Was Einstein lucky when not considering twin paradox as paradox?

[pervect]

Pedagogically, the techniques needed to handle general coordinate systems (aka non-inertial frames) are taught in GR courses, but as far as the physics go, if you have flat space time (i.e. no gravity), you can use the techniques of SR just fine. You could conceivable even do the analysis without tensors, though if you want to compare your results to textbook results, I'm not aware of any textbooks that don't use tensors (not that I've read them all).

A brief outline of one way to go about doing this:

1) Solve the relativistic rocket equation for a constant proper acceleration rocket. Check your solution against the standard ones on wikipedia and/or the relativistic rocket FAQ at http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html .

I will use the following notation: let your reference inertial frame be (t,x,y,z). Let your rocket coordinates be (T,X,Y,Z). Let the momentarily co-moving inertial frame coordinates at some proper time T be represented by (t', x', y', z'). Let the acceleration of the rocket be called g.

So solving the relativistic rocket equation you want to find z(T) and t(T), the inertial coordinate frame position and time as a function of rocket time T, which is just the proper time of the accelerating rocket.2) Consider the momentarily co-moving inertial frame at proper time T on the rocket. Because of linearity, there will be a linear relationship between $\hat{z'}$,$\hat{t'}$, the components of the basis vectors in the momentarily comoving inertial frame at time T, and $\hat{z}$, $\hat{t}$ the basis vectors in the inertial frame. Basically all we really need to do is figure out the components of $\hat{z'}$ in the inertial frame, this vector will have some time components due to the relativity of simultaneity, and some space-component due to relativistic length contraction

3) Find an expression that converts rocket coordinates (T,X,Y,Z) to inertial coordinates (t,x,y,z). We will basically have
t = t(T) + Z * (t-component of $\hat{z'}$)
x = X
y = Y
z = z(T) + Z * (z-component of $\hat{z'}$)

here t(T) and z(T) are the expressions we derived in part 1, and while the relationship between changes in the Z coordinate at time T and the changes in the t and z coordinates is linear, we need to find out what the coefficients are.

At this point we've codified precisely what we mean by "rocket coordinates" by having an expression that gives the inertial coordinates (t,x,y,z) as functions of the rocket coordinates (T,X,Y,Z).

4) Optional. Using the above results from 3, find the metric in the accelerating coordinates by taking dx^2 + dy^2 + dz^2 - c^2 dt^2 in terms of dT,dX,dY, and dZ. This is just algebra (albeit lengthly without computer assistance). Compare it to the textbook results, which say that you should get dX^2 + dY^2 + dZ^2 - c^2 (1+gZ)^2 dT^2

5) Optional, but recommened. Consider whether the mapping derived in part 3 is a 1:1 mapping (hint: we need to insist that 1+gZ is positive if we want to have a 1:1 mapping).

 

[harrylin]

SR as a theory of physics is not about rest frames. That's a way to talk about SR, and a way to derive the Lorentz transformations, but as a theory of physics, it makes claims about the behavior of clocks and rods and light signals and so forth. Those claims can be expressed in any coordinates you like. The fact that they were originally derived for inertial reference frames is irrelevant [..] There is nothing about using K' that requires going beyond SR.

Einstein certainly agreed with that. I promised to leave our disagreement about the issue that you next brought up, as it is irrelevant for my clarification that obviously this was not "retrospectively" understood - instead it was understood right from the start.

 

[stevendaryl]

Einstein certainly agreed with that. I promised to leave our disagreement about the issue that you next brought up, as it is irrelevant for my clarification that obviously this was not "retrospectively" understood - instead it was understood right from the start.

I know, I'm not arguing about that. I'm arguing about something else related to GR/SR and the twin paradox, which is the idea that somehow SR says that the traveling twin isn't in a "valid rest frame" while GR says otherwise. That doesn't make any sense. If Einstein said that, then that was very misleading of him.

 

[stevendaryl]

I know, I'm not arguing about that. I'm arguing about something else related to GR/SR and the twin paradox, which is the idea that somehow SR says that the traveling twin isn't in a "valid rest frame" while GR says otherwise. That doesn't make any sense. If Einstein said that, then that was very misleading of him.

On the other hand, I can't figure out what it could possibly mean to say that "the traveling twin is at rest in a valid rest frame", other than "there is a coordinate system in which the traveling twin is at rest".

 

[harrylin]

I know, I'm not arguing about that.

OK. :)
I though that you were still trying to avoid that conclusion.

I'm arguing about something else related to GR/SR and the twin paradox, which is the idea that somehow SR says that the traveling twin isn't in a "valid rest frame" while GR says otherwise. That doesn't make any sense. If Einstein said that, then that was very misleading of him.

It is misleading to pretend that there is no difference at all between 1916 GR and modern GR... In a nutshell:

1. The frame of the traveling twin is invalid for SR's laws of nature (even the second postulate doesn't work with it!).
That simple fact has also been elaborated many times on this forum.

2. The frame of the traveling twin is valid for GR's laws of nature according to 1916 GR.
The trick: "we can "create" a gravitational field by a simple variation of the co-ordinate system." -E. 1916
I briefly discussed this in the beginning of a post some time ago in a different thread: https://www.physicsforums.com/threa...solution-compatible-with-einsteins-gr.656240/

 

[stevendaryl]

1. The frame of the traveling twin is invalid for SR's laws of nature (even the second postulate doesn't work with it!).
That simple fact has also been elaborated many times on this forum.

What does "invalid" mean? The traveling twin's frame is not an inertial frame. It's not an inertial frame in GR, either. The second postulate states that the speed of light has speed "c" in any inertial frame. That doesn't mean that a noninertial frame is "invalid", it just means that the speed of light doesn't necessarily have speed c in that frame. You don't need a separate law to deal with a noninertial frame, you just need calculus. Calculus plus SR is still SR.

There is no such thing as a valid or invalid coordinate system. There is only valid or invalid reasoning. If you reason about a noninertial coordinate system as if it were inertial, then you've engaged in invalid reasoning.

So it is true that SR, when expressed as laws about inertial reference frames, tempts people into invalid reasoning if they try to apply the laws, as written, in a noninertial reference frame. That means that you need to understand what the laws say in a way that it is independent of coordinate systems. That was not completely understood at the time Einstein wrote SR. He did not know how to formulate laws that worked in any coordinate system. But that's a limitation of his mathematics, not his physics.

2. The frame of the traveling twin is valid for GR's laws of nature according to 1916 GR.

It's neither more nor less valid according to SR than GR. There is NO difference between SR and GR when it comes to noninertial frames. In the limiting case of flat spacetime, they are the SAME physical theory. You are perpetuating a misconception. That's my original point about Einstein's "GR solution to the twin paradox". It introduced a misconception that apparently persists to this very day.

 

[harrylin]

I think there was a specific reason Einstein used GR to resolve the twin paradox. He wanted to construct an analogous scenario (via the equivalence principle) in which the "rocket-twin" could say that he was absolutely stationary and unaccelerated during the whole time that the twins were separated. When he fired his rocket engine, he was doing it strictly to counteract the spatially-uniform gravitational field that is somehow momentarily switched on, so that the rocket-twin would remain stationary and unaccelerated. That momentarily switched-on gravitational field causes the "home twin" (the twin who has no rocket) to accelerate, reverse course, and move toward the "traveler". The resulting conclusion using this GR scenario is that the rocket-twin will say that the "home-twin" suddenly gets much older while that gravitational field is switched on.

