# Thread: Can General Relativity model causation of time dilation?

1. I know all the logic behind General Relativity quite well. However, what seems to be lacking is the model for causation of time dilation. The generic notion is that if an observer accelerates (change of inertia) his aging slows down relative to anyone he left behind. So now after acceleration (and in constant relative motion) the observer is in reality supposed to be aging slower relative to anyone left behind.

So the question is:

Is there any model for the causation of time dilation or is the causation undetermined?
Essentially, how does acceleration affect the relative time dilation between any two observers?

2.

3. It is just the geometry of space time.

In the same way that distance can make something appear smaller, so movement can make something appear slower. (That is a terrible analogy.)

The other way of thinking about it is that there is a constant amount of movement, when you increase movement through space this reduces the movement through time. This can mathematically be described as a rotation between the space and time directions (which is what the Lorentz transform describes).
Lorentz transformation - Wikipedia, the free encyclopedia

4. Originally Posted by van erst
Is there any model for the causation of time dilation or is the causation undetermined?
Essentially, how does acceleration affect the relative time dilation between any two observers?
In the context of GR, the mechanism behind time dilation is space-time curvature. Remember that both space and time are aspects of just one underlying manifold, space-time. An accelerating observer will, in his own frame, experience "curved time", i.e. a curvature along the time axis of the manifold ( but no tidal forces ). This manifests itself physically as time dilation with respect to a stationary, unaccelerated reference frame.

5. Originally Posted by Markus Hanke
Originally Posted by van erst
Is there any model for the causation of time dilation or is the causation undetermined?
Essentially, how does acceleration affect the relative time dilation between any two observers?
In the context of GR, the mechanism behind time dilation is space-time curvature. Remember that both space and time are aspects of just one underlying manifold, space-time. An accelerating observer will, in his own frame, experience "curved time", i.e. a curvature along the time axis of the manifold ( but no tidal forces ). This manifests itself physically as time dilation with respect to a stationary, unaccelerated reference frame.
Thank you for the quick responses.

I suppose I should have been more specific. Space-time curvature is the model explaining the causation of gravitational time dilation. I am looking for the solution for the case where gravitational acceleration is not a factor. Do you know of any such model?

6. Originally Posted by van erst
I suppose I should have been more specific. Space-time curvature is the model explaining the causation of gravitational time dilation. I am looking for the solution for the case where gravitational acceleration is not a factor. Do you know of any such model?
You mean such as an accelerating rocket far away from any massive bodies ?

7. Originally Posted by van erst
I suppose I should have been more specific. Space-time curvature is the model explaining the causation of gravitational time dilation. I am looking for the solution for the case where gravitational acceleration is not a factor. Do you know of any such model?
Special relativity can deal with this case. Curvature itself is not the cause of time dilation. Gravitational time dilation is ultimately caused by the time dilation associated with an accelerated frame of reference. The time dilation associated with acceleration is geometric in origin. For example, consider a spring. If one side along the length of the spring is compressed relative to the opposite side, the spring will curve.

8. Originally Posted by Markus Hanke
You mean such as an accelerating rocket far away from any massive bodies ?
Exactly. So far away that gravity really is not a real factor for any time dilation there.

9. Originally Posted by KJW
Special relativity can deal with this case. Curvature itself is not the cause of time dilation. Gravitational time dilation is ultimately caused by the time dilation associated with an accelerated frame of reference. The time dilation associated with acceleration is geometric in origin. For example, consider a spring. If one side along the length of the spring is compressed relative to the opposite side, the spring will curve.
Special relativity can not deal with this case of the causation as it actually skips the causation. Acceleration is first required to produce the time dilation. Acceleration is the cause. I am seeking for any model for this causation.

10. Originally Posted by KJW
Curvature itself is not the cause of time dilation. Gravitational time dilation is ultimately caused by the time dilation associated with an accelerated frame of reference. The time dilation associated with acceleration is geometric in origin. For example, consider a spring. If one side along the length of the spring is compressed relative to the opposite side, the spring will curve.
Sorry for being picky here, but that seems like a pretty circular and contradictory statement to me; you are saying :

1. Curvature is not the cause of gravitational time dilation
2. Gravitational time dilation is associated with acceleration
3. Time dilation due to acceleration is geometric in origin
4. It is analogous to a spring curving on one side ( now refer back to (1)... )

Do you see what I mean ?

11. Originally Posted by van erst
Originally Posted by KJW
Special relativity can deal with this case. Curvature itself is not the cause of time dilation. Gravitational time dilation is ultimately caused by the time dilation associated with an accelerated frame of reference. The time dilation associated with acceleration is geometric in origin. For example, consider a spring. If one side along the length of the spring is compressed relative to the opposite side, the spring will curve.
Special relativity can not deal with this case of the causation as it actually skips the causation. Acceleration is first required to produce the time dilation. Acceleration is the cause. I am seeking for any model for this causation.
I'm sorry, but I can't see what exactly you are asking. Acceleration is the cause of the time dilation, and if the spacetime is flat, then special relativity is sufficient. If one needs to consider an accelerated frame of reference, then part of the maths of general relativity will be required, but even this can still be dealt with from special relativistic considerations as the spacetime is flat.

12. Originally Posted by van erst
Exactly. So far away that gravity really is not a real factor for any time dilation there.
Ok. For simplicity let us assume that the acceleration is constant at some factor g. In the immediate neighbourhood of the accelerated observer ( distance of order 1/g ), it is then possible to construct a local coordinate frame of the form

To be more technical, the accelerated observer carries with him a tetrad of four orthonormal vectors, each of which is Fermi-Walker transported along his world line, and one of which is along the 4-velocity direction of the observer. You will find an in-depth treatment of this in Misner/Thorne/Wheeler's Gravitation, chapter 6.

13. Originally Posted by Markus Hanke
Originally Posted by KJW
Curvature itself is not the cause of time dilation. Gravitational time dilation is ultimately caused by the time dilation associated with an accelerated frame of reference. The time dilation associated with acceleration is geometric in origin. For example, consider a spring. If one side along the length of the spring is compressed relative to the opposite side, the spring will curve.
Sorry for being picky here, but that seems like a pretty circular and contradictory statement to me; you are saying :

1. Curvature is not the cause of gravitational time dilation
2. Gravitational time dilation is associated with acceleration
3. Time dilation due to acceleration is geometric in origin
4. It is analogous to a spring curving on one side ( now refer back to (1)... )

Do you see what I mean ?
When I said that "curvature itself is not the cause of time dilation", what I mean is that time dilation is caused by being in an accelerated frame of reference, and this is also true for gravitational time dilation. The curvature itself is irrelevant to cause of the time dilation being acceleration.

14. Sorry, I just realised what the confusion was! I was referring to spacetime curvature as being not the cause of time dilation. The curved spring was referring to the curvature of the trajectory. Sorry for not making this distinction with regards to "curvature".

15. Originally Posted by Markus Hanke
Originally Posted by van erst
Exactly. So far away that gravity really is not a real factor for any time dilation there.
Ok. For simplicity let us assume that the acceleration is constant at some factor g. In the immediate neighbourhood of the accelerated observer ( distance of order 1/g ), it is then possible to construct a local coordinate frame of the form

To be more technical, the accelerated observer carries with him a tetrad of four orthonormal vectors, each of which is Fermi-Walker transported along his world line, and one of which is along the 4-velocity direction of the observer. You will find an in-depth treatment of this in Misner/Thorne/Wheeler's Gravitation, chapter 6.
So would you say that the causation of time dilation (without gravity) is such that in reality acceleration must always lead to a slowing relative rate of aging of the accelerator?

16. Originally Posted by KJW
Sorry, I just realised what the confusion was! I was referring to spacetime curvature as being not the cause of time dilation. The curved spring was referring to the curvature of the trajectory. Sorry for not making this distinction with regards to "curvature".
No problem, I get now what you were trying to say...was confused myself

17. Originally Posted by van erst
So would you say that the causation of time dilation (without gravity) is such that in reality acceleration must always lead to a slowing relative rate of aging of the accelerator?
It is much more accurate to say that the world line of an accelerated observer in space-time ( not space ! ) is curved. That is the true reason why his clock ticks at a different rate compared to an unaccelerated observer ( whose world line is straight ).

18. Originally Posted by Markus Hanke
It is much more accurate to say that the world line of an accelerated observer in space-time ( not space ! ) is curved. That is the true reason why his clock ticks at a different rate compared to an unaccelerated observer ( whose world line is straight ).
Well I am trying to speak in terms of the observable reality, which then is modeled. It seems that in reality, all acceleration can not lead into a relatively slowing rate of aging of the accelerator. Would you agree with that?

