Clock postulate and differential aging

[atyy]

Hi atyy,
Just reading the summary of the reference you provided tells me it has nothing to do with the issue I am trying to raise. A lot of posting have been put up and I wonder if the issue I am trying to raise is getting lost.

Yes, the issue you are trying to raise is getting lost. Although in the twin paradox, we can do what you are asking, more generally, about the point of view of various observers in terms of coordinates is irrelevant. The elapsed proper time along a worldline is a geometric invariant, and that is all there is to it.

 

[ThomasT]

If the clocks accumulate less time for longer periods at high speed, then the acceleration has to cause a permanent change in speed, not just a change in speed while the clock is accelerating. Is that what you mean, because your wording doesn't imply that to me.

I don't know what you mean by a permanent change in speed. Do you mean constant or uniform speed for a certain interval? If the speed is changing, then the clock is accelerating, right?

What I meant was that the interval that you run one of the clocks at the higher (constant) speed is longer, so, even though the acceleration histories are identical, the clock that you ran longer at the higher (constant) speed will accumulate less time for the two intervals associated with the two clocks (including the accelerations) being compared.

Did I just confuse myself again?

 

[MikeLizzi]

Yes, the issue you are trying to raise is getting lost. Although in the twin paradox, we can do what you are asking, more generally, about the point of view of various observers in terms of coordinates is irrelevant. The elapsed proper time along a worldline is a geometric invariant, and that is all there is to it.

Hi atyy,

Exactly. So if someone draws a diagram in which the elasped proper time of the astronaut is greater than that of the earthbound twin you have a paradox. Right?.

 

[atyy]

So if someone draws a diagram in which the elasped proper time of the astronaut is greater than that of the earthbound twin you have a paradox. Right?.

What do you mean by "paradox"?

 

[Mentz114]

MikeLizzi:

To the skeptic, trying to work the problem on a Minkowski diagram, it looks like you draw the exact same diagram, only with the twins switch places. And that would be a paradoxical answer.

Yes, that would be wrong.

The accelerating observer carries his own orthogonal frame along his worldline. At any point you can construct a set of axes tangential to the worldline. If you then rotate the space-time so the relevant tangent is vertical, you have a spacetime diagram from the point of view of an inertial observer instantaneously at rest wrt to the accelerating observer.

The accelerating world-line is still curved, and the other observers worldline is tilted.

It's not exactly what you wanted but it's the best you can do.

 

[ThomasT]

I think part of the original problem is the wording of statements like this. All changes in speed involve acceleration, but not all accelerations involve changes in speed. They are not always equivalent.

I think Baez said it well in the link bcrowell provided above, and I would highly recommend a careful reading of it. But in the end, if we use the same equation then in my book we agree on everything important.

Thanks. I skimmed over Baez, but will give it a more careful reading.

 

[Frame Dragger]

MikeLizzi:

Yes, that would be wrong.

The accelerating observer carries his own orthogonal frame along his worldline. At any point you can construct a set of axes tangential to the worldline. If you then rotate the space-time so the relevant tangent is vertical, you have a spacetime diagram from the point of view of an inertial observer instantaneously at rest wrt to the accelerating observer.

The accelerating world-line is still curved, and the other observers worldline is tilted.

It's not exactly what you wanted but it's the best you can do.

Thank you, but I kept saying this and it doesn't seem to make an impact. I think he's reaching for a metaphysical interpretation that just isn't there, or he's seeing a problem that likewise, is not there.

 

[MikeLizzi]

What do you mean by "paradox"?

Hi again,

The canonical analysis of the twins example puts the worldline of the earthbound twin on the vertical ct axis. The astronaut gets two world lines one with positive slope and one negative eventually joining up with the worldline of the earthbound twin. Spacetime intervals must be equal. The astronaut's spacetime interval has a longer space component so he must have a shorter time component. He comes back younger.

So if someone draws the worldline of the astronaut along the ct axis and givess the earthbound twin the two legs of the triangle, the above conclusion would have to be reversed. One would have to declare that the astronaut came back older. And that would be a praradox.

Can we agree on that?

 

[MikeLizzi]

Thank you, but I kept saying this and it doesn't seem to make an impact. I think he's reaching for a metaphysical interpretation that just isn't there, or he's seeing a problem that likewise, is not there.

Frame Dragger.

?? I hate metaphysics. I'm just tying to draw a Minkowski diagram.

