Talk:General theory of relativity
Higher math template?
Question: I know we have this template for sections that get into early-undergrad math:
|This article/section deals with mathematical concepts appropriate for late high school or early college.|
But the quantitative section is going to make (gentle) reference to tensor algebra and some aspects of differential geometry. Do we have a template that somehow conveys "this section is hard-core math, but don't be scared?" Or maybe something that gets across the idea that you don't have to follow everything in this section to understand this article?
I'm going to leave this page half-done for tonight and pick it up tomorrow. Anybody available to give what's there right now a proofread? I'd love any and all feedback. --KSorenson 18:45, 13 November 2009 (EST)
- The math is way out of my depth, but I tried clarifying the section about Eddington's experiment. It now says what I think it said - if I'm way off-base, please correct me. -- EvanW 19:01, 13 November 2009 (EST)
- Yep, everything in there seems okay. Can't wait to read the rest of the article.--DanHutchin 19:14, 13 November 2009 (EST)
- Yeah, I'm struggling with the math. The math of general relativity is hellishly complex, to the point where only a small number of exact solutions have been found in nearly a century. But I think examining the mathematics, even if it's in a superficial way, could really help the motivated student understand what the theory means, and to understand what it does and doesn't predict. I'm not planning to dive into covariant and contravariant tensor transformations or the expansion of the Christoffel symbols, but I'm trying to provide a sort of nickel tour of the equations. To your right, we see the stress-energy tensor; it's like mass. To your left, we see the Ricci tensor; it's a contraction of one of the vector fields of the Riemann with its dual, leaving the fully traceless Weyl tensor to describe gravitational radiation and yeah, I think you see my point. It's challenging.
- But the math is so pretty. Really, it's like a jigsaw puzzle, and every piece has a really clear physical interpretation. This piece describes how the volume of a sphere in curved spacetime differs from the volume of a sphere in flat spacetime; it's a direct representation of the shape of spacetime in a gravitational field. This piece represents the plain, old, warm-and-friendly Pythagorean theorem; it's really no more complicated than that! This slot here is where you put the energy of the vacuum in order to model a vacuum-dominant expanding universe. The math is gorgeous, and I want to find a way to share that with kids without boring them or scaring them off.
- Anyway, the Mercury thing really deserves its own section under tests of general relativity, I think. It's such an important prediction from the points of view of astronomy, theoretical physics and the modern history of science. I'll turn my attention to it after I finish the quantitative section, unless somebody wants to dive into it.
- Thanks for the feedback, you guys! Enjoy your Friday night. --KSorenson 19:25, 13 November 2009 (EST)
The templates are math-e, math-m, math-h, and math-a, for elementary, middle-school, high-school, and advanced. But the detailed content suggests some overlap. (By the way, I made them, based on earlier work by, I think, DanielB.) If anything deserves a "math-a", it's this! Math-a was intended for discussion of things like topology, the axiom of choice, the Riemann hypothesis, etc. That is, stuff really at the outer edges of CP's mission.
I know a fair amount about this subject, and about presenting it in a palatable way. I will look at the page over the weekend. There are many other things on my plate, but this has just gone to the top of the list. I'll deal with the Peano postulates later.
Kate: I will send you the paper on tensor calculus that we discussed.
PatrickD 20:31, 13 November 2009 (EST)
Idea for presenting this
The General Relativity seems to be the place to be this weekend!!!!!
I've done some thinking about how we can explain, semi-adequately, what is going on, at an acceptable educational level. We all know that adequate explanations for lay people just don't exist. But we're making an effort.
Here's an idea that I had: Put something in, probably in front of your sections on T_mu_nu and G_mu_nu, about fictitious forces, and how curved coordinate systems, and curved manifolds, lead to these forces.
First, some kind of spacetime diagram, completely flat, with x going straight left to right, t going straight up, and some events labeled "me, now" at the origin, "me, 5 minutes from now" up the page, "My neighbor's house, now" to the right, and a diagonal line for my neighbor walking to my house and arriving in 5 minutes. An explanation that these are "world lines". Mine goes straight up, and my neighbor's goes diagonally. We show coordinate "graph paper" lines, all rectilinear.
Next, figure 2: a diagram showing the frame of reference of a car traveling uniformly down the street. The formerly vertical lines are now slanted. My neighbor appears to be walking faster in my frame of reference, and he arrives at my house at x=-5 or whatever. He is now behind me, because my car moved.
