Solving Basic SR Confusion for Twin Paradox

[cscott]

I'm trying to understand SR better however I think I've got myself confused in some ways.

Firstly, I can't think of exactly why a mechanical wave like sound should follow gallilean relativity (besides that F = ma is used on the "mechanical pieces" of the wave and we know F = ma obeys gallilean relativity), yet maxwell's equations don't. What's different regarding these two waves?

Secondly is a bit about the twin paradox (Sorry ;]): We assume the standard "paradox" where one twin goes out (straight line, short/quick accelerations) and comes back. One twin will feel accelerations but as I take it, it's the velocity that will "age" the twin, but what is difference between the velocity of the true departing twin perceived by the "stationary twin" (say, on earth) and the velocity of the "stationary twin" as perceived by the twin who feels accelerations?

 

[member 11137]

What do you exactly mean with euclidian relativity? Do you mean Galilean mechanic? And what is your trouble exactly?

 

[Doc Al]

Firstly, I can't think of exactly why a mechanical wave like sound should follow gallilean relativity (besides that F = ma is used on the "mechanical pieces" of the wave and we know F = ma obeys gallilean relativity), yet maxwell's equations don't. What's different regarding these two waves?

The difference is speed. Sound waves, and other things going slowly compared to light speed, appear to obey galilean addition of velocity. But that's only approximately true due to the low speeds involved--everything really obeys relativistic addition of velocities.

Secondly is a bit about the twin paradox (Sorry ;]): We assume the standard "paradox" where one twin goes out (straight line, short/quick accelerations) and comes back. One twin will feel accelerations but as I take it, it's the velocity that will "age" the twin, but what is difference between the velocity of the true departing twin perceived by the "stationary twin" (say, on earth) and the velocity of the "stationary twin" as perceived by the twin who feels accelerations?

I'm sure our experts will give you a more complete answer, but I'll just note that the observations made by the "stationary" twin are all made from a single inertial frame, while those made by the "moving" twin are not.

 

[smallphi]

The situation with the twins is not symmetric.

From the very beginning you have two possible frames - the one in which the Earth twin and the distant Star are not moving and the one in which the 'traveling' twin is not moving. They can't be both inertial since two inertial frames must have a constant relative velocity. Their relative velocity changes in direction when the traveling twin reaches the star and goes back. In order to use the SR formulas you have to decide from the onset which frame is the inertial one and we pick up the frame of Earth and the Star since they both don't have engines and are in a free-fall motion.

On the other hand had we decided from the onset that the traveling twin frame was the inertial one because the Earth and Star have some engines and propel back and forth creating the impression that the 'traveling' twin goes to the Star and back, we would have gotten exactly the opposite result, that the Earth twin is younger than the traveling.

So the asymmetry of the result is a consequence of our asymmetric choice which frame is inertial and writing the SR formulas in that frame or other inertial frames that move with constant velocities with respect to it.

 

[JesseM]

I'm trying to understand SR better however I think I've got myself confused in some ways.

Firstly, I can't think of exactly why a mechanical wave like sound should follow gallilean relativity (besides that F = ma is used on the "mechanical pieces" of the wave and we know F = ma obeys gallilean relativity), yet maxwell's equations don't. What's different regarding these two waves?

It's not so much the waves that are different, it's just that the laws of physics have the property of "Lorentz symmetry" which means they obey the same equations in different inertial coordinate systems related by the Lorentz transformation:

$$x' = \gamma (x - vt)$$
$$t' = \gamma (t - vx/c^2)$$
where $$\gamma = 1/\sqrt{1 - v^2/c^2}$$

On the other hand, the laws of physics would not obey the same equations if you created a similar coordinate system based on some other speed s, like the speed of sound:

$$x' = \gamma (x - vt)$$
$$t' = \gamma (t - vx/s^2)$$
where $$\gamma = 1/\sqrt{1 - v^2/s^2}$$

That's just because the laws of physics don't have the property of being symmetric relative to this coordinate transformation, this is not one of the symmetries of nature.

Secondly is a bit about the twin paradox (Sorry ;]): We assume the standard "paradox" where one twin goes out (straight line, short/quick accelerations) and comes back. One twin will feel accelerations but as I take it, it's the velocity that will "age" the twin, but what is difference between the velocity of the true departing twin perceived by the "stationary twin" (say, on earth) and the velocity of the "stationary twin" as perceived by the twin who feels accelerations?

Each twin's velocity will be different in different inertial coordinate systems, and you can analyze how much each twin ages using whichever inertial coordinate system you want. But as long as you're using inertial coordinate systems, the twin who accelerates will always change velocities when he does so, while the twin who moves inertially will have a constant velocity. For example, you could pick an inertial coordinate system where the twin on the ship is actually at rest during the outbound leg of the trip, and the Earth is moving away from it at relativistic speed--in this frame, the twin on the ship will actually age faster than the Earth twin during the outbound leg, not slower. But when the twin on the ship turns around and returns to Earth for the inbound leg, in this frame the twin on the ship will be moving even faster than the Earth, and thus aging slower, and if you actually calculate how much each twin ages during the inbound leg and add that to how much they age during the outbound leg, you'll still find that the twin on the ship has aged less in total, by exactly the same amount that you'd have predicted if you used the same procedure in the Earth's inertial frame.

