Implications of the statement "Acceleration is not relative"

by GregAshmore
Tags: implications, statement
P: 221
 Quote by ghwellsjr What statement of mine are you referring to in post #161?
It was this one: "So what I did was transform from the IRF in which the black inertial twin is at rest to the IRF in which the blue traveling twin is at rest during the outbound portion of his trip."
However, reading it again I see that it does not necessarily imply that the traveling twin is not at rest while accelerating wtr to the stay-at-home twin. I inferred that from the context. My apologies if I attributed a position to you which you do not hold.
 Mentor P: 15,601 GregAshmore, I think that it is pretty clear from your comments that you do not get the difference between proper acceleration and coordinate acceleration. As PeterDonis mentioned, this distinction is critical and is the source of almost all confusion about acceleration. Let's use the following as the definitions of proper and coordinate acceleration: Proper acceleration is the acceleration that would be measured by an ideal accelerometer. Coordinate acceleration is the second time derivative of the coordinate position. From those definitions, do you understand why proper acceleration must be frame invariant and coordinate acceleration must be frame variant?
P: 221

I understand that the distinction between proper acceleration and coordinate acceleration is important to this discussion. I don't get that distinction yet--I'm not able to point to one example that I am sure is coordinate acceleration and another that I am sure is proper acceleration. I will study it.

[I see that while writing this, DaleSpam has provided definitions for proper acceleration and coordinate acceleration. I'll have to think about what they mean. What follows in this post is unaffected.]

In his book Relativity, Einstein talks about different kinds of gravitational fields. He points out that only gravitational fields of a "quite special form" (his words) can be made to "go away" (my words) by the choice of reference frame. Perhaps there is a correspondence between the various kinds of gravitational fields and proper acceleration versus coordinate acceleration.

However, whatever the kind of acceleration that we are dealing with in the twin paradox, Einstein states unequivocally that it is relative.

At the end of section XVIII in Relativity he says,
 At all events it is clear that the Galileian law does not hold with respect to the non-uniformly moving [railway] carriage. Because of this, we feel compelled at the present juncture to grant a kind of absolute physical reality to non-uniform motion, in opposition to the general principle of relativity.
A few pages later, after developing the principle of equivalence of inertial mass and gravitational mass, he says,
 We can now appreciate why that argument is not convincing, which we brought forward against the general principle of relativity at the end of Section XVIII. It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognises in this the non-uniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a "real" acceleration (retardation) of the carriage. He might also interpret his experience thus: "My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational field which is directed forwards and which is variable with respect to time. Under the influence of this field, the embankment together with the earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continually reduced."
According to Einstein, the rocket twin may be considered at rest throughout the episode. The rest frame is not inertial, of course. But it is a rest frame nonetheless. In that rest frame, the acceleration of the rocket is not "real"; there is no acceleration. What the rocket twin feels is not acceleration, but the force (transmitted through the seat) that holds the rocket still in a gravitational field.

According to Einstein, the acceleration of the rocket is relative. Taken at face value, the statement, "Acceleration is not relative", is at odds with Einstein's statement.
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P: 15,601
 Quote by GregAshmore I understand that the distinction between proper acceleration and coordinate acceleration is important to this discussion. I don't get that distinction yet--I'm not able to point to one example that I am sure is coordinate acceleration and another that I am sure is proper acceleration. I will study it.
This is where we should start then. Please study and come up with questions that we can discuss about the two concepts.

 Quote by GregAshmore In his book Relativity, Einstein talks about different kinds of gravitational fields.
Once you understand the concepts of proper and coordinate acceleration then understanding his writings is easy, but going the other way is not so easy. So let's concentrate on the concepts first.
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P: 4,773
 Quote by GregAshmore I'm not able to point to one example that I am sure is coordinate acceleration and another that I am sure is proper acceleration.
An object in free fall has zero proper acceleration. So if you are standing at rest on the surface of the Earth, and use coordinates in which you are at rest, then if you drop a rock, the rock has zero proper acceleration and nonzero coordinate acceleration; but you have zero coordinate acceleration and nonzero proper acceleration.

