Using General Relativity to analyze the twin paradox

[PeterDonis]

In a previous thread, reference was made to an entertaining "defense" of relativity by Einstein, which can be found here:

https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

One of the arguments Einstein makes in this dialog is that the twin paradox can be analyzed using "pseudo gravitational fields"--the traveling twin can explain the fact that his clock shows less elapsed time when he and the stay-at-home twin meet again, by saying that, when he fires his rockets to turn around, a gravitational field exists, and since the stay-at-home twin is at a much higher altitude in the field, his clock runs much faster. The Usenet Physics FAQ article on the twin paradox includes an analysis based on this same idea:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html

In this post in a recent thread, harrylin mentions a criticism of this argument of Einstein's, by Builder. An actual quote from his criticism is given in that thread, but here I will try to paraphrase it to get at the essence of the argument: the problem with the traveling twin's explanation in terms of a "gravitational field" is that, when he fires his rocket to turn around, this "field" must instantly propagate to the stay-at-home twin, and when he completes his turnaround and turns off his rocket, the disappearance of the "field" must likewise propagate instantly to the stay-at-home twin. But this violates causality: no field can propagate faster than light. So this "field" explanation cannot be physically correct.

The question I want to pose to the forum is: is this criticism correct? Does it invalidate the "GR analysis" of the twin paradox that is presented by Einstein and in the Usenet Physics FAQ? I have my own answer, but I'll put it in a separate post.

 

[PeterDonis]

My response to the question in the OP: no, Builder's criticism does not invalidate the GR analysis of the twin paradox. Here's why:

(1) The "field" that appears in the analysis does not have to propagate, because it is a coordinate effect, not a physical effect. No physical influence has to travel from the traveling twin to the stay-at-home twin. This can be verified by the fact that, if you compute the curvature of spacetime, it does not change: spacetime is flat throughout the scenario. According to the Einstein Field Equation, changes in spacetime curvature are what must propagate, and those changes propagate at the speed of light. But if there is no change in spacetime curvature, as in this case, there is nothing to propagate.

(A similar rebuttal can be made to the related claim that there is no source for the "gravitational field" that the traveling twin claims to exist, and a field cannot exist without a source. According to the EFE, spacetime curvature is what requires a source. There is no spacetime curvature in this scenario, so no source is required.)

(2) The criticism makes an implicit assumption (which, to be fair, Einstein can also be argued to have made) that any entity which appears in any frame's account of events must be "real". So if the traveling twin explains observations by using a "gravitational field", that field must be "real"; if it turns out not to be "real", the explanation cannot be valid.

However, this assumption is not correct. If it were, it would invalidate any explanation that invokes any frame-dependent quantity, since the argument for why the "gravitational field" invoked by the traveling twin is not "real" can be equally well applied to any frame-dependent quantity. So, for example, any explanation involving "time dilation" or "length contraction" would be invalid, since these are frame-dependent. The "gravitational field" that appears in the traveling twin's explanation is the same sort of frame-dependent quantity, and it is just as valid to use it in an explanation for that frame as it is to use time dilation or length contraction in an explanation.

 

[harrylin]

My response to the question in the OP: no, Builder's criticism does not invalidate the GR analysis of the twin paradox.

That implies regretfully a misunderstanding of the issue at hand. Not a modern GR calculation is at stake, nor your interpretation of GR, but Einstein's 1916 interpretation of GR. In a parallel thread I gave a quick impression of the view that he still tried to defend in 1918 (but apparently not in any later publication):

https://www.physicsforums.com/threads/how-does-light-slide-sideways.804112/page-2#post-5058667

In summary, he claimed that acceleration in GR is just as "relative" as velocity in SR, so that it is physically valid to regard an accelerating K' as not accelerating but truly in rest.
He set out to defend that point of view (which was the starting point of his development of GR) by means of his 1918 paper.

[..]
(1) The "field" that appears in the analysis does not have to propagate, because it is a coordinate effect, not a physical effect. [..]

That's a nice way of saying that Einstein's field is fake ("fictitious"). Einstein argued that the field is "induced" by acceleration relative to the stars, and that it is just as "real" and "physical" as for example a magnetic field that can be made to disappear by a suitable choice of coordinate system. The way I look at it, you debunked Einstein's interpretation with your first analytical remark alone.

(2) The criticism makes an implicit assumption (which, to be fair, Einstein can also be argued to have made) that any entity which appears in any frame's account of events must be "real". So if the traveling twin explains observations by using a "gravitational field", that field must be "real"; if it turns out not to be "real", the explanation cannot be valid.

However, this assumption is not correct. If it were, it would invalidate any explanation that invokes any frame-dependent quantity, since the argument for why the "gravitational field" invoked by the traveling twin is not "real" can be equally well applied to any frame-dependent quantity. So, for example, any explanation involving "time dilation" or "length contraction" would be invalid, since these are frame-dependent.

First, that's putting the argument on its head; and second, I don't follow your argument. Any explanation involving "time dilation" would be invalid if it cannot pretend full physical reality. Relativity of "time dilation" (for inertial motion) means that other descriptions are also physically fully valid.

The "gravitational field" that appears in the traveling twin's explanation is the same sort of frame-dependent quantity, and it is just as valid to use it in an explanation for that frame as it is to use time dilation or length contraction in an explanation.

Once more, I don't see how that can be right: if it is supposedly equally "physical" as for example time dilation and magnetism, then it cannot be treated as fictional according to the point of view that it truly exists. That looks like pure Doublethink to me!

 

[PeterDonis]

he claimed that acceleration in GR is just as "relative" as velocity in SR, so that it is physically valid to regard an accelerating K' as not accelerating but truly in rest.

More precisely, he claimed that a non-inertial frame is just as valid for doing physics as an inertial frame, provided you are willing to allow a "gravitational field" to exist in the non-inertial frame that does not exist in the inertial frame. Note that this applies to flat spacetime, i.e., "in SR"; you don't need to have curved spacetime to use non-inertial frames. Rindler coordinates are a simple example of a non-inertial frame in flat spacetime in which a "gravitational field" exists in Einstein's sense.

