Using General Relativity to analyze the twin paradox

[harrylin]

You are using "non-inertial" and "at rest" as if they were mutually exclusive. But in General Relativity, they are not. If you are at rest on the surface of a planet, the natural coordinate system is non-inertial.

The language in Einstein's paper is consistent with classical mechanics and SR; and there is negligible nearby mass in the discussion. However, Einstein does not use the word "inertial" there, so it would perhaps have been clearer if I had replaced the cited "inertial motion" by "uniform motion" or "Galilean motion", as "inertial" has become ambiguous.

In fact, if one considers "in rest" to mean the same as "accelerating" in this context then one mixes up the two different points of view that Einstein distinguishes. Then it's quite impossible to even understand what the discussion was about.

 

[stevendaryl]

Perhaps you use modern jargon that is incompatible with Einstein's 1918 paper and that could hinder a correct understanding.

I don't think so. In the dialog written by Einstein, he says the following: (from the point of view of an accelerated reference frame, K'):https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

1. A gravitational field appears, that is directed towards the negative x-axis. Clock U1 is accelerated in free fall, until it has reached velocity v. An external force acts upon clock U2, preventing it from being set in motion by the gravitational field. When the clock U1 has reached velocity v the gravitational field disappears.

2. U2 moves with constant velocity v up to point B of the positive x-axis. U1 remains at rest.

He's saying, that from the point of view of K', it is U2 that is at rest. But U2 is certainly NOT inertial. So it's a big mistake to conflate "being at rest" with "moving inertially".

I also disagree with you that the two mean the same thing in Newtonian physics, either.

You can write the Newtonian equations of motion in an arbitrary coordinate system as follows:

m \frac{d^2 x^j}{dt^2} = F^j + F_{fict}^j

where F^j is the same force that would be present in an inertial coordinate system, and F_{fict}^j is the extra terms due to curvilinear, noninertial coordinates. Being intertial means that F_{fict}^j = 0, while being at rest means that \frac{dx^j}{dt} = 0. Those aren't the same, at all.

 

[PeterDonis]

you appeared to be saying that K' according to Einstein considers itself to be an accelerating frame,

I said K' was a non-inertial frame. I did not say that means it is "accelerating". The only one who is equating "non-inertial" with "accelerating" is you.

 

[PeterDonis]

the separation between two objects is only an unambiguous vector if parallel transport is path-independent.

This is true for Minkowski spacetime, so a "Euclidean" manifold is not required. But a flat manifold (zero Riemann tensor) certainly is.

 

[stevendaryl]

This is true for Minkowski spacetime, so a "Euclidean" manifold is not required. But a flat manifold (zero Riemann tensor) certainly is.

But in Minkowsky spacetime, the separation vector between two objects is ambiguous if the objects are moving relative to one another. The vector will be frame-dependent. (I don't just mean that the components are frame-dependent---that's always the case.) The separation between EVENTS is unambiguous any Minkowsky spacetime, but not the separation between OBJECTS. That's the reason that in SR, forces can't really be direct interactions between objects; they have to be mediated by fields, which propagate.

 

[harrylin]

[..] But U2 is certainly NOT inertial.

While K' is an at times accelerating frame in SR according to all observers, K' is never accelerating or moving according to an Einstein observer who takes K' as reference; that's what I tried to clarify. It's still not clear to me if I managed to get that point through...

You can write the Newtonian equations of motion in an arbitrary coordinate system as follows:

m \frac{d^2 x^j}{dt^2} = F^j + F_{fict}^j

where F^j is the same force that would be present in an inertial coordinate system, and F_{fict}^j is the extra terms due to curvilinear, noninertial coordinates. Being intertial means that F_{fict}^j = 0, while being at rest means that \frac{dx^j}{dt} = 0. Those aren't the same, at all.

Those "extra terms" are fictional in Newton's mechanics; they correspond to the use of non-Galilean reference systems. In Newton's mechanics and SR, any frame that can be chosen as "rest frame" can also be chosen as "frame in uniform motion"; these together are considered a single class of "Galilean" reference systems (also said to be "preferred" systems as they prevent the need for such fictional terms).

 

[PeterDonis]

in Minkowsky spacetime, the separation vector between two objects is ambiguous if the objects are moving relative to one another. The vector will be frame-dependent.

Yes, agreed. But that's a separate requirement from parallel transport being path-dependent. It looks to me like both requirements would be needed for a "Machian" theory (at least for Mach's interpretation of what "Machian" meant ), and I think it's the second requirement that rules out pseudo-Riemannian manifolds (the first only rules out curved ones).

 

[PeterDonis]

K' is never accelerating or moving according to an Einstein observer who takes K' as reference

This is the same error I pointed out a few posts ago: since "accelerating" and "non-inertial" are not the same, neither are "not accelerating" and "inertial".