 Quote by GregAshmore In his book Relativity, Einstein talks about different kinds of gravitational fields. He points out that only gravitational fields of a "quite special form" (his words) can be made to "go away" (my words) by the choice of reference frame. Perhaps there is a correspondence between the various kinds of gravitational fields and proper acceleration versus coordinate acceleration.
Kind of. Here's a restatement of what Einstein was saying that may help: in flat spacetime, we can choose coordinates so that objects in free fall have nonzero coordinate acceleration; but we are never forced to. In flat spacetime, we can always find coordinates where objects in free fall have zero coordinate acceleration, and objects not in free fall don't. In other words, we can always find coordinates in which coordinate acceleration is present if and only if proper acceleration is present.

In curved spacetime, however, we can never find such coordinates; no matter which coordinates we choose, there will be objects in free fall that have nonzero coordinate acceleration. So we can never find coordinates in curved spacetime that will allow us to clearly distinguish, using coordinates alone, between free fall and proper acceleration.

What Einstein meant by "gravitational fields of a special form" was the apparent "field" produced by choosing coordinates in flat spacetime so that objects in free fall have nonzero coordinate acceleration. You can make them go away by choosing coordinates where objects in free fall never have nonzero coordinate acceleration; but as above, you can only do that in flat spacetime.

 Quote by GregAshmore However, whatever the kind of acceleration that we are dealing with in the twin paradox, Einstein states unequivocally that it is relative.
I'm not sure how you are inferring this from the passages you quote. He is saying that coordinate acceleration is relative, because we can make it disappear by choosing coordinates in which the railway carriage is at rest. But that doesn't make the jerk that the passengers in the carriage feel disappear, and the felt jerk is what corresponds to nonzero proper acceleration. Einstein never says the jerk is only there in one set of coordinates.

Similarly, in the twin paradox, the traveling twin feels a jerk when he fires his rocket to turn around. He feels it regardless of the coordinates we choose.

 Quote by GregAshmore According to Einstein, the acceleration of the rocket is relative. Taken at face value, the statement, "Acceleration is not relative", is at odds with Einstein's statement.
That's why you shouldn't take it at face value. You need to look at what the terms being used actually *mean*. In the statement "acceleration is not relative", "acceleration" means proper acceleration, and the statement is correct.

It's true that Einstein did not use the term "acceleration" to refer to what we have been calling proper acceleration. But as I said before, that's a matter of terminology, not physics. You can't change the physics by changing terminology, any more than you can change it by changing coordinates. By "proper acceleration" we are talking about "the real thing that corresponds to the felt jerk". If you want to know why we use the term "acceleration" for that, we can go into that, but it seems to me to be a separate question.
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 Quote by PeterDonis If you want to know why we use the term "acceleration" for that, we can go into that, but it seems to me to be a separate question.
On second thought, perhaps it isn't a separate question. Here's a quick answer: proper acceleration is the derivative of proper velocity (usually called 4-velocity) with respect to proper time. This is obviously analogous to coordinate acceleration, which is the derivative of coordinate velocity with respect to coordinate time.

The key difference, though, is that proper acceleration is independent of coordinates, because proper velocity and proper time are independent of coordinates. In fact, you don't even need coordinates at all to define any of them. And in relativity, the physical content of the theory is entirely contained in quantities that are independent of coordinates and can be defined without needing coordinates. (Einstein said that, too; I'm pretty sure he said it, or something close to it, in the relativity book that has been quoted from.)
P: 221
I'll respond to this one first.
 Quote by PeterDonis On second thought, perhaps it isn't a separate question. Here's a quick answer: proper acceleration is the derivative of proper velocity (usually called 4-velocity) with respect to proper time. This is obviously analogous to coordinate acceleration, which is the derivative of coordinate velocity with respect to coordinate time. The key difference, though, is that proper acceleration is independent of coordinates, because proper velocity and proper time are independent of coordinates. In fact, you don't even need coordinates at all to define any of them. And in relativity, the physical content of the theory is entirely contained in quantities that are independent of coordinates and can be defined without needing coordinates. (Einstein said that, too; I'm pretty sure he said it, or something close to it, in the relativity book that has been quoted from.)
I don't know what Einstein may have said on the subject of coordinate systems outside of the two documents I have cited, so I do not claim that what follows is the sum total of his views on the subject. It's probably not important at this time to be concerned with the totality of his thought on that subject. I make the following comments mostly out of general interest.

