Time & Gravity: GR in Physics Today?

[jimbobjames]

Thanks Chris -

In units where c and G are not 1, we have:

$$ds^2 = -c^2(1-2GM/rc^2) \, dt^2 + \frac{dr^2}{1-2GM/rc^2}$$

And expanding like you suggested we get:

$$-c^2(1-2GM/rc^2) \, dt^2 + (1+2GM/rc^2) \, dr^2$$

which seems to me to be flat Minkowski space plus a delta. The delta is

$$ds^2 = (2GM/r) \, dt^2 + (2GM/rc^2) \, dr^2$$

The c^2 below the line in the in the second term led me to believe that the spatial curvature was weaker than the time curvature by a factor of c^2. That was my mistake (I think?).

Mentz, that's a coincidence - I ordered a book by him last week - "Gauge Fields, Knots, and Gravity". Sounds like it was a good choice. Any open source software you can recommend?

 

[jimbobjames]

Chris, I've been thinking about what you wrote but I'm still not clear on it.

Are you saying that just because (in the weak field approximation) g00 provides the parallel to Newtonian Gravity, that does NOT mean that g11 is insignificant? But that would mean that, even in the weak field about the earth, there are significant effects (ie everyday noticable effects like "falling") related to the spacial curvature. I'm confused about this. I thought that what we know and love as "gravity" here on earth, was almost entirely related to the time curvature in the Schwarzschild metric only. The curvature related to the dr^2 coefficient is relatively insignificant. Am I wrong with this?

(But thanks for pointing out the danger of confusing the weak field and other more stringent approximations.)

 

[Chris Hillman]

Computing stuff to order O(m)

And expanding like you suggested we get:
$$-c^2(1-2GM/rc^2) \, dt^2 + (1+2GM/rc^2) \, dr^2$$
which seems to me to be flat Minkowski space plus a delta.

To find out, compute the curvature, up to first order in the parameter M. And for gosh sake do set G=c=1; I can see you're getting confused in part by worrying about them and making a minor error.

Let's try a Riemannian analogue. Consider the coframe field
$$\sigma^1 = \sqrt{1+m \, f} \, dx, \; \; \sigma^2 = \frac{1}{\sqrt{1+m\, g}} \, dy$$
where f,g are functions of x,y. The metric is
$$\sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 = (1 + m \, f) \, dx^2 + \frac{dy^2}{1 + m \, g}$$
We compute the curvature a la Cartan, keeping only terms which are first order in the parameter m. So
$$\sigma^1 = \left( 1+ m \, f/2 \right) \, \dx, \; \; \sigma^2 = \left (1- m\, g/2 \right) \, dy$$
Taking the exterior derivative of the cobasis one-forms, we find
$$d \sigma^1 = \frac{m \, f_y \, dy \wedge dx}{2} = \frac{-m \, f_y}{2} \, dx \wedge dy = \frac{-m \, f_y}{2} \, dx \wedge \sigma^2$$
(remember, we are only keeping terms first order in m), and likewise
$$d \sigma^2 = \frac{-m g_x}{2} \, dy \wedge \sigma^1$$
where subscripts denote partial differentiation. Comparing with
$$d\sigma^1 = -{\omega^1}_2 \wedge \sigma^2, \; \; d\sigma^2 = -{\omega^2}_1 \wedge \sigma^1$$
we find
$${\sigma^1}_2 = m \, \frac{f_y \, dx + g_x \, dy}{2}$$
Taking the exterior derivative of this, we find the curvature two-form
$${\Omega^1}_2 = m \, \frac{g_{xx}-f_{yy}}{2} \, dx \wedge dy = m \, \frac{g_{xx}-f_{yy}}{2} \, \sigma^1 \wedge \sigma^2$$
(remember, we are only keeping terms linear in m!), from which we read off the Gaussian curvature
$$K = m \, \frac{g_{xx}-f_{yy}}{2}$$
which is valid to first order in the parameter m.

Chris, I've been thinking about what you wrote but I'm still not clear on it. Are you saying that just because (in the weak field approximation) g00 provides the parallel to Newtonian Gravity

Actually, I didn't say that!

I thought that what we know and love as "gravity" here on earth, was almost entirely related to the time curvature in the Schwarzschild metric only.

