant with with what we know about gravity and time dilation and the fact that gravity and mass are one in the same thing.

Am I missing something here?

Do we see time dilation between the equator and the poles, acounting for any gravitational or centrifugal differences?

 

[Janus]

There are three factors that have to be considered when dealing with the Twin paradox: Time dilation, length contraction and the Relativity of simultaneity.

As far as A in concerned, he can consider B as the one moving, and from his measurement it is B that undergoes time dilation. But also, the distance between B and the turnaround point undergoes length contraction.

So for example, if the relative velocity of A and B is 0.866 c, and the distance to the point where A travels to is .866 ly as measured by B, then according to B, it takes 2 years for A to complete the trip, and A ages 1 year due to time dilation.

By A's measurement however, the same distance measured as 0.866 ly by B is only 0.433 ly due to length contraction. Thus according to A, a relative motion of 0.866c results in only 1 year passing before A and B rejoin.

Relativity of simultaneity explains why even though A measures B as aging more slowly on the outbound and return trips, he determines that B has aged more when they meet up again. In essence it means that when A turns around to head back to B, he changes inertial frames which causes him to determine that B has "jumped forward" in age between the times that they were heading away from each other and when they are heading towards each other.

In the example I gave this means that while they are receding from each other, according to A, B will age 3 mo, and the same will be true when they are approaching each other. So when A accelerated to go from heading away to heading towards B, B ages 1 1/2 years.

If you haven't familiarized yourself with the Relativity of Simultaneity yet, I suggest that you do so.

You can use the equivalent of gravitational time dilation in this problem, but you do have to be careful. Gravitational time dilation is not due to a difference in the gravitational force, but a difference in gravitational potential.

So for instance in your question about clocks running at different speeds at the pole and equator, the answer is no. While the gravitational force differs between these two points, the surface of the Earth is at an equal gravitational potential at all points, which results in no time dilation.

So for the Twin paradox consider this:

We add a third "twin". He travels at the same speed as A, using exactly the same acceleration at all points of the trip. However, after he reaches the point where A turns around, he continues on. He doesn't turn around until he reaches a point twice the distance from B. When he returns he will have aged 2 years while 4 years have passed for B. In addition, A will have aged 3 years. (1 yr during his trip and 2 years waiting with B waiting for the other twin to return.

So even though A and the third twin experienced exactly the same accelerations, they end up aging differently.

 

[Malorie]

I think you missed my point. Or I may have missed yours. :)

With the twins, who's to say which twin should age? Without any outside point of reference and barring changes in velocity, the observation from either twin's point of view is the same in either case. Why would either twin age differently because of perceived speed which could only be measured from the other's point of view and would be exactly the same wheather A traveled away from B or B traveled away from A.

The only difference between each case (ie. A travels from and to B, or B travels from and to A) is the changes in velocity.

 

[DrGreg]

There is one significant difference between A and B. B turns round. A does not. When B turns round he feels a "G-force" of acceleration, and A does not, which proves that it is B who turned round, not A. So the situation is not symmetrical.

If neither turned round and just kept moving apart forever, there is no absolute answer to which of the two is older. And if, instead of B turning round, A decided to chase after B and catch up with him, then it would be A who ended up younger than B.

 

[tiny-tim]

Welcome to PF!

Hi Malorie! Welcome to PF!

Two brief points:

i] Why are you using the word "inertia"??

("inertia" is usually another name for "mass")

One twin experiences acceleration, not inertia, to get up to speed.

ii] yes, gravity and acceleration are more-or-less the same thing …

and if one twin experienced changes in gravity while the other didn't (both remaining stationary), then the different gravity would cause a "time dilation."

This would imply to me that it's not the speed that dilates time, but the inertia used to get up to speed

(if we read "acceleration" instead of "inertia" …)

What you're saying is;

If we analyse it the usual way, then the steady velocity causes the time dilation, while the three periods of acceleration (which can be as short as we like) contribute nothing

But if we analyse it substituting gravity for acceleration, and invoke the equivalence principle, then the gravity (which is standing in for the accelerations) causes the time dilation.​

No, between the affected twin's three changes in gravity (which can be as short as we like) there are two periods of steady gravity at a different strength to that of the unaffected twin.

And these two periods of steady gravity (which is standing in for steady velocity) contribute most of the time dilation … this is exactly equivalent to the effect of steady velocity in the original case.

 

[Malorie]

I guess the confusion then is my use of the word inertia, I was mistakingly using it to describe changes in velocity. Fixed my previous posts. :)

Which would be why I said that the only difference between the two cases was that one of them experiences inertia. You could just exchange the words 'changes in velocity' or 'acceleration/deceleration' for the word 'inertia' in my post. Outside of that, the velocity itself is irrelevant as the pair could possibly be traveling through space at an unknown velocity at the outset of the experiment.

My whole point is that I don't believe it is the velocity that causes the time dilation, but the changes in velocity that do.

