harrylin said:
It is specified that Eve uses a "co-moving inertial reference frame", which is only justified if she thinks that she is not accelerating.
Eve *is* accelerating; she feels a nonzero acceleration, an accelerometer attached to her reads nonzero, if she stood on a scale it would register weight, etc. There is no way in which she "thinks she is not accelerating".
The term "co-moving inertial reference frame" is more precisely stated as "
momentarily co-moving inertial reference frame" (MCIF). You can construct such a frame centered on any event on Eve's worldline; it will be the inertial frame in which Eve is momentarily at rest at the chosen event. This does not mean that Eve is not accelerating; it just means that it is easier to do a lot of the math in an inertial reference frame. The disadvantage of doing this is that, as I said, Eve is only at rest momentarily in any such frame; so if you want to look at the physics along Eve's worldline for any significant period of time, you can't use a single MCIF to do it. So the MCIF isn't a good representation of "Eve's point of view" over any significant length of Eve's time.
An MCIF is also *not* the same as Rindler coordinates; those are non-inertial. The advantage of Rindler coordinates is that they can cover all of Eve's worldline with a single coordinate chart in which Eve is "at rest" (her spatial coordinates in this chart don't change). This makes Rindler coordinates a better candidate for representing "Eve's point of view". But since the coordinates are non-inertial, they behave differently from the inertial coordinates you're used to in SR. For one thing, as has been noted before, Rindler coordinates don't cover the entire spacetime, so if Rindler coordinates describe Eve's "point of view", then as I said before, Eve's point of view is necessarily limited in a way that Adam's is not.
harrylin said:
The author suggests further on in his discussion that she should adopt Adam's perception of reality. I agree that that is a more sensible approach, but my opinion is based on the fact that her "gravitational field" looks fictive to me: there is no physical cause that could allow for a difference from SR.
There is no "difference from SR". Rindler coordinates and Adam's coordinates (Minkowski coordinates) both describe the same spacetime; they describe the same geometric object (or at least a portion of it, in the case of Rindler coordinates). That spacetime is flat, so there is no spacetime curvature present. Whether that counts as a "fictive gravitational field", or no "gravitational field" at all, depends on how you define the term "gravitational field". The physics is the same either way.
This brings up a general comment: you are insisting on centering your reasoning around terms like "gravitational field" that are simply
not fundamental according to GR. That is why you are having all these problems trying to interpret what's going on. There is
no single consistent interpretation of the term "gravitational field" that matches all the physics. There just isn't. To find an interpretation that matches all the physics, you have to give up the term "gravitational field" and change your set of concepts, to include things like "spacetime curvature", "stress-energy", "Einstein Field Equation", etc. instead.
harrylin said:
"They can make the same computations" is very much the "twin paradox". As Einstein explained, a symmetrical interpretation for an asymmetrical physical situation is incompatible with the foundations of GR.
How does making the same computations require a "symmetrical interpretation"? In the case of the twin paradox, both twins can compute that the traveling twin will have aged less when they meet up again. In other words, both twins compute an asymmetrical result. The same is true here; both observers (Eve and Adam) compute that Adam's point of view is not limited, while Eve's point of view is. What's the problem?
harrylin said:
In this illustration it is assumed that Adam uses his newly found rest frame as reference for physical reality. At the moment that Eve starts accelerating away, Adam ascribes the frequency difference that Eve observes to "classical" Doppler; according to him, her rocket has still negligible length contraction so that her clocks go at nearly equal rate.
I didn't say Adam attributed the frequency difference to "length contraction". I said he attributed it to the fact that the nose of Eve's rocket is moving more slowly than the tail, in Adam's rest frame. A more precise description would actually be pretty much identical to "classical Doppler":
(1) A light beam emitted from the nose of Eve's rocket to the tail will look blueshifted at the tail, because the tail is accelerating towards it (in Adam's rest frame).
(2) A light beam emitted from the tail to the nose will look redshifted at the nose, because the nose is accelerating away from it (in Adam's rest frame).
