"Jimbo"'s annointing of "experts"
pervect said:
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,
<br />
ds^2 = -(1-2 \, \phi) \, dt^2 + (1 + 2 \, \phi ) \, (dx^2 + dy^2 + dz^2),<br />
\; \; -\infty < t, \, x, \, y, \, z < \infty<br />
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.
Chris Hillman said:
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.)
jimbobjames said:
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!
pervect said:
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.
- to produce what we know and love as Gravity here on earth.