 

[yogi]

"Secondly is a bit about the twin paradox (Sorry ;]): We assume the standard "paradox" where one twin goes out (straight line, short/quick accelerations) and comes back. One twin will feel accelerations but as I take it, it's the velocity that will "age" the twin, but what is difference between the velocity of the true departing twin perceived by the "stationary twin" (say, on earth) and the velocity of the "stationary twin" as perceived by the twin who feels accelerations?"

That is actually an unanswered question - just what role acceleration plays in time dilation. Einstein says in his 1905 paper that two separated clocks at rest in the same frame will read differently when one is moved toward the other until they meet - this involves a start-up acceleration but the amount of time dilation is given as a function of the velocity (assuming the acceleration is short). On the other hand, as Jesse points out, either twin could initially considered himself to be at rest and calculate how much time the other twin loses during the first part of the trip. In such an experiment, however, the initial conditions do not specify any acceleration - rather, the problem starts with the two twins in relative motion. Some well noted writers insist that a consideration of which twin initally accelerated is critical to avoid the paradox - that is, at the end of the first phase (reaching the turnaround point) two clocks cannot each be slower than the other. Where the times are compared in the same frame (e.g., the twin that accelerated stops and checks his time with a clock in the Earth frame before starting his return trip) at the end of the first leg, any difference must be attributed to asymmetry caused by the initial acceleration. Real age differences always involve an acceleration somewhere - whereas measurements made between two relatively moving inertial frames always result in an apparent slowing of the other guys clock.

 

[JesseM]

That is actually an unanswered question - just what role acceleration plays in time dilation.

This may be a philosophical question you have, but physicists do not consider it an "unanswered question."

Einstein says in his 1905 paper that two separated clocks at rest in the same frame will read differently when one is moved toward the other until they meet - this involves a start-up acceleration but the amount of time dilation is given as a function of the velocity (assuming the acceleration is short). On the other hand, as Jesse points out, either twin could initially considered himself to be at rest and calculate how much time the other twin loses during the first part of the trip. In such an experiment, however, the initial conditions do not specify any acceleration - rather, the problem starts with the two twins in relative motion. Some well noted writers insist that a consideration of which twin initally accelerated is critical to avoid the paradox - that is, at the end of the first phase (reaching the turnaround point) two clocks cannot each be slower than the other.

Which writers say this, specifically? They must be writers who disagree with the relativity of simultaneity, and thus disagree with the theory of relativity itself. If no frame's definition of simultaneity is to be preferred over any others, then there will be frames where the Earth-twin is older than the traveling twin "at the same moment" that the traveling twin turns around, and other frames where the Earth-twin is younger "at the same moment".

Real age differences always involve an acceleration somewhere - whereas measurements made between two relatively moving inertial frames always result in an apparent slowing of the other guys clock.

If one accepts the relativity of simultaneity, one can only talk about a "real" (i.e. frame-independent) age difference if the twins reunite at a single position in space, not when they are far apart.

 

[MeJennifer]

That is actually an unanswered question - just what role acceleration plays in time dilation.

It is fairly straight-forward:

The factor between the future accumulation of proper time for a clock that performs an instant acceleration and the future accumulation of proper time for a clock in the prior (perhaps even instantly comoving) frame of the first clock changes. The factor decreases when the direction of the acceleration is away from this other clock it increases when the direction is towards this other clock. The factor is never larger than one.
Conform the equivalence principle we can see the same effect near a gravitational field. Here the factor decreases as well for a clock that is closer to the center of a gravitational field.

 

[cesiumfrog]

Firstly, I can't think of exactly why a mechanical wave like sound should follow gallilean relativity (besides that F = ma is used on the "mechanical pieces" of the wave and we know F = ma obeys gallilean relativity), yet maxwell's equations don't. What's different regarding these two waves?

Medium.

The difference is that classical or mechanical waves (eg. sound) move at a constant speed with respect to a physical medium (eg. air), and you can devise experiments to measure the velocity of the medium itself. But with light, it travels at a constant and invariant speed with respect to every observer, which doesn't even make sense according to Galilean relativity.

 

[lalbatros]

... why a mechanical wave like sound should follow gallilean relativity ... yet maxwell's equations don't ...

The special relativity applies to both.
Galilean relativity applies approximatively for low-speed objects like sound waves.
Galilean relativity does not apply for high-speed object because it becomes a bad approximation for high speeds.
Galilean relativity is totally wrong very close to the speed of light.

 

[pmb_phy]

I'm trying to understand SR better however I think I've got myself confused in some ways.

Firstly, I can't think of exactly why a mechanical wave like sound should follow gallilean relativity (besides that F = ma is used on the "mechanical pieces" of the wave and we know F = ma obeys gallilean relativity), yet maxwell's equations don't. What's different regarding these two waves?