Whether this is equivalent to saying "acceleration is relative" is, to me, an unimportant matter of terminology, just like the question of what it means to say that "velocity is relative". It's unfortunate that we use the word "relative" to describe things that are, in fact, invariant: for example, we use the term "relative velocity" to describe something that, mathematically, is the inner product of two 4-velocity vectors, and is therefore a Lorentz scalar and is invariant. Similarly, as I remarked in a previous thread, we use the word "acceleration" to describe two things, one of which is frame-dependent and one of which is invariant. With our terminology in such a mess, it's no wonder that we get confused sometimes about what we are talking about.

That's why I would rather focus on the physics and on the mathematical description of the physics, which is unambiguous. Mathematically, my restatement of Einstein above is just the observation that a non-inertial frame can have nonzero connection coefficients.

Einstein argued that the field is "induced" by acceleration relative to the stars

Yes, and the justification for this is that, according to the Einstein Field Equation, the metric in a region of space is determined by the distribution of stress-energy in the past light cone of that region of space. When you fire your rockets and accelerate "relative to the stars", you are judging your acceleration by the light reaching you from those stars--which of course means you are looking at the sources in your past light cone. You are not judging your acceleration by the distribution of stars "right now". And of course the metric in your vicinity is indeed determined by the propagation (to the extent things even have to propagate--see below) of spacetime curvature from those sources in your past light cone, to your current spacetime location.

It is easy to miss the above in many ordinary situations because many ordinary situations are, to a very good approximation, static, so that the distribution of sources in the past light cone is the same as the distribution of sources "right now". For example, the distribution of the "distant stars" around our solar system has been essentially static and spherically symmetric for billions of years, and therefore there has been plenty of time for the spacetime curvature induced by that stress-energy distribution to determine the metric in our vicinity. But when you fire your rocket and feel the acceleration induced, that acceleration is still being proximately caused by the metric in your vicinity, not by the distant stars "instantly"; it is only caused by the distant stars via the propagation of spacetime curvature in your past light cone. The fact that it appears that the inertial force is induced "instantly" by accelerating (or rotating) relative to the stars is only an appearance, due to the fact that the situation is static.

The reason we know the above is the case is that we have measured the small corrections due to the fact that situations are not exactly static. The precession of the perihelion of the planets is an example of such a small correction (this measurement for Mercury was one of the "classic tests" of GR that Einstein made, but it has now been verified for at least all the inner planets, IIRC). This precession is due to the fact that the Newtonian "force" on a planet is not exactly pointed at the Sun "right now"; it only appears to be, to a good approximation, because the field of the Sun is, to a good approximation, static. But there are still small effects due to the non-staticity of the field, i.e., to the fact that changes in curvature can only propagate at the speed of light, and perihelion precession is one of them.

and that it is just as "real" and "physical" as for example a magnetic field that can be made to disappear by a suitable choice of coordinate system.

Einstein did choose an unfortunate comparison here, because a magnetic field is not a coordinate artifact the way connection coefficients are. For some electromagnetic fields, it is possible to find a frame in which the magnetic field is zero; we call these "electrostatic" fields, and they correspond to a very special physical situation. But for any other EM field, it is not possible to find a frame in which the magnetic field vanishes completely. However, it is always possible to find a frame in which the connection coefficients vanish.

As for the terms "real" and "physical", I personally would be just as happy to restrict those terms to direct observables: for example, I would be just as happy to say that length contraction and time dilation are "fictitious" in the same sense that you are saying the "pseudo gravitational field" in Einstein's analysis is "fictitious". Then we could focus on frame-invariant observables in place of those "fictitious" things, such as the observed Doppler shift in place of "time dilation". But if we are going to allow some of those frame-dependent things to be "real and physical", then I don't see any way to pick and choose and say that time dilation, for example, is "real" but the gravitational field in Einstein's example is not.

Relativity of "time dilation" (for inertial motion) means that other descriptions are also physically fully valid.

Why the restriction to inertial motion? Are you saying that, if I am moving non-inertially, "time dilation" somehow becomes unreal, but it's real if I'm moving inertially? On what do you base that distinction? Without that distinction, it seems to me that the same argument you are making for time dilation could be made for the gravitational field in Einstein's example.

if it is supposedly equally "physical" as for example time dilation and magnetism, then it cannot be treated as fictional according to the point of view that it truly exists.

I don't understand what you're trying to say here.

 

[PeterDonis]

Any explanation involving "time dilation" would be invalid if it cannot pretend full physical reality.

But what is "full physical reality" supposed to mean? How do I tell that time dilation has "full physical reality", even though it's frame-dependent, whereas Einstein's "gravitational field" does not?

The only distinction I see being made in Builder's criticism is propagation: something "real" has to propagate no faster than light. But time dilation is not "real" in this sense any more than the "gravitational field" is: time dilation does not propagate. You can change it "instantly" by changing coordinates. That is basically the point of my rebuttal: that without some principled way to distinguish frame-dependent things that are "real" from frame-dependent things that are not, Builder's criticism of Einstein cannot stand.

 

[Dale]

There is nothing wrong with an explanation in terms of a non causal field. For instance, the Coulomb gauge is a perfectly valid gauge in which to describe EM.

The object that Einstein referred to as "the gravitational field" in his piece was the Christoffel symbols. They are a legitimate part of the theory, just like the potentials are a legitimate part of EM, regardless of their "propagation speed".

That said, this presentation by Einstein is a good example of why pop-sci sources are not considered valid references here. Even Einstein couldn't do a pop-sci piece that is also rigorous.

 

[Orodruin]

Another issue is if to really call this a GR analysis or not. Space time is still Minkowski space and the only difference is that we have tried to find a set of coordinates which are locally equivalent to the standard coordinates for the non-accelerated parts of the traveling twin. It is still just SR, but in some set of curvilinear coordinates.

I have to agree with Dale, this is an example of the impossiblility to write accurate pop-sci.

