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from the I-can-really-learn-my-maths-and-sciences-now dept.
If you ever wanted to learn General Relativity, now here's your chance. The caveat is that first you must learn differential geometry. But it's not difficult, really. Only lots of hard work, but not difficult. I was attending this February such a course. This course is fully documented: there are recordings of all lectures, and of tutorials with solutions (also the .pdf files with practice questions). For easier access you can also visit the The WE-Heraeus International Winter School on Gravity and Light YouTube channel.
You should know though that this material on the internet is not everything we were doing there, the biggest omission are the advanced tutorials, which were done in groups and couldn't be filmed. Also their solutions were too difficult to be "quickly" filmed like the tutorials that have videos. However there's hope that advanced tutorials will also be put online some time later this year (as promised by the organizers). In that case I'll submit a follow up story.
I must tell you that attending this course was really a great experience, and Prof. F. P. Schuller is in fact on of the best lecturers I have ever met.
(Score: 2) by Hartree on Tuesday April 07 2015, @04:39PM
"But it's not difficult, really"
For me, probably not. Yes, it's a lot of work (I started a GR course, while I was working full time and going to grad school part time. We lost two computer operators at work and I had to cover their shifts for a few weeks. I made a lot of overtime, but the sleep lack and lack of time meant I had to drop it.). But then I had boatloads of mathematical background.
The prerequisites are fairly steep. For someone who hasn't had some vector calculus and attained the oft mentioned "mathematical maturity" it would be pretty damn hard.
(Score: 2) by maxwell demon on Tuesday April 07 2015, @06:03PM
Well, I'd expect the "mathematical maturity" of typical readers of this site to be above average. At least if you exclude the trolls.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2, Insightful) by Anonymous Coward on Tuesday April 07 2015, @07:00PM
Hey! We ACs must work bloody hard at math & science stuff so our trolling can look semi-plausible.
(Score: 1, Interesting) by Anonymous Coward on Tuesday April 07 2015, @09:29PM
If we didn't, we'd ask for things like 7 non-intersecting, non-parallel red lines. [youtube.com]
(Score: 2) by hendrikboom on Thursday April 09 2015, @03:27PM
Without looking at the video (my computer's streaming video leaves a lot to be desired) I'd guess either hyperbolic geometry or three-dimensional geometry.
(Score: 1, Insightful) by Anonymous Coward on Wednesday April 08 2015, @01:48AM
Yeah, you know, every now and then this kind of statement is made, but when you go and look at any hard science or maths stories that get posted, there aren't any real comments (if you exclude the same three or four people who do understand stuff like this) and zero discussion. All you get is the same, predictable "tech" jokes and puns.
I know that when we form cliques, we tend to think of ourselves as better/smarter/whatever than "other" people, but that doesn't mean that it is true, and this usually leads to derogatory comments about the "stupid" public or "joe six-pack" that make us feel better.
(Score: 2) by khchung on Wednesday April 08 2015, @12:07PM
Too bad I ran out of mod points before seeing the parent comment.
Yeah, it is the same in the green site. While a lot of posters claimed to be good at science and maths, but when you read what they wrote, it became obvious that most had no real understanding at all.
(Score: 2) by DeathMonkey on Tuesday April 07 2015, @06:20PM
But it's not difficult, really. Only lots of hard work, but not difficult.
Merriam-Webster: Difficult [merriam-webster.com]
not easy : requiring much work or skill to do or make
(Score: 2) by SlimmPickens on Tuesday April 07 2015, @08:39PM
However there is context. We are talking about relativity after all.
(Score: 1, Funny) by Anonymous Coward on Wednesday April 08 2015, @05:25PM
Well, in that case … it's relatively difficult.
(Score: 2) by HiThere on Tuesday April 07 2015, @06:41PM
I disagree. I took and passed a differential geometry decades ago, and still can't really wrap my mind around "Well, yes, the manifold is bent, but not in any direction." I find it much easier to accept that "It's bent in the (n+1)th dimension".
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 3, Interesting) by maxwell demon on Tuesday April 07 2015, @07:02PM
Your problem is the word "bending". The spacetime is not bent, it is just curved.