One important aspect of any answer is that light waves don't need a medium to travel through while a mechanical wave does. In the absense of a medium there is no preferred frame while in the presence of a medium there is a preferred frame, i.e. the rest frame of the medium.

Secondly is a bit about the twin paradox (Sorry ;]): We assume the standard "paradox" where one twin goes out (straight line, short/quick accelerations) and comes back. One twin will feel accelerations but as I take it, it's the velocity that will "age" the twin, but what is difference between the velocity of the true departing twin perceived by the "stationary twin" (say, on earth) and the velocity of the "stationary twin" as perceived by the twin who feels accelerations?

Its not the acceleration per se that causes the time dilation effect by the non-symmetry of the motion of the traveling twin. The traveling twin has a bent worldline whereas the stay at home twin travels on a straight world line.

This would actually make a great web page. I've been putting it off for some time but perhaps now is the time to start. Thanks.

Pete

 

[yogi]

This may be a philosophical question you have, but physicists do not consider it an "unanswered question." Which writers say this, specifically? They must be writers who disagree with the relativity of simultaneity, and thus disagree with the theory of relativity itself.


If no frame's definition of simultaneity is to be preferred over any others, then there will be frames where the Earth-twin is older than the traveling twin "at the same moment" that the traveling twin turns around, and other frames where the Earth-twin is younger "at the same moment". If one accepts the relativity of simultaneity, one can only talk about a "real" (i.e. frame-independent) age difference if the twins reunite at a single position in space, not when they are far apart.[/QUOTe

Einstein in his 1918 paper. Lederman, scima, Rindler, a few others, all insist that acceleration is necessary to explain age difference. See Rindler, special relativity at page 31



No - the traveling clock can be compared to any clock that is at rest in the Earth frame at any time. Synch your watch with an Earth clock and quickly accelerate to 0.1c toward Altair...then coast. Assume Altair and Earth have no relative motion - so they are in the same frame - when you arive at Altair compare your clock with one on Altair - there is a time difference - and since the Altair clock must read the same as the Earth clock - you have a one way time difference - the only asymmetry is due to the fact that you accelerated.

Here is what Rindler says: How is it that a large asymmetric effect can arise, and moreover, one that is proportional to the symmetric portions of the trip. The reason is that accelerations, however brief, have immediate and finite effects on A but not B.

 

[JesseM]

Which writers say this, specifically? They must be writers who disagree with the relativity of simultaneity, and thus disagree with the theory of relativity itself.

Einstein in his 1918 paper. Lederman, scima, Rindler, a few others, all insist that acceleration is necessary to explain age difference. See Rindler, special relativity at page 31

I agree acceleration is necessary to explain the age difference, but I was asking about some more specific statements of yours: "Some well noted writers insist that a consideration of which twin initally accelerated is critical to avoid the paradox - that is, at the end of the first phase (reaching the turnaround point) two clocks cannot each be slower than the other." Which writers say that which twin initially accelerated (as opposed to which twin accelerated to turn around once they had moved a large distance apart) is critical to avoid paradox? And which writers say that at the turnaround point "two clocks cannot each be slower than the other"? That latter statement was what I was referring to as a disagreement with the relativity of simultaneity, since the relativity of simultaneity means that different frames disagree about which twin has aged more at the time of the turnaround.

No - the traveling clock can be compared to any clock that is at rest in the Earth frame at any time.

It can't be compared in any absolute way, according to relativity. You can compare the time on the traveling clock with the time on the Earth clock in the Earth's rest frame, or you can compare it in the rest frame of the ship, and you'll get different answers.

Synch your watch with an Earth clock and quickly accelerate to 0.1c toward Altair...then coast. Assume Altair and Earth have no relative motion - so they are in the same frame - when you arive at Altair compare your clock with one on Altair - there is a time difference - and since the Altair clock must read the same as the Earth clock

The Altair clock only reads the same as the Earth clock in the earth/Altair rest frame. There is nothing that makes this frame physically preferred over any other inertial frame, according to relativity.

Here is what Rindler says: How is it that a large asymmetric effect can arise, and moreover, one that is proportional to the symmetric portions of the trip. The reason is that accelerations, however brief, have immediate and finite effects on A but not B.

I entirely agree with Rindler here, but this quote doesn't support the statements I was asking you about above.

 

[yogi]

It is fairly straight-forward:

The factor between the future accumulation of proper time for a clock that performs an instant acceleration and the future accumulation of proper time for a clock in the prior (perhaps even instantly comoving) frame of the first clock changes. The factor decreases when the direction of the acceleration is away from this other clock it increases when the direction is towards this other clock. The factor is never larger than one.
Conform the equivalence principle we can see the same effect near a gravitational field. Here the factor decreases as well for a clock that is closer to the center of a gravitational field.