19. Originally Posted by van erst
It seems that in reality, all acceleration can not lead into a relatively slowing rate of aging of the accelerator. Would you agree with that?
No. Compared to an unaccelerated observer, the clock of a non-inertially moving observer will always tick a slower rate.

20. Originally Posted by Markus Hanke
Originally Posted by van erst
It seems that in reality, all acceleration can not lead into a relatively slowing rate of aging of the accelerator. Would you agree with that?
No. Compared to an unaccelerated observer, the clock of a non-inertially moving observer will always tick a slower rate.
So you say that in reality all acceleration must always lead to a relatively slowing rate of aging? You can not use acceleration to cause your relative aging to speed up again?

21. Originally Posted by van erst
So you say that in reality all acceleration must always lead to a relatively slowing rate of aging? You can not use acceleration to cause your relative aging to speed up again?
Not relative to an inertial object. This is because the longest timelike trajectory in spacetime is the straight trajectory. In Euclidean geometry the straight line is the shortest but due to the Minkowskian metric of spacetime, it is the longest for timelike trajectories.

22. Originally Posted by van erst
So you say that in reality all acceleration must always lead to a relatively slowing rate of aging? You can not use acceleration to cause your relative aging to speed up again?
The most amount of proper time is always experienced by an unaccelerated reference observer at rest; hence any kind of acceleration will lead to relatively less proper time experienced, compared to said observer at rest. You cannot use acceleration to speed up your aging from rest, no.

23. Originally Posted by Markus Hanke
Originally Posted by van erst
So you say that in reality all acceleration must always lead to a relatively slowing rate of aging? You can not use acceleration to cause your relative aging to speed up again?
The most amount of proper time is always experienced by an unaccelerated reference observer at rest; hence any kind of acceleration will lead to relatively less proper time experienced, compared to said observer at rest. You cannot use acceleration to speed up your aging from rest, no.
Suppose you and I are in some space ships beside each other, so we share an inertial frame. I then accelerate away from you and thus begin to age relatively slower than you. Then later I accelerate towards you until we are relatively stationary. Did not my rate of aging speed up again to match yours?

24. Originally Posted by van erst
Suppose you and I are in some space ships beside each other, so we share an inertial frame. I then accelerate away from you and thus begin to age relatively slower than you. Then later I accelerate towards you until we are relatively stationary. Did not my rate of aging speed up again to match yours?
The rate of aging will once again be same once we are back together in the same frame, but the total amount of time you have aged will remain less than mine. Do you see the distinction ?

25. Originally Posted by Markus Hanke
Originally Posted by van erst
Suppose you and I are in some space ships beside each other, so we share an inertial frame. I then accelerate away from you and thus begin to age relatively slower than you. Then later I accelerate towards you until we are relatively stationary. Did not my rate of aging speed up again to match yours?
The rate of aging will once again be same once we are back together in the same frame, but the total amount of time you have aged will remain less than mine. Do you see the distinction ?
Yes. I see the distinction and agree with you on the total accumulated amount of time.

Do you agree with me that all acceleration can not result in the accelerators aging relatively slowing down?

26. Originally Posted by van erst
Do you agree with me that all acceleration can not result in the accelerators aging relatively slowing down?
No - the accelerated observer's clock is always slowed down as compared to an unaccelerated reference observer, so the total amount of time by which he has aged will always be less. That is what we have just explained.

27. Originally Posted by Markus Hanke
Originally Posted by van erst
Do you agree with me that all acceleration can not result in the accelerators aging relatively slowing down?
No - the accelerated observer's clock is always slowed down as compared to an unaccelerated reference observer, so the total amount of time by which he has aged will always be less. That is what we have just explained.
Suppose there was another ship with me that kept going while I accelerated towards you to again become stationary relative to you. I then accelerated towards you and we are now stationary together separated by some distance.

You seem to indicate that in reality upon the second acceleration towards you:
1. my aging relative to the other ship slowed down
2. my aging relative to you sped up

28. Originally Posted by van erst

Suppose there was another ship with me that kept going while I accelerated towards you to again become stationary relative to you. I then accelerated towards you and we are now stationary together separated by some distance.

You seem to indicate that in reality upon the second acceleration towards you:
1. my aging relative to the other ship slowed down
2. my aging relative to you sped up

If I myself remained stationary and unaccelerated during your entire experiment, then no matter how you did your acceleration, it will always be me who experiences the most proper time, both in relation to you, and in relation to the other accelerating ship.

It is not immediately possible to examine your relationship to the other accelerated ship's clock unless we know the exact trajectories and accelerations involved in both ships.

I'll put it again quite generally for you : the more acceleration you experience, the less you age as compared to an unaccelerated reference point. This is always true, no matter what the exact details of the scenario are. If you want to compare two accelerated observers to one another, you need to know the details of their acceleration and their trajectories.

29. Originally Posted by Markus Hanke
It is not immediately possible to examine your relationship to the other accelerated ship's clock unless we know the exact trajectories and accelerations involved in both ships.
The other ship that was with me continued with the same velocity. So we know all about the accelerations of the ships. When my ship and the other ship were traveling away from you we supposedly were aging slower than you. Then I accelerated towards you leaving the other ship stationary in the frame we shared while traveling away from you.

My aging relative to you must have sped up again as I was aging slower while traveling and then was aging at the same rate with you again.

The question now is:
does someone claim that in reality my aging slowed down relative the other ship I left behind to continue its travels?

This is why I am seeking for a specific model for the causation of non-gravitational time dilation.

30. Originally Posted by van erst
The other ship that was with me continued with the same velocity.
If it continues at the same velocity it becomes an inertial reference frame without acceleration.

does someone claim that in reality my aging slowed down relative the other ship I left behind to continue its travels?
Yes, if you accelerate relative to an inertially moving reference frame such as a ship with constant velocity, you experience less proper time, i.e. age at a lesser rate as compared to that ship.

My aging relative to you must have sped up again as I was aging slower while traveling and then was aging at the same rate with you again
Yes, your rate of aging has increased if you come back into my inertial frame, because you decelerated, i.e. experienced negative acceleration. However, because you experienced acceleration, and I did not, you will still have aged a lesser amount in total.

31. I think the problem concerns the notion of the rate of aging. In relativity, what one does is determine the proper time elapsed between two points in spacetime along a specified trajectory. Because is not an exact differential, the path length (elapsed proper time) will depend on the trajectory.

32. Originally Posted by van erst
I am looking for the solution for the case where gravitational acceleration is not a factor. Do you know of any such model?
Sure, there are many such models. In effect, any situation where the two twins experienced different path through spacetime produces differential aging. The proper time experienced by the twins is given by the line integral

where

where are the components of the metric tensor and represents the path taken by the twins. ANY differences in the path taken by the twins result in differences in the proper time measured by them.As KJW just beat me to it, because above is not an exact differential, the value of the integral is path dependent.

33. Originally Posted by KJW
I think the problem concerns the notion of the rate of aging
I am starting to think that, too. It is important to distinguish between the rate of aging, and the total amount of time aged. The latter is straightforward, the former has the potential for much confusion.

34. Originally Posted by Markus Hanke
Originally Posted by van erst
The other ship that was with me continued with the same velocity.
If it continues at the same velocity it becomes an inertial reference frame without acceleration.

does someone claim that in reality my aging slowed down relative the other ship I left behind to continue its travels?
Yes, if you accelerate relative to an inertially moving reference frame such as a ship with constant velocity, you experience less proper time, i.e. age at a lesser rate as compared to that ship.

My aging relative to you must have sped up again as I was aging slower while traveling and then was aging at the same rate with you again
Yes, your rate of aging has increased if you come back into my inertial frame, because you decelerated, i.e. experienced negative acceleration. However, because you experienced acceleration, and I did not, you will still have aged a lesser amount in total.
Well lets use simple numbers.. Say that the initial and your factor of aging is 1.0. When we first accelerate away from you we reach a steady aging relative to you with a factor of 0.9.

Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again, and thus my aging sped up relative to both you and the other ship I left behind.

You don't agree with this?

35. Originally Posted by van erst

Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again, and thus my aging sped up relative to both you and the other ship I left behind.

You don't agree with this?
Yes, I agree with this, as I already stated in my previous post. Your relative rate of aging has sped up to match mine once again as you return into my frame; what doesn't match though is the total aging times, because you experienced acceleration whereas I haven't.

36. Originally Posted by Markus Hanke
Originally Posted by van erst

Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again, and thus my aging sped up relative to both you and the other ship I left behind.