Note that this is because you are using "acceleration" to mean "coordinate acceleration"; if we use it instead to mean "proper acceleration", then we can equate "accelerating" with "non-inertial" and "not accelerating" with "inertial". But that's not the interpretation of "acceleration" you're using.

 

[harrylin]

I said K' was a non-inertial frame. I did not say that means it is "accelerating". [..].

You definitely wrote: "Einstein [..] said you feel the acceleration required to hold yourself at rest in the gravitational field."
I do hope that the mix-up in that sentence is clear now!
But apparently not.
If you use "acceleration" to mean "proper acceleration" then you use it in the contrary meaning of Einstein. That's verbal sabotage...

 

[stevendaryl]

While K' is an at times accelerating frame in SR according to all observers, K' is never accelerating or moving according to an Einstein observer who takes K' as reference; that's what I tried to clarify. It's still not clear to me if I managed to get that point through...

I'm not sure what you mean by an "accelerating frame". I don't consider frames to be objects, so they don't accelerate. To me, the important distinction is an inertial frame versus a non-inertial frame. K' is a noninertial reference frame. But an object that is "at rest" relative to K' can have zero (coordinate) acceleration.

Whether something is moving inertially or noninertially is a frame-independent notion. But whether something is accelerating or not (if by acceleration we mean coordinate acceleration) is frame-dependent.

Those "extra terms" are fictional in Newton's mechanics; they correspond to the use of non-Galilean reference systems. In Newton's mechanics and SR, any frame that can be chosen as "rest frame" can also be chosen as "frame in uniform motion"; these together are considered a single class of "Galilean" reference systems (also said to be "preferred" systems as they prevent the need for such fictional terms).

Yes, the inertial frames are special (in both Newtonian physics and SR) in not having the weird extra terms. But I don't think that it's an oxymoron to call something a noninertial rest frame. If you have a rotating frame, such as the Earth, it's still meaningful to talk about something being at rest relative to the Earth. Buildings and mountains and trees are all at rest relative to the Earth. The difference that a noninertial frame makes is that it requires a force to keep something at rest in a noninertial frame (as opposed to an inertial frame, where an object at rest will remain at rest without any forces applied).

 

[PeterDonis]

I do hope that the mix-up in that sentence is clear now!

Einstein did not really have a consistent term for "proper acceleration", so it's hard to describe what he said about it without using modern terminology. If you have a better term for "proper acceleration" that you think is consistent with Einstein's terminology, by all means suggest it.

 

[PeterDonis]

I am working in Einstein's frame K', with some valid chart, say the "MCIF solution".

This actually won't work, because all of the stars "behind" you will be beyond the Rindler horizon, so the coordinate chart for frame K' won't cover that portion of spacetime. It's actually non-trivial to find a chart for frame K' that does cover all of spacetime, or at least enough of it to include the distant stars. I'll assume that we've found such a chart in what follows, but it won't be the simple "MCIF solution" chart.

Can the "fields" here be explained as having a "physical" cause?

From the standpoint of GR, the appropriate law of physics is the Einstein Field Equation. If we use flat Minkowski spacetime as our solution, we are assuming there are no sources of gravity anywhere in the universe, which isn't really consistent with attributing anything to the distant stars. However, we could assume that the distant stars are distributed in a spherically symmetric manner, and use the GR version of the "shell theorem", which says that spacetime is flat in any vacuum region surrounded by a spherically symmetric matter distribution. So we could account for the fact that spacetime is flat in our local region (assuming we're way out in deep space far from all gravitating bodies) in this manner; and then any "gravitational field" we observe in our vicinity due to acceleration relative to the "distant stars" would just be due to that distant matter distribution making spacetime flat in our vicinity, plus the effects of accelerated motion in flat spacetime.

 

[harrylin]

In the dialog that is linked to in the very first post, Einstein doesn't explicitly use the word "Christoffel symbol", but [..] The equations of motion for a test mass in SR in general, non-inertial, curvilinear coordinates attributes the (coordinate) acceleration due to gravity to the Christoffel symbols:

\frac{d^2 x^j}{dt^2} = - \Gamma^j_{kl} \frac{dx^k}{dt} \frac{dx^l}{dt} - \frac{d log(\gamma)}{dt} \frac{dx^j}{dt}

(The second term is due to using the non-affine parameter t rather than proper time \tau; \gamma is the conversion factor: \frac{dt}{d\tau} = \gamma)

I suppose you meant GR; and you seem there to refer to physical, non-fictional gravitational fields like that of the Earth. But next:

Christoffel symbols are not physical fields. Think about this: you're walking directly away from a tree, and then you decide to turn around and start walking toward the tree. From your point of view, the tree is behind you, moving away from you, then stops and moves in a big circle until it is in front of you. What force could possibly cause a huge tree to behave that way? The answer is that there is nothing happening to the tree, it's only the location of the tree relative to a you-centered coordinate system that is changing.