In Relativity, Einstein is content to speak strictly in terms of the coordinates of a reference body while discussing special relativity. When he gets into general relativity, he dispenses with the reference body and introduces the idea of Gaussian coordinates. These coordinates, he points out, have no physical meaning in themselves. Even so, the principle of general relativity is stated in terms of coordinates: "All Gaussian coordinate systems are essentially equivalent for the formulation of the general laws of nature."

In the magazine article in which he explains the twin paradox, Einstein says that while it is desirable to divorce the laws of physics from coordinate systems, the effort to do so has failed. The relevant text follows. The first paragraph sets the context for the discussion of coordinate systems. The meat of the matter begins with the bold text. The "money quote" is: "Only certain, generally quite complicated expressions, that are constructed out of field components and coordinates [my emphasis], correspond to coordinate-independent, measurable (that is, real) quantities."

From the magazine article:

It should be kept in mind that in the left and in the right section exactly the same proceedings are described, it is just that the description on the left relates to the coordinate system K, the description on the right relates to the coordinate system K'. According to both descriptions the clock U2 is running a certain amount behind clock U1 at the end of the observed process. When relating to the coordinate system K' the behaviour explains itself as follows: During the partial processes 2 and 4 the clock U1, going at a velocity v, runs indeed at a slower pace than the resting clock U2. However, this is more than compensated by a faster pace of U1 during partial process 3. According to the general theory of relativity, a clock will go faster the higher the gravitational potential of the location where it is located, and during partial process 3 U2 happens to be located at a higher gravitational potential than U1. The calculation shows that this speeding ahead constitutes exactly twice as much as the lagging behind during the partial processes 2 and 4. This consideration completely clears up the paradox that you brought up.
Critic:
I do see that you have cleverly pulled away from the noose, but I would be lying if I would declare myself fully satisfied. The stumbling stone has not been removed; it has been relocated. You see, your consideration only shows the connection of the difficulty that was just discussed with another difficulty, that has also often been presented. You have solved the paradox, by taking the influence on the clocks into account of a gravitational field relative to K'. But isn't this gravitational field merely fictitious? Its existence is conjured up by a mere choice of coordinate system. Surely, real gravitational fields are brought forth by mass, and cannot be made to disappear by a suitable choice of coordinate system. How are we supposed to believe that a merely fictitious field could have such an influence on the pace of a clock?
Relativist:
In the first place I must point out that the distinction real - unreal is hardly helpful. In relation to K' the gravitational field "exists" in the same sense as any other physical entity that can only be defined with reference to a coordinate system, even though it is not present in relation to the system K. No special peculiarity resides here, as can easily be seen from the following example from classical mechanics. Nobody doubts the "reality" of kinetic energy, otherwise the very reality of energy would have to be denied. But it is clear that the kinetic energy of a body is dependent on the state of motion of the coordinate system, with a suitable choice of the latter one can arrange for the kinetic energy of the continuous motion of a body to assume a given positive value or the value of zero. In the special case where all the masses have a velocity in the same direction and of the same magnitude, a suitable choice of coordinate system can adjust the collective kinetic energy to zero. To me it appears that the analogy is complete.
Rather than distinguishing between "real" and "unreal" we want to more clearly distinguish between quantities that are inherent in the physical system as such (independent from the choice of coordinate system), and quantities that depend on the coordinate system. The next step would be to demand that only quantities of the first kind enter the laws of physics. However, it has been found that this objective cannot be realized in practice, as has already been demonstrated clearly by the development of classical mechanics. One could for instance consider, and this has actually been attempted, to enter into the laws of classical mechanics not the coordinates, but instead just the distances between the material points; a priori one could expect that in this way the goal of the theory of relativity would be reached most easily. The scientific development has however not confirmed this expectation. She cannot dispense with the coordinate system, and therefore has to use in the coordinates quantities that cannot be construed as results of definite measurements. According to the general theory of relativity the four coordinates of the space-time continuum are entirely arbitrary choosable parameters, devoid of any independent physical meaning. This arbitrariness partially affects also those quantities (field components) that are instrumental in describing the physical reality. Only certain, generally quite complicated expressions, that are constructed out of field components and coordinates, correspond to coordinate-independent, measurable (that is, real) quantities. For example, the component of the gravitational field in a space-time point is still not a quantity that is independent of coordinate choice; thus the gravitational field at a certain place does not correspond to something "physically real", but in connection with other data it does. Therefore one can neither say, that the gravitational field in a certain place is something "real', nor that it is "merely fictitious".
The circumstance that according to the general theory of relativity the connection between the quantities that occur in the equations and the measurable quantities is much more indirect than in terms of the usual theories, probably constitutes the main difficulty that one encounters when studying this theory. Also your last objection was based on the fact that you did not keep this circumstance constantly in mind.
You declared the fields that were called for in the clock example also as merely fictitious, only because the field lines of actual gravitational fields are necessarily brought forth by mass; in the discussed examples no mass that could bring forth those fields was present. This can be elaborated upon in two ways. Firstly, it is not an a priori necessity that the particular concept of the Newtonian theory, according to which every gravitational field is conceived as being brought forth by mass, should be retained in the general theory of relativity. This question is interconnected with the circumstance pointed out previously, that the meaning of the field components is much less directly defined as in the Newtonian theory. Secondly, it cannot be maintained that there are no masses present, that can be attributed with bringing forth the fields. To be sure, the accelerated coordinate systems cannot be called upon as real causes for the field, an opinion that a jocular critic saw fit to attribute to me on one occasion. But all the stars that are in the universe, can be conceived as taking part in bringing forth the gravitational field; because during the accelerated phases of the coordinate system K' they are accelerated relative to the latter and thereby can induce a gravitational field, similar to how electric charges in accelerated motion can induce an electric field. Approximate integration of the gravitational equations has in fact yielded the result that induction effects must occur when masses are in accelerated motion. This consideration makes it clear that a complete clarification of the questions you have raised can only be attained if one envisions for the geometric-mechanical constitution of the Universe a representation that complies with the theory. I have attempted to do so last year, and I have reached a conception that - to my mind - is completely satisfactory; going into this would however take us too far.
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P: 4,773
 Quote by GregAshmore Einstein says that while it is desirable to divorce the laws of physics from coordinate systems, the effort to do so has failed.
I'm not entirely sure he was right even when he wrote the article; I believe that differential geometry even then had developed to the point of being able to write down coordinate-free expressions, similar to vector notation. However that may be, though, it certainly is not true today. There are well-developed formalisms for dealing with physical problems without ever having to choose or deal with coordinates. MTW goes into this in some detail.