You don't agree with this?
Yes, I agree with this, as I already stated in my previous post. Your relative rate of aging has sped up to match mine once again as you return into my frame; what doesn't match though is the total aging times, because you experienced acceleration whereas I haven't.
Lets separate my situation with the other ship for inspection. In the above I am first in the same inertial frame (aging rate 0.9). Then I accelerate away from the ship and I begin to age relatively faster (rate 1.0). Now I spend some time traveling away from the ship and then accelerate back to the same inertial frame with it again (rate 0.9).

In this separate scenario I was the one to accelerate away from the inertial frame and I didn't age less but more.

37. Originally Posted by van erst
Originally Posted by Markus Hanke
Originally Posted by van erst
The other ship that was with me continued with the same velocity.
If it continues at the same velocity it becomes an inertial reference frame without acceleration.

does someone claim that in reality my aging slowed down relative the other ship I left behind to continue its travels?
Yes, if you accelerate relative to an inertially moving reference frame such as a ship with constant velocity, you experience less proper time, i.e. age at a lesser rate as compared to that ship.

My aging relative to you must have sped up again as I was aging slower while traveling and then was aging at the same rate with you again
Yes, your rate of aging has increased if you come back into my inertial frame, because you decelerated, i.e. experienced negative acceleration. However, because you experienced acceleration, and I did not, you will still have aged a lesser amount in total.
Well lets use simple numbers.. Say that the initial and your factor of aging is 1.0. When we first accelerate away from you we reach a steady aging relative to you with a factor of 0.9.

Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again, and thus my aging sped up relative to both you and the other ship I left behind.

You don't agree with this?
This is incorrect. Here is the correct statement:

1. Twins A and B in the same frame, identical aging rates.
2. Twin B accelerates away from the initial frame, his aging rate decreases wrt A
3. Twin B turns around and accelerates towards A, his aging rate is still less than A
4. Twin B decelerates and stops next to A, his aging rate is now identical to A's.

38. Originally Posted by van erst
Originally Posted by Markus Hanke
Originally Posted by van erst

Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again, and thus my aging sped up relative to both you and the other ship I left behind.

You don't agree with this?
Yes, I agree with this, as I already stated in my previous post. Your relative rate of aging has sped up to match mine once again as you return into my frame; what doesn't match though is the total aging times, because you experienced acceleration whereas I haven't.
Lets separate my situation with the other ship for inspection. In the above I am first in the same inertial frame (aging rate 0.9). Then I accelerate away from the ship and I begin to age relatively faster (rate 1.0). Now I spend some time traveling away from the ship and then accelerate back to the same inertial frame with it again (rate 0.9).

In this separate scenario I was the one to accelerate away from the inertial frame and I didn't age less but more.
Incorrect. You have some serious misconceptions on the subject.

39. Originally Posted by xyzt
This is incorrect. Here is the correct statement:

1. Twins A and B in the same frame, identical aging rates.
2. Twin B accelerates away from the initial frame, his aging rate decreases wrt A
3. Twin B turns around and accelerates towards A, his aging rate is still less than A
4. Twin B decelerates and stops next to A, his aging rate is now identical to A's.
You changed the experiment to a totally different one. In mine they do not end up next to each other.

40. Originally Posted by van erst
Originally Posted by xyzt
This is incorrect. Here is the correct statement:

1. Twins A and B in the same frame, identical aging rates.
2. Twin B accelerates away from the initial frame, his aging rate decreases wrt A
3. Twin B turns around and accelerates towards A, his aging rate is still less than A
4. Twin B decelerates and stops next to A, his aging rate is now identical to A's.
You changed the experiment to a totally different one. In mine they do not end up next to each other.
It has become clear that you don't know what you are writing:

Originally Posted by van ernst
Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again

41. Originally Posted by van erst
Well lets use simple numbers.. Say that the initial and your factor of aging is 1.0. When we first accelerate away from you we reach a steady aging relative to you with a factor of 0.9.

Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again, and thus my aging sped up relative to both you and the other ship I left behind.

You don't agree with this?
By what means are we to compare the frames of reference? Are we observing each other's clocks and using Doppler shifts? Are we using the time of an inertial frame as a reference for comparison of the moving clocks of each object? Are we constructing a single-parameter family of orthogonal three-dimensional manifolds for each timelike trajectory and observing the intersections with the other trajectories?

42. Originally Posted by xyzt
Originally Posted by van erst
Originally Posted by xyzt
This is incorrect. Here is the correct statement:

1. Twins A and B in the same frame, identical aging rates.
2. Twin B accelerates away from the initial frame, his aging rate decreases wrt A
3. Twin B turns around and accelerates towards A, his aging rate is still less than A
4. Twin B decelerates and stops next to A, his aging rate is now identical to A's.
You changed the experiment to a totally different one. In mine they do not end up next to each other.
It has become clear that you don't know what you are writing:

Originally Posted by van ernst
Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again
Two observers are not required to be next to each other to share an inertial frame.

43. Originally Posted by KJW
Originally Posted by van erst
Well lets use simple numbers.. Say that the initial and your factor of aging is 1.0. When we first accelerate away from you we reach a steady aging relative to you with a factor of 0.9.

Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again, and thus my aging sped up relative to both you and the other ship I left behind.

You don't agree with this?
By what means are we to compare the frames of reference? Are we observing each other's clocks and using Doppler shifts? Are we using the time of an inertial frame as a reference for comparison of the moving clocks of each object. Are we constructing a single-parameter family of orthogonal three-dimensional manifolds for each timelike trajectory and observing the intersections with the other trajectories?
We can discuss in terms of what GR dictates must occur in reality.

44. Originally Posted by van erst
Originally Posted by KJW
By what means are we to compare the frames of reference? Are we observing each other's clocks and using Doppler shifts? Are we using the time of an inertial frame as a reference for comparison of the moving clocks of each object? Are we constructing a single-parameter family of orthogonal three-dimensional manifolds for each timelike trajectory and observing the intersections with the other trajectories?
We can discuss in terms of what GR dictates must occur in reality.
GR doesn't dictate any particular choice because it is an artificial choice. The notion of rates of aging is an artificial notion because it demands an artificial choice as to how to compare the frames of reference. The reality that GR does provide is the distance (proper time) between two points in spacetime along a specified trajectory.

45. Originally Posted by KJW
Originally Posted by van erst
We can discuss in terms of what GR dictates must occur in reality.
GR doesn't dictate any particular choice because it is an artificial choice. The notion of rates of aging is an artificial notion because it demands an artificial choice as to how to compare the frames of reference. The reality that GR does provide is the distance (proper time) between two points in spacetime along a specified trajectory.
The key is relative rates of aging for which the causation with respect to acceleration should be covered by GR. That is not an artificial notion.

46. Originally Posted by van erst
Lets separate my situation with the other ship for inspection. In the above I am first in the same inertial frame (aging rate 0.9).
If you are in the same inertial frame, you age at the same rate. Your clocks will agree, so your rate of ageing is 1.0.

Then I accelerate away from the ship and I begin to age relatively faster (rate 1.0).
You are not making sense now. If you accelerate away, you age relatively slower, because you experience acceleration, whereas the reference frame doesn't. Your rate of ageing is thus less than 1.

Now I spend some time traveling away from the ship and then accelerate back to the same inertial frame with it again (rate 0.9).
So your ageing rate increases back to 1, and your clocks will once again tick at the same rate. However, your world lines between events (1) and (2) will be of different length; the inertial frame's word line is of length

in his own frame, whereas the accelerated observer's world line is of length ( all other things being equal )

Hence it is obvious that

Hence the inertial observer always experiences the longest world line ( proper time ) through two given events (1) and (2) as compared to accelerated observers. The specifics of the accelerated observer's rate of acceleration and trajectory are not important for this general statement.

47. Originally Posted by van erst
The key is relative rates of aging for which the causation with respect to acceleration should be covered by GR. That is not an artificial notion.
The choice of coordinates is artificial; what is not artificial is the proper time experienced by the observers, since that is an invariant, and given by

in the most general case. However, if there is no gravity, then the only component of the metric tensor which differs between the accelerated and unaccelerated cases is , as demonstrated in the previous post. Note though that both space-times are flat, since the Riemann curvature tensor vanishes in both cases, so the accelerated case can be treated just fine under Special Relativity. The metric I gave for the accelerated observer follows via a simple coordinate transformation from the standard Minkowski metric.

The key is relative rates of ageing
That relative rate is just the ratio between proper time and coordinate time, which is

as seen from your inertial frame, so it depends on the acceleration.