That's all that Christoffel symbols do, is describe the aspects of the motion of objects that are artifacts of your coordinate system.

That's the exact contrary of what Einstein argued about his "induced gravitational fields"! Indeed, the objection of his early critics and later also Builder, was that his "induced fields" are merely fictive: artefacts of using an accelerating coordinate system.

 

[stevendaryl]

I suppose you meant GR; and you seem there to refer to physical, non-fictional gravitational fields like that of the Earth.

Actually, the same equation holds for SR or GR or even Newton-Cartan theory.

That's the exact contrary of what Einstein argued about his "induced gravitational fields"!

I don't know how that could possibly be true. It's just a fact that when you use curvilinear coordinates, you have to include Christoffel symbols in the equations of motion. If the usual equations of SR are valid in inertial coordinates, then the version with Christoffel symbols is valid in noninertial coordinates. That's just a mathematical fact. You can use SR in noninertial coordinates to compute trajectories or elapsed times on clocks. So either the description in terms of "induced gravitational fields" is exactly equivalent, or it's wrong.

Indeed, the objection of his early critics and later also Builder, was that his "induced fields" are merely fictive: artefacts of using an accelerating coordinate system.

I can't see how that could fail to be the case. Once again, SR in inertial coordinates completely determines what things look like in noninertial coordinates. There is no room for any additional physical assumptions. The room for disagreement is what you CALL the various terms. Whether you call something an "induced gravitational field" or a "Christoffel symbol", whether you call something "gravitational time dilation" or not, makes no physical difference.

 

[harrylin]

Einstein did not really have a consistent term for "proper acceleration", so it's hard to describe what he said about it without using modern terminology. If you have a better term for "proper acceleration" that you think is consistent with Einstein's terminology, by all means suggest it.

Happily stevendaryl already did so in post #39
Indeed, if there is a difference of opinion if the force that you feel far away from masses is due to an inertial effect from acceleration or due to "induced gravitation", then "force" is a non-ambiguous and factual term.

 

[harrylin]

[..] I don't know how that could possibly be true. It's just a fact that when you use curvilinear coordinates, you have to include Christoffel symbols in the equations of motion. If the usual equations of SR are valid in inertial coordinates, then the version with Christoffel symbols is valid in noninertial coordinates. That's just a mathematical fact. You can use SR in noninertial coordinates to compute trajectories or elapsed times on clocks. So either the description in terms of "induced gravitational fields" is exactly equivalent, or it's wrong. [..]

I suppose that you don't claim that the Earth's gravitational field is a fiction; and for sure Einstein did not. And it was the assertion of Einstein that he could make the set of Galilean frames non-preferred; in other words, that the laws of nature don't have such fictional terms any more in coordinate systems in arbitrary motion. The consequence of that is just as you say, only much stronger: Either the description in terms of "induced gravitational fields" is exactly equivalent and makes physical sense (i.e. can be looked at as being non-fictional), or it's wrong. Builder and most authors just argue that it is fictional; I go one step further, but I'll start a new thread on my own simple analysis including Doppler. This thread has become too much a thread on what people (Einstein, Builder, peterdonis) really mean.

 

[A.T.]

"force" is a non-ambiguous and factual term.

I prefer "proper acceleration". You don't need forces to proper accelerate a reference frame. And you can determine the proper acceleration in that frame using photons, for which the concept of force doesn't make sense.

 

[harrylin]

[..] K' is a noninertial reference frame. But an object that is "at rest" relative to K' can have zero (coordinate) acceleration.

That is and was already so in SR; Einstein clarified that he was not talking SR here.

Yes, the inertial frames are special (in both Newtonian physics and SR) in not having the weird extra terms. But I don't think that it's an oxymoron to call something a noninertial rest frame. [..]

I do consider that an oxymoron; and I'm certain that Einstein did not use such an oxymoron here. A correct and non-ambiguous term for that is noninertial reference frame.

 

[stevendaryl]

I suppose that you don't claim that the Earth's gravitational field is a fiction;

It depends on what you mean by fiction. I'm claiming that what's normally called "the gravitational field" in Newtonian mechanics correponds to Christoffel symbols in GR. They are coordinate-dependent, but given a choice of coordinates, the Christoffel symbols are objective.

and for sure Einstein did not.

Einstein did not believe that the gravitational field of Newtonian physics corresponds to the Christoffel symbols of GR? That seems like a pretty straight-forward calculation, to start with an exact GR solution, such as the Schwarzschild metric, compute the corresponding Christoffel symbols, and then show that in the limit of weak fields, \Gamma^j_{00} \Rightarrow - g^j, where g^j is the component of the Newtonian gravitational field.

 

[PeterDonis]

That's the exact contrary of what Einstein argued about his "induced gravitational fields"!

No, it isn't. Einstein's critics simply didn't understand that, on the view he was arguing for, a "gravitational field" could be both "real" and coordinate-dependent. You appear to suffer from the same confusion. I have pointed this out before.