"time curvature"?

The curvature related to the dr^2 coefficient is relatively insignificant. Am I wrong with this?

Yes, that's why I mentioned the static wf metric (using a Cartesian type chart, incidently, not a polar spherical chart).

I second the recommendation of any expositions by John Baez, incidently, but I am not sure the particular book you bought is the best book for learning gtr first time around. You might try something like D'Inverno or Schutz first. See http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html for more recommendations.

 

[jimbobjames]

OK thanks Chris.

Incidently by saying that a spacetime has "time curvature" only, I mean a spacetime whose metric is flat apart from the coefficient of dt^2.

The ideas I am trying to relate come from reading Feynman who computes the spacial curvature even at the surface of the sun to be tiny - that's why I am saying relatively insignificant - and from Dirac's introductory lectures on GR from which I understand that the acceleration due to gravity on Earth can be computed by considering the coefficient of dt^2 only (g00, "time curvature"). Dirac explicitly mentions that its when you multiply by c^2 you get from a tiny number to 9,8m/s^2, which we recognize as Newton's acceleration due to gravity on earth. I've lent that Dirac book to a friend - otherwise I would quote from it. Of course I may have misunderstood and will double check that.

Ive looked at D'Inverno and Schutz, thanks. GR has been a hobby of mine for the past year or so - but it is deep and will require more time and study... I think I am making some progress though - in no small part because of the help I am getting from you guys here. This time last year I didnt know any tensor algebra and I did'nt know what a metric was.

 

[Mentz114]

JimBob - have you checked your messages lately ?

As Chris has said, you can't tell much about the curvature by looking at the metric. You need to calculate the curvature scalar.

 

[jimbobjames]

Got it - thanks.

 

[Chris Hillman]

Don't overestimate what you can learn from PF

Incidently by saying that a spacetime has "time curvature" only, I mean a spacetime whose metric is flat apart from the coefficient of dt^2.

As Mentz said, that doesn't really make sense.

The ideas I am trying to relate come from reading Feynman who computes the spacial curvature even at the surface of the sun to be tiny

I don't know what you mean by "spatial curvature" vs. "time curvature", but whatever you mean, you must be mistaken in thinking they have very different magnitudes.

You might have seen reference to such terms as "time-time components" of the Riemann tensor, aka the electrogravitic or tidal tensor in the Bel decomposition taken with respect to a family of observers (for convenience let's assume their world lines are hypersurface orthogonal), which controls tidal forces, or "time-space" components, aka the magnetogravitic tensor, which controls gravitomagnetic effects, or "space-space", aka the topogravitic tensor which controls the curvature of hyperslices orthogonal to our observers.

But the Riemann components behave nothing like what you are assuming. In fact, the nonzero Riemann curvature components (evaluated wrt a frame field, i.e. we consider only "physical components" which can inferred directly from measurements by some observer) in the Schwarzschild vacuum are all of magnitude comparable to the square root of the Kretschmann invariant $48 m/r^6$. If you can grab a copy of Misner, Thorne, and Wheeler, Gravitation, in your local physics library (this book will probably be on reserve in most university library systems), see the first chapter for a discussion of how spacetime curvature relates to falling apples on the surface of the Earth.

the acceleration due to gravity on Earth can be computed by considering the coefficient of dt^2 only (g00, "time curvature").

Or [tex]g_{rr}[/itex], as we keep saying. In the weak field metric, as you now realize, the components [tex]g_{tt}, \; g_{rr}[/itex] differ by comparable magnitudes from $\pm 1$.

Ive looked at D'Inverno and Schutz, thanks. GR has been a hobby of mine for the past year or so

So what is your math/physics background? It is possible to obtain a good knowledge gtr from self-study, but only if you have a solid math background. (A solid physics background is also highly desirable, but oddly enough, not essential if one is careful to always bear in mind the limits of one's knowledge/appreciation of physical issues.) A graduate level course in manifolds and Riemannian geometry is particularly important. Don't overestimate what you can learn from a forum such as this. We are not likely to exhibit the patience required to convey the subtle but essential notion of local versus global structure, for example, because this is best acquired by working problems in a structured environment.

 

[jimbobjames]

"Don't overestimate what you can learn from a forum such as this."

OK.