48. Originally Posted by Markus Hanke
Originally Posted by van erst
Lets separate my situation with the other ship for inspection. In the above I am first in the same inertial frame (aging rate 0.9).
If you are in the same inertial frame, you age at the same rate. Your clocks will agree, so your rate of ageing is 1.0.

Then I accelerate away from the ship and I begin to age relatively faster (rate 1.0).
You are not making sense now. If you accelerate away, you age relatively slower, because you experience acceleration, whereas the reference frame doesn't. Your rate of ageing is thus less than 1.
Well lets expand this back to the whole scenario to try to make some sense into what I was saying

1. all three (you, my ship, the other ship) are in the same inertial frame right next to each other (relative aging factor 1.0)
2. my ship and the other ship accelerate away from you (our relative aging slows down, reaching aging factor 0.9 relative to your 1.0)
3. I accelerate back towards you (my aging speeds up, again factor 1.0)
4. I cruise along some time away from the other ship, aging faster than the ship
5. After some time I accelerate back towards the other ship (my aging factor relative to your 1.0 reference is 0.9 again)

In the above from 3. to 5. I accelerate away from the inertial frame shared with the other ship. When I return to the same inertial frame with the ship I will have aged more.

You don't agree with this?

49. Originally Posted by van erst
The key is relative rates of aging for which the causation with respect to acceleration should be covered by GR. That is not an artificial notion.
The point you are missing is that in order to have a relative notion of anything, you have to have a means to compare them. For rates of aging, this is not provided by the reality itself, and has to be artificially constructed. This is not a flaw of general relativity, but a misunderstanding of it.

50. Originally Posted by van erst
When I return to the same inertial frame with the ship I will have aged more.
Whether you will have aged more or less than the other ship depends on your trajectories, but you will have aged differently, because your world line is of a different length than the other ship ( the acceleration profile is not the same ), which is in turn of a different length than both yours and my own. So what you have demonstrated here is only that these three observers will each record a different proper time. Which is precisely what I am saying to you all along.

Can you - instead of getting lost in specific scenarios - perhaps just simply state what your point is; do you have a specific question with regards to GR or SR, or is this a case where you are trying hard to find some apparent paradox, or what ?

51. If you want to stick to Special Relativity, then perhaps rather than considering a "rate of ageing", a different approach to this problem would be to think in terms of the relativity of simultaneity.

52. Originally Posted by KJW
Originally Posted by van erst
The key is relative rates of aging for which the causation with respect to acceleration should be covered by GR. That is not an artificial notion.
The point you are missing is that in order to have a relative notion of anything, you have to have a means to compare them. For rates of aging, this is not provided by the reality itself, and has to be artificially constructed. This is not a flaw of general relativity, but a misunderstanding of it.
To expand on this point, to compare rates of aging, you have to compare an interval of time of one object to an interval of time of another object. The question is: Which intervals of time for each object are we comparing?

53. Originally Posted by van erst
Originally Posted by xyzt
Originally Posted by van erst
Originally Posted by xyzt
This is incorrect. Here is the correct statement:

1. Twins A and B in the same frame, identical aging rates.
2. Twin B accelerates away from the initial frame, his aging rate decreases wrt A
3. Twin B turns around and accelerates towards A, his aging rate is still less than A
4. Twin B decelerates and stops next to A, his aging rate is now identical to A's.
You changed the experiment to a totally different one. In mine they do not end up next to each other.
It has become clear that you don't know what you are writing:

Originally Posted by van ernst
Now when I accelerate towards you and reach the same inertial frame my factor is 1.0 again
Two observers are not required to be next to each other to share an inertial frame.
While this is true, the contradiction between your two sentences stands. You obviously don't know what you are talking about.

54. Originally Posted by van erst
Originally Posted by Markus Hanke
Originally Posted by van erst
So you say that in reality all acceleration must always lead to a relatively slowing rate of aging? You can not use acceleration to cause your relative aging to speed up again?
The most amount of proper time is always experienced by an unaccelerated reference observer at rest; hence any kind of acceleration will lead to relatively less proper time experienced, compared to said observer at rest. You cannot use acceleration to speed up your aging from rest, no.
Suppose you and I are in some space ships beside each other, so we share an inertial frame. I then accelerate away from you and thus begin to age relatively slower than you. Then later I accelerate towards you until we are relatively stationary. Did not my rate of aging speed up again to match yours?
No, it didn't. The aging rate depends on the absolute value of acceleration, you are under the misconception that it depends on its sense (away vs. towards). That is wrong.

55. Originally Posted by KJW
Originally Posted by van erst
The key is relative rates of aging for which the causation with respect to acceleration should be covered by GR. That is not an artificial notion.
The point you are missing is that in order to have a relative notion of anything, you have to have a means to compare them. For rates of aging, this is not provided by the reality itself, and has to be artificially constructed. This is not a flaw of general relativity, but a misunderstanding of it.
For thought experiments you don't have to produce any comparisons. What the model dictates has to be logically solid.

If relativity dictates something to occur, it is irrelevant if that can or can not be compared. It still has to pass logical tests.

56. Originally Posted by van erst
If relativity dictates something to occur, it is irrelevant if that can or can not be compared. It still has to pass logical tests.
Relativity dictates that the world lines of two observers through the same two events (1) and (2) are of different length if their acceleration profiles differ. That is all there is to it, and that is all that we have tried to explain to you, with both words and maths.
Now I am asking you again - what is your point in all of this ? Are you trying to find logical contradictions ? If that is the case then you are wasting your time, because it can be shown that Special Relativity ( which this falls under ) is internally self-consistent, and that it hence is impossible to arrive at any contradictions. Here is the formal proof, which I did some time ago for reference :

General Proof that Special Relativity is Self-Consistent

This works for all metric tensors which can be obtained from the Minkowski metric via coordinate transformation.

It still has to pass logical tests.
The above is a general argument which rules out all contradictions derived from SR's basic axioms. That is the strongest and most general logical test you can possibly do.

57. Originally Posted by Markus Hanke
Originally Posted by van erst
When I return to the same inertial frame with the ship I will have aged more.
Whether you will have aged more or less than the other ship depends on your trajectories, but you will have aged differently, because your world line is of a different length than the other ship ( the acceleration profile is not the same ), which is in turn of a different length than both yours and my own. So what you have demonstrated here is only that these three observers will each record a different proper time. Which is precisely what I am saying to you all along.
We can clarify the situation so that all relative acceleration between the two ships and you occurs along a single straight line.

So do you agree that in the above example from 3. to 5. I will have aged more than the ship even though I was the one to accelerate away from the ship?

Originally Posted by Markus Hanke
Can you - instead of getting lost in specific scenarios - perhaps just simply state what your point is; do you have a specific question with regards to GR or SR, or is this a case where you are trying hard to find some apparent paradox, or what ?
As said, I want to find a proper description of a model for time dilation causation within relativity. A description for how acceleration is actually supposed to cause relative time dilation between any two observers.

58. Hello everyone,

I wanted to ask a question to make sure I understand the point Markus, KJW, and xyzt are making here:

Is it true that the rest frame represents the "fastest" possible rate of aging, and that there is no way "speed up" this rate by manipulating accelerated reference frames (i.e. spaceships)?

Perhaps the confusion is that while the clock in a decelerating frame may speed up to match the rest frame, the previously compressed time is not changed by deceleration. I'm sorry if this is poorly worded or just plain wrong! I'm studying t'Hooft's Theoretical Physics curriculum, but of course I have years to go.

59. Originally Posted by van erst
I will have aged more than the ship even though I was the one to accelerate away from the ship?
I have already explained this to you - if you compare two non-inertial frames, than the outcomes depends on the explicit acceleration profiles. You cannot say whose world line is shorter unless you know the specific form of their metric tensors; in either case there is no contradiction, since it is always the inertial observer who records the longest proper time, and the non-inertial observer's world lines are of different length.

A description for how acceleration is actually supposed to cause relative time dilation between any two observers.
That has already been answered long ago - the proper time of an observer is the length of his world line between two given events, which is invariant; this length depends on his acceleration profile, hence observers with different accelerations will record different proper times. To put it even simpler - an accelerated observer's world line is curved in space-time. It is that simple. What else do you want to know ?

60. Originally Posted by RobinM
Is it true that the rest frame represents the "fastest" possible rate of aging, and that there is no way "speed up" this rate by manipulating accelerated reference frames (i.e. spaceships)?
Yes, that's the basic idea - an unaccelerated inertial observer always records the longest proper time, i.e. the most "ageing".