The exact same result (regarding the rocket-twin's conclusion about the home-twin suddenly getting much older during the turnaround) is obtained without recourse to GR (and without any gravitational fields), purely from SR, using a non-inertial reference frame for the rocket-twin which is formed by piecing together multiple inertial frames that are each momentarily co-moving with the rocket-twin at different instants of his life. The rocket-twin is always at the spatial origin of his non-inertial reference frame, but he never contends that he doesn't accelerate. He knows that he accelerates, and reverses course, when he turns on his rocket. And he knows that it is the home-twin who is unaccelerated for the whole trip.

There is a difference between being "always absolutely at rest" (Einstein's GR scenario for the rocket-twin) versus "being always at the spatial origin of your own personal reference frame, but accelerating at will using your rocket engine" (the SR scenario). But what the rocket-twin says about the home-twin suddenly getting much older during the turnaround is exactly the same for both scenarios (even though it's a different twin doing the turnaround in the two cases).

I had missed that post of yours. Good summary of Einstein's paper! :)

PS. Note that I don't agree with that paper; but before criticizing an opinion, one has to correctly present it first.

 

[harrylin]

What does "invalid" mean? [..]

I gave the example of invalid application of SR's second postulate on accelerating frames, and the same happens with invalid application of the Lorentz transformations. But of course, you know that:

So it is true that SR, when expressed as laws about inertial reference frames, tempts people into invalid reasoning if they try to apply the laws, as written, in a noninertial reference frame. [..]

Exactly - that was the point Einstein made; and this was at that time rather well understood (but not anymore, see my next post!).

It's neither more nor less valid according to SR than GR. There is NO difference between SR and GR when it comes to noninertial frames. In the limiting case of flat spacetime, they are the SAME physical theory. You are perpetuating a misconception. That's my original point about Einstein's "GR solution to the twin paradox". It introduced a misconception that apparently persists to this very day.

Sorry, the link I gave in my last post clarifies that the misconception here is yours, as you project your understanding of modern GR on Einstein's GR v.1.0 of 1916. And maybe you overlooked, like I did, the clear summary by PhoebeLasa? A few minutes ago I now also commented on that post.

 

[harrylin]

[..] Perhaps many people who did not correctly understand SR, got the wrong impression that GR had to be used for accelerated objects frames and even accelerated objects, because Einstein argued that GR could be used like that?

[..] it now seems plausible to me that many people may have misunderstood Einstein's arguments in his papers from 1916-1918 that GR could be used for accelerated frames and even accelerated objects, so that they misconstrued that according to Einstein GR had to be used. And that's all that I will hypothesize about that. :)

By pure coincidence (for my brother in law I'm checking a book), I stumbled yesterday evening on a case that supports my hypothesis about that misconception in a strong way:

"[SR] accounted for straight-line uniform motion, but it did not account for non-uniform or accelerated motion. And as everybody knows, accelerated motion is common on Earth and throughout the universe.
[..]
The [twin] paradox, as it existed before general relativity, was that you couldn't be sure which of the twins would be the youngest when they got back together after the flight [...] since all motion is relative according to special relativity. But when Einstein formulated his general theory of relativity he showed that there was, indeed, a distinct difference between the twins." -Einstein's Brainchild, Barry Parker.

And note that the author was physics professor from 1967 to 1997! :(

 

[stevendaryl]

I gave the example of invalid application of SR's second postulate on accelerating frames, and the same happens with invalid application of the Lorentz transformations. But of course, you know that:

There is invalid reasoning, but what does an invalid "frame" mean?

Sorry, the link I gave in my last post clarifies that the misconception here is yours

It's a misconception that GR is equivalent to SR in the limit of vanishing spacetime curvature? It's provably the case.

And note that the author was physics professor from 1967 to 1997

Well, the misconception is perpetuated by many people who should know better.

 

[stevendaryl]

Look, tell me what aspect of Einstein's derivation involves GR. If you agree that it is a GR derivation, then point to a step in the derivation that is not valid in SR. The key remark by Einstein is this:

‘according to the general theory of relativity, a clock works faster the higher the gravitational potential at the place where it is situated’

That result is derived from SR. The way you get that result is by:

1. Calculating time dilation in SR using noninertial coordinates.
2. Noting that in these coordinates, there is a "pseudo-gravitational field", and that clocks higher up in this "field" run faster. (This is pure calculus--it follows from SR plus the calculus of coordinate changes).
3. Invoke the equivalence principle, to get the analogous fact about "real" gravitational fields.

Gravitational time dilation was a derivation of SR + the equivalence principle. It was a step toward GR, but this step required almost nothing beyond SR and the insight of the equivalence principle.

Now, when Einstein (or whoever) invokes GR to explain the twin paradox, what is he doing? He's saying:

1. The accelerating twin feels a pseudo-gravitational field.
2. By the equivalence principle, this is like real gravitational fields.
3. By GR, there must be time dilation for clocks that are higher up in this "field".

That is what is going on in the GR explanation for the twin paradox. Use the equivalence principle to transform an accelerated reference frame to one in a gravitational field, then use gravitational time dilation. But when you consider that gravitational time dilation is derived from SR plus the equivalence principle, this explanation is seen to be completely circular! You start with a pure SR problem--acceleration in flat spacetime--then transform to a GR problem, which is then solved by transforming BACK to SR.

It makes sense to view the twin paradox as a way to explain GR in terms of SR concepts. But the other way around is nonsense.

 

[harrylin]

There is invalid reasoning, but what does an invalid "frame" mean?

It means that its use as reference for those laws does not give correct results.

It's a misconception that GR is equivalent to SR in the limit of vanishing spacetime curvature? It's provably the case.

As explained, Einstein's 1916 GR claimed to be more than that.

Well, the misconception is perpetuated by many people who should know better.

That's all you have to say about it? I'll elaborate. I don't think that he is a lone exception, and neither do you.
Thus, I just found a probable cause of the modern "twin paradox" miscomprehension. Apparently, physics professors taught their students for decades that according to SR's first postulate, all motion is relative.

 

[stevendaryl]

It means that its use as reference for those laws does not give correct results.

You have a law saying "The speed of light is c in every inertial reference frame". It is DERIVABLE from this law that in a noninertial reference frame, the speed of light is not c. If a fact is a derivation from SR, how can it be contrary to SR?

As explained, Einstein's 1916 GR claimed to be more than that.

But it's not. That's a misconception. If Einstein believed it, then he was mistaken. In the absence of spacetime curvature, GR is equivalent to SR. From the point of view of the twin paradox in flat spacetime, there is no difference between GR and SR. None.

 

[harrylin]

Look, tell me what aspect of Einstein's derivation involves GR. If you agree that it is a GR derivation, then point to a step in the derivation that is not valid in SR. The key remark by Einstein is this:

‘according to the general theory of relativity, a clock works faster the higher the gravitational potential at the place where it is situated’

That result is derived from SR. The way you get that result is by:

1. Calculating time dilation in SR using noninertial coordinates.
2. Noting that in these coordinates, there is a "pseudo-gravitational field", and that clocks higher up in this "field" run faster. (This is pure calculus--it follows from SR plus the calculus of coordinate changes).
3. Invoke the equivalence principle, to get the analogous fact about "real" gravitational fields.

Gravitational time dilation was a derivation of SR + the equivalence principle. It was a step toward GR, but this step required almost nothing beyond SR and the insight of the equivalence principle.

That part of your argument looks totally correct to me.

Now, when Einstein (or whoever) invokes GR to explain the twin paradox, what is he doing? [..]

Once more: Einstein did not "invoke GR" to "explain the twin paradox" - quite the contrary! Critics of 1916 GR invoked GR's "general principle of relativity" to create the twin paradox. So, please take my advice and don't quickly reply. Instead verify that Phoebelisa's summary of the 1918 paper is correct, next ponder over it, and then re-read the last part of our discussion. Also, recall my earlier clarification:

[..] I don't agree with that paper; but before criticizing an opinion, one has to correctly present it first.