Nowhere in any of this does "length contraction" appear; the reasoning applies equally well at the moment Adam drops off the rocket and at a later time when the rocket is moving at nearly the speed of light relative to Adam. If you work through the math, the observed blueshift/redshift depends only on Eve's proper acceleration; it does not depend on her instantaneous velocity relative to Adam. So it does not depend on the absolute value of her "length contraction" or "time dilation"; it only depends on the *change* in those values during the time of flight of a light beam across her rocket, which depends on her acceleration.
harrylin said:
In contrast, Eve claims to be in rest and ascribes the frequency difference to the effect of a gravitational field which makes her clocks go at a different rate. This is just to illustrate how a different interpretation of gravitational fields and acceleration is both necessary and understood.
Yes, no problem here.
harrylin said:
After some of you brought it up, I elaborated on the equivalence principle because you and several others seem to interpret it as requiring that we can make gravitational fields from matter "vanish" (which is simply wrong)
Agreed; the quote you gave from Einstein on this was apposite. I wasn't intending to argue about that.
harrylin said:
and you seem to deny the physical reality of gravitational fields of matter.
I certainly didn't intend to deny that matter causes gravity; I was only bringing up issues relative to the term "gravitational field" and what it means. See my comments above.
harrylin said:
You thus claimed that in 1916GR, 'gravitation" can be turned into "acceleration" by changing coordinates'.
I suppose I should have included the qualifier "locally", since I was really just trying to affirm the equivalence principle, and the EP only says that you can do this locally (i.e., in a small patch of spacetime centered on a particular chosen event).
harrylin said:
That is the inverse of the equivalence principle that I have seen proposed in GR (of course, I may have just missed it; if so, please cite it!).
There are a number of different ways of stating the EP; the Wikipedia page gives a decent overview:
http://en.wikipedia.org/wiki/Equivalence_principle
The key thing I was trying to focus in on is that, in GR, you can always set up a local inertial frame centered on a particular event, in which "the acceleration due to gravity" vanishes. More precisely, you can always set up a local inertial frame centered on a particular event in which the following is true:
(1) The metric in the local inertial frame, at the chosen event, is the Minkowski metric; i.e., it is
ds^2 = - dt^2 + dx^2 + dy^2 + dz^2
where t, x, y, z are the local coordinates in the local inertial frame, whose origin (0, 0, 0, 0) is the chosen event.
(2) The first derivatives of all the metric coefficients are zero at the chosen event; this means that the metric coefficients are the Minkowski ones not just at the chosen event, but in the entire local inertial frame.
The second condition is what ensures that there is no "apparent gravitational field" in the local inertial frame; i.e., that the worldlines of inertial objects (i.e., freely falling objects) are straight lines in the local inertial frame. But this also means that the worldlines of accelerated objects--for example, the worldlines of objects at rest on the surface of the Earth, if we set up a local inertial frame centered on some event on the Earth's surface--are *not* straight lines in the local inertial frame: in fact they are hyperbolas, just like Eve's worldline in Adam's frame. This is the sense in which, locally, we can "make gravity look like acceleration"; we are making objects that are static in the local gravitational field look like accelerated objects in flat spacetime [Edit: and we are also making objects that are freely falling in the local gravitational field, and hence are "accelerating" from the viewpoint of an observer static in the field, look like objects at rest in an inertial frame in flat spacetime.]
But the local inertial frame only covers a small piece of spacetime around the chosen event; how small depends on how curved the spacetime is and how accurate our measurements of tidal gravity are. Spacetime curvature, i.e., tidal gravity, depends on the *second* derivatives of the metric coefficients, and those *cannot* all be set to zero by any choice of coordinates if the spacetime is curved. This is the sense in which we *cannot* "make gravity vanish" by choosing coordinates; the curvature will always be there, as in Einstein's example of being unable to make the gravitational field of the Earth vanish in its entirety by any choice of coordinates.
harrylin said:
GR is based on the assumption of physical reality of gravitational fields
I would say it is based on the assumption of the physical reality of *spacetime* as a dynamical object. As I said above, the term "gravitational field" is problematic.
harrylin said:
and the equivalence between acceleration and a homogeneous gravitational field.