 Quote by GregAshmore "Only certain, generally quite complicated expressions, that are constructed out of field components and coordinates [my emphasis], correspond to coordinate-independent, measurable (that is, real) quantities."
Again, even if this was true when he wrote the article (which I'm not sure it was, as above), I don't think it's true today.

However, even if we allow for the sake of argument that these statements of Einstein are correct, I don't see how they are relevant to the question at issue, because Einstein agrees that it is "coordinate-independent" quantities which are "measurable (that is, real)". And that is all that is needed to make sense of the statement "acceleration is not relative". Proper acceleration is a coordinate-independent, measurable (that is, real) quantity, and such quantities are not relative.
P: 221
 Quote by PeterDonis An object in free fall has zero proper acceleration. So if you are standing at rest on the surface of the Earth, and use coordinates in which you are at rest, then if you drop a rock, the rock has zero proper acceleration and nonzero coordinate acceleration; but you have zero coordinate acceleration and nonzero proper acceleration.
Okay, I get that. I'm glad you explained why "proper acceleration"--which one has while at rest--is called "acceleration", because otherwise one might suspect that physicists are getting their kicks by playing a kind of nerdy language joke on the general population.

 Quote by PeterDonis Kind of. Here's a restatement of what Einstein was saying that may help: in flat spacetime, we can choose coordinates so that objects in free fall have nonzero coordinate acceleration; but we are never forced to. In flat spacetime, we can always find coordinates where objects in free fall have zero coordinate acceleration, and objects not in free fall don't. In other words, we can always find coordinates in which coordinate acceleration is present if and only if proper acceleration is present.
okay.

 Quote by PeterDonis In curved spacetime, however, we can never find such coordinates; no matter which coordinates we choose, there will be objects in free fall that have nonzero coordinate acceleration. So we can never find coordinates in curved spacetime that will allow us to clearly distinguish, using coordinates alone, between free fall and proper acceleration.
And, presumably, we make the distinction for an object based on whether there is a force on the object.