Perhaps the confusion is that while the clock in a decelerating frame may speed up to match the rest frame, the previously compressed time is not changed by deceleration.
I think the confusion is because the OP is working with rates of change ( which are observer dependent ), instead of simply considering the total length of the world lines ( which are invariant and not observer dependent ).

61. Originally Posted by Markus Hanke
Originally Posted by van erst
I will have aged more than the ship even though I was the one to accelerate away from the ship?
I have already explained this to you - if you compare two non-inertial frames, than the outcomes depends on the explicit acceleration profiles. You cannot say whose world line is shorter unless you know the specific form of their metric tensors; in either case there is no contradiction, since it is always the inertial observer who records the longest proper time, and the non-inertial observer's world lines are of different length.

A description for how acceleration is actually supposed to cause relative time dilation between any two observers.
That has already been answered long ago - the proper time of an observer is the length of his world line between two given events, which is invariant; this length depends on his acceleration profile, hence observers with different accelerations will record different proper times. To put it even simpler - an accelerated observer's world line is curved in space-time. It is that simple. What else do you want to know ?
I am seeking for the link between the theory and actual predicted measurements. In reality we can not observe "world lines". We can only observe that during the experiment I must have recorded a specific amount of accumulated time between 3. and 5. The other ship must have done the same.

My question for this situation in a very specific form is:
Can relativity produce a prediction for which recorded accumulated time from 3. to 5. is greater, mine or the other ships?

62. Originally Posted by KJW
To expand on this point, to compare rates of aging, you have to compare an interval of time of one object to an interval of time of another object. The question is: Which intervals of time for each object are we comparing?
The most reasonable way to select the time-intervals for each object being compared is to base it on a rule. The problem is that the application of the rule by each object may not (and in general won't) lead to the same time intervals being compared. That is, there won't be a symmetrical relationship between the objects. For example, observing each other's clocks will not lead to the same pair of time intervals being compared because the observed clock will always be in the past relative to the observer's clock.

63. Originally Posted by van erst
In reality we can not observe "world lines". We can only observe that during the experiment I must have recorded a specific amount of accumulated time between 3. and 5.
The amount of accumulated time is the length of the world line, so yes, we do observe world lines. I really don't know what else you want.

Can relativity produce a prediction which recorded accumulated time from 3. to 5. is greater, mine or the other ships?
If by "mine" you mean the inertial observer, than his proper time will always be the longest. If "mine" means one of the ships, then the outcome depends on the exact acceleration profiles of both ships ( which are absolute and measurable, and don't depend on the inertial frame ). In either case, the answer is yes - relativity gives exact and specific predictions for exact and specific scenarios; it does so via the line integral I have written down earlier.

So one more time : inertial clocks record the longest proper time. Accelerated clocks record a shorter proper time, the amount of which explicitly depends on the acceleration profile. This is just precisely the length of the clocks' world lines. So the link you are searching for is the length of the world line, which is equal to what a clock that travels between two events physically records. Time dilation is just the ratio between the length of two such world lines, which is determined by their geometries.

64. Originally Posted by RobinM
Is it true that the rest frame represents the "fastest" possible rate of aging, and that there is no way "speed up" this rate by manipulating accelerated reference frames (i.e. spaceships)?
Essentially correct. That is, every non-inertial object will experience less time than the inertial object between the same two points in spacetime (clocks separating from the same location and meeting again at some time in the future).

65. Originally Posted by Markus Hanke
Originally Posted by van erst
In reality we can not observe "world lines". We can only observe that during the experiment I must have recorded a specific amount of accumulated time between 3. and 5.
The amount of accumulated time is the length of the world line, so yes, we do observe world lines. I really don't know what else you want.
Isn't time just one axis in the representation of the world line? The length of the line would include spatial position properties.

Originally Posted by Markus Hanke
Can relativity produce a prediction which recorded accumulated time from 3. to 5. is greater, mine or the other ships?
If by "mine" you mean the inertial observer, than his proper time will always be the longest. If "mine" means one of the ships, then the outcome depends on the exact acceleration profiles of both ships ( which are absolute and measurable, and don't depend on the inertial frame ). In either case, the answer is yes - relativity gives exact and specific predictions for exact and specific scenarios; it does so via the line integral I have written down earlier.

So one more time : inertial clocks record the longest proper time. Accelerated clocks record a shorter proper time, the amount of which explicitly depends on the acceleration profile. This is just precisely the length of the clocks' world lines. So the link you are searching for is the length of the world line, which is equal to what a clock that travels between two events physically records. Time dilation is just the ratio between the length of two such world lines, which is determined by their geometries.
Just to clarify: the participants were you (as the non-accelerating stationary reference), I (with my ship) and the other ship.

If I understand correctly we earlier agreed that when I accelerate back towards you GR predicts that my relative aging will speed up.

This would mean that my accelerated clock is ticking relatively faster than the clock of the other ship. Here you seem to disagree if I understand you correctly (?)

Just for easier reference:
1. all three (you, my ship, the other ship) are in the same inertial frame right next to each other (relative aging factor 1.0)
2. my ship and the other ship accelerate away from you (our relative aging slows down, reaching aging factor 0.9 relative to your 1.0)
3. I accelerate back towards you (my aging speeds up, again factor 1.0)
4. I cruise along some time away from the other ship, aging faster than the ship
5. After some time I accelerate back towards the other ship (my aging factor relative to your 1.0 reference is 0.9 again)

66. Originally Posted by van erst
A description for how acceleration is actually supposed to cause relative time dilation between any two observers.
There are two aspects to consider: Firstly, there is the time dilation associated with the speed of the object. This is sufficient to determine the twin-clock problem. Secondly, there is the position-dependent time dilation associated with an accelerated frame of reference. This occurs because of the change in velocity over the time it takes for light from an object to reach the observer. In this case, objects behind the direction of acceleration appear redshifted while objects in front of the direction of acceleration appear blueshifted. This is the time dilation that the equivalence principle applies to gravitation.

67. Originally Posted by KJW
Originally Posted by van erst
A description for how acceleration is actually supposed to cause relative time dilation between any two observers.
There are two aspects to consider: Firstly, there is the time dilation associated with the speed of the object. This is sufficient to determine the twin-clock problem. Secondly, there is the position-dependent time dilation associated with an accelerated frame of reference. This occurs because of the change in velocity over the time it takes for light from an object to reach the observer. In this case, objects behind the direction of acceleration appear redshifted while objects in front of the direction of acceleration appear blueshifted. This is the time dilation that the equivalence principle applies to gravitation.
Yes, you are absolutely right, here is the math associated with your words:

For two observers, situated at radial values and the above gives the ratios of proper times:

, i=1,2

i.e.

68. Originally Posted by van erst
Originally Posted by KJW
The point you are missing is that in order to have a relative notion of anything, you have to have a means to compare them. For rates of aging, this is not provided by the reality itself, and has to be artificially constructed. This is not a flaw of general relativity, but a misunderstanding of it.
For thought experiments you don't have to produce any comparisons. What the model dictates has to be logically solid.

If relativity dictates something to occur, it is irrelevant if that can or can not be compared. It still has to pass logical tests.
But the requirement that relative quantities be able to be compared is a fundamental part of the logic of relativity. You can't just dismiss it because it's a "thought experiment".

69. Originally Posted by van erst
Originally Posted by Markus Hanke
Originally Posted by van erst
Lets separate my situation with the other ship for inspection. In the above I am first in the same inertial frame (aging rate 0.9).
If you are in the same inertial frame, you age at the same rate. Your clocks will agree, so your rate of ageing is 1.0.

Then I accelerate away from the ship and I begin to age relatively faster (rate 1.0).
You are not making sense now. If you accelerate away, you age relatively slower, because you experience acceleration, whereas the reference frame doesn't. Your rate of ageing is thus less than 1.
Well lets expand this back to the whole scenario to try to make some sense into what I was saying

1. all three (you, my ship, the other ship) are in the same inertial frame right next to each other (relative aging factor 1.0)
correct

2. my ship and the other ship accelerate away from you (our relative aging slows down, reaching aging factor 0.9 relative to your 1.0)
correct

3. I accelerate back towards you (my aging speeds up, again factor 1.0)
Nonsense, aging is not a function of the sense of the acceleration, you have been told this before.

70. If I understand correctly we earlier agreed that when I accelerate back towards you GR predicts that my relative aging will speed up
At no point in your accelerated journey will you ever age faster than the inertial observer, regardless of your direction. As long as you are undergoing acceleration, your rate of time will be slower than that of an observer not undergoing acceleration. If you are de-accelerating with respect to the inertial observer, your rate of aging will approach his, but until you match frameworks and you cease to accelerate, your rate of time will always be slower than his.