This is the only logical conclusion that can be drawn from this paradox.

 

[DrGreg]

Here's an analogy. Consider a triangle PQR. Why is PQ + QR greater than PR?

(a) because PQ and QR are not parallel to PR
(b) because the route P-Q-R is not straight and the route P-R is straight

In this analogy (a) is equivalent to there being a velocity between A and B. (b) is equivalent to B turning round.

 

[JesseM]

Here's an analogy. Consider a triangle PQR. Why is PQ + QR greater than PR?

(a) because PQ and QR are not parallel to PR
(b) because the route P-Q-R is not straight and the route P-R is straight

In this analogy (a) is equivalent to there being a velocity between A and B. (b) is equivalent to B turning round.

Also, if you're dealing with paths on a 2D plane, you can add a cartesian coordinate system with x and y axes and then the "slope" of a path at any point (dy/dx) will be pretty closely analogous to the notion of velocity (dx/dt) in an inertial coordinate system in spacetime--if you have multiple paths between the same pair of points, then whichever path has a constant slope will be the the one with the shortest distance, analogous to how in spacetime, if you have different worldlines between the same pair of events (in the twin paradox, the event of the twins departing from one another and the event of them reuniting), then whichever path has a constant velocity will have the greatest proper time.

You can actually use the slope to find the rate that the length of the path is increasing with each incremental increase of the x-coordinate, and integrate over x to find the total length of the path, in a way that's analogous to how you can use the velocity to find the rate a clock ticks with each incremental increase of the t-coordinate, and integrate over t to find the total time elapsed on a given worldline. I expanded on this in post #64 of this thread:

The time dilation at any given instant depends solely on the the velocity in whatever frame you're using, the factor by which a moving clock is slowed down is always $$\sqrt{1 - v^2/c^2}$$ where v is that clock's instantaneous velocity. However, if you have two worldlines that cross paths at two times t0 and t1, and you know the velocity as a function of time v(t) on each worldline, then you can do the integral $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$ for both of them to find the total time elapsed on each worldline between the two points where they cross. If one worldline is inertial (constant value for v(t)) and the other involves some acceleration (the value of v(t) changes with t), it will always work out that when you do the integral above, you'll find that the total time elapsed is greater on the inertial worldline than the worldline that involved an acceleration. That's just a property of the way the integral works, and it's totally compatible with the idea that the time dilation at each moment depends solely on the velocity at that moment, not the acceleration.

If it helps, there's a direct analogy for this in ordinary Euclidean geometry. Suppose we have two paths on a 2D plane which cross at two points, and one is a straight-line path while the other involves some bending. Since we know a straight line on a 2D plane is the shortest distance between points, we know the straight-line path will have a shorter total length. But suppose we want to measure the length of each path by driving cars along them with odometers running to measure how far the cars have travelled. Suppose we also have an x-y coordinate system on this 2D plane, so we can talk about "the rate a car is accumulating distance as a function of its x-coordinate"--if you think about it, it's not hard to show that this is solely a function of the slope of the path at that point in the coordinate system you're using. If you know the function for the path in this coordinate system y(x), then the slope at x is defined by looking at a small interval from x to (x + dx), and seeing the amount dy that the y-coordinate of the path changes between those points, with the slope defined as dy/dx. Since dx and dy are assumed to be arbitrarily small, the path can be assumed to be arbitrarily close to a straight line between the points (x,y) and (x+dx,y+dy), so the distance accumulated on the car's odometer as it travels between those points is just given by the pythagorean theorem, it'll be $$\sqrt{dx^2 + dy^2}$$, which is equal to $$dx*\sqrt{1 + dy^2/dx^2}$$, and since the "slope" at a given coordinate S(x) is defined to be dy/dx, this means the distance accumulated on the car's odometer as it travels between these points can be written as $$dx * \sqrt{1 + S(x)^2}$$.

So, the ratio of (increment odometer increases)/(increment x-coordinate increases), i.e. "the rate the car is accumulating distance as a function of its x-coordinate", will just be $$\frac{dx*\sqrt{1 + S(x)^2}}{dx}$$ which is just $$\sqrt{1 + S(x)^2}$$, purely a function of the slope. On the other hand, if you want to know how much distance accumulates on the odometer over a non-incremental change in the x-coordinate, say from some value $$x_0$$ to $$x_1$$, then we have to integrate the amount the odometer increases over each increment over the entire range from $$x_0$$ to $$x_1$$, giving the integral $$\int_{x_0}^{x_1} \sqrt{1 + S(x)^2} \, dx$$. Since we know a straight path is the shortest distance between two points, and we know straight implies constant slope, this means that if we have two different paths which cross once at $$x_0$$ and then again at $$x_1$$, and one has a constant S(x) while the other has a varying S(x), that means if we do the above integral for both paths the answer for the constant-slope path is guaranteed to be smaller.

