3 clocks thought experiment - Absolute vs. relative aspects?

[A.T.]

Admittedly, this is a nice analogy but I feel it doesn't quite fit the purpose: I'm not asking for the relative speed of the objects in the experiment (which would be responsible for the speed ticket). I'm interested in their accumulated ageing, and it seems that the acceleration profile is actually key here:

Try this analogy:

The rate at which an odometer reading increases is not directly affected by the steering, just by speed. But the steering on the way between two points affects the total distance accumulated by the odometer, and knowing the complete steering profile allows you to compute it.

odometer distance : proper time interval
steering : proper acceleration

 

[A.T.]

And since gravity is the same as acceleration ...

Make sure not to confuse the acceleration of a frame of reference with the acceleration of the observed clocks. This is very common in the "gravity as acceleration" context.

 

[Friggle]

First, thanks to all of you, that has already helped a lot, overall.
I again stumbled on the above from PeroK:

Here's an example. If you have rockets moving in uniform circular motion about a central space station, then in this case the time dilation is not symmetric and it leads directly to differential ageing. The rockets are accelerating centripetally. The calculation for the time-dilation depends only on the rocket's orbital speed and not on the acceleration.

I now fully agree that it's proper acceleration, since somebody in the rocket clearly could measure it using just a local accelerometer. So, that's just fine.

What I'm struggling with is that there is asymmetric time dilation at all in this setup (a clock in the rocket vs. a clock in the base station). Reason for my issues with this is that there is (from my perspective, you will surely correct me if I'm wrong) no relative motion between the base station and the rocket. I mean: The distance between them never varies, right? So how can there be a velocity between them? May be that is naive thinking, please advise...
Nevertheless, this might be a new approch for me to grasp the idea of the root cause for both time dilation and ageing differently: May be it is not the relative motion between an observer and an observed object which drives time dilation and accumulated relative ageing but rather the movement path of the observed object through the reference spacetime frame of the observer. Because, in the rocket example, clearly the rocket moves in relation to the reference frame of the observer (it performs circular movements within that frame), Although it does not move in relation to the observer. Is this correct or am I totally wrong here?

 

[Sagittarius A-Star]

The distance between them never varies, right? So how can there be a velocity between them?

In his popular book "Relativity: The Special and General Theory" from 1917, Einstein explained this scenario with an inertial reference-frame (velocity-based time-dilation) and with a rotating reference-frame (pseudo-gravitational time-dilation).

Source (see Appendix III, sub-chapter "(c) Displacement of Spectral Lines Towards the Red":
https://en.wikisource.org/wiki/Rela...isplacement_of_Spectral_Lines_Towards_the_Red

 

[jbriggs444]

Reason for my issues with this is that there is (from my perspective, you will surely correct me if I'm wrong) no relative motion between the base station and the rocket. I mean: The distance between them never varies, right? So how can there be a velocity between them? May be that is naive thinking, please advise...

Inertial frames do not rotate. In any inertial frame where the central observer is at rest, the rocket will have a non-zero velocity. We can use a rotating frame and get an answer of zero. But then we are using a non-inertial frame and our result for time dilation will be complicated by this.

There is no inertial frame where the orbitting rocket is at rest. But again, we can use a variety of non-inertial frames where the answer is zero. But then we are using a non-inertial frame and our result for time dilation will be complicated by this.

 

[Dale]

there is (from my perspective, you will surely correct me if I'm wrong) no relative motion between the base station and the rocket. I mean: The distance between them never varies, right? So how can there be a velocity between them?

Velocity is the first derivative of position, not the first derivative of distance. Although the distance is constant the position is not.

However, think of the geometry here. In spacetime one path is a straight line and the other is a helix. The relative velocity is the angle between the tangent vectors. The tangent vector to the straight line points along the axis, and the angle of the tangent vector to the helix points diagonally. So the angle between them is not zero, they are not parallel.

this might be a new approch for me to grasp the idea of the root cause for both time dilation and ageing differently: May be it is not the relative motion between an observer and an observed object which drives time dilation and accumulated relative ageing but rather the movement path of the observed object through the reference spacetime frame of the observer

I think the geometric approach is best, but what you say here is correct. You have to add the caveat that the formula differs for inertial and non-inertial frames.