71. Originally Posted by van erst

1. all three (you, my ship, the other ship) are in the same inertial frame right next to each other (relative aging factor 1.0)
2. my ship and the other ship accelerate away from you (our relative aging slows down, reaching aging factor 0.9 relative to your 1.0)
3. I accelerate back towards you (my aging speeds up, again factor 1.0)
4. I cruise along some time away from the other ship, aging faster than the ship
5. After some time I accelerate back towards the other ship (my aging factor relative to your 1.0 reference is 0.9 again)
All true when considered from the inertial frame of the first ship (the one that does not accelerate).

However, let's consider things from another inertial frame, that of the Third ship after it accelerates. For that, let's consider a ship that is already traveling in this frame before any of the other ships do any accelerating.

In this frame,
1. All three of the first group of ships are aging at 0.9 relative to the fourth ship.
2. Your ship and the other ship accelerate to match speeds with him and now age at the same rate as he does, while the first ship remains aging at 0.9
3. Your ship accelerates to match up with the first ship's speed, and now ages at a rate of 0.9 again.
4. You cruise along some time aging slower than the other ship(the ship you originally accelerated with).
5. After some time, you accelerate back to the speed of the other ship and your time rate goes back to 1.

So after step 5, The ship that has aged the least is the first ship, followed by your ship, and then the other ship.

In your example, the aging is reversed, the other ship ages the least and the first ship the most.

The point is, that since the ships never reunite, there is no absolute answer as to which aged more than the other.

Could even consider a third inertial frame; one where after step 1, Your ship, the other ship and the first trip are all moving at the same speed (two ships in one direction and one in the other.)

In this frame at each stage as listed above:

1. All three ships age at a rate of 0.779
2. All three ships age at a rate of 0.779
3. All three ships age at a rate of 0.779
4. All three ships age at a rate of 0.779
5. All three ships age at a rate of 0.779

If we assume that the acceleration phases are negligible in duration, then after stage five, all ships will have aged the same amount in this frame.

72. Originally Posted by KJW
Originally Posted by van erst
A description for how acceleration is actually supposed to cause relative time dilation between any two observers.
There are two aspects to consider: Firstly, there is the time dilation associated with the speed of the object. This is sufficient to determine the twin-clock problem. Secondly, there is the position-dependent time dilation associated with an accelerated frame of reference. This occurs because of the change in velocity over the time it takes for light from an object to reach the observer. In this case, objects behind the direction of acceleration appear redshifted while objects in front of the direction of acceleration appear blueshifted. This is the time dilation that the equivalence principle applies to gravitation.
I am trying to tackle the causation of relative time dilation as modeled within GR. Without acceleration no relative time dilation (nor velocity) can exist between two observers. Thus acceleration initiates the causation of relative time dilation.

1. acceleration leads to relative velocity
2. acceleration leads to relative time dilation

I would assume everyone agrees with the above (?).

73. Originally Posted by xyzt
2. my ship and the other ship accelerate away from you (our relative aging slows down, reaching aging factor 0.9 relative to your 1.0)
correct

3. I accelerate back towards you (my aging speeds up, again factor 1.0)
Nonsense, aging is not a function of the sense of the acceleration, you have been told this before.
In 2. my relative aging supposedly slowed down when I accelerated away from the stationary observer.

Then I accelerated again towards him to become stationary relative to him again, only separated by some distance. Are you saying that I am stationary relative to him but our rates of aging are different (while in the same inertial frame)?

74. Originally Posted by van erst
In 2. my relative aging supposedly slowed down when I accelerated away from the stationary observer.

Then I accelerated again towards him to become stationary relative to him again, only separated by some distance. Are you saying that I am stationary relative to him but our rates of aging are different (while in the same inertial frame)?
In your 2 your rate of ageing is slowed because you are accelerating (away, but that is irrelevant).

Between 2 and 3 you will be stationary for some time. At that point, your rate of ageing will be the same (1.0) as the stationary observer (but you will have experienced less total time).

In 3 your rate of aging is slowed because you are accelerating (back, but that is irrelevant).

Then you reach the same inertial frame of reference as the stationary observer. At that point, your rate of ageing will be the same (1.0) as the stationary observer (but you will have experienced less total time).

Does that make more sense?

75. Originally Posted by Strange
Originally Posted by van erst
In 2. my relative aging supposedly slowed down when I accelerated away from the stationary observer.

Then I accelerated again towards him to become stationary relative to him again, only separated by some distance. Are you saying that I am stationary relative to him but our rates of aging are different (while in the same inertial frame)?
In your 2 your rate of ageing is slowed because you are accelerating (away, but that is irrelevant).

Between 2 and 3 you will be stationary for some time. At that point, your rate of ageing will be the same (1.0) as the stationary observer (but you will have experienced less total time).

In 3 your rate of aging is slowed because you are accelerating (back, but that is irrelevant).

Then you reach the same inertial frame of reference as the stationary observer. At that point, your rate of ageing will be the same (1.0) as the stationary observer (but you will have experienced less total time).

Does that make more sense?
I didn't quite follow you there. After first acceleration there are then two remaining reference frames and two "stationary observers" between who's inertial frames I am jumping back and forth. These two inertial frames are the very initial inertial frame and the inertial frame in which the other ship is left traveling constantly.

While the reference time dilation factor for the very first frame was 1.0, the other ship is left in the frame where the factor is 0.9. These factors now remain constant and unchanging. I am then just hopping back and forth between these inertial frames by accelerating to opposite directions. So essentially my relative aging factor must oscillate between 1.0 and 0.9.

76. Originally Posted by van erst
Originally Posted by KJW
There are two aspects to consider: Firstly, there is the time dilation associated with the speed of the object. This is sufficient to determine the twin-clock problem. Secondly, there is the position-dependent time dilation associated with an accelerated frame of reference. This occurs because of the change in velocity over the time it takes for light from an object to reach the observer. In this case, objects behind the direction of acceleration appear redshifted while objects in front of the direction of acceleration appear blueshifted. This is the time dilation that the equivalence principle applies to gravitation.
I am trying to tackle the causation of relative time dilation as modeled within GR. Without acceleration no relative time dilation (nor velocity) can exist between two observers. Thus acceleration initiates the causation of relative time dilation.

1. acceleration leads to relative velocity
2. acceleration leads to relative time dilation

I would assume everyone agrees with the above (?).
No.

As far as point 1 is concerned, why are you assuming that all observers start out from the same inertial frame of reference?

As far as point 2 is concerned, all time dilation is ultimately based on relative velocity. In the case of acceleration and gravitation, the time dilation effect specific to acceleration (and gravitation by virtue of the equivalence principle) is the result of the relative velocity between the observer and the same observer at a different time. In the case of gravitation, spacetime curvature does make the notion of acceleration somewhat counterintuitive in that even though a person at rest on the ground is not moving relative to the earth, they are being accelerated upward at . Nevertheless, that acceleration is still equivalent to what is normally regarded as acceleration.

77. Originally Posted by KJW
Originally Posted by van erst
I am trying to tackle the causation of relative time dilation as modeled within GR. Without acceleration no relative time dilation (nor velocity) can exist between two observers. Thus acceleration initiates the causation of relative time dilation.

1. acceleration leads to relative velocity
2. acceleration leads to relative time dilation

I would assume everyone agrees with the above (?).
No.

As far as point 1 is concerned, why are you assuming that all observers start out from the same inertial frame of reference?

As far as point 2 is concerned, all time dilation is ultimately based on relative velocity. In the case of acceleration and gravitation, the time dilation effect specific to acceleration (and gravitation by virtue of the equivalence principle) is the result of the relative velocity between the observer and the same observer at a different time. In the case of gravitation, spacetime curvature does make the notion of acceleration somewhat counterintuitive in that even though a person at rest on the ground is not moving relative to the earth, they are being accelerated upward at . Nevertheless, that acceleration is still equivalent to what is normally regarded as acceleration.
What I meant was that if any two observers are in relative motion, that motion can only have been caused by acceleration. Same applies to time dilation.

Relative velocity can only be achieved via acceleration. As dictated by GR it is acceleration that produces changes in relative time dilation between observers.

78. Originally Posted by van erst

I didn't quite follow you there. After first acceleration there are then two remaining reference frames and two "stationary observers" between who's inertial frames I am jumping back and forth. These two inertial frames are the very initial inertial frame and the inertial frame in which the other ship is left traveling constantly.