Obviously all this is very closely analogous to the situation in relativity, where the rate a clock accumulates time as a function of the t-coordinate is just $$\sqrt{1 - v^2}$$ (in units where c=1, like seconds and light-seconds), while the total time accumulated on a path with a specific v(t) is $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2} \, dt$$, and a path with constant v is guaranteed to have a longer total time than a path with a v that changes (the reason a straight path in SR is guaranteed to have the largest time while a straight path in geometry is guaranteed to have the shortest distance has to do with the fact that there's a plus sign in front of the geometric slope but a minus sign in front of the velocity in the two square roots).

 

[D H]

One way to look at the twin paradox is to determine what each twin sees.

Assume acceleration is essentially infinite (e.g., some future scientist has found a way to instantaneously transfer momentum to/from some parallel universe). With this technology, the space twin can instantly start moving toward some remote star at a large fraction of the speed of light. For instance, suppose the space twin moves at 112/113 the speed of light and that the star is 12.656 light years away in the frame of the Earth twin.

Suppose the twins remain in constant communication and that each twin regularly broadcasts time as measured in their own rest frames. During the outbound journey, each twin will see the other twin's clock running slow. Fifteen seconds will pass between each tick of the clock in the received signal due to the relativistic doppler effect.

This symmetric relation would continue if the traveling twin was the Energizer Bunny twin (she just kept on going and going and going). That is not what happens. The traveling twin turns around. A symmetric condition exists shortly before she returns to Earth: Each twin will see the other twin's clock as running 15 times *faster* than their own clock.

Some asymmetry must exist to have the traveling twin age less than the Earth twin. This asymmetry is the point in time at which the received signal switches from running 15 times slower to 15 times faster. For the traveling twin, this transition point occurs at the turnaround point. From her perspective, her Earth-bound twin ages 1/15*1/2+15*1/2=113/15 times as fast as she does.

For the Earth twin, the transition from slow time to fast time occurs shortly before the traveling twin returns. The traveling twin ages at a rate of 1/15*225/226+15*1/226=15/113 his aging rate.

(1/15*x+

 

[Malorie]

First I would like to keep Earth or any other objects in the universe out of this. Using them tends to muddy up the analogy. The twins are in a region of space where the only reference they have outside their own inertial frame of reference is the other twin. If we involve other objects, then there is a tendency to think of one twin as if they were in some sort of rest state with the rest of the universe which we know is irrelevant.

Assume acceleration is essentially infinite (e.g., some future scientist has found a way to instantaneously transfer momentum to/from some parallel universe).

Do you mean that we are assuming that neither twin feels the affects of acceleration or deceleration due to this awsome scientist? (they ROCK!) ;)
OK, but remember that without any acceleration/deceleration fealt, there is no way to know which twin is doing the travelling.

This symmetric relation would continue if the traveling twin was the Energizer Bunny twin (she just kept on going and going and going). That is not what happens. The traveling twin turns around. A symmetric condition exists shortly before she returns to Earth: Each twin will see the other twin's clock as running 15 times *faster* than their own clock.

OK. Still assuming neither would feel the acceleration or deceleration. How do we know which twin is doing the travelling?

Some asymmetry must exist to have the traveling twin age less than the Earth twin.

In order to have one age differently than the other it is certainly reasonable to assume there has to be an asymmetry. :) But again, how do we know which twin is doing the traveling without any acceleration/deceleration fealt?

This asymmetry is the point in time at which the received signal switches from running 15 times slower to 15 times faster. For the traveling twin, this transition point occurs at the turnaround point. From her perspective, her Earth-bound twin ages 1/15*1/2+15*1/2=113/15 times as fast as she does.

For the Earth twin, the transition from slow time to fast time occurs shortly before the traveling twin returns. The traveling twin ages at a rate of 1/15*225/226+15*1/226=15/113 his aging rate.

I guess this is where I'm getting totally lost. Why would either twin experience anything different from the other. We have no way to know which twin is doing the travelling.

Without any acceleration/deceleration, it could be that;
Twin A moves away from twin B then stops moving in relation to twin B, Then twin B moves to rejoin twin A.

Or twin A moves away from twin B, turns around and comes back.

Or both twins move away from each other an equal distance and come back the same distance to rejoin each other.

Or any other of another million other possabilities.

Without the acceleration/deceleration, there is no reason to believe that there would be any time dilation whatsoever.

Again, the only logical conclusion here is that time dilation is only due to the acceleration/decelleration and has nothing to do with the velocity (speed of travel).

 

[D H]

Do you mean that we are assuming that neither twin feels the affects of acceleration or deceleration due to this awsome scientist? (they ROCK!) ;)
OK, but remember that without any acceleration/deceleration fealt, there is no way to know which twin is doing the travelling.