 

[Frame Dragger]

Hi again,

The canonical analysis of the twins example puts the worldline of the earthbound twin on the vertical ct axis. The astronaut gets two world lines one with positive slope and one negative eventually joining up with the worldline of the earthbound twin. Spacetime intervals must be equal. The astronaut's spacetime interval has a longer space component so he must have a shorter time component. He comes back younger.

So if someone draws the worldline of the astronaut along the ct axis and givess the earthbound twin the two legs of the triangle, the above conclusion would have to be reversed. One would have to declare that the astronaut came back older. And that would be a praradox.

Can we agree on that?

No, because that isn't the reverse of the twin problem. What you're doing is a bit like expecting binoculars to work regardless of which end you look through, ignoring relative orientation. That would be a metaphorical, metaphysical interpreation of the very sound answer that Mentz gave you.

 

[MikeLizzi]

MikeLizzi:

Yes, that would be wrong.

The accelerating observer carries his own orthogonal frame along his worldline. At any point you can construct a set of axes tangential to the worldline. If you then rotate the space-time so the relevant tangent is vertical, you have a spacetime diagram from the point of view of an inertial observer instantaneously at rest wrt to the accelerating observer.

The accelerating world-line is still curved, and the other observers worldline is tilted.

It's not exactly what you wanted but it's the best you can do.

Hi Mentz114,

I understand exactly what you are saying. May I conclude that you agree with my belief that one cannot draw a Minkowski with the astronaut as an observer?

 

[atyy]

Hi again,

The canonical analysis of the twins example puts the worldline of the earthbound twin on the vertical ct axis. The astronaut gets two world lines one with positive slope and one negative eventually joining up with the worldline of the earthbound twin. Spacetime intervals must be equal. The astronaut's spacetime interval has a longer space component so he must have a shorter time component. He comes back younger.

So if someone draws the worldline of the astronaut along the ct axis and givess the earthbound twin the two legs of the triangle, the above conclusion would have to be reversed. One would have to declare that the astronaut came back older. And that would be a praradox.

Can we agree on that?

The canonical analysis says 4>3, but if I write 3=6, and since 6>4, then 3>4, which would be a paradox?

 

[MikeLizzi]

No, because that isn't the reverse of the twin problem. What you're doing is a bit like expecting binoculars to work regardless of which end you look through, ignoring relative orientation. That would be a metaphorical, metaphysical interpreation of the very sound answer that Mentz gave you.

Frame Dragger,

I didn't say it was the reverse of the Twins problem. I know it is not. I'm asking how to draw a Minkowski diagram with the astronaut as the observer.

 

[MikeLizzi]

The canonical analysis says 4>3, but if I write 3=6, and since 6>4, then 3>4, which would be a paradox?

Hi aty,

I agree. I didn't intend to include my second paragraph as canonical analysis, only the first paragraph. Sorry if I didn't make that clear.

 

[Dmitry67]

Which one?
In twin paradox astronaut has 2 different diagrams: on the way away and when he returns. You can not have just one, as astronaut changes the frame

 

[MikeLizzi]

Which one?
In twin paradox astronaut has 2 different diagrams: on the way away and when he returns. You can not have just one, as astronaut changes the frame

Hi Dmitry67,

Thank you for your post. If I may restate what you wrote... (I think english is not your native language)

In the twins paradox (with the astronaut as the observer) one must use 2 different diagrams. One for the case where the Earth is moving away and one where the Earth returning.

OK?

 

[Dmitry67]

yes, correct.

 

[MikeLizzi]

yes, correct.

Hi Dmitry67,

Thank you. I put down some numbers for the first diagram. I assume gamma is 2. The first column is the time on the astronaut's clock. The second column is the calculated time on an Earth clock. OK?

First Diagram:

Astronaut Time____Earth Time
(observer)
0 _______________ 0
1 _______________ .5
2 _______________ 1.0
3 _______________ 1.5
4 _______________ 2.0
5 _______________ 2.5
Turn around

 

[Mentz114]

No single space-time diagram can represent the view-point of the accelerated observer, you need a different one for every instant along the worldline.

But what is the importance of that ? We know that elapsed time is equated with the proper length, so I don't really see that it matters.

 

[Frame Dragger]

No single space-time diagram can represent the view-point of the accelerated observer, you need a different one for every instant along the worldline.

But what is the importance of that ? We know that elapsed time is equated with the proper length, so I don't really see that it matters.

Again, I've been asking this question several times without any meaningful effect. *shrug*

EDIT: Al68: I pointed out 2 pages ago that there is an implied period of accelleration and braking, but it didn't seem to faze him. :/ I'm sticking with my "looking the wrong way through binoculars" metaphor. This is the use of a tool in the manner it wasn't intended, then drawing conclusions that a paradox exists, or a skeptic requires satisfaction in this particular form.