In SR, the role acceleration plays in determining real age difference is not agreed upon - many writers insist that SR alone is sufficient to un-paradox the twin paradox. Others, which I have quoted above, insist that acceleration is necessary. Both get the same answer but by entirely different process - can you say there is no difference between 1) observing a spaceship passing Earth at 0.1c and 2) starting with the rocket clock at rest on Earth and initially synchronized with Earth clocks, then quickly accelerating to 0.1c - we assume here that both rockets travel to a distant point P and stop and we compare the accumulated time on the rocket clock with the Earth clock (e.g., by sending signals that indicated how many hours have been logged on the rocket clock). In the first case there is no bases for assuming any age difference inasmuch as we do not have a way to distinguish between the inertial frame of the rocket and the earth-P frame. In the second case there is an asymmetry introduced by the initial acceleration.

 

[yogi]

"It can't be compared in any absolute way, according to relativity. You can compare the time on the traveling clock with the time on the Earth clock in the Earth's rest frame, or you can compare it in the rest frame of the ship, and you'll get different answers"

Yes - I am using the Earth frame as the convenient frame in which to make comparisons - we start with all clocks at rest in the Earth frame and we end up with all clocks in the Earth frame at rest - but not necessarily close together

My point in raising the issue is directed at the questions posed by cscott - the literature is divided between those that resolutely insist SR is sufficient to explain the clock paradox and those that assert it is really a problem that must be treated by GR.

 

[yogi]

Jesse: "Which writers say that which twin initially accelerated (as opposed to which twin accelerated to turn around once they had moved a large distance apart) is critical to avoid paradox? And which writers say that at the turnaround point "two clocks cannot each be slower than the other"? That latter statement was what I was referring to as a disagreement with the relativity of simultaneity, since the relativity of simultaneity means that different frames disagree about which twin has aged more at the time of the turnaround. "

I am drawing a distinction between measurements made while the clocks are in relative motion (since each will see the other running slow, and two clocks cannot each be running slower than the other) and the age difference that will be measured when the traveling clock reaches the end of the outward journey, and all clocks are temporarily at rest in the Earth frame. In the latter case, the traveling clock will have accumulated less time assuming initial synchronization of all clocks in the Earth rest frame. I see no ambiguity so long as the comparisons are made in the same frame in which all clocks were initially synchronized. The amount of time lost on the one way trip is 1/2 that for the round trip assuming the other factors are the same (speed, path length etc)

 

[robphy]

My point in raising the issue is directed at the questions posed by cscott - the literature is divided between those that resolutely insist SR is sufficient to explain the clock paradox and those that assert it is really a problem that must be treated by GR.

I believe that part of this apparent division relies on the what is meant by "SR" and by "GR".

Does "SR" mean "the geometry of zero-curvature spacetime with R⁴ topology" [what I will call the modern geometric interpretation] or does "SR" mean that only algebraic,trigonometric, and geometric methods of [a single] Minkowski vector space (akin to methods of Euclidean geometry) [what I will call the older Einsteinian interpretation or SR-methods]?

Certainly one can use GR-methods (with, say, non-Cartesian coordinates and related Christoffel symbols) with the flat SR-spacetime [as one can use Riemannian methods with flat Euclidean space].

The classical twin paradox is a problem on a flat SR-spacetime with R⁴ topology. While algebraic,trigonometric, and geometric methods of [a single] Minkowski vector space are quite capable to explain it, one can (if desired) go further with GR-methods [on this flat SR-spacetime] to discuss additional details not easily done with those SR-methods (for example, how to generalize to non-SR spacetimes).

 

[JesseM]

I am drawing a distinction between measurements made while the clocks are in relative motion (since each will see the other running slow, and two clocks cannot each be running slower than the other)

Well, anyone who accepts relativity would disagree that "two clocks cannot each be running slower than the other" (in different inertial frames), I don't have a problem with this any more than I have a problem with the idea that in each clock's frame the other clock's velocity is higher, and there is no absolute truth about which clock "really" has a higher velocity.

Again, this is the thing I was asking you to back up when you said that "Some well noted writers insist that a consideration of which twin initally accelerated is critical to avoid the paradox - that is, at the end of the first phase (reaching the turnaround point) two clocks cannot each be slower than the other." Here it makes it sound as though these "well noted writers" agree with your claim that "the paradox" which must be avoided is the idea that two clocks are each running slower than the other. Did you not mean to imply that, and were you just adding the part about there being a problem with the rate of the clocks as your own opinion, not one shared by any of these "well noted writers"? If it's not just your personal opinion, than please show me a quote where any "well noted writer" suggests there must be an absolute truth about which of two separated clocks is running slower.

and the age difference that will be measured when the traveling clock reaches the end of the outward journey, and all clocks are temporarily at rest in the Earth frame. In the latter case, the traveling clock will have accumulated less time assuming initial synchronization of all clocks in the Earth rest frame. I see no ambiguity so long as the comparisons are made in the same frame in which all clocks were initially synchronized. The amount of time lost on the one way trip is 1/2 that for the round trip assuming the other factors are the same (speed, path length etc)

But it's totally arbitrary to choose the Earth/Altair frame and say that this frame's view of which clock accumulated less time is the "true" one. There is no physical motivation for it, it's just a matter of your own prejudices. As I recall, you also admitted that you had no general way to pick which frame's view is the "true" one on post #187 of this thread. As an example of the problem here, imagine that Earth and Altair pass by two other star systems, Murth and Faltair, which are moving at a high velocity relative to them...if Earth first passes next to Murth and the time is noted on Murth's clock, then Earth passes next to Faltair and the time is noted on Faltair's clock, then according to the synchronized clocks in the Murth/Faltair frame, less time has passed on Earth than passed in their frame...but during exactly the same flyby, if Murth first passes by Altair and then passes by Earth, then according to synchronized clocks in the Earth/Altair frame, less time passed by on Murth's clock than passed in the Earth/Altair frame. If you think there must be a truth about whether Murth's clock or Earth's clock is running slower, how would you decide it?