Sure there is. Suppose the twins start at rest with respect to the target star. This means the star will have zero transverse velocity when viewed from the perspective of a non-rotating frame and the star's hydrogen alpha line will be at 656.281 nanometers. Note that other nearby stars might show some proper motion and their Hα line might vary a bit from 656.281 nanometers due to a non-zero radial velocity. However, this observed proper motion and radial velocity will be small compared to the speed of light.

Now the traveling twin presses the magic button. From her perspective, the target star's Hα line will blue-shift to 43.7521 nanometers, well into the ultraviolet. The Hα lines of the stars directly aft will red-shift to about 9.8442 microns, well into the infrared. The stars off to the side will show a huge proper motion. The stationary twin sees exactly what he saw before his sister pushed the button.

Again, the only logical conclusion here is that time dilation is only due to the acceleration/decelleration and has nothing to do with the velocity (speed of travel).

That is just wrong.

 

[Malorie]

Suppose the twins start at rest with respect to the target star.

There is no target star in this scenario. As I said at the start of my last post, other items in the scenario will just confuse the subject. So the twins are in a section of the universe with no frame of reference other than the other twin.

You are implying that the twins are starting out in some sort of universal rest state by putting in a 'target star' and other stars. Then you are using that target star as proof of who is travelling.

What if the target star and everything except the other twin was matching the movements of a twin traveling away from and to the other twin? We end up with the same thing you discribed except now the star and everything else is traveling as well and the stationary twin sees the blue and red shifts. Again, I want to keep the outside celestial stuff out of the scenario to reduce the confusion.

The scenario as it was in my last post illustrates my point that there is no way to know which twin is traveling without any other reference than the other twin or the perceived acceleration/deceleration. Which follows that without knowing which twin does the travelling, there is no way to imply which twin should age faster. And that leads us logically to the fact that the velocity has nothing to do with the time dilation.

Stating that something is 'just wrong' is not an answer to anything it is just your assurtion.

 

[JesseM]

There is no target star in this scenario. As I said at the start of my last post, other items in the scenario will just confuse the subject. So the twins are in a section of the universe with no frame of reference other than the other twin.

You don't need any external objects to determine who moved inertially and who accelerated and changed velocities relative to all inertial reference frames. In flat SR spacetime with no gravity, whichever one moved inertially will have felt weightless throughout, while the one who accelerated will have felt G-forces during the acceleration, which could be measured with an accelerometer.

 

[Malorie]

You don't need any external objects to determine who moved inertially and who accelerated and changed velocities relative to all inertial reference frames. In flat SR spacetime with no gravity, whichever one moved inertially will have felt weightless throughout, while the one who accelerated will have felt G-forces during the acceleration, which could be measured with an accelerometer.

As DH stated

Assume acceleration is essentially infinite (e.g., some future scientist has found a way to instantaneously transfer momentum to/from some parallel universe). With this technology, the space twin can instantly start moving toward some remote star at a large fraction of the speed of light.

Remember the future scientists (who ROCK!)?
Without any acceleration/deceleration there is no way to tell which twin is the one travelling.

This is the whole point of my post and my confusion with the twins paradox. Without the affects of acceleration and deceleration all we are left with is realative speed and there is no way to assign that speed to either twin and therefor sensless to assume that either twin would age differently than the other. This implys that the affects of acceleration and deceleration are the only thing that cause time dilation and velocity has nothing to do with it.

 

[JesseM]

Remember the future scientists (who ROCK!)?
Without any acceleration/deceleration there is no way to tell which twin is the one travelling.

Even if the change in velocity is instantaneous, you can still detect that you've accelerated--for example, if you have a ball floating in a vacuum in the middle of the ship which isn't connected to the rest of the ship in any way, then when the ship instantaneously accelerates the ball will continue to move inertially, so observers on the ship will see it appear to suddenly change speed relative to themselves.

 

[D H]

Remember the future scientists (who ROCK!)?
Without any acceleration/deceleration there is no way to tell which twin is the one travelling.

The whole point of that supposition was to simplify the math. I could just as easily have said that science has developed some new technique that allows the traveling twin to withstand arbitrarily high accelerations.

It is important to remember that neither the ability to withstand high acceleration nor the ability to transfer momentum to another universe does not exist. In comparison, time dilation and length contraction are very real.

This is the whole point of my post and my confusion with the twins paradox. Without the affects of acceleration and deceleration all we are left with is realative speed and there is no way to assign that speed to either twin and therefor sensless to assume that either twin would age differently than the other. This implys that the affects of acceleration and deceleration are the only thing that cause time dilation.

You appear to be intentionally misunderstanding the twin paradox.

Acceleration does not affect the rate at which a clock ticks. This is the clock hypothesis, and this hypothesis (along with other aspects of relativity) has been tested multiple times.

 

[Malorie]

I'm not trying to argue that you could or couldn't detect the acceleration.

What I am saying is that without any way to detect acceleration, all you are left with is the relative speed. And relative speed doesn't logically cause any time dilation because you can't assign that speed to either twin if you can't tell who is doing the travelling.