 

[Friggle]

Velocity is the first derivative of position, not the first derivative of distance. Although the distance is constant the position is not.

Great! Think I was confusing speed with velocity!? Is relative speed the first derivative of distance between objects, while velocity is the first derivative of position? Then SR and time dilation are all about velocity and not about speed. True?

 

[Friggle]

However, think of the geometry here. In spacetime one path is a straight line and the other is a helix.

You mean, a straight line vs. a helix in direction of the time coordinate. Correct?

 

[Sagittarius A-Star]

Great! Think I was confusing speed with velocity!? Is relative speed the first derivative of distance between objects, while velocity is the first derivative of position? Then SR and time dilation are all about velocity and not about speed. True?

Velocity means the vector. Speed is its magnitude.

Source:
https://www.physicsforums.com/threads/special-relativity-and-velocity.994372/post-6400115

 

[jbriggs444]

Great! Think I was confusing speed with velocity!? Is relative speed the first derivative of distance between objects, while velocity is the first derivative of position? Then SR and time dilation are all about velocity and not about speed. True?

"Closing speed" is a better term for what you are describing. I would interpret "relative speed" as the magnitude of relative velocity.

Time dilation is all about inertial coordinate systems in relative motion. Yes, that means that direction matters, so it is velocity, not speed.

 

[Friggle]

"Closing speed" is a better term for what you are describing. I would interpret "relative speed" as the magnitude of relative velocity.

Got it. Thanks.

 

[Friggle]

This is surprisingly insightful to me: Much about the apparent contradictions and misinterpretations about SR out there to me seem to stem from exactly this often incorrectly or imprecisely perceived picture: I was always thinking of relative speed in terms of a varying distance between objects. That might have been triggered by the initially simple picture of SR using inertial relative motions only.
Now I see: It's the overall path through the (inertial) reference frame of the observer which makes the deal.
Really, really, thanks a lot! I will have to go through the classical experiments on relativity now over again with this new thinking (e.g. Hafele-Keating).

 

[Dale]

Cool, I am glad this helped.

You mean, a straight line vs. a helix in direction of the time coordinate. Correct?

Yes. An object undergoing uniform circular motion is represented in spacetime as a helix with the axis of the helix parallel with the $t$ axis in spacetime.

 

[Nugatory]

Is relative speed the first derivative of distance between objects...?

Relative velocity is the first derivative of the position when using coordinates in which the object to which the speed is relative is at rest. It is a vector. Relative speed is the magnitude of the relative velocity and is a scalar. Thinking in terms of "distance" just gets in the way except when working with objects that are at rest relative to one another.

Be aware that just about everyone is occasionally careless about the distinction between speed and velocity when it is clear from the context which one is being discussed.

 

[Friggle]

I think the geometric approach is best, but what you say here is correct. You have to add the caveat that the formula differs for inertial and non-inertial frames.

The reason why I prefer to think about the path through spacetime in the frame of reference of the observer is that this allows me to think of time and ageing being an effect caused by an object moving through some background space. Even if it is not a space in an absolute sense but rather some kind of "private" spacetime area of the observer. I find this easier to accept than the idea of time dilation being an effect of two objects moving relative to each other, expressed by a relative velocity vector. I kind of think "how does the observed object know in which way it has to age and experience time dilation in relation to an observer?"
But it may be that I just can't really get the 4-dimensional idea of spacetime in my head, it still stays abstract to me. I'll give it a few more tries, however, it seems that it may be worth struggling with it ;-)

 

[PeterDonis]

I kind of think "how does the observed object know in which way it has to age and experience time dilation in relation to an observer?"

It doesn't. The observed object does not feel any change at all in its aging or experience of time. You, right now, are moving at more than 99% of the speed of light relative to cosmic ray particles coming in to Earth from outer space. Do you feel any change in your aging or experience of time?