 

[harrylin]

Another issue is if to really call this a GR analysis or not. Space time is still Minkowski space and the only difference is that we have tried to find a set of coordinates which are locally equivalent to the standard coordinates for the non-accelerated parts of the traveling twin. It is still just SR, but in some set of curvilinear coordinates.

I have to agree with Dale, this is an example of the impossiblility to write accurate pop-sci.

From those remarks it appears that Einstein's 1916 interpretation of GR as well as the objection to it are not sufficiently rehashed in this thread. First of all, your remark is the essence of the early criticism on Einstein's views. Moreover, you are sharing Builder's criticism here. And also Baez claims that this is not really a GR analysis.

Next, Einstein's claims as well as the objections against them have nothing to do with "pop-sci". As referenced (admittedly not one, but two clicks away), he made those claims in his 1916 overview paper on GR. The objection was that -contrary to Einstein's claims- a rigorous and self consistent analysis in which a frame that is in arbitrary motion can be pretended to be truly in rest all the time, is not possible.

Once more:

- Einstein main goal was not to predict the effects of gravitation, although that very important achievement resulted from his work. He held that SR's Minkowski space (which is still "Galilean space" for rotation) should be rejected and replaced by a Machian space ("The Galiliean space, which is here introduced is however only a purely imaginary cause, not an observable thing"). While that also has been the subject of discussions, it's still not the issue that became the topic of this thread. He went even one step further: just as it is the case that in SR, the laws of physics are valid in any Galilean frame so that one may treat any such frame as a true rest frame, "The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner."
He claimed that GR achieves this, so that an observer at rest in a frame in arbitrary motion can always consider that frame as a true rest frame for the description of phenomena according to the established laws of physics. Einstein tried to kill three flies with one blow.

- The main objections focused on this last mentioned claim, which was in direct contradiction with Langevin's claim about his example of the traveling astronauts. The accelerating astronaut can not just as well claim to be not accelerating but all the time truly in rest; that is not a valid point of view for describing physical phenomena if one wants the laws of nature to hold with respect to that frame. But Einstein now claimed exactly the contrary. That created the "twin paradox", and most modern authors don't know or don't acknowledge the existence of Einstein's proposed solution (however, Moller provides the calculation that Einstein did not include in his 1918 paper).

Perhaps I should have clarified in my first reply that it is simply not correct that "One of the arguments Einstein makes in this dialog is that the twin paradox can be analyzed using "pseudo gravitational fields" " (emphasis mine), just as in SR a capacitor is not argued to have a "pseudo magnetic field" according to a frame in which the capacitor is in motion.

I'll get back to peterdonis' comments later.

 

[Ben Niehoff]

I've only read the OP, so this may have been said. But...

I think the best criticism of this argument is that it assumes that one can define a global notion of simultaneity for observers in arbitrary motion. For observers in inertial motion in Minkowski space, it is obvious that the spacelike plane perpendicular to their four-velocity vector is the appropriate surface of simultaneity. But for observers in accelerated motion, there is no natural way to extend local notions of simultaneity to global ones. We can certainly agree that a "surface of simultaneity", if one should exist, should be perpendicular to the observer's worldline; however, this does not give us enough information to define a unique surface.

The implication of this is when either observer is undergoing accelerated motion, there is in fact no unambiguous way to compare the rates at which their clocks tick. So the question of considering "Twin A in the pseudo-gravitational field experienced by Twin B" is nonsensical to begin with. The time difference when the twins meet again does not come from comparing local rates of time along their worldlines; it comes from integrating the proper time along the worldlines and concluding that their paths through spacetime have different lengths.

Secondly, I think arguments about the speed of field propagation are subtle, and may not be correct here. For example, in standard (Lorentz-invariant!) electrodynamics, the monopole part of the electric field always manages to point directly toward the current location of the source charge (in arbitrary motion), due to a mathematical conspiracy with the magnetic field.

Finally, as probably others have mentioned, in Minkowski space there is of course no gravitational field. That one observer is undergoing acceleration does not allow you to use the equivalence principle to say there is a gravitational field, let alone one that applies to the inertial observer. The equivalence principle does not actually equate gravitational fields and acceleration; it equates acceleration with uniform fields. But the gravitational field is precisely a non-uniform acceleration field, because the Riemann tensor specifically measures the non-uniformity.

 

[stevendaryl]

(2) The criticism makes an implicit assumption (which, to be fair, Einstein can also be argued to have made) that any entity which appears in any frame's account of events must be "real". So if the traveling twin explains observations by using a "gravitational field", that field must be "real"; if it turns out not to be "real", the explanation cannot be valid.

Just to add to what you've said: In GR, gravitational fields have exactly the same ontological status as fictitious forces such as "centrifugal force". Centrifugal forces seems completely real in the sense that you can reason about them using force diagrams: Anything that doesn't have a centrally directed force (such as the tension force of a string) will be flung away by centrifugal force. This includes buildings and planets infinitely far away; they are all affected equally by centrifugal force. But centrifugal force ISN'T real, in the sense that it has no source, and it propagates instantly; it has very different behavior from real fields that have sources and propagate at the speed of light.

 

[bcrowell]

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?

 

[PeterDonis]

It has probably been 70 to 90 years since this was considered a viable interpretation of GR.

I don't understand. Doesn't GR say that the laws of physics must look the same in all valid coordinate charts, inertial or not?

 

[PeterDonis]

The accelerating astronaut can not just as well claim to be not accelerating but all the time truly in rest; that is not a valid point of view for describing physical phenomena if one wants the laws of nature to hold with respect to that frame.

This is not correct. The Einstein Field Equation, along with all other laws of nature in proper tensor form, holds in all valid coordinate charts. So an accelerating astronaut can use a chart in which he is always at rest and call that his "rest frame". He may find that the calculations he has to do are more complex, but that doesn't make his claim invalid.

 

[PeterDonis]

The implication of this is when either observer is undergoing accelerated motion, there is in fact no unambiguous way to compare the rates at which their clocks tick. So the question of considering "Twin A in the pseudo-gravitational field experienced by Twin B" is nonsensical to begin with.