DH,
So previously, you accepted my interpritaion of your scientists and now you don't?

I said;

Do you mean that we are assuming that neither twin feels the affects of acceleration or deceleration due to this awsome scientist? (they ROCK!) ;)
OK, but remember that without any acceleration/deceleration fealt, there is no way to know which twin is doing the travelling.

You appear to be intentionally misunderstanding the twin paradox.

No, you are missing my point.

 

[pervect]

It does not follow logically that "relative speed can't cause time dilation", unless you make additional assumptions about time.

I can probably even guess what additional assumptions you are making, but it might be more instructive for you to work them out for yourself. Logically, what properties does time need to have so that your argument follows? You have made an intuitive leap here, you'll need to fill in the missing parts of your arguments to proceed to finding the error. (I suppose I should be diplomatic and call it a difference of thinking, but I'm feeling a bit grumpy today, so I'll be straightforwards and call it an error.).

 

[Malorie]

Thanks for the insight on this pervect. NOT!

If all you are going to add is that you think I'm playing a game or something then don't.

I'm being as clear as I can think to be.

 

[JesseM]

I'm not trying to argue that you could or couldn't detect the acceleration.

What I am saying is that without any way to detect acceleration, all you are left with is the relative speed. And realative speed doesn't logically cause any time dilation.

But in relativity there is an objective physical truth about who really accelerated, which is why it's important that you can in fact always detect it physically. Your argument seems equivalent to "if there was no objective truth about who accelerated, all you'd be left with is relative speed"--well, there is an objective truth, and that's why there can also be an objective truth about which twin aged more?

Did you read through the analogy with the two paths on a 2D plane, with slope standing in for speed and change in slope standing in for acceleration? Do you agree there's an objective truth about which path is straight (constant slope) and which is bent (change in slope), and that the straight path between a pair of points always has a shorter distance than a bent path between the same pair of points? If I and a friend are driving down the two paths measuring our distance with our cars' odometers, I suppose you could argue "if there was no objective truth about whose slope/direction changed, then all you'd be left with would be the relative angle between the direction the two cars are moving at each instant, and since this is symmetrical there'd be no basis for saying one car's odometer would have measured a greater distance than the other when the two cars reunited". But it's a moot point, because geometrically there is an objective truth about which path is a straight line and which isn't, and it will always be the straight path which has a smaller distance.

 

[Malorie]

If you all want to get hung up on acceleration, let's say the stationary twin is in a centrifuge that mimics the acceleration and deceleration affects that the traveling twin feels.

Yes I did read though your analogy. It didn't apply to what my point is.

Your argument seems equivalent to "if there was no objective truth about who accelerated, all you'd be left with is relative speed"

Right!

And relative speed doesn't cause the time dilation.

I'm trying to get you to separated speed and acceleration. They are after all two separate things. Speed is purely relative and acceleration is not.

 

[JesseM]

If you all want to get hung up on acceleration, let's say the stationary twin is in a centrifuge that mimics the acceleration and deceleration affects that the traveling twin feels.

Do you understand that it's the overall shape of the path through spacetime that matters, not the magnitude of the G-forces felt? If the "stationary" twin is accelerated briefly but in such a way that after the acceleration his velocity relative to any inertial frame remains the same as it was before the acceleration (so that his worldline looks like two straight line segments at the same angle with a small curved section between them), he can determine this by paying attention to the magnitude and direction of the G-forces during the accelerated period. Similarly, the twin that turns around can use the magnitude and direction of the G-forces to determine that relative to any inertial frame, his velocity after the acceleration is quite different from his velocity before the acceleration, so that his worldline looks like two straight line segments at different angles joined by a curved segment (like the letter "V" with a slightly rounded bottom).

You haven't really responded to my questions about the geometric analogy...please tell me, do you agree that a straight line between two points will always have a shorter distance than a bent path between the same two points, and that there is an objective truth about the geometry of different paths? If so, what's the problem with accepting the idea that there's also an objective truth about the geometry of different paths through spacetime?

 

[Malorie]

Do you understand that it's the overall shape of the path through spacetime that matters

From who's frame of reference?

You haven't really responded to my questions about the geometric analogy

I did respond. I agree with your analogy completely. Accept the objective truth part, which I'll explain.

do you agree that a straight line between two points will always have a shorter distance than a bent path between the same two points, and that there is an objective truth about the geometry of different paths?

This is two questions:
1. Do you agree that a straight line between two points will always have a shorter distance than a bent path between the same two points?

Yes!

2. Do you agree that there is an objective truth about the geometry of different paths?

No! I don't agree that there is an objective truth about any path through spacetime without a frame of reference. A frame of reference is by definition completely subjective.

If I shoot a laser beam from here to the alpha centuri is it objectively traveling a straight line?
Who says so?
Why is their truth the objective truth?