I don't have a graduate level course in manifolds and Riemannian geometry, but I learned enough math during my Engineering undergraduate years (almost 20 years ago now) to be able to read those introductory GR books and follow the derivations. Sure some parts are'nt crystal clear yet but I'm getting there. I am pleased with what I've managed to learn from books (and the web) so far. Sean Caroll's teaching is also excellent. I love the 3 hour introduction to GR (video lecture) by him.

I was pleased to learn what a manifold is, what tensor calculus on a manifold is, what the covariant derivative is and why it is useful, and in the end I was pleased to learn how to go from a metric to Christoffel Symbols to the Riemmann Tensor to Ricci Tensor and Scalar and finally to the field equation. And to learn how the equivalence principle provides the physical motivation for the theory. But there's a lot more to GR, I know.

Ive always wanted to better understand why things seem to fall to the ground. To learn what people mean when they say that "spacetime" is curved. I am fascinated by it.

For me its a real luxury to delve into physics whenever I get a chance - but the available time is limited to a few hours per week unfortunately. And you are right - without having a teacher or mentor to ask questions, it is easy to make mistakes and to misunderstand some of the ideas.

Anyway in this case I see I made the mistake of

1) interpreting too much into the coefficients of the metric directly - I need to compute the Riemann before talking about curvature
and
2) thinking there are real metrics where gtt is the only significant non-flat component

I think the world would be a better place (!) if some GR expert took the time to tape his or her lectures on GR and put them on the web. The few video lectures I could find so far (from Kip Thorne and Sean Carroll) were excellent, but unfortunately only skim the surface.

 

[Chris Hillman]

Appreciate the info, although I fear you may be prone to vastly overestimate your grasp of what you read in the textbooks. In particular, just because most textbooks avoid discussing the local versus global distinction (Carroll actually gives this more attention than most authors) doesn't mean this won't prevent you from understanding gtr until you appreciate it. Unfortunately, there are quite a few more stumbling blocks.

I don't think taped lectures would be helpful. You might want to look for the problem book edited by Lightman et al., since working problems can be a good way to assess your understanding.

BTW, if this sounds discouraging, from time to time I like to point out that there are billions and billions (nod to CS again) of mathematically beautiful things one could try to learn, many far more accessible but no less important than gtr. One could even argue that gtr is far less timely than many other phenomena currently of interest to mathematicians.

 

[Antman]

Couldn't gravity be a direct result of the time distorsion created by masses? If we assume that all mass is comprised of elementary, vibrating particles. All these particles are vibrating at certain frequencies. If atoms are vibrating at different frequencies we say they have different temperatures and if they have higher temperature they have a higher pressure. If an object comprised of more than one elementary particle is within the time distortion field of a mass, the particles in the object would slow down because time is passing slower closer to the mass. Some of the particles in the object must be closer to the mass than some others and thus move slower. The particles further away from the mass will be moving faster than the ones closer and thus put a pressure inside the object towards the mass. That is if these particles were to work the same as atoms when vibrating at different temperatures. I think...

What do you think?

 

[jimbobjames]

I thought that too Antman - a way to test it would be to heat something on the space shuttle and check if it moves in the direction of the temperature gradient.

I had typed the same thing last week and deleted it before posting. The reason I didnt post it was - among other reasons - that you can check that this effect cannot be the cause of gravity on Earth by simply heating any object and realizing that even when the base of the object is hotter than the top (and the atoms and molecules are vibrating faster at the bottom than at the top) the object does not start to move away from the surface of the earth. (A rocket lifts off of course for other reasons, and not just because the bottom is hotter than the top, as you know).

I think others here will point out that the object is just following a straight line path through curved spacetime but like yourself, I would like to better understand what is happening in the immediate vicinity of the object itself - ie to understand the "contact" between the local spacetime and the object - if that is understandable!

Good idea though.

 

[Antman]

I don't think that the atoms movements cause gravitation but rather some smaller particles. If you say something that enters the gravitational field of a mass and gets "pulled" downwards the mass actually is proceeding in a straght line but the spacetime is curved I agree that might be the case. But what if something is stationary and starts to accelerate towards a mass because of gravity? That couldn't be because of bent spacetime.