 Quote by PeterDonis What Einstein meant by "gravitational fields of a special form" was the apparent "field" produced by choosing coordinates in flat spacetime so that objects in free fall have nonzero coordinate acceleration. You can make them go away by choosing coordinates where objects in free fall never have nonzero coordinate acceleration; but as above, you can only do that in flat spacetime.
Yes, except that Einstein put it in terms of the object which is experiencing the force, not the object in free fall. The gravitational field explains why the object, which is being acted on by a force, remains at rest.

 Quote by PeterDonis I'm not sure how you are inferring this from the passages you quote.
Because that's what he says. I don't quarrel with your definitions of proper acceleration and coordinate acceleration. It's just that Einstein makes no such distinction in this text. He may have in other places. But in this text, even in the appendix added in 1952, he simply says "acceleration". Indeed, in the 1952 appendix he says, "The following concept is thus compatible with the observable facts: S2 [a non-inertial system] is also equivalent to an "inertial system", but with respect to S2 a (homogenous) gravitational field is present (about the origin of which one does not worry in this connection)."

To say it in the simplest language I can think of, Einstein was obviously proud of his having eliminated the absoluteness of acceleration, "relativising it", if you will. It's hard to imagine him doing anything but bristle at the unqualified statement, "Acceleration is not relative."

 Quote by PeterDonis He is saying that coordinate acceleration is relative, because we can make it disappear by choosing coordinates in which the railway carriage is at rest. But that doesn't make the jerk that the passengers in the carriage feel disappear, and the felt jerk is what corresponds to nonzero proper acceleration. Einstein never says the jerk is only there in one set of coordinates.
Nor did I say that the jerk is only present in one set of coordinates. What Einstein actually says is that the jerk is the force which keeps the carriage/rocket at rest in a gravitational field. The "at rest" part is significant; see below.

 Quote by PeterDonis That's why you shouldn't take it at face value. You need to look at what the terms being used actually *mean*. In the statement "acceleration is not relative", "acceleration" means proper acceleration, and the statement is correct.
Again, I don't quarrel with your definition of proper acceleration, nor do I suggest that Einstein would (or did) quarrel with it. In fact, I'm sure that I will come to appreciate it as I move forward with the math of relativity.

Now that I know what you [all] meant when you said acceleration is not relative, and what you meant when you said that the rocket ship cannot be considered at rest, there is not so much of a shock factor as at first.

There remain yet two issues in my mind.

First, with regard to treating the twin paradox as a problem of special relativity, it is my opinion that you do damage to the concept of relativity. According to the principle of relativity, every observer can legitimately consider himself to be at rest; there is no preference in principle for one frame over another. In terms of coordinate systems, the laws of nature have the same form in all coordinate systems, including the coordinate system in which any arbitrary observer is at rest. You have chosen to resolve the twin paradox by saying that the rocket twin cannot be considered at rest while undergoing proper acceleration. True, he cannot be considered to be at rest in an inertial frame while accelerating. Well, then, discuss the problem in terms of the non-inertial frame in which he is at rest. Until you do so, you have not satisfied the principle of relativity, and you have not resolved the paradox.

Second, it is not clear to me that Einstein successfully resolves the paradox in terms of general relativity. I understand that the math works out so that the traveling twin is younger. I do not challenge the calculation. I do wonder at the validity of the premise on which the calculations are based, though I do not go so far as to contradict it outright. The doubt is with regard to the physical reality of the gravitational field. Einstein himself felt the need to address that issue; hence his attempt to explain the field as the result of inductance originating in the distant stars. That explanation, lacking further detail, is unconvincing.
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P: 15,601
 Quote by GregAshmore I don't quarrel with your definitions of proper acceleration and coordinate acceleration. It's just that Einstein makes no such distinction in this text. He may have in other places.
I don't think that he ever did use those terms. I think that they were invented after his death. However, with the advantage of hindsight and more sophisticated terminology it is clear that the concept he was describing as relative was coordinate acceleration.

 Quote by GregAshmore To say it in the simplest language I can think of, Einstein was obviously proud of his having eliminated the absoluteness of acceleration, "relativising it", if you will. It's hard to imagine him doing anything but bristle at the unqualified statement, "Acceleration is not relative."
Which is why I did qualify it, at length, in the post you referenced earlier.

 Quote by GregAshmore it is my opinion that you do damage to the concept of relativity. According to the principle of relativity, every observer can legitimately consider himself to be at rest; there is no preference in principle for one frame over another.
I think that your opinion is wrong in this case. The first postulate of special relativty is expressly stated in terms of inertial frames. That postulate was later generalized for general relativity, but for problems in special relativity it is reasonable to treat inertial frames as priveliged according to the first postulate.