 

[stevendaryl]

That is and was already so in SR; Einstein clarified that he was not talking SR here.

If there are two ways of deriving something and they give exactly the same answers in all situations, then it's hard for me to see how they could fail to be the same thing in different language.

Einstein's so-called GR analysis of the twin paradox has no physical content beyond the use of SR with noninertial coordinates. If you say it's not the same as SR in noninertial coordinates, you'll have to tell me why not. Saying that in the one case, certain terms are "regarded" as real forces, and in the other case they are "regarded" as Christoffel symbols is just a language choice. There is no difference, physically.

I do consider that an oxymoron; and I'm certain that Einstein did not use such an oxymoron here. A correct and non-ambiguous term for that is noninertial reference frame.

I think that's quibbling. "At rest" means "not moving". But in light of relativity, whether something is moving or not is relative to a coordinate system. Einstein himself uses the word "at rest" to describe the "traveling" clock. So I don't know why you want to say that K' is not a rest frame. It gives a standard of "rest". "Rest frame" and "reference frame" seem like synonyms to me.

My complaint about much of what you're saying is that you seem to be insisting that there are differences that make no difference, whatsoever. "Christoffel symbol" or "gravitational field" they both come into play in exactly the same way in problems involving noninertial observers. So why insist that they aren't the same thing?

 

[harrylin]

It depends on what you mean by fiction. I'm claiming that what's normally called "the gravitational field" in Newtonian mechanics correponds to Christoffel symbols in GR. They are coordinate-dependent, but given a choice of coordinates, the Christoffel symbols are objective.

Einstein intended gravitational fields, including the ones he invented, to be seen as just as physically "real" as magnetic fields. And I have never seen discussions in the literature about "fictive magnetic fields" or "pseudo magnetic fields".

Einstein did not believe that the gravitational field of Newtonian physics corresponds to the Christoffel symbols of GR?

I wrote that for sure Einstein did not claim that the Earth's gravitational field is a fiction.

 

[PeterDonis]

You don't need forces to proper accelerate a reference frame.

Really? Can you give an example of a "proper accelerated reference frame" in which an object at rest in the frame is not subject to a (non-gravitational) force?

 

[PeterDonis]

Einstein did not claim that the Earth's gravitational field is a fiction.

Sure, but he didn't claim that Christoffel symbols were a "fiction" either. You believe they should be viewed as a fiction (because they're coordinate-dependent), but Einstein did not take that view. Please stop mixing up your beliefs with Einstein's.

 

[stevendaryl]

Einstein intended gravitational fields, including the ones he invented, to be seen as just as physically "real" as magnetic fields. And I have never seen discussions in the literature about "fictive magnetic fields" or "pseudo magnetic fields".

I keep trying to steer away from terms that are emotionally-laden, but have no clear meaning, towards terms that are less emotionally laden, and have very clear meanings.

I don't care whether you call something "fictional" or "real". The distinction doesn't come into play in the physics. What does come into play is the terms in the equations of motion that are dependent on the choice of coordinates, and can be made to vanish at a point through a choice of coordinates (the Christoffel symbols). I don't know why you keep wanting to bring up terms that cause disagreement but shed no light on the problem.

I wrote that for sure Einstein did not claim that the Earth's gravitational field is a fiction.

There is no content that I can see to claiming that they are real or are fictional. There is content to saying that they are Christoffel symbols, since that tells you that:

1. The additional "forces" are strictly proportional to mass (which is another way of saying that the acceleration due to them is mass-independent)
2. The additional "forces" are necessarily quadratic in the 4-velocity.
3. The additional "forces" are the same for any test particle, regardless of its composition.
4. The additional terms can be made to vanish at a point through a choice of coordinates.
5. The additional forces do not obey Newton's third law (in a noninertial frame of reference, the "induced gravitational forces" are not paired with any "equal and opposite" force).

So there's a lot of information in calling them "Christoffel symbols" (and the term "fictional forces" pretty much means the same thing--the word "fictional" is just a technical term here, there is no implication other than 1-5 above) Calling them "real, induced gravitational fields" doesn't imply anything.

 

[stevendaryl]

Really? Can you give an example of a "proper accelerated reference frame" in which an object at rest in the frame is not subject to a (non-gravitational) force?

Yeah, to me, that's the big distinction between "proper acceleration" and "coordinate acceleration". Coordinate acceleration does not require any physical force, but proper acceleration always does.

 

[A.T.]

Really? Can you give an example of a "proper accelerated reference frame" in which an object at rest in the frame is not subject to a (non-gravitational) force?

I said nothing about forces on an object at rest in the frame. I said that the proper accelerated reference frame itself, as an immaterial object, doesn't require a force to be proper accelerated. That's why the concept of force is not usefull to describe proper acceleration of frames in general.