While the reference time dilation factor for the very first frame was 1.0, the other ship is left in the frame where the factor is 0.9. These factors now remain constant and unchanging. I am then just hopping back and forth between these inertial frames by accelerating to opposite directions. So essentially my relative aging factor must oscillate between 1.0 and 0.9.
If you are measuring the time dilation in a single frame, then as you change velocities with respect to that frame your relative aging factor oscillates between 1 and 0.9, as measured in that frame compared to its time rate.

However, if you are hopping back and forth between frames( your "viewpoint" from which time dilation is measured stays with you, then the other ship's relative aging factor is the one that oscillates between 0.9 and 1.

In other words, if you have accelerated away from me and are now at rest in a different rest frame, according to me, you are aging slower, but according to you, I am aging slower.

79. Originally Posted by Janus
In other words, if you have accelerated away from me and are now at rest in a different rest frame, according to me, you are aging slower, but according to you, I am aging slower.
That is only an illusion so why even talk of such when discussing natural science. Both aging slower than the other is not possible. This is why I am only speaking of what GR predicts must occur in reality.

80. Originally Posted by van erst
What I meant was that if any two observers are in relative motion, that motion can only have been caused by acceleration.
Why?

81. Originally Posted by van erst
As dictated by GR it is acceleration that produces changes in relative time dilation between observers.
I disagree, so let me ask you this: How does acceleration produce time dilation?

82. Originally Posted by van erst
Originally Posted by Janus
In other words, if you have accelerated away from me and are now at rest in a different rest frame, according to me, you are aging slower, but according to you, I am aging slower.
That is only an illusion so why even talk of such when discussing natural science. Both aging slower than the other is not possible. This is why I am only speaking of what GR predicts must occur in reality.
It's no more of a illusion than the fact that I could be standing to your left while you were standing to my left. This only requires us to be facing in different directions. With Relativity the measurement of time is frame dependent just like "left" is dependent on which way you are facing. There is no "absolute time". Without which, the term "both aging slower than the other" has no meaning.

One of the cornerstones of Relativity is that there is no preferred inertial frame, or frame of "absolute rest". Thus any inertial frame can lay equal claim to being "at rest". Thus for any frame, it is always the "other" frame that is moving and undergoing time dilation.

If you try to claim otherwise, that for instance, frame 1 runs at a 0.9 time rate compared to frame 2 according to both frames, then you are introducing an absolute preferred frame. Either frame 2 is this preferred frame, or it is moving slower with respect to it than frame 1. Since Relativity denies such a frame's existence, you are not discussing Relativity but denying it.

83. Originally Posted by van erst
Originally Posted by xyzt
2. my ship and the other ship accelerate away from you (our relative aging slows down, reaching aging factor 0.9 relative to your 1.0)
correct

3. I accelerate back towards you (my aging speeds up, again factor 1.0)
Nonsense, aging is not a function of the sense of the acceleration, you have been told this before.

Then I accelerated again towards him to become stationary relative to him again, only separated by some distance. Are you saying that I am stationary relative to him but our rates of aging are different (while in the same inertial frame)?
No. when you become stationary your aging rates are the same. What you said wrong was the part about accelerating towards the stationary twin while having the same aging factor 1.0. Look, you clearly don't understand these things but I think I am not the only one who gets the impression that you aren't interested in learning, you are interested in "challenging the mainstream dogma".

84. Originally Posted by van erst
Originally Posted by KJW
Originally Posted by van erst
A description for how acceleration is actually supposed to cause relative time dilation between any two observers.
There are two aspects to consider: Firstly, there is the time dilation associated with the speed of the object. This is sufficient to determine the twin-clock problem. Secondly, there is the position-dependent time dilation associated with an accelerated frame of reference. This occurs because of the change in velocity over the time it takes for light from an object to reach the observer. In this case, objects behind the direction of acceleration appear redshifted while objects in front of the direction of acceleration appear blueshifted. This is the time dilation that the equivalence principle applies to gravitation.
I am trying to tackle the causation of relative time dilation as modeled within GR. Without acceleration no relative time dilation (nor velocity) can exist between two observers. Thus acceleration initiates the causation of relative time dilation.

1. acceleration leads to relative velocity
2. acceleration leads to relative time dilation

I would assume everyone agrees with the above (?).

1. acceleration leads to relative velocity
2. relative velocity (and differences in the gravitational potential) lead to relative time dilation

85. Originally Posted by van erst
Isn't time just one axis in the representation of the world line? The length of the line would include spatial position properties.
Yes, of course, as you can see in the line integrals I gave earlier. The important thing to remember here is that we connect the same two events (1) and (2) in 4-dimensional space-time with different world lines, and then compare the length ( = proper recorded time ) along those world lines. In other words - all observers must start in the same frame of reference at the beginning of the experiment, and reunite in a common frame of reference at the end of the experiment, or else comparison of proper times is meaningless.

The other thing to remember is that we are dealing with Minkowski space-time, not Euclidean space, so the time coordinate has a minus sign in front of it, as you may have noticed in the path integral. This gives us a geometry which is quite different from what we are used to in our everyday "Euclidean" world.

Now look at the line integrals again, and consider this - if you choose a world line which has a spatial component ( i.e. you choose to move in space when going between events ), you can do so only by introducing acceleration ( i.e. change of speed and direction of travel ); this reduces the total length of the word line, because the time coefficient has a minus sign. The further you move, the more acceleration is required, so the shorter the world line gets, and the more time dilation you experience. The one option that stands is the one of not moving at all - the spatial component vanishes, and the coefficient of the time coordinate becomes exactly 1. It can be shown formally that this represents the maximum length one can achieve for a world line between (1) and (2); in other words, if you stand perfectly still you travel only through time between the events, and the geometry of Minkowski space-time dictates that this is the option with the longest proper time experienced. Anything else you do results in a shorter world line, purely due to space-time geometry, and hence in time dilation.

Everything here boils down to simple geometry; that is why I don't understand your insistence on analysing specific and increasingly complicated scenarios involving "rates of ageing" and such like, when all you really need to consider is simple world lines in Minkowski space-time. I think you are just confusing yourself by making things unnecessarily complicated.

86. Originally Posted by KJW
Originally Posted by van erst
What I meant was that if any two observers are in relative motion, that motion can only have been caused by acceleration.
Why?
How do you mean?

Suppose you and I are stationary right next to each other. You want there to be relative motion between us. Can you achieve that without involving acceleration?

Originally Posted by KJW
Originally Posted by van erst
As dictated by GR it is acceleration that produces changes in relative time dilation between observers.
I disagree, so let me ask you this: How does acceleration produce time dilation?
You disagree that General Relativity dictates that it is acceleration that produces time dilation?

As to how the Universe works by nature we of course can not observe and thus know. What is known is that there is no relative velocities without acceleration.

Or can you produce relative motion between two stationary objects without accelerating one or both of them?

87. Originally Posted by Janus
Originally Posted by van erst
Originally Posted by Janus
In other words, if you have accelerated away from me and are now at rest in a different rest frame, according to me, you are aging slower, but according to you, I am aging slower.
That is only an illusion so why even talk of such when discussing natural science. Both aging slower than the other is not possible. This is why I am only speaking of what GR predicts must occur in reality.
It's no more of a illusion than the fact that I could be standing to your left while you were standing to my left. This only requires us to be facing in different directions. With Relativity the measurement of time is frame dependent just like "left" is dependent on which way you are facing. There is no "absolute time". Without which, the term "both aging slower than the other" has no meaning.

One of the cornerstones of Relativity is that there is no preferred inertial frame, or frame of "absolute rest". Thus any inertial frame can lay equal claim to being "at rest". Thus for any frame, it is always the "other" frame that is moving and undergoing time dilation.

If you try to claim otherwise, that for instance, frame 1 runs at a 0.9 time rate compared to frame 2 according to both frames, then you are introducing an absolute preferred frame. Either frame 2 is this preferred frame, or it is moving slower with respect to it than frame 1. Since Relativity denies such a frame's existence, you are not discussing Relativity but denying it.
What sorcery is this?

If we stand side by side next to each other you either stand to my right and I to your left or vice versa. Both can not be true at the same time.
If we stand facing each other (or back to back) then we by definition do not stand to either left or right with respect to each other.

In reality the recorded observations can only show that one of the two aged less. It would be quite silly to argue against that.

88. Originally Posted by van erst
Or can you produce relative motion between two stationary objects without accelerating one or both of them?
No, and that is exactly the point - if you introduce motion to connect the same two events in space-time as a stationary reference observer travels through, you can do so only by having acceleration; due to the geometry of Minkowski space-time this automatically shortens your world line, i.e. it takes you less proper time. Hence time dilation.