 

[jimbobjames]

"but rather some smaller particles"

If you mean some as yet undetected hypothetical particles inside the object, then I think you might need to come up with (or derive) a description of their properties and an explanation as to why they have evaded detection until now.

"But what if something is stationary and starts to accelerate towards a mass because of gravity? That couldn't be because of bent spacetime."

Do you mean, for example, holding an apple and letting go and watching it appear to fall to the ground?

Why do you think that could'nt be because of curved spacetime?

The curved spacetime way of describing gravitation is the best model available and it has been tested to high precision.

In the curved spacetime view of the world, the apple was actually accelerating until you let go of it! (You can calculate the acceleration by putting dr=0 in the field described by the Schwarzschild metric. Just like a rocket hovering at a fixed distance above the Earth is actually accelerating, just to stay the same height above the earth.)

And only when you let go of the apple and when it was in free fall, did it actually stop accelerating, because it then started to follow the straight line path through the curved spacetime around the earth!

(I'm not an expert though either - that's just my understanding of it).

 

[pervect]

Appreciate the info, although I fear you may be prone to vastly overestimate your grasp of what you read in the textbooks. In particular, just because most textbooks avoid discussing the local versus global distinction (Carroll actually gives this more attention than most authors) doesn't mean this won't prevent you from understanding gtr until you appreciate it. Unfortunately, there are quite a few more stumbling blocks.

While I wouldn't want anyone to think that they can become an "expert in GR" just by reading a forum like this, I wouldn't underestimate what one can learn in a forum like this either. I would, however, generally agree with that most of the work has to be done outside the forum, that one has to read textbooks and work problems to learn GR, just reading a forum is not going to be enough.

I would also agree that it is easy to "go off the rails". I'm not sure, though, if this "local vs global" issue has been well-defined enough to count as "going off the rails", or whether it is just a philosophical disagreement. If something has not been well-defined enough to have been published in textbooks, or to have a particular reference in the literature along with a proof, I'm not sure that I agree it should be put in the classification of being necessary to understand GR.

I don't think taped lectures would be helpful. You might want to look for the problem book edited by Lightman et al., since working problems can be a good way to assess your understanding.

Different people work in different ways as to how they learn. I would agree that the ability to work problems is a good way to asses one's understanding, though, making worked problems very valuable. In fact, I would argue that being able to get the correct answer to problems more or less operationally defines what it means to "understand GR".

 

[jimbobjames]

Chris, Ill have to disagree with you.

I'm convinced that a set of video lectures from a good GR teacher would provide a great service to the community.

And it wouldn't have to be 100 hours of lectures - Even if Sean Carroll did another 3 hours in order to provide the next level of detail, I know it would accelerate my own learning and probably that of many other newcomers to the subject.

And as to the accessibility of GR - Sean Carroll also disagrees with you - he starts his lectures with - General Relativity is easy! And then he goes on to show that it really isn't as difficult as its reputation (and some experts) would have the world believe.

Even the author of one of the books you recommended - D'Inverno - begins his book with words of encouragement to the less able student:

"I will not deny that the book contains some very demanding ideas (indeed I do not understand every facet of all these ideas myself)... Take heart from my story. I am certain that if you persevere you will consider it worth the effort in the end".

I do agree however that I need to work though the exercises. Thats key. Its only when you try to work it out yourself that you realize how much you did'nt yet understand. I will check out that problem book you recommended.

 

[jimbobjames]

In one of my earlier posts in this thread I wrote that I thought that only the coefficient of $$dt^2$$ in the Schwarzschild metric was relevant when aproximating to Newtonian Gravity as we experience it here on Earth and that "The curvature related to the $$dr^2$$ coefficient is relatively insignificant."

And an expert here wrote back telling me that I was wrong with this assertion.

Later I wrote again that:

"I understand that the acceleration due to gravity on Earth can be computed by considering the coefficient of $$dt^2$$ only".

But this was also deemed incorrect.

I just read the following from Jim Hartle in his Introduction to GR:

"You may have noticed that the factor $$(1-2\phi/c^2)$$ in the spatial part of the line element:

$$ds^2 = -c^2(1+2\phi/c^2) \, dt^2 + (1-2\phi/c^2) \, dr^2$$

played no role to leading order in $$1/c^2$$ in reproducing either the relativistic relation between time intervals on clocks or the Newtonian equation of motion. Any factor there that is unity to leading order in $$1/c^2$$ would have worked, including 1. There are therefore many spacetimes that will reproduce the predictions of Newtonian Gravity for low velocities."