With the qualifier "locally", and subject to reservations about the term "gravitational field", yes, this is OK.
harrylin said:
What Einstein originally denied was the physical reality of acceleration, which he thought could be "relativised" by pretending that instead a homogeneous gravitational field is induced.
This is only true if "acceleration" is interpreted to mean "coordinate acceleration". Einstein never, AFAIK, claimed that *proper* acceleration (i.e., feeling weight, registering nonzero on an accelerometer, etc.) could be relativised. With the proper terminology, Einstein was correct: coordinate acceleration *can* be relativised (again, with the qualifier "locally"), and proper acceleration cannot (which is good since it's a direct observable).
harrylin said:
If we hold that GR was wrong on that last point
It wasn't and isn't. See above.
harrylin said:
Einstein warned for a misconception that you seem to hold
I don't. See above.
harrylin said:
I wonder what you mean with "physical"; certainly nothing measurable!
You don't think proper time is measurable? What do you think your watch measures?
harrylin said:
But now that also you talk of "the foundation of the physical interpretation of relativity": I may have overlooked it but I do not find the word "proper" in either "The Foundation of the Generalised Theory of Relativity" or "Relativity: The Special and General Theory".
If you want to learn about how a physical theory actually works, you can't depend on popular books, even if they're written by the person who invented the theory.
harrylin said:
One should expect something that is at the foundation of the physical interpretation of relativity to be easy to find. So: reference please!
Try any relativity textbook. MTW talks extensively about proper time. So does Taylor & Wheeler's Spacetime Physics, which may be a better starting point since it is only about the fundamentals of relativity; MTW has a *lot* of other material.
harrylin said:
"Mere mathematical" in the sense of Applied Mathematics? Not only I do I think so, GR is based on such thinking:
Einstein used the term "the world of physical phenomena". He wasn't talking about a mathematical abstraction; he was saying that *the real, actual universe* is a four-dimensional thing, which we call "spacetime".
harrylin said:
That is completely wrong: he makes no such implicit assumption. Following your misunderstanding, Einstein would have meant with "For v=c all moving objects—viewed from the “stationary” system—shrivel up into plane figures" that it is *possible* for an object to move at c.
I have no idea what you're trying to say here. What you are claiming "follows my misunderstanding" does not follow from what I said at all, as far as I can see.
Perhaps I should belabor this some more, since it is an important point. Here's what you quoted from Einstein: "a clock kept at this place [i.e., at the horizon] would go at rate zero". I can interpret this one of two ways:
(1) Einstein is claiming that a clock can be kept at the horizon, and saying that it would go at rate zero.
(2) Einstein is claiming that a clock *cannot* be kept at the horizon, because if it could, it would go at rate zero, and that doesn't make sense.
Are you interpreting him as saying #1 or #2? If it's #1, the refutation is pretty easy: the clock would have to go at the speed of light, and no clock can do that. So let's look at the other possibility; #2 leads to one of the following:
(2a) A clock can't be kept at the horizon, because if it could, it would go at rate zero, and that doesn't make sense. Therefore the horizon can't exist, and neither can any spacetime inside it.
(2b) A clock can't be kept at the horizon, because if it could, it would go at rate zero, and that doesn't make sense. Therefore the horizon (i.e., a curve of constant r = 2m) can't be a timelike curve, because if it were, a clock could follow it as a worldline. But the horizon could still exist if it were some other type of curve, such as a null curve; and if so, there could also be a region of spacetime inside the horizon, where curves of constant r < 2m are also not timelike.
Einstein, as far as I can tell, believed #2a; but "modern GR" says #2b. I can't tell for sure what Oppenheimer and Snyder thought, since they didn't address the question in their paper; but everyone who has extended their model has come up with #2b as well.
I'll put comments on the "metaphysical" aspects of all this in a separate post.