 

[harrylin]

[...] Einstein's so-called GR analysis of the twin paradox has no physical content beyond the use of SR with noninertial coordinates. If you say it's not the same as SR in noninertial coordinates, you'll have to tell me why not.

I told you but you did not hear me. The whole debate that that paper accounts would have been a farce. The physics papers testify that that debate was (and still is) real.

[...] "At rest" means "not moving". But in light of relativity, whether something is moving or not is relative to a coordinate system. Einstein himself uses the word "at rest" to describe the "traveling" clock. So I don't know why you want to say that K' is not a rest frame. It gives a standard of "rest". "Rest frame" and "reference frame" seem like synonyms to me. [..]

"Rest frame" inherited the assumption of "true rest": no artefacts or "funny things" in the description of nature by means of that reference system. That notion is not necessarily the case with "reference frame", which means simply what you want to say with "rest frame".

Suppose that someone has been brought up with the credo "War is Peace".
How can one possibly explain to that person, about someone who tried to argue that peace can be regarded as a form of war in some situations, that he wasn't just discussing warfare, and that the debate wasn't a farce but a true debate?

 

[stevendaryl]

I told you but you did not hear me.

It's not that I didn't hear, but that what you said made no sense. To say that the difference between a GR analysis and an SR analysis using noninertial coordinates is whether you call the additional terms "Christoffel coefficients" or "induced gravitational fields" seems completely trivial to me. Call the additional terms "Monkey dancing terms", that doesn't change the physics.

"Rest frame" inherited the assumption of "true rest":

How does it have that assumption? Newtonian physics has rest frames, but does not have any notion of "true rest".

no artefacts or "funny things" in the description of nature by means of that reference system. That notion is not necessarily the case with "reference frame", which means simply what you want to say with "rest frame".

It seems to me that "inertial" versus "noninertial" already takes into account those differences. You don't need the term "rest" to make that distinction.

Suppose that someone has been brought up with the credo "War is Peace". How can one possibly explain to that person, about someone who tried to argue that peace can be regarded as a form of war in some situations, that he wasn't just discussing warfare, and that the debate wasn't a farce but a true debate?

To make something into a true debate, as opposed to quibbling over words, you have to show a difference between the two points of view that is more than just terminology.

In the war versus peace scenario, maybe somebody considers economic sanctions to be a form of warfare, and so will disagree with the claim that country X is at peace with country Y. But you can clarify the situation by saying: "Okay, let's talk about bombs and invading armies. Can we at least agree that country X is not bombing country Y, and that it has not sent an army to invade it?"

There is an objective difference between the situation in which X is bombing Y and the situation in which X is not bombing Y. It's not merely a matter of terminology.

But the difference between "the extra terms are christoffel coefficients" and "the extra terms are induced gravitational fields" has NO consequences, other than terminology. So it's not a true debate, it's quibbling over terminology.

 

[stevendaryl]

The whole debate that that paper accounts would have been a farce. The physics papers testify that that debate was (and still is) real.

You have a different expectations about "debate" than I do. In my experience, the fact that a debate has been raging for years is not in any sense proof that it's a real debate, and not a farce.

 

[PeterDonis]

the proper accelerated reference frame itself, as an immaterial object, doesn't require a force to be proper accelerated

I guess this is true, strictly speaking, but I don't see how it's very useful. Any object at rest in the frame will have nonzero proper acceleration, and that's what is of interest as far as the physics is concerned.

 

[A.T.]

I guess this is true, strictly speaking, but I don't see how it's very useful.

It wasn't supposed to be useful insight, just an argument against using the concept force to describe proper accelerated frames in general, because it is too specific. In general you don't need an object at rest in the such frames in order to define them, or to measure their proper acceleration.

 

[Jimster41]

I just wanted to come back and confirm after thinking about this and trying to remember it.

Are these correct?

The question of "which twin ages slower" is determined by the proper acceleration. The traveling twin experiences proper acceleration (because he's on the rocket and is physically subject to the non-gravitational force applied by it). Proper acceleration causes length contraction and time dilation and that's why he ages slower.

The bowling ball and the feather, when dropped (or allowed to become inertial in a g-field), although they do experience "coordinate acceleration", experience no "proper acceleration". However, they do experience Time Dilation and Length Contraction determined by how much proper time they spend in the inertial frame of the g-field. If they were to each spend different amounts of proper time inertial (at rest) in the g-field they would accrue different total effects of time dilation and length contraction (assuming for the sake of the thought experiment one could be removed magically before the other one - without proper acceleration).

Also, in both methods of "exposure" to space-time distortion (proper acceleration and proper time spent in a g-field), the effects on physical objects are irreversible. More proper acceleration (regardless of direction) always adds to the length contraction and time dilation effect. In other words, the traveling twin (if he decelerated, turned and re-accelerated to head home) was also exposed to the time dilation, and length contraction effects in each of those steps (decell,turn,accell).

