Using General Relativity to analyze the twin paradox

[vanhees71]

2. The standard explanation that is given for the asymmetry is that the laws of nature of SR relate to inertial frames; it is faulty to apply the Lorentz transformations from an accelerating frame. According to SR only the "stay-at-home" may pretend to be "truly in rest"; the "traveler" may not claim that it is the "stay-at-home" who accelerates instead.

Why shouldn't you be allowed to use non-inertial frames within SR? It's also allowed in Newtonian mechanics which also obeys the special relativity principle as does SR. With your argument you'd even be forbidden to describe non-uniform (free-particle) motion at all. This is not right for sure, since SRT works well for accelerated particles. Otherwise LHC and other accelerators wouldn't work.

It doesn't matter, who is "truly at rest" or not. This doesn't make sense already in Newtonian physics. The usual hypothesis is that aging is given by the proper time of the object under consideration. It has been proven for unstable particles to very high accuracy ("age" = "mean lifetime"). Whether it has ever been checked for living organisms, I don't know, and I guess, it's hard to invent an experiment.

 

[harrylin]

Why shouldn't you be allowed to use non-inertial frames within SR? It's also allowed in Newtonian mechanics which also obeys the special relativity principle as does SR. With your argument you'd even be forbidden to describe non-uniform (free-particle) motion at all. [..]

Evidently you misunderstand "my" (Einstein's) argument. The papers that I referred to in post #90 show that SR is perfectly capable of describing non-uniform motion. There is also no issue with mapping to non-inertial frames in SR, nor can there have been such an issue; I gave the example of how common and accepted that was in Newtonian mechanics in post #34.

The "twin paradox" is a common textbook example of misapplying non-inertial frames so that some students wrongly conclude that clocks in rest in the inertial frame S will age less. Another example of misapplying accelerating frames was given in a parallel thread by me and Nugatory here :
#49 https://www.physicsforums.com/threa...-spaceship-paradox.804582/page-3#post-5054605
#54 https://www.physicsforums.com/threa...-spaceship-paradox.804582/page-3#post-5054628

 

[stevendaryl]

Although I agree with you, Einstein fully disagreed with you at the time that he wrote his 1918 paper. If this is not clear, I'll gladly contrast his statements with yours.

I think that early on, Einstein was hoping to develop a theory of gravity that was consistent with Mach's principle--that acceleration of one object was only meaningful relative to other objects. However, General Relativity doesn't actually have this property, in general. Empty spacetime still has a notion of acceleration. (I suppose you could consider spacetime itself to be an object, but unlike material objects, there is no notion of being at "rest" relative to it.)

 

[vanhees71]

I haven't followed all these arguments. But, again, proper time and proper distance are independent of the chosen coordinates, it's a Minkowski invariant (scalar). So there's no difference in the result, if I choose non-inertial coordinates. Of course, the same holds even true in general relativity, which is covariant wrt. to general diffeomorphisms. It's so by construction!

 

[harrylin]

I think that early on, Einstein was hoping to develop a theory of gravity that was consistent with Mach's principle--that acceleration of one object was only meaningful relative to other objects. However, General Relativity doesn't actually have this property, in general. Empty spacetime still has a notion of acceleration. (I suppose you could consider spacetime itself to be an object, but unlike material objects, there is no notion of being at "rest" relative to it.)

That empty space still has a "notion" of acceleration is just what Einstein denied - and it's that denial that led to the criticisms that he tried to counter with his 1918 paper.

 

[maline]

I think that early on, Einstein was hoping to develop a theory of gravity that was consistent with Mach's principle--that acceleration of one object was only meaningful relative to other objects. However, General Relativity doesn't actually have this property, in general. Empty spacetime still has a notion of acceleration. (I suppose you could consider spacetime itself to be an object, but unlike material objects, there is no notion of being at "rest" relative to it.)

How can this make sense? If we mean coordinate acceleration, then it is only relative, as Einstein explained in our article. If we mean proper acceleration, how could anyone have thought it relative? Any one of us standing on Earth is undergoing proper acceleration that isn't relative to any object!

 

[harrylin]

I haven't followed all these arguments. But, again, proper time and proper distance are independent of the chosen coordinates, it's a Minkowski invariant (scalar). So there's no difference in the result, if I choose non-inertial coordinates. Of course, the same holds even true in general relativity, which is covariant wrt. to general diffeomorphisms. It's so by construction!

Here we are dealing with something entirely different! What would you think if I claimed that one may equally well hold that your "non-inertial" coordinate frame K' is in fact not "non-inertial" but in rest? And that as a consequence, the "inertial" coordinate frame K can be considered to be an accelerating frame (in other words, the clock in rest in K has non-inertial coordinates)? That's what Einstein did.

 

[stevendaryl]

How can this make sense? If we mean coordinate acceleration, then it is only relative, as Einstein explained in our article. If we mean proper acceleration, how could anyone have thought it relative? Any one of us standing on Earth is undergoing proper acceleration that isn't relative to any object!

Well, Mach thought that acceleration should be relative. He didn't actually have a theory that worked that way, though. Mach's reasoning was that there should be no observable difference between:

1. Hopping into rocket ship and accelerating in a straight line in the x-direction.
   
2. Somehow contriving to get all the masses in the universe except for the rocket to accelerate in the negative x-direction.

As I said, General and Special Relativity are not Machian in this sense, because a rocket that is alone in the universe can still feel acceleration. (Actually, that's a bad example, since you can't accelerate without throwing mass behind you, in which case, there is some other mass that you can be accelerating relative to. A better example is rotation. If you are on a space station that is rotating, you can feel the rotation, even in the case where the space station is the only object in the universe, and so there is nothing that it is rotating relative to.)

 

[stevendaryl]

Here we are dealing with something entirely different! What would you think if I claimed that one may equally well hold that your "non-inertial" coordinate frame K' is in fact not "non-inertial" but in rest? And that as a consequence, the "inertial" coordinate frame K can be considered to be an accelerating frame (in other words, the clock in rest in K has non-inertial coordinates)? That's what Einstein did.

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.

 

[stevendaryl]

Well, Mach thought that acceleration should be relative. He didn't actually have a theory that worked that way, though. Mach's reasoning was that there should be no observable difference between:

1. Hopping into rocket ship and accelerating in a straight line in the x-direction.
   
2. Somehow contriving to get all the masses in the universe except for the rocket to accelerate in the negative x-direction.

As I said, General and Special Relativity are not Machian in this sense, because a rocket that is alone in the universe can still feel acceleration. (Actually, that's a bad example, since you can't accelerate without throwing mass behind you, in which case, there is some other mass that you can be accelerating relative to. A better example is rotation. If you are on a space station that is rotating, you can feel the rotation, even in the case where the space station is the only object in the universe, and so there is nothing that it is rotating relative to.)

It seems to me that for a theory to be Machian, space can't be a Riemannian manifold, but must be Euclidean. A machian theory would have to be formulated in terms of relative position vectors: the vector separation between objects. But the separation between two objects is only an unambiguous vector if parallel transport is path-independent. I'm not sure if that uniquely would make space Euclidean, but it surely constrains the geometry considerably. Also, if what's relevant is the separation at a particular time, it would seem to require that simultaneity be absolute, and not relative.

So based on a few moments thought, it seems to me that a machian theory would have to be formulated in something like Galilean spacetime, rather than a general pseudo-Riemannian spacetime.

 

[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.