Point out that a "flat coordinate system" is one in which the coordinate lines are straight. That means they make boxes that are either parallelograms or rectangles. They don't have to be rectangles. The coordinate system of figure 2 is still flat. Point out also, that both the stationary observer in front of my house (figure 1) and the observer in the car (figure 2) don't feel any fictitious forces. They are in inertial frames of reference.
Now do figure 3. This is a car that starts out at rest at t=0, and accelerates. We show the formerly vertical coordinate lines as being curved. (They're parabolas, I believe.) This is a "curved coordinate system", characterized by curved coordinate lines. (You and I know that it's not that simple -- curved? What does that mean? It means they aren't geodesics. We can't get into that. It involves Christoffel symbols. We just draw curved lines and leave it at that.) We also point out that someone in the accelerating car feels a [fictitious] force, even though he is "at rest" in his coordinate system.
Then we state the fundamental principle: If you are in a curved coordinate system, you will feel fictitious forces. A fictitious force is traditionally something that arises from some kind of acceleration, such as an accelerating vehicle, or a centrifugal force, or a Coriolis force.
Then we state the equivalence principle -- gravity perhaps can be explained as a fictitious force. We can point out that this principle had sort of been known for a long time before Einstein.
Now let's look at curved coordinate systems. Put up a piece of polar graph paper. This is a curved coordinate system. But, no problem -- it's a flat piece of paper. The manifold (we're using that term, right?) is flat; it just happens to have a curved coordinate system. This is equivalent the the accelerating car. By pressing down on the accelerator of the car, we intentionally put ourselves into a curved coordinate system, but the manifold itself is flat. The observer standing on the sidewalk can attest to that.
Now let's look at curved manifolds. We say, in a hand-wavey sort of way, that there are "curved manifolds", and that the surface of a sphere is an example of a curved 2-dimensional manifold. And we state (we can't possibly prove it, or even rigorously define the terms, at this level of exposition) that curved manifolds have only curved coordinate systems. There is no flat coordinate system on the surface of the sphere. Flat manifolds, like the spacetime diagrams of figures 1, 2, and 3, have both flat and curved coordinate systems. People living in the curved coordinate systems will feel fictitious forces.
We make some hand-wavey statement that the curvature of a manifold arises from somethng called the "metric tensor", "Riemann's tensor", "Ricci's tensor", and "Einstein's tensor". If we know the metric tensor, we can tell whether the manifold is curved. If it is (that is, Einstein's tensor is nonzero), then the manifold is curved, all possible coordinate systems are curved, and there is no global coordinate system that is free of fictitious forces. (Of course, we can make a coordinate system that is locally flat -- just jump down an elevator shaft.)
Then we say that the essence of General Relativity, and its explanation of gravity, is that gravity appears to be a fictitious force for which no flat coordinate system exists, and hence it arises from the curvature of the manifold itself, as given by Einstein's tensor. Which is the left side of the equation.
I really think this will be better than the exhibits of balls rolling around on curved surfaces that we see in science museums. A lot of handwaving, but we just may be able to do this a bit better than most elementary treatments.
PatrickD 15:37, 14 November 2009 (EST)
- By luck (whether good or bad is left as an exercise for the reader) that's pretty much the curriculum for the first six weeks of my intro to general relativity course. I mean beat for beat: we start out reviewing special relativity and transformations from orthonormal coordinates to oblique coordinates, which takes us into the hyperbolic trig interpretation, then we do curvilinear coordinates and the equivalence principle. I use the unit 2-sphere and tidal forces to introduce the concept of an obstruction, which gets us into the metric and parallel transport. That's usually when I say, "Okay, just assume everything I'm about to tell you is true and trust me," and I do a couple hours of tensor algebra so we can talk Riemann.
- I'm not a good candidate to boil all that down into something suitable for a precocious high-schooler. That's why I used the same story I told my niece when she asked me what I do for a living. Except we actually went out to the back yard trampoline with a tennis ball. (Her adorable and patient aunt played the part of the sun in our little model solar system.)