Think outside the box a bit here. Acceleration is a force and speed is purely subjective.

 

[D H]

My crackpot meter is going off scale high.

 

[Malorie]

So what? You think I'm being a crackpot now?

I'm just trying to have a discussion here.

I just answered your questions and posed a couple of logical questions back. Why is that being a crackpot?

 

[JesseM]

From who's frame of reference?

Geometry is frame-independent, all frames agree on which path between a pair of points is the "straight" one which maximizes the proper time, and they also agree on the proper time between any pair of points on a non-straight path.

This is two questions:
1. Do you agree that a straight line between two points will always have a shorter distance that a bent path between the same two points?

Yes!

2. Do you agree that there is an objective truth about the geometry of different paths?

No! I don't agree that there is an objective truth about any path through spacetime without a frame of reference. A frame of reference is by definition completely subjective.

Well, in relativity this is incorrect. A "frame of reference" is just a spacetime coordinate system, so it's analogous to a Cartesian coordinate system in 2D space. For a given set of paths through 2D space, you can pick different Cartesian coordinate systems with their xy axes oriented at different angles, and they will disagree on things like the slope of a given path at a given point or how fast a path is accumulating distance relative to an incremental increase in the x-coordinate near some point, but they'll all agree on geometric questions like which path between points has the shortest distance, and what the length of a given path is. It's exactly analogous with inertial coordinate systems in SR--they can disagree on frame-dependent things like the velocity of a worldline at a given point or how fast a worldline is accumulating proper time relative to an incremental increase in the t-coordinate near some point, but they'll all agree on geometric questions like which path between points has the greatest proper time, and what the proper time along a given path is. And just as different Cartesian coordinate systems can disagree on the distance along the x-axis $$\Delta x$$ and the distance along the y-axis $$\Delta y$$ between a given pair of points on the plane, yet they all agree on the value of the formula $$\sqrt{\Delta x^2 + \Delta y^2}$$ (since by the Pythagorean theorem this is just the distance between the points, a coordinate-independent geometric quantity), so it is true that different inertial frames in SR can disagree about the distance $$\Delta x$$ and time $$\Delta t$$ between a given pair of events, but they all agree on the value of the formula $$\sqrt{\Delta t^2 - \Delta x^2 / c^2}$$ (this will be the proper time between the events for an inertial observer whose worldline crosses both of them).

The analogy is really quite precise, pretty much anything you can say about coordinate-dependent vs. geometric quantities for paths on a plane has a direct analogue with worldlines in SR spacetime.

 

[matheinste]

If you all want to get hung up on acceleration, let's say the stationary twin is in a centrifuge that mimics the acceleration and deceleration affects that the traveling twin feels.

Yes I did read though your analogy. It didn't apply to what my point is.


Right!

And relative speed doesn't cause the time dilation.

I'm trying to get you to separated speed and acceleration. They are after all two separate things. Speed is purely relative and acceleration is not.

The fact that relative speed causes length contraction and time dialtion appears in the first couple of chapters of most textbooks on relativity. I say most, just in case you have actually read one and it happened to be one of those which take the first few chapters to explain the historical background to Einsteins theory and how he came to postulate his postulates, the most immediate consequences of which are time dilation, length contraction and loss of simultaneity.

Matheinste.

 

[cfrogue]

So what? You think I'm being a crackpot now?

I'm just trying to have a discussion here.

I just answered your questions and posed a couple of logical questions back. Why is that being a crackpot?

I can show you the integral from the calcs of the stay at home twin that the traveling twin is younger.

Further, I can show you a paper cited by wiki that uses the Cauchy-Schwarz inequality in the frame of the traveling twin to prove the stay at home twin is older and thus the traveling twin is younger in the context of the traveling twin.

 

[Malorie]

Sorry JesseM,
I was going to deconstruct your whole explanation but it was getting to be to much chopping up and not enough input from me.

I'll just agree with the whole thing and add a question.

You used the words "proper time" throughout your post. Could you clarify this for me? Maybe it's just a terminology issue for me.

One issue I'm having here is that I tend to be more of a visual person and verbal/literal discussions take a bit to sink in. A picture is definitely worth a thousand words for me.

Let me try it this way:
Both twins start out in the same inertial frame.

The traveling twin changes their inertial frame by accelerating up to travel speed. Are we still together here?

After the traveling twin finishes accelerating and is again at rest in a now moving frame of refrence, both twins are now opposed in what they see (both see the other as moving through time slowly from their own perspecive). This state is no different than if the twins were moving away from each other instead of just one of them moving. Are we still together here?

Now the traveling twin decelerates and accelerates back toward the other twin. Changing their frame of reference from departing to aproaching.

Once the traveling twin reaches their approach speed, again the twins are opposed in their view of the other (again they both see the other moving slowly though time from their own perspective). Again we reached a relative speed state that would be equivalent to both twins moving towards each other instead of just one of them moving. How am I doing, have I lost it yet?