And so the metric I was proposing:

$$ds^2 = -c^2(1-2M/rc^2) \, dt^2 + dr^2 = -c^2(1+2\phi/c^2) \, dt^2 + dr^2$$

is one of those spacetimes that will reproduce the predictions of Newtonian Gravity, as I was saying.

Italics by Jim Hartle.

The metric I proposed above may not satisfy Einstein's field equation - nor will it predict the correct bending of light near a star - but it is certainly close enough to the home cosmos - and certainly closer than the original metrics I was playing with - to produce what we know and love as Gravity here on earth.

 

[pervect]

In one of my earlier posts in this thread I wrote that I thought that only the coefficient of $$dt^2$$ in the Schwarzschild metric was relevant when aproximating to Newtonian Gravity as we experience it here on Earth and that "The curvature related to the $$dr^2$$ coefficient is relatively insignificant."

And an expert here wrote back telling me that I was wrong with this assertion.

I think you probably confused the expert by saying "curvature" instead of "metric coefficient".

Most of the time "curvature" means the Riemann curvature tensor or some component thereof.

 

[jimbobjames]

"Most of the time "curvature" means the Riemann curvature tensor or some component thereof."

Right pervect, got it. I'm getting there. I appreciate your help along the way.

 

[intel]

Certainly not! Intel and Dravish, I think you both need to be very very careful about drawing conclusions from verbal descriptions. These can play a valuable role, but to understand gtr you'll need to understand the math. That said, I think the book by Geroch does a fabulous job at getting absolutely the most out of the least mathematical background. Also, posters like pervect and robphy are far more reliable sources than some other regular posters here. As I think they will agree, ultimately, good textbooks (and perhaps local faculty, especially if they specialize in gtr) are still your most reliable sources of information...

Thanks for the reference I am getting hold of it - should be here in a couple of days!

But i would still like to know if such a question (maybe not this one in particular) is askable of maths/physics and if I can expect a definitive yes, no or don't know.

The question I asked was:

Originally Posted by MeJennifer
...time and space are emergent properties of the gravitational field...

"Can I take this and say that spacetime, space and time are the direct result of particles or any matter which came into existence at t=0... ?"

 

[Dravish]

Some interesting posts here, and also some posts with rather condescending tones.
I would firstly like to make a small point: mathematics is a descriptive tool.

ok, now let me break things down a little;
1. Assume that nothing exists; no matter, no space, no time; nothing
2. Now add an imaginary mass; we now have mass and nothing
3. Now add a secondary mass at some undefined distance away from the first; we now have mass and space (as space is simply a description of the nothing between the two masses).
4. Now allow the masses to move in any direction; we now have space and time (as time is simply the description for changing states)
5. In the QM view of reality, in order to introduce gravity we would need to add 'gravitons' to this setup so that our two masses could be attracted to one another.
6. Ignoring the QM view and using the GR view that gravity is the curvature of space-time, gravity is a property of mass existing within space.
7. From the above we can deduce that time = the movement of each (undefined) mass, gravity = an undefined attractive force (movement of each mass towards the other) between the two (undefined) masses.

Is this or is this not the fundamental view of things from the GR point of view?

 

[Mentz114]

Now add a secondary mass at some undefined distance away from the first

Hi Dravish. How do you do that if there's no space ? Surely you need 2a. Add space ?

It's certainly true that to define space we need at least two locations, and only matter can have a location. It seems that space and matter must have been born simultaneously, and by being born caused change and therefore time.

 

[Dravish]

Hi Dravish. How do you do that if there's no space ? Surely you need 2a Add space ?

Adding the second mass at an undefined distance from the first creates spaces, because (as I mentioned) space is simply a description of between masses

 

[Mentz114]

OK, so space comes along with the second mass. I disagree about time's birthday, though. As soon as you introduce the first mass, you've caused change - and hence time now exists?

[sorry, I keep dropping letters and having to edit]

 

[Dravish]

you miss understand what I'm trying to say (or perhaps I am being unclear), I have presented a sequence of events, but they do not take place in that sequence; I am simply trying to show a complete picture one step at a time.