 

[Al68]

Now somebody's clock can go slower or faster than yours but it can't jump...What is really happening is that the astronaut determines the earthbound twin's clock is running FASTER than his during the spaceship turnaround. Skipping that calculation is why this artificial jump needs to be added.

Yep, exactly. Of course time doesn't jump, it only appears to jump as an artifact of the simplifying assumption of instantaneous turnaround, which equally can't happen. In the accelerated frame of the ship, Earth's clock runs fast. The greater the acceleration, the faster Earth's clock runs in the ship's frame. It's only modeled as a jump associated with instantaneous turnaround to simplify the math.

In reality there must be a finite interval for the turnaround acceleration and for Earth's clock to advance. They are both only treated as instantaneous to simplify the math, not to claim it's physically possible.

I've never seen a Minkowski diagram for the ship twin either, but it couldn't be in the standard form. It would have to be modified to allow for gravitational time dilation of clocks in an accelerated reference frame, but I doubt it could then be called a Minkowski diagram.

Einstein's own 1918 Twins paradox resolution is unique in that it does consider the non-inertial reference frame in which the ship is "stationary" the entire trip. It skips the math, presumably because his intended audience wouldn't need it to be shown. You can find it here: http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

 

[Dale]

Your comment would seem to support my belief that I have not yet see a correct Minkowski diagram with the astronaut as the observer. (In fact I believe it can't be done) The references given by Frame Dragger give the astronaut the orthagonal axis and then offer a kind of compensation for the ensuing contradiction. Yet those references are considered canon.

It is certainly ok to use non-inertial coordinates in which a non-inertial observer is given a constant coordinate position, but then those diagrams are not Minkowski diagrams because the Minkowski metric would not be valid in such coordinates. That would be a more general class of coordinate charts where general metrics are permitted rather than limiting it to the Minkowski metric. You certainly can use such coordinate systems, provided you are careful with your math and if you do so you always get the same result for the twins.

 

[Frame Dragger]

It is certainly ok to use non-inertial coordinates in which a non-inertial observer is given a constant coordinate position, but then those diagrams are not Minkowski diagrams because the Minkowski metric would not be valid in such coordinates. That would be a more general class of coordinate charts where general metrics are permitted rather than limiting it to the Minkowski metric. You certainly can use such coordinate systems, provided you are careful with your math and if you do so you always get the same result for the twins.

...Which brings the whole thing back to... why?! The original statement that a "skeptic" would somehow require a valid reversal of the diagram is still baffling.

 

[MikeLizzi]

Yep, exactly. Of course time doesn't jump, it only appears to jump as an artifact of the simplifying assumption of instantaneous turnaround, which equally can't happen. In the accelerated frame of the ship, Earth's clock runs fast. The greater the acceleration, the faster Earth's clock runs in the ship's frame. It's only modeled as a jump associated with instantaneous turnaround to simplify the math.

In reality there must be a finite interval for the turnaround acceleration and for Earth's clock to advance. They are both only treated as instantaneous to simplify the math, not to claim it's physically possible.

I've never seen a Minkowski diagram for the ship twin either, but it couldn't be in the standard form. It would have to be modified to allow for gravitational time dilation of clocks in an accelerated reference frame, but I doubt it could then be called a Minkowski diagram.

Einstein's own 1918 Twins paradox resolution is unique in that it does consider the non-inertial reference frame in which the ship is "stationary" the entire trip. It skips the math, presumably because his intended audience wouldn't need it to be shown. You can find it here: http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

Hi Al68,

Yours is the point I was trying to make. Thank you very much for stating it.

To those who think I was being unscientific:

Are you now going to declare Al68 unscientific too?

Just adding the following: I am thu with this thread.

 

[Dmitry67]

Hi Dmitry67,

Thank you. I put down some numbers for the first diagram. I assume gamma is 2. The first column is the time on the astronaut's clock. The second column is the calculated time on an Earth clock. OK?

First Diagram:

Astronaut Time____Earth Time
(observer)
0 _______________ 0
1 _______________ .5
2 _______________ 1.0
3 _______________ 1.5
4 _______________ 2.0
5 _______________ 2.5
Turn around

Yes, then

5 (after turnaround) __ 17.5
6 _______________ 18
7 _______________ 18.5
8 _______________ 19
9 _______________ 19.5
10 ______________ 20

You are probably surprised by the gap in "Earth time"? It is not something really physical. It is just a a position of Earth at 4D miskovski spacetime based on the calculations of the spaceship. When you turn around, you rotate that diagram, and positions of the points change.