- I'm all for putting it in terms of geometry rather than balls and surfaces, if you think you can explain it in a way that gives a good intuitive grasp of the ideas. --KSorenson 16:29, 14 November 2009 (EST)
- Oh, also I'm going to move your advanced-math-ahead warning sign down to the quantitative section and replace the some-math-ahead warning sign. --KSorenson 16:30, 14 November 2009 (EST)
- I'm always on the lookout for math articles that would actually be useful to write, so let me know if there's anything that could help give background for this article -- I can write about geodesics or curvature or whatever else comes in here (these both have pages already, but they're rather silly). --MarkGall 19:51, 14 November 2009 (EST)
- Oh, thank you so much for offering. I spent some time this afternoon thinking about the next section; it's obviously more math-intensive than the first. The subjects I'm going to have to at least touch on in order to write it are obviously curvature as a concept, parallel transport as a way to quantify curvature and the metric tensor as an extension of the Pythagorean theorem. I don't want to get into tensor contraction (the Riemann->Ricci thing) unless I have to to explain that the Einstein tensor only represents part of the curvature of spacetime, and the other part is where gravitational radiation is expected to appear.
- Any of that sound like ripe picking for math articles? --KSorenson 20:06, 14 November 2009 (EST)
- Sure, I'll work on curvature (pretty sure I can improve what's on there now!). I'm thinking I'll start off talking about curvature of hypersurfaces in R^3 and the idea of parallel transport (illustrated on the round 2-sphere, probably), then define the Riemann curvature tensor (roughly -- we'll see about actually defining tensors) via holonomy. My thinking is that it probably then makes most sense to define sectional curvatures next and then say a bit about scalar and Ricci curvature. This all comes with the disclaimer that my expository abilities aren't really up to par, but I'll try to at least get the groundwork in place. I probably won't really start until Monday, but I'll see if I can't get a start tomorrow. --MarkGall 20:23, 14 November 2009 (EST)
- HA! Yeah, I'd say we can do better than that for curvature. ;-) If it were me writing it (and I'm glad it's not; thank you) from the perspective of a physicist, the point I'd emphasize is that it is not possible to put a consistent set of Cartesian coordinates onto a curved surface. If the surface is curved, then it cannot be flattened by a coordinate transform, period. So you have to deal with the curvature. That's the point I emphasize when I do "differential geometry for dummies." But of course, your article won't be for budding physicists, it'll be for budding mathematicians or anybody who's interested. So I'll support whatever choice you make. --KSorenson 20:46, 14 November 2009 (EST)
(Unindent) My take on this, and what I'm thinking of doing for my work on this article (feel free to steer me in a different direction if you disagree) is:
- There are flat coordinate systems. (We know this as Christoffel symbols = 0, but that's beyond our scope for the article.) When dealing with spacetime, if you are in a flat coordinate system, you don't feel any "fictitious forces" (accelerations). Your system is intertial.
- There are curved coordinate systems. (Nonzero Christoffel symbols, geodesics are hairy.) When dealing with spacetime, you feel fictitious forces.
- There are flat manifolds (Riemann=0.)
- There are curved manifolds (Riemann !=0.)
- A flat manifold may have flat or curved coordinate systems. In physics, we could be in a rotating or accelerating frame of reference. But we can always find a flat coordinate system. That is, the curvature of the coordinate system can be "transformed away" (for example, by stepping off the amusement park ride.)
- A curved manifold has NO flat coordinate systems. At least not globally. You can't step outside.
Then we say that gravity is really just a fictitious force (equivalence principle), and it arises because the spacetime manifold itself is curved. It can't be globally transformed away.
Obviously, I come at this from a physicist's perspective (though my major was in math.) It sounds as though Mark is planning the mathematicians' approach for the curvature article. The two articles will make interesting, and excellent, reading.
PatrickD 21:32, 14 November 2009 (EST)
- These suggestions sounds good to me -- I don't think the two approaches are mutually exclusive, either. I can start off talking about "flat coordinate systems" and maybe even throw in something about map projections (the Mercator kind, I mean) and the theorema egregium (this is supposed to be written for Majoring in Mathematics anyway, though JacobB who suggested it seems to have disappeared). This can all be integrated with what I'll say about surfaces in R^3. I'm a bit wary of saying much about Christoffel symbols since then we'd have to say a bit about connections, which I'd rather avoid -- I think geodesics are a more intuitive thing, and require a lot less other stuff to define if we're willing to wave our hands a bit (but if they're needed for the relativity article I can certainly define). Then I can start babbling about parallel transport and then actually define the curvature tensor. It's probably better if one of you write the physics-y stuff about fictitious forces here, but I can take a shot at it. --MarkGall 21:42, 14 November 2009 (EST)
Yay! New intro!
Patrick, love the new intro section! Got a question though, stylistically. I'm really unaccustomed to seeing theories (like general relativity or quantum mechanics, not proper-named ones) capitalized. With, of course, the anomalous exception of the Standard Model, but I think we write it that way 'cause that's how it's pronounced. Do we want to lowercase theories as the rule, or uppercase them?