 

[Mentz114]

But 'one step at a time' means 'in sequence' to me.

Anyhow, it's good to think analytically about these things. I must disagree with a couple of your points though.

6. Ignoring the QM view and using the GR view that gravity is the curvature of space-time, gravity is a property of mass existing within space.

From my understanding of GR, "matter tells space-time how to curve, and space-time tells matter how to move". Gravity isn't a property of mass, it's the result of mass.

7. From the above we can deduce that time = the movement of each (undefined) mass, gravity = an undefined attractive force (movement of each mass towards the other) between the two (undefined) masses.

I don't follow this because your use of the word 'undefined' seems to rob that sentence of meaning. I'm sure you actually mean something else.

 

[Chris Hillman]

"Jimbo"'s annointing of "experts"

I think you probably confused the expert by saying "curvature" instead of "metric coefficient".

Most of the time "curvature" means the Riemann curvature tensor or some component thereof.

Jimbo, in future, if you think two sources are disagreeing with one another, you should pause and consider the possibility that you misunderstood something. In many cases simply considering this possibility will reveal the actual misunderstanding without your having to ask for help.

In this case, because I have said all this a thousand times before, I may have been a bit less clear than I would have been in ideal world in which no-one ever repeated the same mistake :-/

In the weak-field approximation, we require the curvature to be everywhere small. This can be a bit tricky to formulate in complete generality (those who know about gravitational plane waves can consider an ultraboosted observer as treated in the usual Rosen chart--- compare the invariants of the Riemann tensor, which all vanish identically!), but physically speaking, the idea is that anything such as matter or an electromagnetic field which contributes to the source term on the RHS of the EFE, plus any gravitational radiation coming in "from infinity", should all be "small". This approximation was used by Einstein to study how rearranging a configuration of matter can generate gravitational radiation, for example. It is also used in the GEM formalism.

The metric form which I quoted, which is due to Einstein himself,
$$ds^2 = -(1-2 \, \phi) \, dt^2 + (1 + 2 \, \phi ) \, (dx^2 + dy^2 + dz^2), \; \; -\infty < t, \, x, \, y, \, z < \infty$$
is valid in the static case of the weak-field approximation. Here, a static spacetime is by definition one in which we have a timelike Killing vector field. In the line element I just wrote down, this Killing vector field is of course the coordinate vector field $\partial_t$.

A closely related formalism is the far-field approximation, in which we study the gravitational field of an isolated massive object sufficiently far from the object that we can treat the fields as weak. This is used in defining the mass and angular momentum of an isolated object in gtr, for example. "Isolated" means that far from the object, the spacetime curvature is very small, in fact decreasing to zero as we move away from the object. In the far-field approximation, it is natural to adopt some kind of "comoving center-of-mass" coordinate chart, in which a world tube containing our massive object is excised, but is static and "centered".

Neither of these approximations is the Newtonian limit of gtr, however. To recover Newtonian gravitation, we need to restrict ourselves to studying only test particles which are moving slowly with respect to the only thing of physical importance in view, the afore-mentioned massive object. Then we have the field equation of (Laplace's field theoretic formulation of) Newton's theory of gravity (the Laplace equation) together with the slow motion approximation of special relativity, at the level of tangent spaces, which yields Galilean kinematics with Newton's gravitational dynamics.

It might help to point out that most papers/books dealing with gtr use units in which G=c=1. This almost always is a good idea, but in this case it does obscure the Newtonian limit, which can expressed imprecisely but vividly (?) by the slogan "expand about m=0, 1/c=0 and neglect all terms of higher than first order".

Now, let's go back to "curvature". As I showed you in my very detailed computation, it makes perfect sense to compute the curvature tensor in the weak-field approximation--- we just follow the same rules as for any Lorentzian manifold, except that we expand in a kind of power series about $m=0$ where m is a kind of "maximal curvature parameter" (if you like, a mass parameter), and neglect any terms of higher than order $O(m)$. However, if we want to compute curvature in Newtonian field theory, we need to do something rather different. Following Cartan, we need to use an entirely different model, not Lorentzian manifolds, in which the tangent spaces have a "hyperbolic trigonometry" defined by a certain nondegenerate but indeterminate bilinear form, but Galilean manifolds, in which the tangent spaces have a "parabolic trigonometry", defined by a certain degenerate bilinear form. See for example the chapter on "Newtonian spacetime" in MTW, and then see the textbook by Sharpe; full citations are at http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html

I think everyone here should be more careful about a number of things. One trend which worries me is the current trend towards redefining the term "expert" to mean something quite different from the definition which I think is most useful to physics/math students.