 

[George Jones]

You are probably surprised by the gap in "Earth time"? It is not something really physical.

Right. Visual appearances are given by the Doppler effect, so moving clocks can to run fast or slow, depending on the relative direction of travel. I worked a couple of examples in

https://www.physicsforums.com/showthread.php?p=2186296#post2186296.

As the astronaut watches (with a telescope) a clock on Earth, the astronaut does not see the visual image of the Earth clock make a discontinuous jump in time.

 

[MikeLizzi]

Yes, then

5 (after turnaround) __ 17.5
6 _______________ 18
7 _______________ 18.5
8 _______________ 19
9 _______________ 19.5
10 ______________ 20

You are probably surprised by the gap in "Earth time"? It is not something really physical. It is just a a position of Earth at 4D miskovski spacetime based on the calculations of the spaceship. When you turn around, you rotate that diagram, and positions of the points change.

Hi Dmitry67,
Thanks for your reply. No, I am not surprised by the gap. I know about it. And I know that it is not really physical. That's what I wanted to focus on.

If you have a solution that is not physically real then there is one of two possibilities:
1. You solved the problem wrong.
2. The problem is not physically real.

Your solution is correct. So that means the problem, as traditionally described, is not physically real. I tried to get the other members of this forum to recognize that but I was not successfull. Poster "Al68" said the same thing. Since he is a senior member of this forum, other members apparently accepted his statement without challange.

There is more that can be discovered be examining the Twins Paradox but I have accomplished as about as much as I think I can with this thread so I will not bother you anymore.

 

[Dale]

So that means the problem, as traditionally described, is not physically real. I tried to get the other members of this forum to recognize that but I was not successfull.

I don't know what you are talking about. The majority of the regulars on this forum already understand and recognize that. That is why most of us prefer the spacetime geometric approach.

 

[Frame Dragger]

I don't know what you are talking about. The majority of the regulars on this forum already understand and recognize that. That is why most of us prefer the spacetime geometric approach.

I guess he really likes things a particular way?

 

[DanRay]

In a recent thread about differential aging in the archetypal twin scenario, I suggested that the periods of oscillators are affected by accelerations, or in other words that a clock's tick rate is affected by changes in its speed.

Time dilatation is a consequence of Special Relativity where acceleration is not allowed. It is demonstrated by solving the Lorentz Transformation equation specific to time as the unknown when considering extream relative speeds from one frame of reference to another.

The inertial reference systems used in Special Relativity are alway straight line movement at a constant speed. That is the definition of an inertial reference system.

 

[bcrowell]

Time dilatation is a consequence of Special Relativity where acceleration is not allowed.

This is incorrect. Here is a cut and paste that may be useful, from a FAQ I maintain at http://www.lightandmatter.com/cgi-bin/meki?physics/faq .

====

Does special relativity apply when things are accelerating?

Yes. There are three things you might want to do using relativity: (1) describe an object that's accelerating in flat spacetime; (2) adopt a frame of reference, in flat spacetime, that's accelerating; (3) describe curved spacetime. General relativity is only needed for #3. The reason you'll see statements to the contrary is historical. Einstein published special relativity in 1905, general relativity in 1915. During that ten-year period in between, nobody really knew what the boundaries of applicability of special relativity were. This uncertainty made its way into textbooks and lectures, and because of the conservative nature of education, some students are still hearing, a century later, incorrect assertions that SR can't handle #2, or even #1 (which would make SR a useless theory for describing interactions!).

This issue often comes up in discussions of the twin paradox. A good way to see that general relativity is totally unnecessary for understanding the twin paradox is to pose a version in which the four-vector equation a=b+c represents the unaccelerated twin's world-line a and the accelerated twin's world-line consisting of displacements b and c. The accelerated twin is subjected to (theoretically) infinite accelerations at the vertices of the triangle. The triangle inequality for flat spacetime is reversed compared to the one in flat Euclidean space, so proper time |a| is greater than proper time |b|+|c|.

In an accelerating frame (#2), the equivalence principle tells us that measurements will come out the same as if there were a gravitational field. But if the spacetime is flat, describing it in an accelerating frame doesn't make it curved. (Curvature is invariant under any smooth coordinate transformation.) Thus relativity allows us to have gravitational fields in flat space --- but only if the gravitational field is a certain special configuration, such as a uniform field. SR is capable of operating just fine in this context. For example, Chung et al. did a high-precision test of SR in 2009 using a matter interferometer in a vertical place, specifically in order to test whether there was any violation of Lorentz invariance in a uniform gravitational field. Their experiment is interpreted purely as a test of SR, not GR.