 Quote by GregAshmore Second, it is not clear to me that Einstein successfully resolves the paradox in terms of general relativity. I understand that the math works out so that the traveling twin is younger. I do not challenge the calculation. I do wonder at the validity of the premise on which the calculations are based, though I do not go so far as to contradict it outright. The doubt is with regard to the physical reality of the gravitational field. Einstein himself felt the need to address that issue; hence his attempt to explain the field as the result of inductance originating in the distant stars. That explanation, lacking further detail, is unconvincing.
You are right to be concerned about this. I think that the modern resolution has been to just leave it alone. The problem is that there are many quantities which could reasonably be called the "gravitational field" and none of them are so important as to clearly demand that they and not the others be called thus.

My personal preference is to call the Christoffel symbols the gravitational field, others prefer to use the Riemann curvature tensor or the Einstein tensor. Still others like to refer to the metric as the gravitational field. Before you can even discuss the "reality" of the field you need to decide what it is that you are talking about. If you have a preference then I would be glad to use your preference in the discussion.
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 Quote by GregAshmore I'm glad you explained why "proper acceleration"--which one has while at rest--is called "acceleration", because otherwise one might suspect that physicists are getting their kicks by playing a kind of nerdy language joke on the general population.
That could be true in any case.

 Quote by GregAshmore And, presumably, we make the distinction for an object based on whether there is a force on the object.
We make it based on whether an accelerometer attached to the object reads zero or not; that's the actual observable. Equating that with a "force" being present is fine, but once again, that's terminology. I suspect it's terminology Einstein would have preferred; see below.

 Quote by GregAshmore It's just that Einstein makes no such distinction in this text. He may have in other places. But in this text, even in the appendix added in 1952, he simply says "acceleration". To say it in the simplest language I can think of, Einstein was obviously proud of his having eliminated the absoluteness of acceleration, "relativising it", if you will. It's hard to imagine him doing anything but bristle at the unqualified statement, "Acceleration is not relative."
This might be true; Einstein might indeed have preferred to say "force is not relative". I think he was indeed proud of having "relativised" acceleration, because I think he saw that as the logical extension of "relativising" velocity in SR. In other words, I think he saw the equivalence principle as the logical extension of the principle of relativity.

However, trying to say that force is not relative (instead of acceleration) still raises the same kind of definitional issues; you have to define "force" properly. The kind of force which is not relative is the derivative of the object's 4-momentum with respect to its proper time; in other words, it's defined the same way proper acceleration is, just using 4-momentum instead of 4-velocity.

 Quote by GregAshmore According to the principle of relativity, every observer can legitimately consider himself to be at rest
Note that this is the *generalized* principle of relativity, the one that Einstein was trying to reach by "relativising" acceleration. It is different from the principle of relativity that was first enunciated (so far as I know) by Galileo and was used in both Newtonian mechanics and special relativity.

 Quote by GregAshmore the coordinate system in which any arbitrary observer is at rest.
Note that there is not one such coordinate system; so the word "the" is not really appropriate here. Given any observer, we can construct an infinite number of coordinate systems in which that observer is "at rest".

 Quote by GregAshmore You have chosen to resolve the twin paradox by saying that the rocket twin cannot be considered at rest while undergoing proper acceleration.
I think this is also a matter of terminology; it depends on how you define "at rest". The physics, as I've pointed out several times, is that the traveling twin feels a force while the stay-at-home twin does not. That's an invariant physical difference, and it is sufficient to "resolve" the paradox without talking at all about coordinates or which twin is "at rest".

 Quote by GregAshmore Well, then, discuss the problem in terms of the non-inertial frame in which he is at rest. Until you do so, you have not satisfied the principle of relativity, and you have not resolved the paradox.
I thought we had already done this; in the non-inertial frame in which the traveling twin is at rest, there is a gravitational field present while his rockets are firing. But here's a quick elaboration of that, if you like:

While the traveling twin's rockets are firing, in his non-inertial rest frame, there is a gravitational field present. He feels a force, and the force he feels holds him static in the gravitational field; but the stay-at-home twin feels no force, so he falls freely in the field. That does two things: one, it reverses their relative motion (they were moving away from each other before, now they are moving towards each other--this is shown, for example, by the switch from Doppler redshift to Doppler blueshift in light signals emitted by the stay-at-home twin and received by the traveling twin); and two, it causes the stay-at-home twin to age much faster while the field is present, because he is at a much higher "altitude" in the field.