There doesn't need to be a unique way to do this for the analysis to be valid; there only needs to be some valid coordinate chart that covers the required region of spacetime and has nonzero connection coefficients (which are the "gravitational field"). This is certainly true for the twin paradox scenario under discussion.

 

[Dale]

From those remarks it appears that Einstein's 1916 interpretation of GR as well as the objection to it are not sufficiently rehashed in this thread.

harrylin, this thread (as posed by Peter Donis in the OP) is about a specific objection (superluminal propagation of the gravitational field) to a specific paper by Einstein ("Dialog about Objections ..." 1918). There is no need to rehash everything on the subject, and I don't know why you are bring in 1916.

Next, Einstein's claims as well as the objections against them have nothing to do with "pop-sci".

The specific paper by Einstein which we are discussing is "pop-sci". The fact that it is "pop-sci" is not the specific objection that this thread is discussing, but a different objection that I personally have to the paper under discussion. It is not really relevant to the topic at hand, but I didn't want to leave the impression that I agreed with the paper simply because I disagreed with the specific objection under discussion.

a rigorous and self consistent analysis in which a frame that is in arbitrary motion can be pretended to be truly in rest all the time, is not possible.

Such an analysis is certainly possible, although that is not the topic of this thread and was certainly not something that was done in the pop-sci work in question.

 

[PeterDonis]

it is simply not correct that "One of the arguments Einstein makes in this dialog is that the twin paradox can be analyzed using "pseudo gravitational fields" " (emphasis mine),

If it's just the "pseudo" that you object to, feel free to ignore it. I was simply referring to the fact that Einstein himself, in places, points out the difference between this field, in flat spacetime, and the field that is present in a curved spacetime.

 

[PeterDonis]

Another issue is if to really call this a GR analysis or not.

Yes, in the modern view, "GR" means "curved spacetime", rather than "curvilinear coordinates" or "gravitational field" (nonzero connection coefficients), which was how Einstein was (sort of) thinking of it. So in the modern view, this analysis is not a "GR" analysis, it's just "SR in curvilinear coordinates".

However, calling it a "GR analysis" also draws attention to the essential unity between the flat and curved spacetime cases, from the standpoint of GR: flat spacetime is just one particular solution to the EFE, not some separate domain of physics.

 

[bcrowell]

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.

I don't understand. Doesn't GR say that the laws of physics must look the same in all valid coordinate charts, inertial or not?

There are many problems with Einstein's early interpretation. The biggest problem with it is that SR has absolutely no problem dealing with accelerated frames of reference (or with arbitrary coordinate charts). The modern way of defining the distinction between SR and GR is that SR assumes zero curvature.

 

[Orodruin]

There are many problems with Einstein's early interpretation. The biggest problem with it is that SR has absolutely no problem dealing with accelerated frames of reference (or with arbitrary coordinate charts). The modern way of defining the distinction between SR and GR is that SR assumes zero curvature.

This is something I never really thought about: The flatness requirement would not exclude some strange geometries (such as a torus world) which are flat, but clearly distinct from Minkowski space. Do we consider this SR? It could include things such as closed timelike curves etc.

 

[bcrowell]

This is something I never really thought about: The flatness requirement would not exclude some strange geometries (such as a torus world) which are flat, but clearly distinct from Minkowski space. Do we consider this SR? It could include things such as closed timelike curves etc.

I don't think there is any widespread agreement on whether topologically nontrivial SR is still referred to as SR. It does introduce some issues that are not present in ordinary SR. As you point out, there can be CTCs. Other examples: (1) there can be a preferred frame, e.g., in 1+1 dimensions where the spacelike dimension wraps around, there is a frame in which the circumference is maximized; (2) you may be unable to cover all of spacetime with a single coordinate chart.

 

[PAllen]

Though it is often said that SR is a subset of GR, I do not agree with this. With GR, if you have any field (e.g. EM, which leads, by itself, to non-vanishing curvature) or any mass (however small), you do not have SR. On the other hand, flat spacetime with non-trivial topology has fundamentally different group structure than conventional SR. Thus I would argue that flat spacetime with non-trivial topology is in the domain of mathematical physics, and is neither SR nor GR.

As to this thread, IMO, I have no problem with Einstein's point of view in the 1918 paper, except that I would call the analysis SR if you choose to ignore the mass of the observers.

 

[Jimster41]

It will be homework to read this whole thread.

My Theery (you guys have seen Monty Python right?):

Because the second twin wanted to head "home" eventually, he had to keep an exact history of what his frame had done at every proper instant since leaving. Thank god for the nav computer.

By subtracting that history from his current frame (using his thrusters) he effectively creates a consistent "field" between himself and his twin at home.

The actions of the thrusters at the turn, contain the information (and energy) to turn some far off location into a spot on the home coordinate "field" - a long ways down it's side. The distant location of the final turn for home could be randomly accelerated nine ways from Sunday for all the traveling twin knows - but his nav computer better have kept track. Either that or he has to be able to see home.

Seems kind of interesting to imagine the twin on the ship is in hypersleep, and the twin at home is responsible for bringing his bro home, either with a tractor beam, that was waiting to get turned on, or by sending a signal to the ship. The implications w/respect to energy and information seem consistent.

That's probly wrong.

 

[bcrowell]

Though it is often said that SR is a subset of GR, I do not agree with this. With GR, if you have any field (e.g. EM, which leads, by itself, to non-vanishing curvature) or any mass (however small), you do not have SR.

Whenever a new theory of physics replaces an old one, the old one is always found to have been in error in all cases. However, the error will have been negligible in the experiments that originally led to the acceptance of the old theory. (This is the correspondence principle.) Therefore it is true, but only in a very trivial and uninteresting sense, that no theory of physics can ever be a subset of a newer and more general theory. In a nontrivial and interesting sense, SR is indeed a special case of GR. For example, particle physicists use SR, not GR, and the GR corrections are negligible for them.