In academia and scholarly circles generally, an expert is someone who has written a (mainstream) textbook or two. In Wikipedia, an "expert" is someone who has read a textbook or two.

This is in fact a good way to begin to appreciate why scholars are generally apalled by the prospect of university students relying on WP as a source of "information" simply because it is more convenient than walking to the college library. The typical Brittanica article is written by an expert in the scholarly sense. The typical WP article is written in chaotic "collaboration"--- or very possibly, via "edit warring"--- between trolls, cranks, ignoramuses, and "experts" in WP sense. That said, at the time of this post, some of the specialized math articles in WP are very good, but this could change the moment some new mathematical crank appears. (Those who remember sci.math in pre-Plutonium days might appreciate my point.)

Sean Carroll also disagrees with you - he starts his lectures with - General Relativity is easy! And then he goes on to show that it really isn't as difficult as its reputation (and some experts) would have the world believe.

Actually, that sounds very much like statements I wrote in sci.physics.relativity long before that textbook appeared. Again, I claim that any apparent contradiction disappears once you place these remarks in their proper context. In fact, I think that Carroll would probably agree with me that the things many students starting a gtr course expect will be hard to understand are not in fact so hard, but there are many issues (such as local versus global distinction) which they cannot possibly anticipate on the basis of past experience (unless they've had a really solid course in manifold theory!) which really are difficult to explain in a few words. But there's no point in emphasizing that at the beginning of a textbook which is written to invite students who might otherwise be afraid of gtr's scary "rep" to join the fun!

I'm not sure, though, if this "local vs global" issue has been well-defined enough to count as "going off the rails", or whether it is just a philosophical disagreement.

Local versus global structure is one of the great themes of mathematics from the twentieth century onwards. See for example the textbooks by Jack Lee cited in http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html#mathback and then read Michael Monastyrsky, Modern Mathematics in the Light of the Fields Medals. If anyone follows this program and posts followups in the Math forum, no doubt I and/or mathwonk can explain how many Field's Medals have involved this distinction in an essential way. In a less elevated manner, my own Ph.D. diss concerned generalized Penrose tilings, in which one is interested in how local rules (such as Penrose's inflation rule) can enforce unanticipated global behavior (such as almost periodicity). Having said that much, I can't resist adding my suspicion that the best way to approach this, as elsewhere in mathematics, may be by treating spaces of tilings as non-Hausdorff sheaves, as I tried to do in my discussion of Conway's "empires", and not as branched manifolds. In the sheaf formulation, a complete tiling is a global section and Conway, Thurston, and other luminaries have indeed studied "cohomological obstructions" to completing a locally valid Penrose tiling (local section) to a tiling of the complete plane (global section). Sheaves are in fact more "fashionable" (at least in mathematical circles) than the kind of gtr stuff we are perenially hung up on in this forum. One reason is easy to appreciate: appropriate categories of sheaves form a topos (roughly speaking, a category sufficiently rich that one can use objects in this category to model any situation which can be discussed in mathematical terms). This means that the foundations of mathematics itself can be treated in terms of sheaves; the appropriate "pointwise" structure on stalks is then an intuitionistic logic, and quantifiers arise from adjunction. This seems worth remarking upon as we remember the career of the late Paul Cohen.

Of course, at a higher level, when someone mentioned boundary conditions for PDEs, that is right on the money, as Einstein himself was well aware. These days, gtr is of continuing interest mostly in mathematical circles, not physical circles, and mostly because of the challenge of understanding the space of global solutions.

As for "well-defined", well, I don't try to define global structure mostly out of laziness, but anyone who has a good grasp of the idea that a tensor field is a global section of a suitable fiber bundle over a smooth manifold will probably understand sufficiently well what this distinction is and why it matters so much by considering extending a vector field from a disk on the sphere or projective plane to the full manifold.