[stevendaryl:] when you consider that gravitational time dilation is derived from SR plus the equivalence principle, this explanation is seen to be completely circular! You start with a pure SR problem--acceleration in flat spacetime--then transform to a GR problem, which is then solved by transforming BACK to SR. [..]

That is absolutely correct; the same was argued in a paper on the clock paradox by Builder, in the fifties.

You have a law saying "The speed of light is c in every inertial reference frame". It is DERIVABLE from this law that in a noninertial reference frame, the speed of light is not c. If a fact is a derivation from SR, how can it be contrary to SR?

Exactly: it is contrary to SR to use that law with non-inertial frames.

But it's not. That's a misconception. If Einstein believed it, then he was mistaken. [..].

It's inherent in his postulate of GR. I don't think that you actually pretend that he was mistaken about his belief of what he postulated. ;)

 

[stevendaryl]

Once more: Einstein did not "invoke GR" to "explain the twin paradox" - quite the contrary! Critics of 1916 GR invoked GR's "general principle of relativity" to create the twin paradox.

Well that was a big mess that is at least partly Einstein's fault. You don't need a theory of physics to permit you to use arbitrary coordinates. And there is no content to the claim "all motion is relative" above and beyond the claim that you can use arbitrary coordinates. The paradox-mongers went from

Coordinates in which the traveling twin is at rest are just as valid as coordinates in which the stay-at-home twin is at rest​

which is true, to

Therefore, there is no way to say that one twin should be older than the other when they reunite.​

which is false. The whole argument has nothing to do with GR, really. It has to do with the use of arbitrary coordinates. The question is: how to describe the source of the asymmetry between the twins in general (noninertial) coordinates.

The modern resolution is that computations of numeric quantities such as elapsed times for spacetime paths involve tensor quantities (the metric tensor in this case), and the components of a tensor have different values for different coordinate systems. So it's a MATHEMATICAL mistake to transform from inertial coordinates to noninertial coordinates without making the corresponding change to the metric tensor.

The terminology of "gravitational fields due to acceleration" is simply a nonmathematical way to talk about the metric tensor in noninertial coordinates (technically, the pseudo-gravitational field corresponds to the connection coefficients, which are computed from the metric tensor).

So both invoking GR to create the twin paradox and invoking GR to resolve it are misconceptions. Neither involves GR. The whole discussion is about general coordinate systems, so it's not GR at all.

 

[JesseM]

So both invoking GR to create the twin paradox and invoking GR to resolve it are misconceptions. Neither involves GR. The whole discussion is about general coordinate systems, so it's not GR at all.

It's a misconception under the modern definition of the distinction between GR and SR, which revolves around whether spacetime has inherent curvature or not. But it seems to be a historical fact that Einstein and other physicists didn't always define the distinction that way. And while defining it differently may be less elegant, it's really only a "misconception" about terminology (and only relative to the modern usage of the terms, so it can't be labeled a 'misconception' on the part of Einstein), and not a misconception about physics--it's just about what theory you say you're using when you do a particular analysis, not about any differences in any actual physical predictions (although obviously it would be a genuine misconception about physics if someone claimed that the SR time dilation formula should still work in a non-inertial coordinate system).

This section of the twin paradox FAQ entry on the site of physicist John Baez has a good discussion of this history:

Einstein worked on incorporating gravitation into relativity theory from 1907 to 1915; by 1915, General Relativity had assumed pretty much its modern form. (Mathematicians found some spots to apply polish and gold plating, but the conceptual foundations remain the same.) If you asked him to list the crucial features of General Relativity in 1907, and again in 1915, you'd probably get very different lists. Certainly modern physicists have a different list from Einstein's 1907 list.

Here's one version of Einstein's 1907 list (without worrying too much about the fine points):

General Principle of Relativity

All motion is relative, not just uniform motion. You will have to include so-called pseudo forces, however (like centrifugal force or Coriolis force).

Principle of Equivalence

Gravity is not essentially different from any pseudo-force.

The General Principle of Relativity plays a key role in the Equivalence Principle analysis of the twin paradox. And this principle gave General Relativity its name. Even in 1916, Einstein continued to single out the General Principle of Relativity as a central feature of the new theory. (See for example the first three sections of his 1916 paper, "The Foundation of the General Theory of Relativity", or his popular exposition Relativity.)

Here's the modern physicist's list (again, not sweating the fine points):

Spacetime Structure

Spacetime is a 4-dimensional riemannian manifold. If you want to study it with coordinates, you may use any smooth set of local coordinate systems (also called "charts"). (This free choice is what has become of the General Principle of Relativity.)

Principle of Equivalence

The metric of spacetime induces a Minkowski metric on the tangent spaces. In other words, to a first-order approximation, a small patch of spacetime looks like a small patch of Minkowski spacetime. Freely falling bodies follow geodesics.

Gravitation = Curvature

A gravitational field due to matter exhibits itself as curvature in spacetime. In other words, once we subtract off the first-order effects by using a freely falling frame of reference, the remaining second-order effects betray the presence of a true gravitational field.

The third feature finds its precise mathematical expression in the Einstein field equations. This feature looms so large in the final formulation of GR that most physicists reserve the term "gravitational field" for the fields produced by matter. The phrases "flat portion of spacetime", and "spacetime without gravitational fields" are synonymous in modern parlance. "SR" and "flat spacetime" are also synonymous, or nearly so; one can quibble over whether flat spacetime with a non-trivial topology (for example, cylindrical spacetime) counts as SR. Incidentally, the modern usage appeared quite early. Eddington's book The Mathematical Theory of Relativity (1922) defines Special Relativity as the theory of flat spacetime.

 

[stevendaryl]

It's a misconception under the modern definition of the distinction between GR and SR, which revolves around whether spacetime has inherent curvature or not. But it seems to be a historical fact that Einstein and other physicists didn't always define the distinction that way.

I suppose that there is an ambiguity about what constitutes a "physical theory" and when one physical theory is the same or different from another. My inclination is to think that developing more sophisticated mathematical way of working with a theory is still the same theory. So Lagrangian mechanics is still Newtonian mechanics, and GR in flat spacetime is still SR.

 

[PhoebeLasa]

The Dolby & Gull SR answer to the question "How does the home twin's age vary during the traveler's turnaround, according to the traveler" seems to be very popular on this forum, but I've never seen a GR solution (via the equivalence principle) that gets the same answer that the D&G SR method gets. The only GR solution that I've ever seen agrees with the co-moving inertial frames SR solution, which is very different from the D&G SR solution. Has a GR solution that agrees with the D&G SR solution been given anywhere?

 

[Dale]

What is a GR solution in your view? Can you provide a reference for this "GR solution" that agrees with the SR comoving frames as an example?

I am just not sure what you are asking since to me all twin paradox solutions are inherently SR solutions.

 

[Nugatory]

but I've never seen a GR solution (via the equivalence principle) that gets the same answer that the D&G SR method gets

There's no reason that you expect that you would. The equivalence principle says that gravity can be modeled locally as acceleration; it does not say that all acceleration can be modeled as gravity, especially not globally.

Whether we attack the twin paradox in flat spacetime using the methods of special or general relativity, we're going to end up computing the same coordinate-independent quantities, namely the proper times along the worldlines of the two twins. The GR machinery just gives us a bit more latitude in choosing coordinates on our way to the solution.

 

[PeterDonis]

the question "How does the home twin's age vary during the traveler's turnaround, according to the traveler"

Which is a question that doesn't have any unique "right" answer, because it's not a question about physics, it's a question about which simultaneity convention you choose. Different "answers" to this question are just different choices of simultaneity convention. These choices have no effect on any actual observables.