Chung -- http://arxiv.org/abs/0905.1929

 

[yuiop]

...
I've never seen a Minkowski diagram for the ship twin either, but it couldn't be in the standard form. It would have to be modified to allow for gravitational time dilation of clocks in an accelerated reference frame, but I doubt it could then be called a Minkowski diagram.

Correct, it would not be a Minkowski diagram. If the ship twin has an initial velocity as he passes the Earth the first time and then decelerates so that he eventually comes to rest wrt the Earth and then continues decelerating until he returns to the Earth, with constant acceleration as measured on the ship throughout the trip, then the path of the Earth could be plotted on a Rindler diagram and the Earth would appear to follow a curved path. However this is a curved path through Rindler spacetime which is different animal from a curved path through Minkowski spacetime. A careful analysis of the elapsed proper time of the Earth would still show that more proper time passes on the Earth than onboard the ship by the time they meet again.

 

[DanRay]

This is incorrect. Here is a cut and paste that may be useful, from a FAQ I maintain at http://www.lightandmatter.com/cgi-bin/meki?physics/faq .

Dear bcrowell,

The scientific world never heard of time dilation until Einstein intorduced it along with his explanations about the consequences of Special Relativity. His explanation is in his book "Relativity" starting with section 7 and continuing through section 17 where he talks about "Minkowski's Four-Dimentional Space." In this section is his first use of the word continuum. But nowhere until Part II which pertains to General Relativity does he deal with anything that includes acceleration. Time dilation was well understood as a consequence of Special Relativity long before he published General Relativity and the Twin Paradox arose before General Relativity as an effort to debunk Special Relativity and Einstein's time dilatation, not to support it.

The most common English translation of Einstein's first Special Relativity Postulate is:
"The laws of physics are the same in all inertial reference systems." The definition of an inertial system is that all motion is in Einstein's own words "in uniform translation " which indisputably disallows acceleration. That doesn't mean that time dilation that includes acceleration can't exist. it simply means you can't atribute that to Special Relativity.

 

[Frame Dragger]

This is incorrect. Here is a cut and paste that may be useful, from a FAQ I maintain at http://www.lightandmatter.com/cgi-bin/meki?physics/faq .

Dear bcrowell,

The scientific world never heard of time dilation until Einstein intorduced it along with his explanations about the consequences of Special Relativity. His explanation is in his book "Relativity" starting with section 7 and continuing through section 17 where he talks about "Minkowski's Four-Dimentional Space." In this section is his first use of the word continuum. But nowhere until Part II which pertains to General Relativity does he deal with anything that includes acceleration. Time dilation was well understood as a consequence of Special Relativity long before he published General Relativity and the Twin Paradox arose before General Relativity as an effort to debunk Special Relativity and Einstein's time dilatation, not to support it.

The most common English translation of Einstein's first Special Relativity Postulate is:
"The laws of physics are the same in all inertial reference systems." The definition of an inertial system is that all motion is in Einstein's own words "in uniform translation " which indisputably disallows acceleration. That doesn't mean that time dilation that includes acceleration can't exist. it simply means you can't atribute that to Special Relativity.

Nothing about TD violates SR's "The laws of physics are the same in all intertial reference systems. If one person is subjected to a force, or not, in one inertial frame, that doesn't require that everyone else be subjected to that force.

Everyone observing the process of the astronaut taking his TD trip would agree on the physics, and causality, and the outcome. That can be formulated within SR. GR extends the concept, but it was there in SR.

 

[Al68]

So that means the problem, as traditionally described, is not physically real. I tried to get the other members of this forum to recognize that but I was not successfull. Poster "Al68" said the same thing. Since he is a senior member of this forum, other members apparently accepted his statement without challange.

That's only because I pointed out that the "impossible time jump" was an artifact of the corresponding impossible instantaneous turnaround.

The fact that an instantaneous turnaround is often stipulated to simplify the math doesn't mean that anyone thinks it's physically possible. Everyone already agreed that an instantaneous turnaround, and therefore the "time jump", couldn't be "physically real". They just thought it was too obvious to be a legitimate point of discussion.

But an apparent time gap could be physically real: Suppose that the turnaround is very short compared to the interval between observations of the Earth clock by the ship twin. In that case, the ship twin would physically observe a "time jump".

Of course that's just the equivalent of "don't blink or you'll miss it".