 

[MeJennifer]

Well, anyone who accepts relativity would disagree that "two clocks cannot each be running slower than the other" (in different inertial frames)

Sorry but I cannot possibly agree with that.

I think you are mixing up the light signals from the clocks with the clocks themselves. When two clocks are moving with respect to each other the light signals make it appear that each clock is running slower.

It is a simple fact that due to the relativity principle we cannot detect which clock is actually running slower, if any. If one of them has accelerated away from the other in the past then that one is certainly accumulating less time compared to the one that did not accelerate.

 

[JesseM]

Sorry but I cannot possibly agree with that.

Agree with what? I said 'anyone who accepts relativity would disagree that "two clocks cannot each be running slower than the other" (in different inertial frames)'. So are you saying that you agree with yogi it's impossible for two clocks to each be running slower than the other, even if we analyze the situation from two different inertial frames? Are you agreeing with him that there must be an absolute truth about which clock is 'really' running slower?

I think you are mixing up the light signals from the clocks with the clocks themselves. When two clocks are moving with respect to each other the light signals make it appear that each clock is running slower.

But the rate of ticking of a clock in a given inertial frame does not depend on what you see with light signals, it depends on measurements made on a network of clocks at rest in that frame which have been "synchronized" in that frame using the Einstein synchronization convention. As I'm sure you know, the relativistic Doppler effect indicates that a clock moving towards you will actually appear to be ticking faster than your own clock if you just look at it with your eyes (or a telescope), but in your rest frame it is ticking slower than your own clock, and measurements on a network of clocks at rest and synchronized in your frame would confirm this.

It is a simple fact that due to the relativity principle we cannot detect which clock is actually running slower, if any.

Yes, I agree. But that's what yogi was disagreeing with.

If one of them has accelerated away from the other in the past then that one is certainly accumulating less time compared to the one that did not accelerate.

You're not saying it's "accumulating less time" in any absolute sense, I hope? If a clock accelerated away from the Earth and then moved away at constant velocity, in the frame where the clock was at rest after its acceleration, the Earth's clock would be accumulating less time than the traveling clock, and this frame is just as good as any other inertial frame in SR.

 

[MeJennifer]

You're not saying it's "accumulating less time" in any absolute sense, I hope?

I am saying that. Acceleration is not relative but absolute.

A accelerates away from B. From that moment on A will accumulate less time than B. As soon as A accelerates in the opposite direction with the same amount then their accumulation of time with be again equal. If they both accelerate, then the higher A accelerates with respect to B, the less time A will accumulate compared to B.

A very simple example:

Two planets X and Y at rest with respect to each other.
A and B residing on X synchronize their stopwatches and accelerate momentarily in the direction of Y. A accelerates much higher than B. When they arrive on X, everybody in the universe will agree that A's clock has accumulated less time than B's.

If a clock accelerated away from the Earth and then moved away at constant velocity, in the frame where the clock was at rest after its acceleration, the Earth's clock would be accumulating less time than the traveling clock, and this frame is just as good as any other inertial frame in SR.

No the clock on Earth would not accumulate less time. Feel free to give a scenario where that would be the case. In flat space it is simply impossible. For the Earth to accumulate less time it would have to top the clock's acceleration.

It is important to distinguish between what the light signals tell us and the actual situation. If gets worse in general relativity, gravitational lensing can give us multiple copies of objects in space, that obviously does not mean that they are there.

 

[JesseM]

I am saying that. Acceleration is not relative but absolute.

Of course, but that doesn't justify your subsequent statements.

A accelerates away from B. From that moment on A will accumulate less time than B. If they both accelerate, then the higher A accelerates with respect to B, the less time A will accumulate compared to B.

In SR there is no absolute truth about which of two separated clocks is "accumulating more time", that's entirely frame-dependent--only when the clocks reunite at a single point in spacetime can there be a single frame-independent truth about which has accumulated more time. Suppose a probe accelerates from being at rest with respect to the Earth to moving away from it at 0.8c. Do you deny that in the inertial frame where the Earth was initially moving at 0.8c and the probe came to rest after accelerating, the Earth is from then on steadily accumulating less time than the probe after the acceleration? Of course if the probe later accelerates again so it's returning to Earth, in this frame the probe will be moving towards the Earth even faster than 0.8c so its clock will run slower than Earth's after this point, and this frame will agree that when the probe finally catches up to the Earth it has accumulated less time in total (all frames must agree on predictions about local events like the times on two clocks at the moment they meet at a single location). Still, this frame's analysis of the entire trip is no less valid than any other frame's, and in this frame the probe's clock was ticking faster than the Earth's clock on the outbound leg of the trip (accumulating more time), and slower than the Earth's clock on the inbound leg (accumulating less time). Do you disagree? If so, are you disagreeing with my statements about how time dilation works in this frame, or are you agreeing with that but somehow claiming this frame's perspective is less valid than the Earth's frame?