 Quote by GregAshmore The doubt is with regard to the physical reality of the gravitational field. Einstein himself felt the need to address that issue; hence his attempt to explain the field as the result of inductance originating in the distant stars. That explanation, lacking further detail, is unconvincing.
In a way Einstein was going down a blind alley here, I think. If you accept that every coordinate system is valid, then you accept that there are going to be "fictitious" entities in some coordinate systems. The "gravitational field" that exists in the traveling twin's non-inertial rest frame while his rockets are firing has the same sort of existential status as "fictitious forces" like centrifugal force. So asking whether it is "physically real" is a mistake, in my view.

There is another way to look at this, though. The standard twin paradox is set in flat spacetime; it is really that which makes the gravitational field in the traveling twin's rest frame (while the rockets are firing) "fictitious", because flat spacetime, as a solution to the Einstein Field Equation, requires that there are no masses present anywhere in the universe. Obviously that's not really true. However, there is an interesting theorem that says that, if we have a region of vacuum (no masses present) that is surrounded by a spherically symmetric mass distribution, the vacuum region is flat; that is, spacetime in that region looks just like the flat spacetime in which the standard twin paradox is set.

So suppose we set the standard twin paradox inside such a vacuum region of spacetime--a very large region with no mass present, surrounded by a spherically symmetric mass distribution. Then we could reason as follows: the gravitational field in the traveling twin's rest frame is "fictitious" because we can make it vanish by changing coordinates; but we can do that because spacetime in that region is flat. But spacetime in that region is flat because the region is surrounded by a spherically symmetric mass distribution; so really the gravitational field seen by the traveling twin is due to that mass distribution.

This is basically what Einstein was thinking of when he talked about inductance originating in the distant stars. To a first approximation, the distant stars are a spherically symmetric mass distribution, so an observer that accelerates in the vacuum spacetime region that is surrounded by the distant stars will see a "fictitious" gravitational field that is ultimately due to the distant stars, because they create the flat spacetime region. This is a perfectly valid way of interpreting the equations of General Relativity as applied to such a scenario.
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 Quote by DaleSpam My personal preference is to call the Christoffel symbols the gravitational field, others prefer to use the Riemann curvature tensor or the Einstein tensor. Still others like to refer to the metric as the gravitational field.
The Christoffel symbol is the one that most directly relates to the Newtonian concept of a gravitational field. The relationship between the Riemann or Einstein tensors and Newtonian gravity is very indirect. Of course, people don't have to care about the correspondence with Newtonian gravity, but since the term "gravitational field" had a meaning before GR, it's kind of strange to completely change the meaning and keep the same term.
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 Quote by stevendaryl The Christoffel symbol is the one that most directly relates to the Newtonian concept of a gravitational field.
It's also the sense in which Einstein was using the term in the quotes GregAshmore gave.
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 Quote by GregAshmore First, with regard to treating the twin paradox as a problem of special relativity, it is my opinion that you do damage to the concept of relativity. According to the principle of relativity, every observer can legitimately consider himself to be at rest; there is no preference in principle for one frame over another. In terms of coordinate systems, the laws of nature have the same form in all coordinate systems, including the coordinate system in which any arbitrary observer is at rest. You have chosen to resolve the twin paradox by saying that the rocket twin cannot be considered at rest while undergoing proper acceleration. True, he cannot be considered to be at rest in an inertial frame while accelerating. Well, then, discuss the problem in terms of the non-inertial frame in which he is at rest. Until you do so, you have not satisfied the principle of relativity, and you have not resolved the paradox.
I think you're mixing up two different things. I have to say that Einstein himself was a little unclear about them, also, but they are, I think, understood better today.

The equivalence of all inertial frames is an empirical fact (or I should say, claim) about the physical world.
The equivalence of all coordinate systems is a mathematical fact about the way your theory was written.