 

[harrylin]

[Sunday at 12:39 PM]
Once more: Einstein did not "invoke GR" to "explain the twin paradox" - quite the contrary! Critics of 1916 GR invoked GR's "general principle of relativity" to create the twin paradox. So, please take my advice and don't quickly reply. Instead verify that Phoebelisa's summary of the 1918 paper is correct, next ponder over it, and then re-read the last part of our discussion. [..]

[Sunday at 1:19 PM]

If the time stamps are right, you could not possibly have done so! Not surprisingly:

[..] there is no content to the claim "all motion is relative" above and beyond the claim that you can use arbitrary coordinates. [..] The whole argument has nothing to do with GR, really. It has to do with the use of arbitrary coordinates. The question is: how to describe the source of the asymmetry between the twins in general (noninertial) coordinates. [..]

As explained in the summary which was posted twice here: the challenge that was thrown at his feet, was how to produce the same prediction about the twins as in SR, while using coordinates in which the "traveler" is claimed to be in rest all the time. That should be possible according to Einstein's GR postulate, as he there also acknowledged. As a reminder, earlier I stated:

..] It's inherent in his postulate of GR. I don't think that you actually pretend that he was mistaken about his belief of what he postulated. ;)

Regretfully, that is increasingly how it appears:

[..] invoking GR to create the twin paradox and [..] are misconceptions.

I suppose that there is an ambiguity about what constitutes a "physical theory" and when one physical theory is the same or different from another. My inclination is to think that developing more sophisticated mathematical way of working with a theory is still the same theory. So Lagrangian mechanics is still Newtonian mechanics, and GR in flat spacetime is still SR.

Following your inclination, I'm unable to discern Maxwell's theory of electrodynamics from Einstein's. That is due to the fact that the difference is not in Maxwell's equations; the difference is primarily in the postulates about the coordinate systems in which the equations are claimed to be valid. That differs in a fundamental way from developing a more sophisticated mathematical way of working with Maxwell's theory.

There's no reason that you expect that you would. The equivalence principle says that gravity can be modeled locally as acceleration; it does not say that all acceleration can be modeled as gravity, especially not globally.

The equivalence principle of 1916 says that acceleration can be modeled as gravity:
"Can any observer, at rest relative to [PLAIN]https://upload.wikimedia.org/math/4/f/4/4f45bf1507f5ace45ff25334e53fece4.png, then conclude that he is in an actually accelerated reference-system? This is to be answered in the negative; the above-named behaviour of the freely moving masses relative to

can be explained in as good a manner in the following way. The reference-system https://upload.wikimedia.org/math/4/f/4/4f45bf1507f5ace45ff25334e53fece4.png has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to [PLAIN]https://upload.wikimedia.org/math/4/f/4/4f45bf1507f5ace45ff25334e53fece4.png."
- https://en.wikisource.org/wiki/The_...siderations_about_the_Postulate_of_Relativity.

 

[harrylin]

[..] it's really [..] not a misconception about physics--it's just about what theory you say you're using when you do a particular analysis [..]

Yes, indeed.

However, several authors (Moller, Builder, ...) criticized the physics for good reasons (IMHO), and probably most of us agree with the them (or would agree if they knew it).

 

[stevendaryl]

As explained in the summary which was posted twice here: the challenge that was thrown at his feet, was how to produce the same prediction about the twins as in SR, while using coordinates in which the "traveler" is claimed to be in rest all the time. That should be possible according to Einstein's GR postulate, as he there also acknowledged.

My point is that both the challenge and the response are based on the misconception that GR is any more (or less) capable of using coordinates in which the traveling twin is at rest than SR. As I said, GR in the limit of vanishing spacetime curvature simply IS SR. There is no difference, as a physical theory. The differences in practice are simply that GR demands more sophisticated mathematics than SR, and so that mathematics is often thought of as part of GR, and not part of SR, but that's not really a physical difference, it's only a difference of how things are described, mathematically. SR described in arbitrary coordinates is still SR. There is no additional hypothesis required to rewrite SR in arbitrary coordinates, it's purely mathematics.

Following your inclination, I'm unable to discern Maxwell's theory of electrodynamics from Einstein's.

Maxwell's theory was already invariant under Lorentz transformations. Einstein's contribution was to develop an analogous theory of mechanics. Before Einstein, we had Maxwell's equations, which were invariant in form under Lorentz transformations, and Newton's laws of motion, which were invariant in form under Galilean transformations. Einstein united the theories by modifying Newton's theory to get one that was invariant under Lorentz transformations, as well. He didn't need to modify Maxwell's equations.

That is due to the fact that the difference is not in Maxwell's equations; the difference is primarily in the postulates about the coordinate systems in which the equations are claimed to be valid.

I think that's a misconception. If you know the equations of motion in one coordinate system, then you know the equations of motion in every coordinate system. You don't need an additional postulate that they are invariant under such and such a transformation, it's just a fact of the equations. It's a fact that could be discovered through a more sophisticated mathematical analysis.

 

[stevendaryl]

The equivalence principle of 1916 says that acceleration can be modeled as gravity:

That's a muddled way of thinking about it.

You don't need an additional principle to know that when you do mechanics (this is true of Newtonian mechanics as well as SR) in a noninertial coordinate system, the equations of motion contain additional terms that you can think of as "gravitational forces". That's pure mathematics. There is no additional physics involved in interpreting "inertial forces" as gravity. It's just what you name terms in the equations of motion.

The significance of the equivalence principle is the other way around--that REAL gravity can be interpreted as inertial forces due to the use of noninertial coordinates. That's the direction that has physical content.

You don't need the equivalence principle to deduce that on board an accelerating rocket, light beams appear to curve downward, and that clocks run at different speeds in different locations of the rocket. That is derivable from pure SR.

What you need the equivalence principle for is to predict that clocks and light beams work in a similar way on the surface of a massive planet. That's the physical content of the equivalence principle, that light curves near a massive planet, and that clocks run at different speeds at different altitudes.

"Can any observer, at rest relative to [PLAIN]https://upload.wikimedia.org/math/4/f/4/4f45bf1507f5ace45ff25334e53fece4.png, then conclude that he is in an actually accelerated reference-system? This is to be answered in the negative; the above-named behaviour of the freely moving masses relative to

can be explained in as good a manner in the following way. The reference-system https://upload.wikimedia.org/math/4/f/4/4f45bf1507f5ace45ff25334e53fece4.png has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to [PLAIN]https://upload.wikimedia.org/math/4/f/4/4f45bf1507f5ace45ff25334e53fece4.png."
- https://en.wikisource.org/wiki/The_...siderations_about_the_Postulate_of_Relativity.

I consider that paragraph to be misleading, if it encourages people to think that you need GR to be able to reason as if the accelerated observer is at rest. That is purely SR + a coordinate transformation, and GR adds nothing. Well, what it adds is that you may need the inspiration of GR to get you thinking about the use of noninertial coordinate systems.

 

[stevendaryl]

To illustrate how confused it is to think that GR is needed to use coordinates in which an accelerated observer is at rest, take a look at Rindler coordinates. It's a coordinate transformation from

(x,t) \Rightarrow (X,T)

where T = tanh^{-1}(\frac{ct}{x}) and X = \sqrt{x^2 - c^2 t^2}

In terms of the coordinates (X,T), you find that

1. Clocks at "rest" (that is, X is constant) run faster the higher up they sit (larger values of X)
2. Light rays bend downwards (toward negative values of X).
3. An object dropped from "rest" will accelerate downward (decreasing X)

These facts don't require GR, they are derivable from SR alone.