If I leave the feather in the inertial frame in the g-field longer than the bowling ball, but then I accelerate the bowling ball (imaging for a minute I magically took it over to my g-field isolation chamber) I could even up it's Time Dilation and Length Contraction "exposure" and the effect so that it is equal to the feather's.

As I think about the reversibility question, the irreversibility of time dilation seems intuitive, but length contraction - not so much?

 

[Jimster41]

Thinking about this some more (in the act of writing it out to you all) I believe I understand what I was missing. Let me know if this is still not right. The time dilation effect is a change to the rate of time, and the length contraction effect is a change to the metric of the accelerated frame. Once the twin comes home, he is back in the same "time-frame" and "metric-frame" as his sister. It's just that his body (because it's a proper-time dependent irreversible-process) has "processed" at a different irreversible rate...

 

[stevendaryl]

The question of "which twin ages slower" is determined by the proper acceleration. The traveling twin experiences proper acceleration (because he's on the rocket and is physically subject to the non-gravitational force applied by it). Proper acceleration causes length contraction and time dilation and that's why he ages slower.

I would not put it that way. It's true that in the twin paradox, there is a distinction between the two twins, in that one twin has nonzero proper acceleration, and the other does not. And it's also true in Special Relativity (but NOT in General Relativity) if one twin has zero proper acceleration than he will age more than the twin that has nonzero proper acceleration. But since we're talking about the "Using General Relativity to Analyze the Twin Paradox", you should realize that in General Relativity, it's not always the case that the unaccelerated twin ages the most. For example, standing on the ground, you throw one clock straight up in the air. A second clock remains on the ground. The one on the ground has nonzero proper acceleration the whole time. The thrown clock has zero proper acceleration the whole time except for the initial throw. But the thrown clock will have the greatest proper time.

The correct way to compute elapsed time is not by asking which clock experienced proper acceleration. You just use the proper time formula.
[/QUOTE]

 

[stevendaryl]

Thinking about this some more (in the act of writing it out to you all) I believe I understand what I was missing. Let me know if this is still not right. The time dilation effect is a change to the rate of time, and the length contraction effect is a change to the metric of the accelerated frame. Once the twin comes home, he is back in the same "time-frame" and "metric-frame" as his sister. It's just that his body (because it's a proper-time dependent irreversible-process) has "processed" at a different irreversible rate...

That is not how I would put it. Time dilation is not an absolute measure of "rate of time". Different coordinate systems will give different answers to the question: What rate is that clock ticking?

I like to make the comparison with roads on a flat section of the Earth. A road is geometrically a path through 2-D space in a similar way that the trajectory of a clock is a path through 4-D spacetime. You can set up road markers to measure your progress down a road--say one marker every 100 meters--in the same way that a ticking clock measures progress down a path through spacetime (say one tick every second). If there are two different roads connecting point A to point B, the roads can have different lengths (as measured by the number of markers encountered along each road). That doesn't mean that one of the roads had markers that were closer together, it just means that the path it took was longer.

In the same way, two clocks can take different paths from spacetime point A (that is, a point in space at a specific moment in time) to spacetime point B. The paths can have different elapsed times (as measured by the number of ticks encountered along each path). That doesn't mean that one path had ticks that were closer together, it just mean that the path it took was longer (as measured in spacetime geometry).

 

[harrylin]

How does it have that assumption? Newtonian physics has rest frames, but does not have any notion of "true rest".

It does, but many people don't know; and it's the same with Maxwell's theory, see next!

It seems to me that "inertial" versus "noninertial" already takes into account those differences. You don't need the term "rest" to make that distinction. [..]

Perhaps the following reminder will be helpful to clarify what Einstein was talking about:

"if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet.
[...]
Examples of this sort, together with the unsuccessful attempts to discover any motion of the Earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possesses no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.
[...]
Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good.2 In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the “stationary system.”

He later calls such frames also "Galilean" frames. Using other frames results in "funny" or fictional terms, in classical mechanics and SR alike. In the 1918 paper he defends his claim that with GR the frame that is all the time co-moving with the traveling twin may be treated as stationary frame, on equal footing with Galilean frames.

Builder and others argued that Einstein failed to achieve that equal footing; there remain fictional terms in his description with magical effects. However, peterdonis and you seem to argue in this thread that Einstein was effectively talking about SR in other words. My reply was that then it's all a farce.

In the war versus peace scenario, maybe somebody considers economic sanctions to be a form of warfare, and so will disagree with the claim that country X is at peace with country Y. But you can clarify the situation by saying: "Okay, let's talk about bombs and invading armies. Can we at least agree that country X is not bombing country Y, and that it has not sent an army to invade it?"

There is an objective difference between the situation in which X is bombing Y and the situation in which X is not bombing Y. It's not merely a matter of terminology.