Also, the question of who accelerated initially is generally pretty irrelevant in any analysis of the twin paradox, completely irrelevant if the initial acceleration was instantaneous. Suppose ship A and ship B are next to each other and initially share the same rest frame, then A accelerates so it is moving away from B at 0.8c, then after a while accelerates again so it is moving back towards B at 0.8c. If B measures 10 years between the time A departs and the time A returns, A will have measured about 6 years (exactly 6 if the accelerations were instantaneous). Now compare this with a situation where it is B who accelerates initially so they are moving apart at 0.8c, then some time later A accelerates in the direction of B so that they are moving towards one another at 0.8c. The total time accumulated will be pretty much the same, exactly the same if the accelerations were instantaneous--If B measured 10 years between the moment B began to accelerate away from A and the moment A returned to B, then A will have measured about 6 years.

Two planets X and Y at rest with respect to each other.
A and B residing on X synchronize their stopwatches and accelerate momentarily in the direction of Y. A accelerates much higher than B. When they arrive on X, everybody in the universe will agree that A's clock has accumulated less time than B's.

If you draw the worldlines for both, then this looks just like the twin paradox, where B's worldline is straight between the point of A and B's worldlines diverging and the point where they reunite, while A's worldline has a bend in the middle between between these events, because A had to accelerate in the direction of B in order to come to rest on the planet Y when arriving prior to B. It's this middle acceleration that's important in explaining why A has accumulated less time when they reunite. You'd get the same answer if there was no initial acceleration at all, if A and B had been traveling through space at constant velocity since the beginning of time and their worldlines happened to cross at Earth, then when A arrived at planet Y it accelerated for the first time in its history to come to rest relative to Y, while B never accelerated and passed Y later.

 

[MeJennifer]

You clearly understand the principle of relativity and the implications of the constancy of the speed of light for all inertial frames but we seem to disagree on the interpretations.

I do not interpret that clocks run slower just because they are moving with respect to each other. It is an effect of relative motion. When a beacon in space radiates a blue light every second, then it physically radiates a blue light every second. How I move with respect to this beacon has no bearing on this.

But acceleration is the cause of real time differentials. The relative rate of accumulation of time for two objects is directly related to how they accelerate with respect to each other.

Anyway that's how I interpret it, you seem to have another interpretation.

 

[JesseM]

You clearly understand the principle of relativity and the implications of the constancy of the speed of light for all intertial frames but we simply disagree on the interpretation.

I do not interpret that clocks run slower just because they are moving with respect to each other. It is an effect of relative motion. When a beacon in space radiates a blue light every second, then it really radiates a blue light every second. How I move with respect to this beacon has no bearing on this.

But acceleration is the cause of real time differentials. The relative rate of accumulation of time for two objects is directly related to how they accelerate with respect to each other.

Anyway that's how I interpret it, you seem to have another interpretation.

Well, I agree with the statement that "acceleration is the cause of real time differentials", it's just that I think it only makes sense to talk about "real time differentials" when you bring two clocks together to compare them locally, at which point all frames agree on the time on each clock. I think the question of which of two clocks has "accumulated more time" when they are a large distance apart cannot have a single frame-independent answer, precisely because different frames disagree owing to their different definitions of simultaneity. If we start out at the same age and then I quasi-instantaneously accelerate and coast away from you 0.6c for a few years, are you claiming there is a single absolute truth about whether you are "really" older or younger than me when I have counted 4 years since we departed? If so, does this mean you think there is a single "correct" definition of simultaneity in this scenario?

 

[yogi]

Jesse - you have always been in denial about the examples Einstein gives in part IV of his 1905 paper. What I am saying, and as a read Jennifer's posts, is that there is an asymmetry introduced by the initial acceleration - I agree that without some initial conditions, all frames are equivalent and there is no way to make a sensible statement about one frame being different or preferred e.g., when two bodies pass each other we cannot say which was accelerated at some distant time in the past - but if you start with the same objects at rest in the same frame and one accelerates - you will, in general, get a real time difference when they are later brought to rest in the same frame - and they do not have to be adjacent to determine which has accumulated more time.

The person who started this thread raised questions about acceleration - I think the Rindler quote i gave is a simple statement as to the effect of acceleration on the rate at which clocks run - we cannot determine which is impacted and by how much so long as they are in relative motion (at least not easily) but we can make determinations when the two clocks are brought to rest in the same frame at a later time - this tells us that there is in fact a real difference in the rate at which the two clocks were accumulating time during the coasting phase

 

[JesseM]

Jesse - you have always been in denial about the examples Einstein gives in part IV of his 1905 paper.

No, I just deny the implications you draw from these examples, implications which Einstein never remotely states himself, and which in order to endorse would require him to deny the basic premises of his own theory.