There is no need for a "principle of equivalence" to allow us to use these coordinates, any more than there is a need for a principle of equivalence to use Newtonian physics in polar coordinates.

 

[PhoebeLasa]

Can you provide a reference for this "GR solution" that agrees with the SR comoving frames as an example?

I've seen it done in lots of places, over the years. And I've never seen its solution NOT agree with the co-moving inertial frames solution. I've never seen it agree with the Dolby&Gull solution. It always says that, anytime the traveler's rocket is off, the traveler concludes that the home twin is aging more slowly, as given by the standard time-dilation result. I.e., the GR solution always says that the traveler's perspective is the same as an inertial observer whenever the rocket is off. And that ALL of the fast aging of the home twin, according to the traveler, occurs only while the rocket is firing. That is different than the D&G solution. If there are multiple "equally good" SR solutions for the traveler's perspective, why aren't there multiple "equally good" GR solutions for the traveler's perspective?

I don't remember all the places I've seen that GR solution given, but I do remember that it is quite common. I think the Wikipedia page on the twin paradox does it (and it may give some specific references). And I think that the link given by JessM to John Baez's webpage on the twin paradox gives it (perhaps also with references). I think I might have also seen it in MTW's "Gravitation" book, and in Born's "Einstein's Theory of Relativity" book.

 

[Dale]

I've seen it done in lots of places, over the years.

I need an example so that I can understand what you are referring to. I simply don't know what you mean by a "GR solution". If it has been done in lots of places then I am sure that I have read one, but I don't recognize it as what you are calling "a GR solution". Every solution that I am aware of is what I would call "an SR solution" since the spacetime is flat.

I'm not saying that you are wrong, I am just saying that I don't know what you are referring to. I suspect that we are talking about the same things and just using different words. Also, are you talking about a quantitative solution, or simply a hand-waving qualitative explanation?

 

[A.T.]

These facts don't require GR,

Why is it so important whether some explanation is called a "SR-explanation" or a "GR-explanation"? Isn't SR a special case of GR, and therefore any SR-explanation also a GR-explanation?

As I said, GR in the limit of vanishing spacetime curvature simply IS SR.

Doesn’t this make the equivalence principle part of SR, since it only applies when spacetime curvature is negligible?

 

[stevendaryl]

Why is it so important whether some explanation is called a "SR-explanation" or a "GR-explanation"? Isn't SR a special case of GR, and therefore any SR-explanation also a GR-explanation?

Sure. Every SR explanation is also a GR explanation. But not the other way around. So some problems require GR, and others don't. My point is that no problem involving flat spacetime requires GR.

Doesn’t this make the equivalence principle part of SR, since it only applies when spacetime curvature is negligible?

If spacetime is perfectly flat, then you don't need the equivalence principle, because everything can be done using SR alone. What the equivalence principle allows is to solve problems in curved spacetime by breaking spacetime into small regions and then approximating those regions by flat sections of spacetime.

To me, the equivalence principle is really the same, as far as physical content, to the two claims below:

1. Spacetime is curved.
   
2. "Gravitational forces" are actually connection coefficients due to the choice of a noninertial coordinate system. (If spacetime is curved, then there are no inertial coordinate systems except in the limit of small regions of spacetime).

It's a guess about the nature of gravity. You don't need the equivalence principle to get "inertial forces" (connection coefficients) for a noninertial coordinate system; that's derivable from the equations of motion in inertial coordinates.

 

[harrylin]

(With some rearranging of comments):

harrylin said: "As explained in the summary which was posted twice here: the challenge that was thrown at his feet, was how to produce the same prediction about the twins as in SR, while using coordinates in which the "traveler" is claimed to be in rest all the time. That should be possible according to Einstein's GR postulate, as he there also acknowledged. [..] Following your inclination, I'm unable to discern Maxwell's theory of electrodynamics from Einstein's."

My point is that both the challenge and the response are based on the misconception that GR is any more (or less) capable of using coordinates in which the traveling twin is at rest than SR.
[..]
Maxwell's theory was already invariant under Lorentz transformations. Einstein's contribution was to develop an analogous theory of mechanics. Before Einstein, we had Maxwell's equations, which were invariant in form under Lorentz transformations, and Newton's laws of motion, which were invariant in form under Galilean transformations. Einstein united the theories by modifying Newton's theory to get one that was invariant under Lorentz transformations, as well. He didn't need to modify Maxwell's equations.
[..]
If you know the equations of motion in one coordinate system, then you know the equations of motion in every coordinate system. You don't need an additional postulate that they are invariant under such and such a transformation, it's just a fact of the equations.[..]

Maxwell's theory assumed the use of Newton's transformations. I think that this is a good example to explain the difference between a theory and the equations of a theory, but apparently it is necessary to elaborate:

- the equations are the same
- the same coordinate transformations are mathematically possible with both theories

Nevertheless they were not the same theory because their physical assumptions differed on an essential point.
For MMX they even made contrary predictions!

Thus I clarified:
" the difference is not in Maxwell's equations; the difference is primarily in the postulates about the coordinate systems in which the equations are claimed to be valid."

"harrylin said: The equivalence principle of 1916 says that acceleration can be modeled as gravity"

That's a muddled way of thinking about it. [..]
[..] What you need the equivalence principle for is to predict that clocks and light beams work in a similar way on the surface of a massive planet. That's the physical content of the equivalence principle, that light curves near a massive planet, and that clocks run at different speeds at different altitudes. [..]

I certainly agree with that; however my dislike for GR "vintage 1916" (as the FAQ calls it) doesn't bring me to deny the facts about it. It appears that you even deny the meaning of the term "GR"! The new theory of gravitation was a fantastic spin-off of Einstein's theory of the general relativity of motion. He put it as follows:

"From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can "create" a gravitational field by a simple variation of the co-ordinate system." (emphasis mine)

However, in the light of this discussion it strikes me that his phrasing here lacks precision, so that it is bound to be misunderstood if one does not already understand what he means. No doubt, if the issue had been brought up, he would have agreed that a mere coordinate transformation will not create a gravitational field. The creation of a gravitational field only occurs in his theory when one reinterprets the accelerating system as a non-accelerating system - one that is "in rest".

A similar situation occurs in SR when one Lorentz transforms from a "stationary" system S to a "moving" system S'. The transformation itself does not yet impose on an observer who is co-moving with S' to pretend not to be moving but to be in rest; however, the observer may do that if he/she so desires. This adaptation of interpretation to the used coordinate system is implied in most discussions.

There is no additional physics involved in interpreting "inertial forces" as gravity. It's just what you name terms in the equations of motion.

A sudden inertial force on a test particle doesn't create a gravitational wave, while - I think - a suddenly "induced gravitational field" certainly must do so. That is the central point of my criticism of Einstein's 1918 paper (which I did not yet present), and with that, of 1916 GR.

https://en.wikisource.org/wiki/The_...siderations_about_the_Postulate_of_Relativity
I consider that paragraph to be misleading, if it encourages people to think that you need GR to be able to reason as if the accelerated observer is at rest. That is purely SR + a coordinate transformation, and GR adds nothing. Well, what it adds is that you may need the inspiration of GR to get you thinking about the use of noninertial coordinate systems.

That does not merely encourage people to think that, it's what SR as well as 1916GR assert. The claim that the accelerated twin is not changing velocity so that the traveler can be considered to be all the time in rest, leads in SR to the twin paradox! We and others discussed this earlier in this same thread and you even brought it up in your reply. The following is according to SR not true in an SR "rest system":
"light beams appear to curve downward, and [..] clocks run at different speeds in different locations of the rocket."

To illustrate how confused it is to think that GR is needed to use coordinates in which an accelerated observer is at rest, take a look at Rindler coordinates. [..]