The ultimate causal link between acceleration and time dilation is quite simply the peculiar geometry of Minkowski space-time itself.

89. Originally Posted by xyzt
Originally Posted by van erst
Originally Posted by xyzt
2. my ship and the other ship accelerate away from you (our relative aging slows down, reaching aging factor 0.9 relative to your 1.0)
correct

3. I accelerate back towards you (my aging speeds up, again factor 1.0)
Nonsense, aging is not a function of the sense of the acceleration, you have been told this before.

Then I accelerated again towards him to become stationary relative to him again, only separated by some distance. Are you saying that I am stationary relative to him but our rates of aging are different (while in the same inertial frame)?
No. when you become stationary your aging rates are the same. What you said wrong was the part about accelerating towards the stationary twin while having the same aging factor 1.0. Look, you clearly don't understand these things but I think I am not the only one who gets the impression that you aren't interested in learning, you are interested in "challenging the mainstream dogma".
You appear to be discussing completely past the point I am making.

There are the two separate inertial frames:
- the original reference with 1.0 aging factor
- the accelerated one with 0.9 aging factor

These aging factors are relative. Now if I jump between these two inertial frames by the means of acceleration my aging rate/factor relative to these two will oscillate between these two aging factors. So essentially one acceleration results in my aging relatively slowing down and the opposite acceleration results in my aging relatively speeding up.

This is what logic dictates. Or do you suppose that for all accelerations my aging just keeps slowing down?

90. Originally Posted by Markus Hanke
Originally Posted by van erst
Or can you produce relative motion between two stationary objects without accelerating one or both of them?
No, and that is exactly the point - if you introduce motion to connect the same two events in space-time as a stationary reference observer travels through, you can do so only by having acceleration; due to the geometry of Minkowski space-time this automatically shortens your world line, i.e. it takes you less proper time. Hence time dilation.

The ultimate causal link between acceleration and time dilation is quite simply the peculiar geometry of Minkowski space-time itself.
Well does this peculiar geometry allow for acceleration resulting in aging of the accelerator relatively either slowing down or speeding up?

If the answer is yes, then I would like to better understand the causation produced by that geometry.

91. Originally Posted by van erst
Well does this peculiar geometry allow for acceleration resulting in aging of the accelerator relatively either slowing down or speeding up?
We have already explained to you - multiple times, actually - that time dilation is not modelled as "rates of ageing", but as total proper time, i.e. as length of world lines. The reason is of course simple - "rates of ageing" are observer dependent, whereas proper time is not. The former leads to lots of confusion and unncessary difficulty, whereas the latter is clear and unambiguous.

92. Originally Posted by van erst
If the answer is yes, then I would like to better understand the causation produced by that geometry.
Why do you keep asking the same question over and over again ? This has already been answered multiple times now !

93. Originally Posted by Markus Hanke
Originally Posted by van erst
Well does this peculiar geometry allow for acceleration resulting in aging of the accelerator relatively either slowing down or speeding up?
We have already explained to you - multiple times, actually - that time dilation is not modelled as "rates of ageing", but as total proper time, i.e. as length of world lines. The reason is of course simple - "rates of ageing" are observer dependent, whereas proper time is not. The former leads to lots of confusion and unncessary difficulty, whereas the latter is clear and unambiguous.
To which I already responded with the back and forth acceleration between the two inertial frames.

Say that the frame with the relatively faster rate of time is FRAME A and the one with the slower rate of time is FRAME B.

1. I am now in frame B with with some other person
2. I accelerate to get to frame A
3. I end up in frame A and stop the acceleration
4. during my pause in frame A my time is ticking faster than in frame B
5. then I accelerate back to frame B and stop the acceleration

In the above my accumulated total amount of proper time is greater than that of the person who stayed in frame B. Thus acceleration lead into a relatively faster rate of time.

94. Originally Posted by van erst
In the above my accumulated total amount of proper time is greater than that of the person who stayed in frame B. Thus acceleration lead into a relatively faster rate of time.
No, you are wrong - you accumulated less proper time, because you experience acceleration whereas the inertial observer did not. For the umpteeth time : inertial observers always experience the longest proper time between the same two events as compared to accelerated ones. Acceleration is equivalent to a shortening of your world line between two events.

95. Originally Posted by Markus Hanke
Originally Posted by van erst
In the above my accumulated total amount of proper time is greater than that of the person who stayed in frame B. Thus acceleration lead into a relatively faster rate of time.
No, you are wrong - you accumulated less proper time, because you experience acceleration whereas the inertial observer did not. For the umpteeth time : inertial observers always experience the longest proper time between the same two events as compared to accelerated ones. Acceleration is equivalent to a shortening of your world line between two events.
Lest narrow this down to something extremely simple.

You are in inertial frame A and I am in inertial frame B. In my inertial frame B time is ticking considerably slower than in your inertial frame.

What do I need to do to get to your inertial frame where the rate of time is faster?

96. Originally Posted by van erst
You are in inertial frame A and I am in inertial frame B. In my inertial frame B time is ticking considerably slower than in your inertial frame.
I think you are confusing coordinate time and proper time. If you connect two spatially separated events (1) and (2) in space-time, you will accumulate the exact same amount of proper time no matter how "fast" you go from event (1) to event (2).

Is this where your confusion lies ?

97. Originally Posted by Markus Hanke
Originally Posted by van erst
You are in inertial frame A and I am in inertial frame B. In my inertial frame B time is ticking considerably slower than in your inertial frame.
I think you are confusing coordinate time and proper time. If you connect two spatially separated events (1) and (2) in space-time, you will accumulate the exact same amount of proper time no matter how "fast" you go from event (1) to event (2).

Is this where your confusion lies ?
No, I am discussing in terms of verifiable measurements observed in reality.

Assume that we by testing come to find out that in your inertial frame A time is in reality ticking faster than in my inertial frame B. Between us this is a confirmed measured result that proper time in your frame really is ticking faster than in my frame.

How do I in reality get to your inertial frame and achieve the same rate of proper time?

98. Originally Posted by van erst
No, I am discussing in terms of verifiable measurements observed in reality.
Verifiable measurements are clock readings taken by an observer.

Assume that we by testing come to find out that in your inertial frame A time is in reality ticking faster than in my inertial frame B
And how do you propose to conduct such a test, exactly ?

Between us this is a confirmed measured result that proper time in your frame really is ticking faster than in my frame.
No, proper time is the exact same for both of us if we cross through the same events while always moving inertially. What you are talking about is coordinate time, which is not the same thing.

How do I in reality get to your inertial frame and achieve the same rate of proper time?
You can't. If you are in relative motion, you need to apply acceleration in order to get into the other frame. This of course means that you experience less proper time than the purely inertial observer.

99. Originally Posted by van erst
Assume that we by testing come to find out that in your inertial frame A time is in reality ticking faster than in my inertial frame B. Between us this is a confirmed measured result that proper time in your frame really is ticking faster than in my frame
That doesn't sound right. A will see B's clock ticking slower and B will see A's clock ticking slower. If you are dealing with inertial frames, then the situation must be symmetrical.

100. Originally Posted by Markus Hanke
Assume that we by testing come to find out that in your inertial frame A time is in reality ticking faster than in my inertial frame B
And how do you propose to conduct such a test, exactly ?
There are many ways to do this. As is GPS satellite time dilation measured and correction produced for accurate GPS service. Or arranging a third inertial frame to measure against. We can easily achieve these measurements.

Originally Posted by Markus Hanke
Between us this is a confirmed measured result that proper time in your frame really is ticking faster than in my frame.
No, proper time is the exact same for both of us if we cross through the same events while always moving inertially. What you are talking about is coordinate time, which is not the same thing.
Incorrect. I am talking about actual recorded verifiable results. Are you indicating that it is not possible to measure the real time dilation between two observers in separate inertial frames?

Originally Posted by Markus Hanke
How do I in reality get to your inertial frame and achieve the same rate of proper time?
You can't. If you are in relative motion, you need to apply acceleration in order to get into the other frame. This of course means that you experience less proper time than the purely inertial observer.
Here you say that I cannot, and then proceed to say that I can by using acceleration.

101. Originally Posted by van erst
Originally Posted by KJW
Originally Posted by van erst
What I meant was that if any two observers are in relative motion, that motion can only have been caused by acceleration.
Why?
How do you mean?

Suppose you and I are stationary right next to each other. You want there to be relative motion between us. Can you achieve that without involving acceleration?
Which gets back to my original question: Why are you assuming that all observers start out from the same inertial frame of reference?

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