The person who holds that War is Peace will say that bombing is a form of maintaining peace. The task is huge, it cannot be solved with a few clarifications like that.

But the difference between "the extra terms are christoffel coefficients" and "the extra terms are induced gravitational fields" has NO consequences, other than terminology. So it's not a true debate, it's quibbling over terminology.

I fully agree with you about christoffel symbols, thanks to your clarification. It was not me who suggested that they could be helpful for the discussion. Neither Einstein, nor Moller, nor Builder brought them up in this context.

PS. I had overlooked the post of bcrowell, #11. I think that he gave a striking sketch of the discussion here!

 

[harrylin]

Einstein originally wanted to interpret GR as a generalization of SR in which all frames of reference, including accelerated ones, were equally valid. It has probably been 70 to 90 years since this was considered a viable interpretation of GR. So to me the question posed here seems analogous to something like this:

In the Ptolemaic cosmology, the planets' cycles and epicycles are organized around the position of the earth. Is this consistent with special relativity, under which we would expect the Earth's influence to propagate at a velocity no greater than c?

Regretfully it has had little publicity that that interpretation was abandoned, perhaps as an after match of Builder's paper of 1957 (he writes about it as if at that time it was still popular). Moller's textbook which supports Einstein's original interpretation is also of the 1950s.

 

[stevendaryl]

He later calls such frames also "Galilean" frames. Using other frames results in "funny" or fictional terms, in classical mechanics and SR alike. In the 1918 paper he defends his claim that with GR the frame that is all the time co-moving with the traveling twin may be treated as stationary frame, on equal footing with Galilean frames.

Builder and others argued that Einstein failed to achieve that equal footing; there remain fictional terms in his description with magical effects. However, peterdonis and you seem to argue in this thread that Einstein was effectively talking about SR in other words. My reply was that then it's all a farce.

Well, as I said, there is no evidence of any physically meaningful content to Einstein's "GR analysis" that isn't captured by "SR in noninertial coordinates". The difference is just language. The only connection with GR that I can see is that GR fully exploits the idea that any coordinate system can be used. To me, the whole thing is just an exercise in noninertial coordinates.

In SR, it seems perverse to use noninertial coordinates, since everything is more complicated using them. But in GR, they are unavoidable; there is no global inertial coordinates. So you might as well get used to the weirdness. Technically, the difference between SR and GR, is that in SR, the metric tensor is static (the same everywhere), while in GR, the metric tensor is dynamic (it's affected by mass and energy). That's the ONLY difference, when it comes to analyses involving proper time, fictitious forces, etc. The "SR analysis in curvilinear, noninertial coordinates" is EXACTLY the same as the "GR analysis", mathematically. The only thing that is different is the exact form of the Christoffel symbols or "fictitious forces" or "induced gravitational forces".

So saying that Einstein's "GR analysis" is just "SR in noninertial coordinates" is not a complaint about the superficiality of his analysis. It's merely a description of how GR works. GR is SR in noninertial coordinates (with the additional feature that the metric tensor is affected by mass/energy, but that doesn't come into play in the twin paradox).

 

[Ben Niehoff]

I just wanted to come back and confirm after thinking about this and trying to remember it.

Are these correct?

The question of "which twin ages slower" is determined by the proper acceleration. The traveling twin experiences proper acceleration (because he's on the rocket and is physically subject to the non-gravitational force applied by it). Proper acceleration causes length contraction and time dilation and that's why he ages slower.

No, this is not a good way to think of it. As has been mentioned, proper time is just the "length" of one's worldline through spacetime. The twin that follows the longer path ages more. Full stop.

In flat spacetime, it happens to be that the twin on the longer path is undergoing proper acceleration in order to follow that path. But the path is what's important. In curved spacetime, you can have twins traveling on paths where they experience zero proper acceleration (i.e. geodesic paths), and yet they can meet up again having aged by unequal amounts.

 

[Dale]

Builder refers to several follow-ups in the literature that corroborate Einstein's calculation.

I don't know how it is possible to corroborate something that doesn't exist.

 

[Jimster41]

How is the mass/energy component of the GR description of the "virtual field", accounted for in the SR flat s.t. non-inertial coordinates description? I got locked onto "proper acceleration" and the idea that in the flat SR description, the connection term had to step-wise distribute/account for the same amount of mass/energy as the source of a field in which twins were seen to be stationary, as it "became curved" - that was the part I thought was intriguing about looking at the paradox from that latter perspective. Playing with the velocity of two masses in flat s.t over some set of steps has to account for the same mass energy of two stationary objects in a field that is curved due to mass energy. Take that black hole (or whatever mass) away, s.t goes flat, now you put that mass back piece by piece, but in a noticeably different "form" by creating a path length difference between the twins.

Thanks for the earlier corrections by the way. Focusing on path length helped. If it sounds like I'm trying to contradict stevendaryls or Ben niehoff's recent posts then I'm not getting my question across. I buy the argument in post 139.