Once more: Einstein did not pretend to need GR in order to use coordinates in which an accelerated observer is at rest - quite the contrary, he used such SR coordinates as input for GR!
Perhaps the phrasing "at rest in an accelerated coordinate system" causes confusion. While one certainly can sit down and "rest" in an accelerating rocket, that does not change the physical interpretation of an accelerating system. It may be better to speak of "co-moving" with an accelerated coordinate system.

 

[stevendaryl]

Nevertheless they were not the same theory because their physical assumptions differed on an essential point.
For MMX they even made contrary predictions!

This is sort of a complicated subject, but the way I understand it is this:

• Maxwell's equations are invariant under Lorentz Transformations.
• Newton's equations are invariant under Galilean Transformations.
• Therefore, there is only one frame in which they both take on their usual forms.

What that means is that either Newton's equations or Maxwell's equations have to be modified if you change frames. I think that everyone assumed that it was Maxwell's equations that had to be modified; that they were only valid as is in the frame in which the "ether" was at rest.

But is it really true that the prediction of a nonnull result from the Michaelson-Moreley experiment was a prediction from the theory of electromagnetism? I don't think it was. I think it was a prediction from Newtonian physics. Newtonian physics says that if in one frame, you have an object that travels at speed c, then in another frame moving at speed v relative to the first, that object travels at some speed between c-v and c+v. The prediction of a nonnull result follows from Newtonian mechanics, not Maxwell's equations.

In any case, could you give a reaction to the following argument:

Assume that the usual equations of Special Relativity holds when written in terms of inertial coordinates (x,y,t). Introduce new coordinates (X,Y,T) related to the first two via:

• X = \sqrt{x^2 - c^2 t^2}
• T = tanh^{-1}(\frac{ct}{x})
• x = X cosh(T)
• t = \frac{X}{c} sinh(T)
• Y = y

Then when described in terms of these new coordinates,

• When an object is dropped from "rest", it accelerates downwards (towards smaller values of X)
• Clocks at "rest" that are "higher up" (at larger values of X) run faster.
• Light signals that are initially emitted in the Y direction bend down (towards smaller values of X)

So the "general relativistic" effects of "gravitational time dilation" and "bending of light rays" and "objects falling under gravitational fields" seem to me to be purely effects of SR in noninertial coordinates. Not only do you not need the principle of equivalence to treat an accelerated observer as if he were at rest, the principle of equivalence plays no role, whatsoever. (Except maybe it allows you to use the word "gravity" when describing the above effects.)

So my questions are: (1) In what sense is the above discussion of Rindler coordinates not an "SR" argument? (2) How would the treatment be any different in a "GR" derivation?

 

[harrylin]

[..] is it really true that the prediction of a nonnull result from the Michaelson-Moreley experiment was a prediction from the theory of electromagnetism? I don't think it was. I think it was a prediction from Newtonian physics. [..]The prediction of a nonnull result follows from Newtonian mechanics, not Maxwell's equations.

I referred to Maxwell's theory, in contrast with Maxwell's equations (in case you did not know it, Maxwell proposed MMX). And that was exactly my point: emphasizing the important difference between a theory of physics, and equations that are used as part of the theory.
Equations without the theory are like a piece of complicated equipment without the manual. :)

In any case, could you give a reaction to the following argument:

I will later, if it appears to still be useful (although I already provided a link to my reply, before you asked!). For could you please first react to :
- the remainder of my last post? (you apparently stopped reading at 1/3 of my post, while the next 2/3 gave a more elaborate explanation of the same)
- post #84? (it may well be that Phoebelasa explained it in a way that is clearer than the way I explained it; but apparently you overlooked it)

 

[stevendaryl]

I referred to Maxwell's theory, in contrast with Maxwell's equations (in case you did not know it, Maxwell proposed MMX). And that was exactly my point: emphasizing the important difference between a theory of physics, and equations that are used as part of the theory.

Right, but as I said, the prediction of a nonnull result has almost nothing to do with Maxwell's theory. It follows from the fact that:

1. Light has speed c in at least one frame.
2. According to Newton's laws, velocity transforms as c \Rightarrow c\pm v

So a nonnull result for MMX was mostly a prediction about Newtonian physics, it seems to me.

 

[harrylin]

Right, but as I said, the prediction of a nonnull result has almost nothing to do with Maxwell's theory. It follows from the fact that:

1. Light has speed c in at least one frame.
2. According to Newton's laws, velocity transforms as c \Rightarrow c\pm v

So a nonnull result for MMX was mostly a prediction about Newtonian physics, it seems to me.

Newton's theory would have predicted a null result, assuming his corpuscular light hypothesis.
Maxwell's theory predicted a positive result, because he assumed the validity of the Galilean transformations.

 

[stevendaryl]

Here's another take on describing what was going on in the MMX:

If you know the equations of physics in one coordinate system x^\alpha and you know how to transform from that coordinate system to a second x'^\mu, then you know the equations of physics in the second coordinate system. There is no additional physical assumption required to be able to use the new coordinate system. There is no empirical test as to whether the new coordinate system is "valid" or not.

But here's where physical assumptions come into play: Suppose you have two frames, F and F'. You set up corresponding coordinate systems x^\alpha and x'^\mu using some physical convention for measuring distances and times. So you use standard clocks and metersticks at rest in F to define the coordinate system x^\alpha, and you use standard clocks and metersticks at rest in F' to define the coordinate system x'^\mu. Then you don't know what is the mathematical relationship between the primed and unprimed coordinate systems, without physical assumptions about the nature of clocks and metersticks. So you actually don't know what the equations look like in the new coordinate system until you perform empirical tests.

 

[stevendaryl]

Newton's theory would have predicted a null result, assuming his corpuscular light hypothesis.
Maxwell's theory predicted a positive result, because he assumed the validity of the Galilean transformations.

Nobody is talking about Newton's theory of light--I'm talking about his laws of motion.

Anyway, I have to vigorously protest the phrase "the validity of the Galilean transformations". That is a meaningless phrase without some additional stipulations. There is no such thing as a valid or invalid coordinate transformation. You can use whatever coordinates are convenient; any are as "valid" as any other.

The real issue is the specific conventions for setting up coordinates in a frame. If you use physical objects, such as clocks, rods, light signals, or whatever, to measure distance and times, and you use those distances and times as the basis for a coordinate system, then it's an empirical question how such a coordinate system depends on the frame of rest of those clocks and rods. So it's not a matter of the Galilean transformations being "valid", but a matter of whether they correctly describe the relationship between two operationally defined coordinate systems.

 

[PAllen]

If you know the equations of physics in one coordinate system x^\alpha and you know how to transform from that coordinate system to a second x'^\mu, then you know the equations of physics in the second coordinate system. There is no additional physical assumption required to be able to use the new coordinate system. There is no empirical test as to whether the new coordinate system is "valid" or not.
.

Don't you also have to assume transformation laws for the objects of the equations of physics? Or are you bundling that into what you mean by 'equations of physics'?

 

[harrylin]

[..] I have to vigorously protest the phrase "the validity of the Galilean transformations". That is a meaningless phrase without some additional stipulations. [..].

The stipulations are the ones that the reader is supposed to know; else we can't even write "SR" in a discussion, because it is meaningless without a clarification of what it is an abbreviation.

 

[harrylin]

[..] Einstein did not even consider the twin paradox as problematic at all [..] ?

I will now expand on my earlier comments.