Now the traveling twin decelerates on arrival back at the non-travelling twin.

Now where did the difference in time dilation happen? It wasn't while they were both at rest in their own reference frame. It was during the acceleration deceleration phases of the journey.

So where am I loosing it?

Let me add this for cfrouge. I am not arguing that the stay at home twin ages more quickly. I believe that fully.

And sorry Jesse for the late edit on this one. I wanted to clarify it just a tiny bit.

 

[cfrogue]

Here is the wiki calculation of the integral for the proper time calculation of the elapsed proper time of the traveling twin from the at rest twin.
http://en.wikipedia.org/wiki/Twins_...lt_of_differences_in_twins.27_spacetime_paths

This is consistent with SR acceleration calculations.

This paper shows also the traveling twin is younger in the calculations of the traveling twin.

http://arxiv.org/PS_cache/physics/pdf/0411/0411233v1.pdf

See theorem II.1

 

[JesseM]

Sorry JesseM,
I was going to deconstruct your whole explanation but it was getting to be to much chopping up and not enough input from me.

I'll just agree with the whole thing and add a question.

You used the words "proper time" throughout your post. Could you clarify this for me? Maybe it's just a terminology issue for me.

The "proper time" along a given worldline is just the time that would be measured by a clock moving along that worldline. It's a coordinate-invariant quantity which is directly analogous to the length along a given path in 2D space.

One issue I'm having here is that I tend to be more of a visual person and verbal/literal discussions take a bit to sink in. A picture is definitely worth a thousand words for me.

Are you familiar with spacetime diagrams, like these one for the twin paradox?

http://www.csupomona.edu/~ajm/materials/twinparadox/twins.jpg

The diagram is taken from http://www.csupomona.edu/~ajm/materials/twinparadox.html ...on the left side we see a diagram drawn from the perspective of a frame where the Earth is at rest, on the right is one where the traveler was at rest during the outbound phase of the journey (up until t=4) but not afterwards. In both cases, the frame's time coordinate is the vertical axis and the frame's space coordinate is the horizontal axis. You can see that regardless of which frame you use, the Earth's worldline is a straight line going from the event of the traveler departing at the bottom to the event of the traveler returning at the top, while the traveler's worldline is has a bend in it (at t=5 in the left diagram, and t=4 in the right diagram). These sorts of diagrams make the geometry of the situation more apparent. Note that in the left diagram, if you just focus on the traveler's outbound leg, the $$\Delta x$$ is 3 and $$\Delta t$$ is 5, whereas if you look at the traveler's outbound leg in the right diagram, $$\Delta x$$ is 0 and $$\Delta t$$ is 4, which means in both cases you get the same value for the proper time along the outbound leg with the formula $$\sqrt{\Delta t^2 - \Delta x^2 }$$ (which as I mentioned is analogous to the Pythagorean theorem that gives distance along a straight line in a Cartesian coordinate system).

Let me try it this way:
Both twins start out in the same inertial frame.

The traveling twin changes their inertial frame by accelerating up to travel speed. Are we still together here?

After the traveling twin finishes accelerating and is again at rest in a now moving frame of refrence, both twins are now opposed in what they see (both see the other as moving through time slowly from their own perspecive). This state is no different than if the twins were moving away from each other instead of just one of them moving. Are we still together here?

Now the traveling twin decelerates and accelerates back toward the other twin. Changing their frame of reference from departing to aproaching.

You have to take into account the fact the relativity of simultaneity here--different inertial frames can disagree about whether two events at different locations in space are simultaneous or not. Suppose as in the example above in the graphic above, the traveling twin's speed relative to the Earth is 0.6c, and the traveling twin turns around after 4 years of proper time have passed on his own clock. In the frame of the Earth, the event of the Earth clock showing that 5 years have passed is simultaneous with the event of the traveling twin's clock reading 4 years, which is when the traveling twin turns around--so in the Earth's frame the traveling twin's clock was slowed down by a factor of 0.8. In the frame where the traveling twin was at rest during the outbound leg of the trip, it was the Earth's clock that was slowed down by a factor of 0.8, so the event of the traveling twin's clock reading 4 years was simultaneous with the event of the Earth's clock reading 4*0.8 = 3.2 years. But if we then look at a third frame where the traveling twin was at rest during the inbound leg of the trip, in this frame the event of the traveling twin's clock reading 4 years was simultaneous with the event of the Earth's clock reading 6.8 years. So you can't just jump from the outbound rest frame to the inbound rest frame without taking into account that they have different definitions of simultaneity and thus different opinions about the time on the Earth clock at the moment the traveling twin turned around.