 

[PeterDonis]

The biggest problem with it is that SR has absolutely no problem dealing with accelerated frames of reference (or with arbitrary coordinate charts).

Ah, I see; you are reading Einstein's original interpretation as saying that, if you are using a non-inertial chart, you are using GR. Ok, yes, I agree that's not a valid claim.

The modern way of defining the distinction between SR and GR is that SR assumes zero curvature.

Agreed.

 

[Ben Niehoff]

The distinction between GR and SR is that SR assumes global Lorentz invariance. This means not only that it is flat, but also rules out nontrivial topologies.

 

[martinbn]

If the space-time is flat, simply-connected and maximal, then it is Minkowski space-time (well, it is a if and only if statement). I suppose when people say flat the other two are understood and not explicitly mentioned.

 

[AshUchiha]

In a previous thread, reference was made to an entertaining "defense" of relativity by Einstein, which can be found here:

https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

One of the arguments Einstein makes in this dialog is that the twin paradox can be analyzed using "pseudo gravitational fields"--the traveling twin can explain the fact that his clock shows less elapsed time when he and the stay-at-home twin meet again, by saying that, when he fires his rockets to turn around, a gravitational field exists, and since the stay-at-home twin is at a much higher altitude in the field, his clock runs much faster. The Usenet Physics FAQ article on the twin paradox includes an analysis based on this same idea:.

I guess what you're saying is related to this?

 

[stevendaryl]

The distinction between GR and SR is that SR assumes global Lorentz invariance. This means not only that it is flat, but also rules out nontrivial topologies.

It's true that people lump considerations of curvature and nontrivial topologies to GR. Dealing with curved spacetime requires a lot of mathematical machinery that flat spacetime does not, but conceptually it doesn't seem that big a leap beyond SR. Conceptually, you break spacetime into little regions, and make sure that SR holds (approximately) in each region, and that solutions in neighboring regions are consistent in the overlap.

To me, the transition from SR to GR has a number of steps:

1. SR in Cartesian, inertial coordinates.
2. SR in curvilinear, noninertial coordinates.
   
3. SR in curved spacetime and nontrivial topologies.
   
4. The field equations relating curvature to the stress/energy tensor.

The transition from 1 to 2 is just mathematics, not physics, even though it's kind of difficult mathematics. But once you've got to step 2, you've already got most of the machinery needed to go on to step 3. Once you've allowed the components of the metric tensor to be nonconstant (which is what you need for curvilinear, noninertial coordinates), allowing spacetime to be curved is not a big leap. I think that what took Einstein so long in developing GR was the final step.

 

[Jimster41]

Id like to know how there can be any two separate things in flat Minkowski space (if that's what space is). If two things are separate, even if they are inertial, at some time they had to have been "separated". Acceleration is the only path from any state, with the exception of "separate" itself, to "separate", isn't it? And I've never heard anyone propose that all the separate things started as the things they are.

Separateness implies acceleration somewhere in the history of "things", and through the equivalence principle (curvature due to acceleration can't be distinguished from curvature due to mass) GR is introduced?

 

[stevendaryl]

Id like to know how there can be any two separate things in flat Minkowski space (if that's what space is). If two things are separate, even if they are inertial, at some time they had to have been "separated". Acceleration is the only path from any state, with the exception of "separate" itself, to "separate", isn't it? And I've never heard anyone propose that all the separate things started as the things they are.

Separateness implies acceleration somewhere in the history of "things", and through the equivalence principle (curvature due to acceleration can't be distinguished from curvature due to mass) GR is introduced?

I don't understand exactly what you're asking or claiming, but I want to correct your terminology slightly: There is no "curvature due to acceleration". What the use of noninertial coordinates does is not to create curvature (because curvature is independent of what coordinates you are using). Using noninertial (or curvilinear) coordinates results in "fictitious forces" appearing in the equations of motion for test masses. The equivalence principle says basically that these fictitious forces are indistinguishable (sort of) from "gravitational forces". More precisely, the equivalence principle is the claim that gravitational forces ARE fictitious forces due to the use of noninertial coordinates.
But gravitational forces are not due to curvature.

 

[PeterDonis]

flat Minkowski space (if that's what space is).

It's not. The Minkowski geometry is a spacetime geometry (a flat one), not a space geometry. More generally, "space" does not have a unique meaning, because any spacetime (Minkowski or otherwise) can be split up into "space" and "time" in multiple different ways.

So the correct way to formulate concepts like "separate things" is not to look at whether they're separate in space, but whether they're separate in spacetime. For example, a given object is modeled as a worldline, or more generally a "world tube" in spacetime--a region of spacetime occupied by the object. Different objects occupy different regions of spacetime. This definition is independent of whether the objects are "accelerated" or not; it's purely in terms of spacetime geometry and which portions of it are occupied by different objects.

 

[harrylin]

More precisely, he claimed that a non-inertial frame is just as valid for doing physics as an inertial frame, provided you are willing to allow a "gravitational field" to exist in the non-inertial frame that does not exist in the inertial frame. Note that this applies to flat spacetime, i.e., "in SR"; you don't need to have curved spacetime to use non-inertial frames.

Others including me and you clarified how a gravitational field cannot explain the observed phenomena.
And one should not confound SR's use of non-inertial frames with Einstein's claims about a physical explanation by means of induced gravitational fields.

Whether this is equivalent to saying "acceleration is relative" is, to me, an unimportant matter of terminology, just like the question of what it means to say that "velocity is relative".

If you are not interested to try to understand Einstein's early interpretation of relativity, which happened to be the topic of his 1918 paper; then it's totally useless to discuss that paper here.

when you fire your rocket and feel the acceleration induced, [...]

I agree of course. In contrast, Einstein: or when you fire your rocket and feel the induced gravitational field!

Einstein did choose an unfortunate comparison here, because a magnetic field is not a coordinate artifact the way connection coefficients are. For some electromagnetic fields, it is possible to find a frame in which the magnetic field is zero; we call these "electrostatic" fields, and they correspond to a very special physical situation. [..]