Although the twin example was not an issue for him in connection with what he named the "special" theory of relativity, it became an issue for him in connection with his "general" theory. He discussed that in his 1918 paper https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity , and PhoebeLasa gave a summary of that section as follows:

I think there was a specific reason Einstein used GR to resolve the twin paradox. He wanted to construct an analogous scenario (via the equivalence principle) in which the "rocket-twin" could say that he was absolutely stationary and unaccelerated during the whole time that the twins were separated. When he fired his rocket engine, he was doing it strictly to counteract the spatially-uniform gravitational field that is somehow momentarily switched on, so that the rocket-twin would remain stationary and unaccelerated. That momentarily switched-on gravitational field causes the "home twin" (the twin who has no rocket) to accelerate, reverse course, and move toward the "traveler". The resulting conclusion using this GR scenario is that the rocket-twin will say that the "home-twin" suddenly gets much older while that gravitational field is switched on.

The exact same result (regarding the rocket-twin's conclusion about the home-twin suddenly getting much older during the turnaround) is obtained without recourse to GR (and without any gravitational fields), purely from SR, using a non-inertial reference frame for the rocket-twin which is formed by piecing together multiple inertial frames that are each momentarily co-moving with the rocket-twin at different instants of his life. The rocket-twin is always at the spatial origin of his non-inertial reference frame, but he never contends that he doesn't accelerate. He knows that he accelerates, and reverses course, when he turns on his rocket. And he knows that it is the home-twin who is unaccelerated for the whole trip.

There is a difference between being "always absolutely at rest" (Einstein's GR scenario for the rocket-twin) versus "being always at the spatial origin of your own personal reference frame, but accelerating at will using your rocket engine" (the SR scenario). But what the rocket-twin says about the home-twin suddenly getting much older during the turnaround is exactly the same for both scenarios (even though it's a different twin doing the turnaround in the two cases).

The only glitch that I notice in the otherwise excellent summary above, is the following:
Einstein did not really "want to construct" a scenario in which the "rocket-twin" could say that he was absolutely stationary and unaccelerated during the whole time; instead it was asserted by critics that this must be possible according to Einstein's theory.
Also - and this is essential - he did not "use GR to resolve the twin paradox".
Instead he presented the twin scenario as one of the "Objections against the Theory of Relativity", that is, vintage 1916 GR. And this was his playful defence against accusations that 1916 GR is self-contradictory.

That criticism targeted the General postulate of relativity, according to which "The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner." - https://en.wikisource.org/wiki/The_...ain_the_extension_of_the_relativity-postulate.

Einstein accepted the challenge with the comment that the critic's 'assertion is of course indisputable'. And near the end of his defence, Einstein states that the theory 'means for a man who maintains consistency of thought a great satisfaction to see that the concept of absolute motion, to which kinematically no meaning can be attributed, does not have to enter physics'. No precise references are given at al, but almost certainly the objection was triggered by Langevin's 1911 paper which gives the first full "twin" example (using SR) to argue that a change of velocity is qualitatively "absolute". And this is just what GR was meant to make "relative".

In an earlier post I imprecisely (sorry!) stated that Moller and Builder criticized Einstein's 1918 paper. Moller's criticism is positive and in his 1952 textbook "The theory of relativity" (which I now again have at hand) he provides the calculation details that are missing in Einstein's paper. Professor Moller thus taught his students GR "vintage 1916". He surely understood* the issue, as he there explains the general principle of relativity and refers to the succession of key papers such as Einstein 1905, Langevin 1911, Einstein 1918.

Notwithstanding that great defence, I think that Builder's 1957 objection turns it into wood wreck - although he apparently never saw Einstein's paper and misunderstood the reason for the calculation; apparently he based his argument on his readings of Tolman and Moller. Historically, the "twin paradox" discussion is a continuous succession of misunderstandings.

Thus, in his paper "The resolution of the clock paradox" (G. Builder, Aust. J. Phys. 10, 246–262, 1957), Builder argues that GR can add nothing to the solution that SR already provides. And he argues that 'any application of the principle of equivalence [..] to such cases would be quite trivial", simply because the calculated fields predict effects from acceleration that were used to calculate those fields in the first place. But he next adds a different objection, in disagreement with his earlier triviality argument(!), and this one I consider pertinent:

The [accelerated] reference system S_m does not correspond to any physical system that is realizable even in principle. This conclusion is not affected by the introduction of the concept of the equivalent gravitational field. On the contrary, nothing could demonstrate more clearly the artificiality of the reference system S_m, than the statement that its physical equivalent is a gravitational field which is everywhere zero until the instant t_m=T', has the potential gx_m(1+gx_m/2c²) from t_m=T' to t_m=T’+tau’₂, and becomes zero everywhere again at t_m=T’ +tau'₂.
The concept of such a field is completely incompatible with the limiting value c for all velocities measured in inertial reference systems; [..] so that the specified field would have to be created simultaneously at all points in S' and be destroyed simultaneously at all points in S_o.
Thus the principle of equivalence [..] only accentuates the artificiality of the description of our hypothetical experiment in terms of the coordinates of the accelerated reference system S_m.

Indeed, according to GR any "induced gravitational field" must propagate at the speed of light. On top of that, what Builder overlooked or didn't bother to mention: an infinite speed of induction is also not allowed in a gravitational field according to GR, and that still does not suffice to match SR's predicted Doppler effect, as also the speed of starlight is finite - all the stars Doppler-shift instantly at turnaround.

As far as I know, none of the involved authors (Einstein, Tolman, Moller, ...) addressed that self-contradiction.

*Note: Moller makes the statement that SR "only allows treatment of the physical phenomena in frames of reference in uniform motion". Perhaps he means that SR's laws of physics are only valid in those reference frames, but it is easy to see how it can bring unaware readers to the misunderstanding that SR cannot describe observations from accelerating rockets!

 

[stevendaryl]

The stipulations are the ones that the reader is supposed to know

I would say that this discussion is all about what it means for a coordinate system to be "valid", and that a lot of misconceptions about SR result from not being clear about what that means.

 

[stevendaryl]

Don't you also have to assume transformation laws for the objects of the equations of physics? Or are you bundling that into what you mean by 'equations of physics'?

I realize that I'm on a little shaky grounds here. I'm not positive that what I said was wrong, but I'm not positive that what I said was correct, either.

The question, which I don't know the answer to, is: What goes wrong if you make the incorrect assumption about the transformation properties of scalars, vectors, tensor, etc.?

It might be worth while to work out an example.

Suppose you have an equation that is correct in one coordinate system, for example, the geodesic equation:

m \frac{d^2 x^\mu}{ds^2} = F^\mu

Let me introduce U^\mu = \frac{dx^\mu}{ds}. Then it's

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

The difficulty is that although I've written F^\mu as a vector, it may actually be a combination of a vector force together with connection coefficients.

So, change coordinates to x'^\alpha and define L^\mu_\alpha = \partial_\alpha x^\mu. Then

U^\mu = L^\mu_\alpha U^\alpha and we get an equation for U^\alpha:

m (L^\mu_\alpha \frac{dU^\alpha}{ds} + (\partial_\beta L^\mu_\alpha) U^\alpha U^\beta) = F^\mu

Multiplying by the inverse transformation matrix gives:
m (\frac{dU^\alpha}{ds} + (L^{-1})^\alpha_\mu (\partial_\beta L^\mu_\gamma) U^\gamma U^\beta) = (L^{-1})^\alpha_\mu F^\mu

So, we can define the "effective 4-force" in the new coordinates to be:

F'^\alpha = (L^{-1})^\alpha_\mu F^\mu - m((L^{-1})^\alpha_\mu (\partial_\beta L^\mu_\gamma) U^\gamma U^\beta)

So in the new coordinate system, we have the equation of motion:

m \frac{dU^\alpha}{ds} = F'^\alpha

So now I'm not sure--where is it necessary to know which things are vectors, tensors or scalars? I think the problem comes in with the physical meaning of F'^\alpha. Even if F^\mu had some simple interpretation, in terms of scalar or vector fields, the new "force" F'^\mu would have a very complicated definition. But the equations of motion would still work, wouldn't they, even though you've misidentified a connection coefficient term as a 4-vector?