All of this has parallels once again to the 2D analogy--if you have two paths between points in space, one straight and the other a bent path consisting of two segments at different angles (so together the two paths form a triangle), then you could pick one Cartesian coordinate system with its x-axis parallel to the bottom segment of the bent path, and see what point on the straight path has the same x-coordinate as the bend in the bent path; then you could pick a different Cartesian coordinate system with its x-axis parallel to the top segment of the bent path, and see what point on the straight path has the same x-coordinate as the bend in the bent path. This would yield two totally different points on the straight path, as you can should be able to see if you draw a diagram. Does that lead you to conclude that all the extra distance along the bent path accumulated at the moment of the bend, so that if you drove a car along the bent path the odometer would suddenly jump when you reach the bend? Of course not--the greater length of the bent path has to do with its overall shape and the fact that neither of the two straight segments is parallel to the straight path, the bend itself can be made arbitrarily short so it contributes almost nothing to the overall length of the bent path.

Once the traveling twin reaches their approach speed, again the twins are opposed in their view of the other (again they both see the other moving slowly though time from their own perspective). Again we reached a relative speed state that would be equivalent to both twins moving towards each other instead of just one of them moving. How am I doing, have I lost it yet?

Now the traveling twin decelerates on arrival back at the non-travelling twin.

Now where did the difference in time dilation happen? It wasn't while they were both at rest in their own reference frame. It was during the acceleration deceleration phases of the journey.

So where am I loosing it?

The problem here is that you are not sticking to a single inertial frame throughout the journey, you are thinking of one inertial frame for the outbound leg and another for the inbound leg. If you stick to a single inertial frame, you will find that there is no sudden change in either twin's age at the moment of acceleration, you can find how much each twin has aged in total just by looking at how much coordinate time went by on each leg of the journey, and figuring out how much each twin's clock would be slowed down relative to coordinate time based on their velocity during that leg. For example, in the right hand diagram above, drawn from the perspective of the inertial frame where the traveling twin was at rest during the outbound leg, the outbound leg lasted 4 years of coordinate time while the inbound leg lasted 8.5 years of coordinate time. In this frame the traveling twin was at rest during the outbound leg so his clock was keeping pace with coordinate time, meaning he aged 4 years of proper time during the outbound leg; then the traveling twin was traveling at 15/17c during the inbound leg, so during this leg his clock was slowed down by a factor of $$\sqrt{1 - (15/17)^2}$$ = 0.470588, meaning during the 8.5 years the inbound leg lasted the traveling twin aged 0.470588*8.5 = 4 years of proper time. This means that if we add the traveling twin's proper time during the outbound and inbound legs we find he has aged a total of 4 + 4 = 8 years between leaving Earth and returning. Meanwhile in this frame the Earth had a constant velocity of 3/5c during both the outbound and inbound leg, so its clock was slowed down by a factor of $$\sqrt{1 - (3/5)^2}$$ = 0.8, so during the 4 + 8.5 = 12.5 years of both legs combined the Earth's clock only ticked forward by 0.8*12.5 = 10 years of proper time. So here we have calculated how much each twin aged throughout the journey using only their velocities in each phase of the trip, as I said there were no jumps during the acceleration. You could do exactly the same thing from the perspective of a different inertial frame and you'd still conclude the traveling twin aged 8 years while the Earth twin aged 10.

 

[Malorie]

THANK YOU JESSE! :)
I get it now.

The problem here is that you are not sticking to a single inertial frame throughout the journey

Between this and the diagrams, the lights went on. :)

Thanks for your patience on this little journey.

 

[clamtrox]

It is still true though that the only way you can have a twin paradox is to have one of the twins accelerate at some point, and the accelerating twin is always the one who's getting old slower.

 

[JesseM]

THANK YOU JESSE! :)
I get it now.



Between this and the diagrams, the lights went on. :)

Thanks for your patience on this little journey.

Cool, glad I could help!

 

[Malorie]

It is still true though that the only way you can have a twin paradox is to have one of the twins accelerate at some point, and the accelerating twin is always the one who's getting old slower.

Yes.
The twin that will age more slowly is the one that changes inertial frames of reference.

It isn't the acceleration that causes the time dilation, it is the time spent in the traveling frame of reference. Those frames of refrence are high speed frames relative to the twin that is stationary.

Because the traveling twin is returning to the stationary twin's frame of refrence, then the stationary twin's frame of refrence is the clock that you have to go by, because ultimately they both end in that frame of refrence.

This might be a confusing example, but it's one that shows an odd view that still works out correctly.

If both twins are traveling past their home planet at high speed, and one twin decelerates to stop and visit their mom. Then accelerates to catch up to the twin that continued on their trip. When the one that stopped catches up and decellerates to match speed with the one that continued, they will find that the one who continued has aged more quickly.

The one that continued, traveled in only one frame of refrence. While the twin that stopped spent time in three, two of which were at high speed relative to the traveling twin. Their ages are then compared to the clock that was always with the traveling twin. So it is the traveling twin's frame of refrence that is correct in this case.

How did I do? ;)

DOH!
I guess I should read more closely. I thought you were asking a question instead of making a statement. I feel silly now.