Not unfortunate, but exactly what he meant. In SR such fields are totally valid; in no way are they fictitious. Fictitious and relative are not to be confounded.

As for the terms "real" and "physical", I personally would be just as happy to restrict those terms to direct observables: for example, I would be just as happy to say that length contraction and time dilation are "fictitious" in the same sense that you are saying the "pseudo gravitational field" in Einstein's analysis is "fictitious".

That would be a serious mistake. As in a parallel thread on Bell's Spaceship was explained, length contraction and time dilation can be treated as perfectly real in the chosen inertial frame: all SR's laws of nature work perfectly.

Then we could focus on frame-invariant observables in place of those "fictitious" things, such as the observed Doppler shift in place of "time dilation". But if we are going to allow some of those frame-dependent things to be "real and physical", then I don't see any way to pick and choose and say that time dilation, for example, is "real" but the gravitational field in Einstein's example is not.

Unwittingly you actually did so, as I pointed out, right at the start. A real gravitational field must obey GR. In GR, cause and effect is assumed and gravitational fields propagate at local speed c. Moreover, light from distant stars also propagates at local speed c.

Why the restriction to inertial motion? Are you saying that, if I am moving non-inertially, "time dilation" somehow becomes unreal, but it's real if I'm moving inertially?

Certainly not. If one mistakenly treats an accelerating frame as an inertial frame, this creates nonsense and paradoxes as discussed in the parallel thread on Bell's Spaceship.

 

[stevendaryl]

One should not confound SR's use of non-inertial frames with Einstein's claims about a physical explanation by means of induced gravitational fields

I don't agree. The "induced gravitational field" that Einstein was talking about is EXACTLY SR in noninertial coordinates. There is no difference.

 

[harrylin]

I don't agree. The "induced gravitational field" that Einstein was talking about is EXACTLY SR in noninertial coordinates. There is no difference.

In that case Einstein would have been debating about nothing - apart of complexity, nobody has or had a problem with non-inertial coordinates! It's even commonly used in classical mechanics. Mapping to the geoid by means of Newton's mechanics is right at the start of many textbooks.

 

[stevendaryl]

I don't agree. The "induced gravitational field" that Einstein was talking about is EXACTLY SR in noninertial coordinates. There is no difference.

The SR equations of motion for a test mass are simplest when you use inertial cartesian coordinates and parametrize using proper time. Then it's just:

m \frac{d U^\mu}{d\tau} = F^\mu

where F is the 4-force. So it looks just like Newton's F=ma.

If you switch to using curvilinear, noninertial coordinates and use some other parameter s besides proper time, you have, instead:

m (\frac{d \tilde{U}^\mu}{ds} + \Gamma^\mu_{\nu \lambda} \tilde{U}^\nu \tilde{U}^\lambda + \frac{d log(f)}{ds} \tilde{U}^\mu) = \tilde{F}^\mu

where \tilde{U} and \tilde{F} is rescaled versions of U and F, and where f = \frac{ds}{d\tau}

The "induced gravitational field" due to acceleration just amounts to moving terms from the left-hand side to the right-hand side, and writing:

m \frac{d \tilde{U}^\mu}{d\tau} = F_{eff}^\mu = \tilde{F}^\mu + F_{grav}^\mu

where

F_{grav}^\mu = - m(\Gamma^\mu_{\nu \lambda} \tilde{U}^\nu \tilde{U}^\lambda + \frac{d log(f)}{ds} \tilde{U}^\mu)

Whether you put the terms F_{grav} on the left side, and call them connection coefficients, or put them on the right side, and call them gravitational forces, is just a matter of taste, but it doesn't change the physics.

Is F_{grav} a real force, or not? Well, it's not real, in that it's not due to any source. People talk about it being "induced by acceleration", but that's not true, really. They are induced by the choice of the noninertial coordinate system. That choice isn't forced on you by the fact that you're in an accelerating rocket. A person inside a rocket can use inertial coordinates just as well as someone floating inertially. Anybody can use any coordinates they like; you don't have to use coordinates in which you, personally, are at rest.

On the other hand, F_{grav} is real, in the sense that it is measurable, to the same extent that coordinate acceleration is.

 

[stevendaryl]

In that case Einstein would have been debating about nothing - apart of complexity, nobody has or had a problem with non-inertial coordinates! It's even commonly used in classical mechanics. Mapping to the geoid by means of Newton's mechanics is right at the start of many textbooks.

Well, I don't see any content to the "induced gravitational field due to acceleration" above and beyond what was already known in Newtonian physics in noninertial coordinates. I really do think that Einstein's GR resolution to the twin paradox had no content above and beyond SR in noninertial coordinates. Now, I think that the discussion was useful, in that it shows how the same situation can be viewed as velocity-dependent time dilation in one set of coordinates, and "gravitational" time dilation in another set of coordinates. But that doesn't actually provide any new insight about the twin paradox. Instead, it provides insight about GRAVITY -- real gravity due to masses. To me, the usefulness of equating "fictitious forces" with "gravitational field" is not that it provides any new insight about SR, but that it provides insight about the nature of gravitational fields. Using GR to solve an SR problem is ridiculous, in my opinion. But using SR to solve (approximately) a problem involving clocks at different altitudes on Earth is a big deal. The problem can't be solved without the equivalence principle, unless you go all the way to full GR.

 

[Dale]

A real gravitational field must obey GR. In GR, cause and effect is assumed and gravitational fields propagate at local speed c.

You are starting with a valid premise but making an incorrect conclusion here.

In GR whatever you mean by the term "gravitational field" clearly must obey GR, however what that term refers to has changed over time. Einstein used the term "gravitational field" to refer to the Christoffel symbols. According to GR the Christoffel symbols are not required to "propagate at local speed c". Thus the statement that the "gravitational fields propagate at local speed c" is false using Einstein's terminology.