 

[PAllen]

I realize that I'm on a little shaky grounds here. I'm not positive that what I said was wrong, but I'm not positive that what I said was correct, either.

The question, which I don't know the answer to, is: What goes wrong if you make the incorrect assumption about the transformation properties of scalars, vectors, tensor, etc.?

It might be worth while to work out an example.

Suppose you have an equation that is correct in one coordinate system, for example, the geodesic equation:

m \frac{d^2 x^\mu}{ds^2} = F^\mu

Let me introduce U^\mu = \frac{dx^\mu}{ds}. Then it's

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

The difficulty is that although I've written F^\mu as a vector, it may actually be a combination of a vector force together with connection coefficients.

So, change coordinates to x'^\alpha and define L^\mu_\alpha = \partial_\alpha x^\mu. Then

U^\mu = L^\mu_\alpha U^\alpha and we get an equation for U^\alpha:

m (L^\mu_\alpha \frac{dU^\alpha}{ds} + (\partial_\beta L^\mu_\alpha) U^\alpha U^\beta) = F^\mu

Multiplying by the inverse transformation matrix gives:
m (\frac{dU^\alpha}{ds} + (L^{-1})^\alpha_\mu (\partial_\beta L^\mu_\gamma) U^\gamma U^\beta) = (L^{-1})^\alpha_\mu F^\mu

So, we can define the "effective 4-force" in the new coordinates to be:

F'^\alpha = (L^{-1})^\alpha_\mu F^\mu - m((L^{-1})^\alpha_\mu (\partial_\beta L^\mu_\gamma) U^\gamma U^\beta)

So in the new coordinate system, we have the equation of motion:

m \frac{dU^\alpha}{ds} = F'^\alpha

So now I'm not sure--where is it necessary to know which things are vectors, tensors or scalars? I think the problem comes in with the physical meaning of F'^\alpha. Even if F^\mu had some simple interpretation, in terms of scalar or vector fields, the new "force" F'^\mu would have a very complicated definition. But the equations of motion would still work, wouldn't they, even though you've misidentified a connection coefficient term as a 4-vector?

What if you misinterpret your initial equation as a collection of 4 scalar equations and transform it? Then you get the same force at any point as you did in the original coordinates. I guess you could still try to argue that this 'works' with a force definition that says all forces must be measured by instruments at rest in in the starting coordinates. Do you really want to argue that?

 

[stevendaryl]

What if you misinterpret your initial equation as a collection of 4 scalar equations and transform it? Then you get the same force at any point as you did in the original coordinates. I guess you could still try to argue that this 'works' with a force definition that says all forces must be measured by instruments at rest in in the starting coordinates. Do you really want to argue that?

That's sort of what I was getting at: You can do a coordinate change without understanding the nature of the terms involved to get equations in the new coordinate system, but it's the physical interpretation of terms in the new coordinate system that are obscure (or extremely convoluted).

 

[stevendaryl]

What if you misinterpret your initial equation as a collection of 4 scalar equations and transform it? Then you get the same force at any point as you did in the original coordinates. I guess you could still try to argue that this 'works' with a force definition that says all forces must be measured by instruments at rest in in the starting coordinates. Do you really want to argue that?

That particular mixup doesn't make sense to me. I'm assuming that what IS known is the coordinates in the two systems. If you don't know that x^\mu is a coordinate, then I don't know what it would mean to say that you know what the coordinate transformation is.

 

[PAllen]

That particular mixup doesn't make sense to me. I'm assuming that what IS known is the coordinates in the two systems. If you don't know that x^\mu is a coordinate, then I don't know what it would mean to say that you know what the coordinate transformation is.

Sure it does. Consider Fx is a function of all 4 coordinates, so is Fy, so is Fz. Now, you just assume you substitute that new coordinate definitions, treating Fx, Fy, Fz as scalar functions. If you consider them as vector something else; as a covector, something else; as a density, something else.

 

[stevendaryl]

Sure it does. Consider Fx is a function of all 4 coordinates, so is Fy, so is Fz. Now, you just assume you substitute that new coordinate definitions, treating Fx, Fy, Fz as scalar functions. If you consider them as vector something else; as a covector, something else; as a density, something else.

My derivation didn't make any assumptions about the nature of F^\mu.

Once again, if U^\mu satisfies

m \frac{dU^\mu}{ds} = F^\mu

and

U^\mu = L^\mu_\alpha U^\alpha

then

m (L^\mu_\alpha \frac{dU^\alpha}{ds} + (\partial_\beta L^\mu_\alpha) U^\alpha U^\beta) = F^\mu

is an equation of motion for U^\alpha. You don't have to know anything about how F^\mu transforms. Whether it's 4 scalars, or a 4-vector, or what not doesn't seem to matter.

 

[PAllen]

My derivation didn't make any assumptions about the nature of F^\mu.

Once again, if U^\mu satisfies

m \frac{dU^\mu}{ds} = F^\mu

and

U^\mu = L^\mu_\alpha U^\alpha

then

m (L^\mu_\alpha \frac{dU^\alpha}{ds} + (\partial_\beta L^\mu_\alpha) U^\alpha U^\beta) = F^\mu

is an equation of motion for U^\alpha. You don't have to know anything about how F^\mu transforms. Whether it's 4 scalars, or a 4-vector, or what not doesn't seem to matter.

But I'm considering F as a field, no expression in terms of U. For example, something like Maxwell's equations. You have these functions of coordinates. To transform the equation to other coordinates, at all, you have to make some assumptions.

 

[harrylin]

I would say that this discussion is all about what it means for a coordinate system to be "valid", and that a lot of misconceptions about SR result from not being clear about what that means.

I thought that it was sufficiently clarified in this thread. And I don't recall ever having seen misconceptions about SR because people did not understand what it means for a coordinate system to be "valid" for the laws of physics.

In this context, it's basically the consequence of applying the principle of relativity (or of relative motion) as formulated by Einstein and earlier by Poincare.
- https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_7
For good understanding: it was also expressed as the "impossibility to detect absolute motion", because the same laws of physics are observed in systems that are moving relative to each other.

According to the theory of Special Relativity, this principle is true for the inertial reference systems of classical mechanics, and SR's laws of physics are expressed with respect to a system of that class; similarly, its transformation equations are specified for systems of that class. Consequently the use of a reference system K' in arbitrary motion is "at your own risk". The twin "paradox" in SR illustrates nicely that non-inertial reference systems are not valid systems for application of the Lorentz transformations with SR.

However, according to 1916 GR, reference systems in any state of motion must be valid in that sense, with the physics of that theory - as Einstein defended in 1918:
"It is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinate system K' as coordinate system K."
I discussed that issue in post #140 in this thread.

PS I came across a strange remark in Moller's textbook: he states that according to the special theory of relativity 'the special principle of relativity is valid for all physical laws', and adds the footnote: 'With the exception of the laws of gravitation [..]'.
I do think that gravitation works the same according to SR independent of the month of the year - the Earth's gravitational field does not act differently in March as in September!

 

[stevendaryl]

I thought that it was sufficiently clarified in this thread. And I don't recall ever having seen misconceptions about SR because people did not understand what it means for a coordinate system to be "valid" for the laws of physics.

I would say that's exactly what's going on when people claim that the twin paradox is not resolved by SR, or when they claim that it requires GR to use a coordinate system in which an accelerating observer is at rest. So I would say that this very thread is an example.