Hopefully I'm at least back on topic, even if wrong. Sorry guys but I'm learning something here.

 

[harrylin]

I don't know how it is possible to corroborate something that doesn't exist.

It's certainly possible to verify if a calculation method that is claimed to work indeed works. I provided two references that verified that Einstein wasn't bluffing, for those who hold that Einstein's description is too vague to verify.

 

[harrylin]

[..] The question of "which twin ages slower" is determined by the proper acceleration. The traveling twin experiences proper acceleration (because he's on the rocket and is physically subject to the non-gravitational force applied by it). Proper acceleration causes length contraction and time dilation and that's why he ages slower. [..]

In addition to the other answers: it's not generally relevant what the person feels, what matters are velocity and gravitational potential. In the first space travel "twin example" by Langevin, the traveler didn't even feel any force at turnaround as he was traveling in a sling shot around a star.

 

[PeterDonis]

Neither Einstein, nor Moller, nor Builder brought them up in this context.

Einstein did bring them up, just not explicitly. As has been pointed out to you multiple times now, when Einstein uses the term "gravitational field", he means "Christoffel symbols". If you look at his actual technical papers (DaleSpam linked to a good one), this is obvious. He just didn't use the term "Christoffel symbols" (or the mathematical equivalent in the terminology of the time) in the non-technical article you linked to because it was for a general audience that might not understand what the mathematical term meant.

 

[PeterDonis]

for those who hold that Einstein's description is too vague to verify

Einstein's description in the non-technical article is too vague to verify. His description in technical articles (like the one DaleSpam linked to) is not. The paper you linked to has to make assumptions about what Einstein meant in the non-technical article in order to "verify" it.

(I don't think that paper is a very good source anyway; it seems to me to exhibit a number of confusions and to make some statements that are just false.)

 

[stevendaryl]

How is the mass/energy component of the GR description of the "virtual field", accounted for in the SR flat s.t. non-inertial coordinates description?

GR doesn't say that "gravitational fields" have mass/energy as the source. It says that the curvature tensor has mass/energy as the source. The "induced gravitational field" that results from switching to a noninertial reference frame has zero curvature, so it has no mass/energy associated with it.

 

[Jimster41]

GR doesn't say that "gravitational fields" have mass/energy as the source. It says that the curvature tensor has mass/energy as the source. The "induced gravitational field" that results from switching to a noninertial reference frame has zero curvature, so it has no mass/energy associated with it.

I take your first sentence to be an encouragement to be precise (per the wiki below), and I think I understand. The second sentence I'm still trying to get.
There is some energy required to curve the world line of the "traveling twin" through flat space-time?The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.[6] First published by Einstein in 1915[7] as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).[8]

The Einstein Field Equations can be written as

where

is the Einstein tensor and

is the stress–energy tensor.

This implies that the curvature of space (represented by the Einstein tensor) is directly connected to the presence of matter and energy (represented by the stress–energy tensor).

 

[PeterDonis]

There is some energy required to curve the world line of the traveling twin through flat space-time?

Curvature of a worldline is not the same as curvature of spacetime. You can have a curved worldline in flat spacetime, and a straight (geodesic) worldline in curved spacetime. They're two different concepts.

It is true that the traveling twin must have some means of curving his worldline (accelerating) in order to change course, such as a rocket, and this requires energy, and energy can in principle curve spacetime. However, the standard twin paradox scenario assumes that the energy required for the traveling twin to change course is too small to have any significant effect on the geometry of spacetime, so spacetime can be assumed to be flat. As a practical assumption, this works very well for all ordinary objects; to significantly curve spacetime, you need a very large object, like a planet or a star.

 

[Jimster41]

Curvature of a worldline is not the same as curvature of spacetime. You can have a curved worldline in flat spacetime, and a straight (geodesic) worldline in curved spacetime. They're two different concepts.

It is true that the traveling twin must have some means of curving his worldline (accelerating) in order to change course, such as a rocket, and this requires energy, and energy can in principle curve spacetime. However, the standard twin paradox scenario assumes that the energy required for the traveling twin to change course is too small to have any significant effect on the geometry of spacetime, so spacetime can be assumed to be flat. As a practical assumption, this works very well for all ordinary objects; to significantly curve spacetime, you need a very large object, like a planet or a star.

Thanks for seeing the part I'm confused about. The above definitely highlights it for me. So that's good.

I know you can't possibly teach me this here, so I'll let it go, and keep studying my Relativity textbooks (just now getting to curvature so... maybe all will be made clear for me soon) The part I'm missing is what physical thing is happening to the "traveling twin" in a single acceleration step, which is dependent on energy, by which time dilation and length contraction occur, but by which no other local physics is affected, that is not a local space-time "bending" moment... and to what is it happening?

Anyway, sincere thanks for your efforts.