More modern usage would be to either not use the term "gravitational field" at all or to use it to refer to the Riemann curvature tensor. The statement that the "gravitational fields propagate at local speed c" would be correct using that terminology. But that is not the argument that Einstein is making nor the terminology that he was using.

 

[PeterDonis]

Others including me and you clarified how a gravitational field cannot explain the observed phenomena.

Huh? Where did I say that?

one should not confound SR's use of non-inertial frames with Einstein's claims about a physical explanation by means of induced gravitational fields.

Since by "gravitational field" Einstein meant "Christoffel symbols", and since those are only nonzero in flat spacetime in a non-inertial frame, I don't see how this is "confounding" at all; it's just matching up the math with Einstein's ordinary language explanation.

In contrast, Einstein: or when you fire your rocket and feel the induced gravitational field!

No, Einstein did not say you "feel" a gravitational field. He said you feel the acceleration required to hold yourself at rest in the gravitational field. The gravitational field is introduced in the non-inertial frame in which you are at rest, in order to explain how you can be at rest while at the same time feeling acceleration. But the field itself is not "felt", any more than the field of the Earth is "felt"--the field of the Earth is introduced to explain how you can be sitting at rest on the Earth while at the same time feeling acceleration. The fact that a gravitational field, by itself, is not felt, is the whole point of Einstein's "happiest thought", that a person falling freely will not feel his own weight.

A real gravitational field must obey GR.

Minkowski spacetime does obey GR: the stress-energy tensor is zero and the Einstein tensor is zero. This is true whether you use inertial or non-inertial coordinates, since tensor equations are covariant.

In GR, cause and effect is assumed and gravitational fields propagate at local speed c.

No, changes in spacetime curvature propagate at local speed c. But in flat spacetime, the curvature is always zero, so there are no changes to propagate.

light from distant stars also propagates at local speed c.

Yes, it does. So what?

 

[stevendaryl]

I agree of course. In contrast, Einstein: or when you fire your rocket and feel the induced gravitational field!

This is perhaps a subtle distinction, but you don't "feel" any induced gravitational field on board an accelerating rocket, and you don't "feel" a gravitational field when standing on the Earth. What you feel is the force of the floor pushing up against the bottom of your feet. Gravity doesn't explain this force, it explains why, in spite of the force, you stay in the same location (relative to Earth-fixed coordinates).

 

[Jimster41]

The SR equations of motion for a test mass are simplest when you use inertial cartesian coordinates and parametrize using proper time. Then it's just:

m \frac{d U^\mu}{d\tau} = F^\mu

where F is the 4-force. So it looks just like Newton's F=ma.

If you switch to using curvilinear, noninertial coordinates and use some other parameter s besides proper time, you have, instead:

m (\frac{d \tilde{U}^\mu}{ds} + \Gamma^\mu_{\nu \lambda} \tilde{U}^\nu \tilde{U}^\lambda + \frac{d log(f)}{ds} \tilde{U}^\mu) = \tilde{F}^\mu

where \tilde{U} and \tilde{F} is rescaled versions of U and F, and where f = \frac{ds}{d\tau}

The "induced gravitational field" due to acceleration just amounts to moving terms from the left-hand side to the right-hand side, and writing:

m \frac{d \tilde{U}^\mu}{d\tau} = F_{eff}^\mu = \tilde{F}^\mu + F_{grav}^\mu

where

F_{grav}^\mu = - m(\Gamma^\mu_{\nu \lambda} \tilde{U}^\nu \tilde{U}^\lambda + \frac{d log(f)}{ds} \tilde{U}^\mu)

Whether you put the terms F_{grav} on the left side, and call them connection coefficients, or put them on the right side, and call them gravitational forces, is just a matter of taste, but it doesn't change the physics.

Is F_{grav} a real force, or not? Well, it's not real, in that it's not due to any source. People talk about it being "induced by acceleration", but that's not true, really. They are induced by the choice of the noninertial coordinate system. That choice isn't forced on you by the fact that you're in an accelerating rocket. A person inside a rocket can use inertial coordinates just as well as someone floating inertially. Anybody can use any coordinates they like; you don't have to use coordinates in which you, personally, are at rest.

On the other hand, F_{grav} is real, in the sense that it is measurable, to the same extent that coordinate acceleration is.

This was helpful. I really appreciate this kind of relatively gentle use of formal symbolism in context, to illuminate the ambiguity of perception being discussed. I'd love to have that more often, though I can see why it's a bit painful.

When you say gravity is not real because it's not due to any "source", There is a temperamental rub or me, (if I am following) in terms of what metaphors or adjectives adhere...for me the bizarre elasticity of inertial perspective in space time, counts as a "source" of a pretty interesting sort. It's not like when you change coordinate frames the physics of acceleration changes. They can be described differently, but when you accelerate, (or are near mass) spacetime elasticity and its distortion is causing things that are recognizably "real" to happen (maybe even "most real"). It seemed for a moment you were suggesting space-time distortion was "not real".

Just watched Interstellar last night. Lots of chuckles and cringes at the physics references, but some of the visualizations were lovely. One sequence about GR, did a good job of giving me the willies.

 

[PhoebeLasa]

I really do think that Einstein's GR resolution to the twin paradox had no content above and beyond SR in noninertial coordinates.

I think Einstein's use of GR to explain the twin "paradox" has an additional importance that has not yet been mentioned here: Einstein's GR solution gives the same result, for the rocket-twin's "point-of-view" about how the home twin's current age varies during the trip, as is given by the co-moving-inertial-frames solution in SR. Einstein's GR solution does not agree with the "Radar Solution" of Dolby & Gull. And Einstein did not appear to regard the rocket-twin's conclusions about the aging of the home twin to be some kind of arbitrary choice of simultaneity "convention".

 

[Jimster41]

I think Einstein's use of GR to explain the twin "paradox" has an additional importance that has not yet been mentioned here: Einstein's GR solution gives the same result, for the rocket-twin's "point-of-view" about how the home twin's current age varies during the trip, as is given by the co-moving-inertial-frames solution in SR. Einstein's GR solution does not agree with the "Radar Solution" of Dolby & Gull. And Einstein did not appear to regard the rocket-twin's conclusions about the aging of the home twin to be some kind of arbitrary choice of simultaneity "convention".