The principle of relativity is just the claim that no experiment can distinguish between being at rest and moving at a constant velocity, that the only kind of velocity that is detectable is relative velocity. Newton's equations of motion and Newton's theory of gravity are both consistent with this principle. However, Newton's equations + Maxwell's equations are not consistent with the relativity principle. That's because Maxwell's equations (at least in the modern form) defines a universal speed of light, which by the relativity principle must be the same in every inertial reference frame. That's not consistent with Newton's laws of motion, which require all velocities to change when you change reference frames. So the point of Einstein's theory of Special Relativity was to come up with a combined theory of mechanics and light which again satisfies the relativity principle.

The equivalence of all coordinate systems is, as I said, just a fact about the way your theory is written. Newton's equations in their original form only apply in an inertial Cartesian coordinate system. Their form is preserved by Galilean transformations, but not by more general coordinate transformations. Einstein's equations of SR are also only valid in an inertial Cartesian coordinate system. Their form is preserved by Lorentz transformations, but not by more general transformations. On the other hand, General Relativity is generally covariant; it has the same form in any coordinate system whatsoever.

But having the same form under a coordinate transformation is not really a statement about the physics. Any theory of physics can be rewritten in a form that is generally covariant, and that makes no difference to the physical predictions of the theory.

 Second, it is not clear to me that Einstein successfully resolves the paradox in terms of general relativity. I understand that the math works out so that the traveling twin is younger. I do not challenge the calculation. I do wonder at the validity of the premise on which the calculations are based, though I do not go so far as to contradict it outright. The doubt is with regard to the physical reality of the gravitational field. Einstein himself felt the need to address that issue; hence his attempt to explain the field as the result of inductance originating in the distant stars. That explanation, lacking further detail, is unconvincing.
Einstein's theory of General Relativity really doesn't make any mention of the distant stars. That's Mach's principle, that the concepts of rotation and acceleration should be relative to the distant stars. Einstein hoped that his theory would satisfy Mach's principle, but it doesn't.
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 Quote by GregAshmore You have chosen to resolve the twin paradox by saying that the rocket twin cannot be considered at rest while undergoing proper acceleration. True, he cannot be considered to be at rest in an inertial frame while accelerating. Well, then, discuss the problem in terms of the non-inertial frame in which he is at rest. Until you do so, you have not satisfied the principle of relativity, and you have not resolved the paradox.
As I said in another post, making a theory so that it works in any coordinate system is just an exercise in mathematics.

To compute the elapsed time on a clock, just pick any coordinate system $x^\mu$. Pick absolutely any real-number quantity $s$ that constantly, smoothly increases for the clock. (It could be the time, according to your coordinate system, or it could be some weird function of the time, like $s = log(t)$, or absolutely anything, as long as $s$ increases continuously.) Then in terms of $x^\mu$ and $s$, give the clock's position as a function of $s$: $x^\mu(s)$. Then the elapsed time on the clock will be given by:

$\tau = \int \sqrt{\sum g_{\mu \nu} \dfrac{dx^\mu}{ds} \dfrac{dx^\nu}{ds}} ds$

where $g_{\mu \nu}$ is the components of the metric tensor for your coordinate system, and where the sum is over all possible values of $\mu$ and $\nu$.

This works in any coordinate system whatsoever, but the values of the components $g_{\mu \nu}$ are different in different coordinate systems.
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 Quote by GregAshmore As a result of the discussion which ensues from this post I hope to understand the implications of this statement: "Acceleration is not relative." [..] What are the broader implications of the statement that acceleration is not relative? Does this mean, as it certainly would appear to mean, that modern relativity is in this very important respect not Einsteinian relativity? Are there other implications as to the meaning of the principle of relativity?
That's quite correct; that acceleration isn't as "relative" in the way Einstein suggested when he developed GR in 1907-1918 is perhaps one of the best "publicly known secrets" of modern science.
And if I correctly understand it, in a somewhat obscured way Einstein admitted this himself in 1920, by saying that "acceleration or rotation" is to be "looked upon as something real" - http://en.wikisource.org/wiki/Ether_..._of_Relativity

For some older comments by myself on this topic see:

- in Langevin's "twin" example the accelerator reading is zero during turn-around; in early SR there was no "twin paradox". http://en.wikisource.org/wiki/The_Ev...Space_and_Time
- as far as I could trace back from reading old papers, the "twin paradox" came with Einstein's attempt to make acceleration "relative" - as you saw in his 1918 paper (which, it appears, you didn't fully understand).
- coordinate acceleration is "absolute" in a qualitative way: at the turn-around all inertial reference systems measure that the traveler accelerates.
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