What I am saying, and as a read Jennifer's posts, is that there is an asymmetry introduced by the initial acceleration

Well, MeJennifer didn't address my point that you can switch which object accelerated initially and as long as everything else is the same it'll have no effect on who has aged more when they reunite--perhaps you would care to address this?

- I agree that without some initial conditions, all frames are equivalent and there is no way to make a sensible statement about one frame being different or preferred e.g., when two bodies pass each other we cannot say which was accelerated at some distant time in the past

And if we did know which was accelerated at some distant time in the past, would this affect your answer about which clock was running faster, even if the acceleration happened millions or billions of years ago and they've both been coasting since? This is something I've never been clear on about your arguments, so please give me a yes or no answer.

- but if you start with the same objects at rest in the same frame and one accelerates - you will, in general, get a real time difference when they are later brought to rest in the same frame

You get a real difference of the times on both clocks, but that doesn't tell you anything about which actually accumulated more time since the moment the clock accelerated, since different frames disagree about the initial time on the non-accelerating clock at the moment the first clock accelerated (since they disagree about simultaneity), and there is no reason to prefer the point of view of the frame where they were initially at rest over other inertial frames except your own prejudices which you never provide the slightest reasoned argument for.

The person who started this thread raised questions about acceleration - I think the Rindler quote i gave is a simple statement as to the effect of acceleration on the rate at which clocks run - we cannot determine which is impacted and by how much so long as they are in relative motion (at least not easily) but we can make determinations when the two clocks are brought to rest in the same frame at a later time - this tells us that there is in fact a real difference in the rate at which the two clocks were accumulating time during the coasting phase

You have a knack for reading between the lines of statements by mainstream physicists to make them out to be relativity-disputers like yourself, even when their words don't state anything like that explicitly. Where in the Rindler quote does he suggest or even imply that "there is in fact a real difference in the rate at which the two clocks were accumulating time during the coasting phase?" Can you point to the specific words that you think imply this, and explain why you think they must be interpreted this way as opposed to given an interpretation consistent with relativity's principle that all inertial frames are equally valid?

 

[yogi]

I believe that part of this apparent division relies on the what is meant by "SR" and by "GR".

The classical twin paradox is a problem on a flat SR-spacetime with R⁴ topology. While algebraic,trigonometric, and geometric methods of [a single] Minkowski vector space are quite capable to explain it, one can (if desired) go further with GR-methods [on this flat SR-spacetime] to discuss additional details not easily done with those SR-methods (for example, how to generalize to non-SR spacetimes).

I would concur Rob, the distinction between GR and SR gets blurred in this example. If we say that whenever we introduce a consequence of acceleration, we are tacitly relying upon GR - then I pose the question of how to do the problem in SR (and assume in this puzzle, there is no affect upon the moving clock due to an initial acceleration) Specifically how do you distinguish which clock accumulated the greater amount of time in the one-way trip? If one denies the impact of the initial acceleration, the two frames will be forever indistinguishable - how can it be that the hi speed pion lives longer than the pion at rest in the lab? As Jesse seems to suggest, its equally valid to consider the Earth and the lab in motion, in which case from the perspective of the pion, it is the pion at rest in the lab that lives longer

 

[JesseM]

I would concur Rob, the distinction between GR and SR gets blurred in this example. If we say that whenever we introduce a consequence of acceleration, we are tacitly relying upon GR

No, we are only relying on GR if we try to consider things from the non-inertial "frame" of the accelerating object. But we can perfectly well use inertial frames to calculate the elapsed time on an accelerating clock and figure out the age difference.

how can it be that the hi speed pion lives longer than the pion at rest in the lab? As Jesse seems to suggest, its equally valid to consider the Earth and the lab in motion, in which case from the perspective of the pion, it is the pion at rest in the lab that lives longer

If you measure the life span of a pion by just noting how far it travels before decaying, then in the pion's frame this is explained by the fact that the lab has Lorentz-contracted so what seems like a long distance according to rulers at rest in the lab is actually a much shorter distance as seen by the pion. And if you could actually use synchronized clocks to measure the time the pion was emitted at the location it was emitted and the time the pion decayed at the location it decayed, then in the pion's frame these clocks would be out-of-sync, with the clock at the location of the decay having been significantly ahead of the clock at the location of emission, so in the pion's frame they're not giving an accurate measure of the time interval.

 

[MeJennifer]

Sorry Jesse, I seem to have deleted my posting while you replied to it.

The theory of relativity shows that:

(1) Clocks go out of sync due to the effects of the gravitational field and due to the effects of accelerations.
(2) Accelerations are absolute with respect to the gravitational field, even in a zero gravitational field as is the case in a Minkowski spacetime.

 

[pervect]

Sorry Jesse, I seem to have deleted my posting while you replied to it.

The theory of relativity shows that:

(1) Clocks go out of sync due to the effects of the gravitational field and due to the effects of accelerations.
(2) Accelerations are absolute with respect to the gravitational field, even in a zero gravitational field as is the case in a Minkowski spacetime.