Yeah, now I'm confused again. If acceleration is only a figment due to non-inertial choice of reference frame why does the "travelling" twin age more slowly. If gravitational force due to acceleration is "fictitious" and only due to choice of reference frame, Just because he had a rocket firing at his brother doesn't mean he was the one accelerating.

I had been picturing that he was, then you convinced me that one could just as easily imagine that his brother could feel like the one zooming away, if the right frame was chosen, but now I'm confused as to how the physical effects of acceleration were assigned to one twin and not the other (regardless of rockets). I realize now I had actually been imagining a physical substance, a sort of geometry-ether made of little tets of space-time rest-frames, both twins are composed of these real objects with identity, relation to each other, and some sense of sequential "history". The twin with the rockets was distorting his using energy - applied to those objects, so his clock slowed down.

 

[PeterDonis]

Einstein's GR solution gives the same result, for the rocket-twin's "point-of-view" about how the home twin's current age varies during the trip, as is given by the co-moving-inertial-frames solution in SR.

Yes, but obtaining that result required choosing a particular non-inertial coordinate chart. See below.

Einstein did not appear to regard the rocket-twin's conclusions about the aging of the home twin to be some kind of arbitrary choice of simultaneity "convention".

Not explicitly, but he did implicitly when he specified the "gravitational field" that appears when the rocket twin chooses to use (non-inertial) coordinates in which he is always at rest (or, equivalently, when he specified the home twin's aging as a function of the rocket twin's time). When he did that, he was implicitly assuming a particular choice of non-inertial coordinates, which in turn implies a particular choice of simultaneity convention. Different choices of non-inertial coordinates (such as Dolby & Gull's), with different simultaneity conventions, would also give different "gravitational fields" (i.e., different connection coefficients), and different behavior of the home twin's aging as a function of the rocket twin's time. The fact that Einstein didn't spell all those implicit assumptions out does not mean he wasn't making them.

 

[PeterDonis]

If acceleration was only a perspective change, why does the "travelling" twin age more slowly.

Acceleration in the sense of proper acceleration--acceleration that you feel--is not a "perspective change". It's a direct observable.

The "perspective change" is the choice of coordinate chart; but coordinates in themselves have no physical meaning. All the physics is in the direct observables. And "relative rate of aging" is not a direct observable. There is no direct observable the traveling twin can use to tell him "how fast the home twin is aging at this moment". The home twin's "rate of aging" depends on which coordinates the traveling twin chooses to use.

The direct observable related to "aging" is the fact that, when the two twins meet up again, the traveling twin's clock shows less elapsed time. But that is a property of the two twins' respective paths through spacetime as a whole; it is not a property of any particular point on the twins' paths.

It's just as if you and I both started out from New York City with cars whose odometers read zero, and met up again in Los Angeles to find that your odometer read more miles than mine, because you took a longer route than I did. The difference in odometers is a direct observable, but it is meaningless to ask at what point on our respective paths the difference in odometers "happened". Nor is it meaningful to ask, during either of our journeys, what the other's odometer reading is "at the same point" in his journey. There is no unique mapping between points on the two paths; there is only the comparison of the total path lengths.

Similarly, the two twins follow different paths through spacetime, and those paths have different lengths. But there is no unique mapping between points on the two paths, so there's no way to tell "which twin is older" in any invariant sense at any particular point. You can only compare the total path lengths.

 

[Jimster41]

Ah that helps.

So this is the problem with intrinsic curvature... It can't be measured from some exterior perspective. It can only be evaluated by comparing end results of closed paths. Is it incorrect to say that it is a property of the points on their path (what else could it be caused by)... but that property is not observable - because for us it is intrinsic?

 

[PeterDonis]

So this is the problem with intrinsic curvature

With path curvature, not spacetime curvature. We are considering a scenario in flat spacetime; spacetime curvature is zero. But the traveling twin's path is curved, whereas the home twin's path is straight. (In the idealized case where the traveling twin's turnaround is instantaneous, his path is composed of two straight legs plus a "corner", and all the path curvature is at the corner. In a more realistic case, the corner would be "rounded off" to be smooth because the twin's acceleration is limited to some finite amount--a "corner" would mean infinite acceleration.)

 

[PeterDonis]

Is it incorrect to say that it is a property of the points on their path (what else could it be caused by)

Path curvature requires the path to be a curve in a higher-dimensional manifold; there is no such thing as completely "intrinsic" path curvature. (This is in contrast to spacetime curvature; for a manifold of two or more dimensions, intrinsic curvature is meaningful, and when we talk about spacetime curvature, we are talking about intrinsic curvature.)

but that property is not observable

Sure it is. Path curvature is just proper acceleration, which is directly observable.

 

[Dale]

Einstein's GR solution gives the same result, for the rocket-twin's "point-of-view" about how the home twin's current age varies during the trip, as is given by the co-moving-inertial-frames solution in SR. Einstein's GR solution does not agree with the "Radar Solution" of Dolby & Gull.

Do you have a reference for any of this?

As far as I am aware none of it is correct and you are again simply pushing an odd personal agenda that you have been repeatedly told is wrong.

 

[PeterDonis]

As far as I am aware none of it is correct

I was interpreting this as meaning simply that the Dolby & Gull simultaneity convention is different from the "comoving inertial frame" simultaneity convention, which is true, and that the simultaneity convention assumed by "Einstein's GR solution" is the same as the latter one, which is true as far as I know (at least to the extent that Einstein specified a coordinate chart at all).

This does not imply that I agree with the poster's interpretation of what all that means, of course.

 

[Dale]

the simultaneity convention assumed by "Einstein's GR solution" is the same as the latter one

I don't even think that that much is clear. In Einstein's "pop-sci" work there is not enough math provided to identify what simultaneity convention he is using, and in his non "pop-sci" work it is clear that he allows for all coordinate choices.