Time dilation is something the observer calculates (note that I did not say "sees", because what the observer actually sees, in the light signals coming to him from the observed object, is described by the relativistic Doppler shift formula, not the time dilation formula). The observed object doesn't even "know" that it's being observed.

 

[Dale]

I kind of think "how does the observed object know in which way it has to age and experience time dilation in relation to an observer?"

Interesting. That seems to me to be exactly the problem with your observer-frame centric approach.

What I like about the spacetime approach is that it focuses on invariants and geometry that are independent of the reference frame or any observer.

 

[Friggle]

What I like about the spacetime approach is that it focuses on invariants and geometry that are independent of the reference frame or any observer.

Does that mean you are talking about an absolute spacetime geometry, independent of an observer? That sounds great to me, actually, I just thought that such an absolute spacetime was explicitly negated by the relativity theories. There is no one-and-only reference frame, right? Wouldn't an absolute spacetime idea be the same as an absolute frame of reference?

 

[Nugatory]

That sounds great to me, actually, I just thought that such an absolute spacetime was explicitly negated by the relativity theories. There
is no one-and-only reference frame, right? Wouldn't an absolute spacetime idea be the same as an absolute frame of reference?

Absolute space and absolute time are rejected. But we’re talking about four-dimensional spacetime; relativity tells us that there is no one-and-only right way of dividing spacetime into three-dimensional space and one dimensional time. It’s somewhat analogous to how there is no unique way of dividing three-dimensional space into one “up” dimension and two-dimensional planes of constant height; likewise, there is no unique way of dividing four dimensional spacetime into one time dimension and three-dimensional spaces of constant time. It depends on which line we choose to be the time axis, and indeed that choice is pretty much equivalent to choosing a reference frame.

If you are not familiar with Minkowski spacetime diagrams, it’s worth spending some time with them to see how different frames can be represented just by drawing different x and t axes to divide the two dimensional (one space, one time) surface into space and time in different ways. These are among the most powerful tools for visualizing special relativity.

 

[Dale]

Does that mean you are talking about an absolute spacetime geometry, independent of an observer?

Yes, although we use the word invariant or covariant because the observer independent spacetime geometry does not include an observer independent concept of rest. And such a concept of rest is part of Newton’s absolute space and time.

There is no one-and-only reference frame, right?

Yes.

Wouldn't an absolute spacetime idea be the same as an absolute frame of reference?

Yes, but again that is why we use the words invariant or covariant to distinguish the concepts. The geometrical model of spacetime does not include these specific concepts.

The geometrical model of spacetime includes observer independent geometric concepts of the spacetime variants of lines, points, distances along lines, vectors, tensors, angles, and curvature.

 

[Dale]

Does that mean you are talking about an absolute spacetime geometry, independent of an observer?

I thought it might be worthwhile to go a little further into what the spacetime geometry entails.

Imagine a sheet of paper. This paper represents spacetime (1D space and 1D time).

Absolute spacetime says that there is one and only one unique way to draw a grid where one set of grid lines is space and the other is time. Relative spacetime says you can take any straight line and draw parallel and perpendicular lines to form a valid grid.

The geometric approach says that the grid lines are entirely optional. You can use them if you like, for convenience, but the geometric approach looks for things that are independent of the grid lines altogether.

Think of Euclidean geometry. Points, lines, angles, and distances were all concepts developed without coordinate grids. Straight, parallel, perpendicular, all of these concepts do not require coordinates in Euclidean geometry. Furthermore, if you choose to use a coordinate grid, doing so does not change any of those geometrical concepts.

The difference between spacetime geometry and Euclidean geometry is the metric. In Euclidean geometry to find the distance from $A$ to $B$ we draw a set of circles centered on $A$ and then the distance to $B$ is the radius of the circle that contains $B$. In spacetime geometry we draw a set of hyperbolas centered on $A$ and the interval is the semi major axis of the hyperbola that contains $B$.

This change in geometry contains all of the rules of special relativity. The most important can be stated in terms of the geometry, without reference to grids. But if one desires to use coordinate grids for convenience, the underlying geometry establishes the rules for that too.