I'm not sure what this second statement is supposed to mean, exactly. Anyway, I think it's simpler to just say that accelerations are absolute.

If you measure your velocity, you have to measure it relative to some other object.

In contrast, if you measure your acceleration, you can carry and read an accelerometer and measure your acceleration without reference to any other object.

 

[robphy]

I would concur Rob, the distinction between GR and SR gets blurred in this example. If we say that whenever we introduce a consequence of acceleration, we are tacitly relying upon GR

No, we are only relying on GR if we try to consider things from the non-inertial "frame" of the accelerating object. But we can perfectly well use inertial frames to calculate the elapsed time on an accelerating clock and figure out the age difference.

In both of these quoted statements, "GR" is being referenced with respect to its methods (e.g., the use of noninertial coordinates systems), akin to methods in Riemannian geometry (e.g., the use of non-Cartesian coordinates), which can be applied to both flat spacetimes (like Minkowski spacetime) and curved spacetimes. Thus, these uses of the term "GR" are not the modern geometrical interpretation [as found in Wald] which is more focused on geometrical structures rather than merely its methods.

 

[yogi]

Jesse: "And if we did know which was accelerated at some distant time in the past, would this affect your answer about which clock was running faster, even if the acceleration happened millions or billions of years ago and they've both been coasting since? This is something I've never been clear on about your arguments, so please give me a yes or no answer"

That is a question that is pondered in many articles that attempt to find an answer to real time dilation - it implies there are intrinsic clock rates tied to every object, and the rate at which time passes relative to other objects depends upon their relative velocity -

What I would say again, is that a careful reading of Einstein's examples in part IV of the 1905 paper can only lead to the conclusion that an asymmetry is created when one of two synchronized clocks is accelerated. The peculiar results Einstein refers to are real age differences as measured by two clocks at rest in the same frame - not apparent time dilations observed between two relativly moving frames while they are in motion. When one of two spaced apart synchronized clocks A at rest in the same frame is briefly accelerated to a velocity v toward the other clock B, it will be found that the A clock, after coasting for a time that is long compared with the acceleration duration, A clock will have accumulated less time when it reaches B. He tells you the clocks will be out of sync when compared in the same frame. There is no ambiguity is what is being conveyed - although you always want to weasel word it to introduce a different experiment to avoid the consequences.

I will answer some of the other questions you have posed later - right now I cannot give you a yes or no to your question because there are different ways to arrive at the results predicted by Einstein.

 

[yogi]

Jesse - there is an aspect of what I have said in post 32 that deserves more comment. Einstein interpreted the peculiar results based upon a manipulation of transforms that were derived by considering inertial frames in relative motion as equivalent - so his conclusion that there would be an age difference in the case of one of two accelerated clocks is fiat - it does not follow from the derivation - that bothers me and a lot of people - maybe you also. But Einstein was intuitive - he knew the result had to conform with the experiments and his own view of the impossibility of detecting motion wrt space - at some point time dilation had to be real - so he simply derives the result based upon an inconsistent premise and labels it as peculiar. This is the way things stood for 13 years - and after thousands of critical words had been written on the subject - then in 1918 he attempted to rationalize the 1905 result based upon GR - introducing a pseudo G force, he sanctified the 1905 result by providing a cause that explained the asymmetry.

 

[pervect]

This reasoning in #33 is really baroque.

The easy route is this: clocks always tick at 1 second per second. Philosophically, clocks do not tick at "different rates" at all. (I don't generally like to get into philosophy too much, but this is basically a large part of the problem here).

This is where yogi is confused. By taking the philosohical view that clocks always tick at 1 second per second, it becomes clearer that in order to compare different clocks, one needs to specify a means by which the comparison is to be made.

The apparent rates of the clocks are not dependent on the clocks themselves, the apparent rates are dependent on the comparison process

This philosphical confusion on yogi's part has been going on for a long time, so I suppose this post won't clear it up, but it's worth another shot. Yogi insists that the apparent rate difference between clocks depends only on the clocks. This is wrong. The apparent rate difference depends both on the clocks, and the means by which they are compared.

When in #32 yogi asks

And if we did know which was accelerated at some distant time in the past, would this affect your answer about which clock was running faster, even if the acceleration happened millions or billions of years ago and they've both been coasting since?

The answer is "it depends on how you compare them".

 

[yogi]

Pervect - you are misreading what has been said - your quoting from post 32 was a question posed by Jessee - not me. I have not answered the question because it is undefined as presented. Secondly, there is no mystery in comparing clocks brought to rest in the same frame to see if one has accumulated more time in a particular experiment than another. At no time do I suggest we make comparisions of clocks while they are in relative motion. What is at issue in Einstein's examples in Part IV is why two clocks initially in sync are out of sync at the end of the travel period. Einstein in 1905 does not discuss acceleration as factor - in fact he is dismissive of its influence by relegating it to a short duration relative to the overall travel time. Real age differences occur in the one way travel experiment - clocks measure the age difference - where do I imply that the clocks are affected - Einstein uses the clock as the thing that measures time, and so have I. We all know that clocks tick at one second per second in their own frame - where did I say otherwise?