I support all the suggestions y'all have floated here; it all sounds good to me. As I wrote the section today on calculating the stress-energy tensor of an ideal dust, I kept asking myself, "If I were a freshman and somebody was telling me about this, what would I want them to omit?" I'm not saying you guys should stick to that guideline too, just sharing what mine was for the edification of all. --KSorenson 21:52, 14 November 2009 (EST)
- OK, I'm convinced. While I was writing that, I kept thinking "Why am I capitalizing this all over the place? Because that's how it's done in other places. But it's kind of dumb." So the policy ought to be: special rel and general rel are lower case, but article titles are upper case. I think. Is that the CP policy? Title case? Initial case? I'll look around. And I'll fix it. Give me a few minutes.
- As far as "CP can't explain", I meant this in the sense of "It's out of the scope of CP to cover this at the full postgrad level." I think an encyclopedia could, in principle, give a full treatment, but I doubt that even Britannica does so. Could someone come up with a way of saying "it's out of the scope/charter of CP" without appearing to have CP diss itself? PatrickD 22:39, 14 November 2009 (EST)
- I'm really not sure what house style is here; I looked around for a guide briefly but didn't stumble across one. The title of this article is sentence-case, but that's just one data point. I'm really fine either way, I just like consistency.
- Yeah, I knew exactly what you meant with the "CP can't" part. I just thought it could be misinterpreted, y'know? I had no problem with the sentiment, just the possible interpretation of it. If we made it more verbose and said something like "Conservapedia is not a textbook," you know, along those lines, I think that could work. But at the same time, a single textbook can't explain all of general relativity. (Though I bet Misner, Thorne and Wheeler would slap my mouth for saying that. Or worse, they'd just hit me with their enormous, enormous book.) --KSorenson 23:30, 14 November 2009 (EST)
As I reread the intro over my morning coffee, a couple thoughts spring to mind. I'd like to hear what you guys think.
- One of the light-bulb moments for me, as an undergrad, was realizing the qualitative difference between metric and field theories. A field theory says that a field exists in space, and that objects within that field will take on some property as a consequence of being in that field. For gravity, it's a vector field and objects in it will accelerate. A metric theory says that spacetime itself takes on certain properties, and objects always behave the same ways; their behavior only looks different because we can't directly observe the properties of the spacetime around them. I'm explaining this poorly (see above, re: coffee), but it was a real "oh I get it now" moment for me as an undergrad. Maybe this distinction deserves some attention?
- Which brings me to my second point: What do you guys think of having one article each for vector field theory, metric theory, quantum field theory and gauge theory? I'm not talking about in-depth articles like this one; just simple, qualitative definitions. A vector field theory says there's an unobservable field there, and things in it will take on some potential energy from it. A metric theory says that spacetime is not uniform (curvature, torsion, etc) and that differences in apparent motion are caused by this observable distortion. A quantum field theory says fields work through the exchange of force particles, like little footballs being thrown around. A gauge theory is … actually pretty hard to explain simply. Anyway, I'm thinking in terms of basic physics concepts right now.
- Speaking of which: Lagrangian mechanics. We do have Lagrangian; it's just a stub, but I think we should do a qualitative introduction to the subject. We should make the point that the classical, Newtonian mechanics taught in high school is basically never used because it's not practical, but that the Lagrangian reformulation is the foundation of all of modern mechanics. (Yes, I know, Hamiltonians in quantum mechanics, but quantum mechanics makes my head hurt. I'm a macro-scale girl.) The principle of least action basically says, roughly speaking, that nature is lazy, and will only do what is absolutely required to get a particle from state A to state B, and that's a basic premise of physics.
Anyway, just some thoughts. I'd like a student who reads one of the physics articles here to come away with a basic understanding of what the subject of the article means, and I think it'd be cool if we could present the information in such a way that their understanding of the subtleties of the subject depends linearly on how far into the article they read.
- Please, create the articles you mentioned above. Karajou 12:50, 15 November 2009 (EST)
Do you guys (and girls, if any are out there) think anybody's reading this stuff? It's a lot of work, and after doing a little research I've begun to have doubts about whether this material will ever actually reach the target audience. I'd love to hear others' thoughts on that. Feel free to email me if you prefer. Thanks. --KSorenson 19:05, 15 November 2009 (EST)