Maybe the following example helps:
Imagine you're measuring a plane with rods. Now the plane is hotter at some area, and therefore the length of the rods increases due to thermal expansion. And this causes you to measure consistently shorter distances across that area than you would expect from measuring around that area and applying Euclidean geometry. Thus you are measuring a curved space; indeed, you can apply the complete machinery of differential geometry to your measurement results. However it doesn't bend into the third dimension; after all, all what physically happens in that scenario is that your rods expand when entering that region, which means that, when measured with those rods (and those rods are all you have to measure it), there's less space there.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 3, Informative) by FatPhil on Tuesday April 07 2015, @07:27PM
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 2) by maxwell demon on Tuesday April 07 2015, @07:45PM
Yes, that's indeed another problem; the rubber sheet is probably the most misleading "illustration" of GR in existence.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by hendrikboom on Thursday April 09 2015, @08:41PM
The rubber sheet sure misled me for a while way back in the sixties.
(Score: 2) by HiThere on Wednesday April 08 2015, @06:19PM
Actually the word used may have been curved. It doesn't make things any easier for me.
And, WRT those who previously replied to you, this was a class in Differential Geometry in the Math department, not Physics. This class was dissociated from any contact with General Relativity.
I still feel the need for a dimension for things to be curved into. (Now perhaps if they'd talked about compression and expansion of the metric it would have been easier. I *now* think that was what they were trying to get at. But it took me *years* after taking the class to decide that.
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 5, Interesting) by boristhespider on Tuesday April 07 2015, @07:37PM
It's very helpful to drum into your mind the difference between intrinsic and extrinsic curvature. What you're describing is extrinsic curvature, and I can easily get an n-d surface which has zero intrinsic curvature -- is a flat sheet -- and embed it in an (n+1)-d space in a horrendously contorted manner. The intrinsic curvature effectively describes the Pythagoras theorem observed by things living on that surface, and the extrinsic curvature describes how it's been embedded in a (potentially fictional) higher-dimensional space.
The rest of this comment is purely for the sake of interest...
A nice example is a torus -- the *surface* of the torus it has zero intrinsic curvature given that it can be made from just a sheet of paper, but it has a distinctly non-zero extrinsic curvature given that it's, well, a torus. And a nice way of seeing *this* is to draw a triangle on it. On a surface with zero (intrinsic) curvature, the angles of a triangle will add up to 180 degrees or, to phrase it a different way, the simple Pythagorean theorem holds (in this instance, ds^2 = dx^2 + dy^2 and I'm writing them as differentials since this is differential geometry and we may as well play along).
The surface of a sphere, on the other hand, has a non-zero intrinsic curvature and a non-zero extrinsic curvature. You can't make a sphere by playing around with a sheet of paper without cutting bits out and warping it. On a surface with non-zero intrinsic curvature, the angles of a triangle will not add up to 180 degrees (and in this case that curvature is positive and the angles add up to more than 180 degrees) or, to phrase it a different way, we're in a Riemannian geometry and the Pythagorean theorem becomes something more complicated (in this case, ds^2 = d\theta^2 + sin^2 \theta d\phi^2 for the surface of the unit sphere).
If you're interested, a saddle forms a good example of a surface with a non-zero negative intrinsic curvature and a non-zero extrinsic curvature. On this kind of surface, the angles of a triangle add up to less than 180 degrees, as you've probably guessed, and the Pythagorean theorem becomes something I shamefully can't actually remember off the top of my head but isn't much more complicated than that of the surface of a sphere.
In GR both concepts are widely used. In the usual formulation, (3+1)d space, we just actually look at the intrinsic curvature, represented by the Ricci scalar which is a complicated little beast formed from the gradients of the components of Pythagorean theorem (which themselves are lumped into an object called the "metric" which we can pretend is a matrix; for instance, the metric of the surface of a sphere is a diagonal matrix with 1 and sin^2 theta as its entries). But if we want to do numerical relativity, for instance, we have to perform what is known as a 3+1 split by identifying a time coordinate, by whatever means we choose, which then forms a vector orthogonal to a three-dimensional spatial surface. This way we can "foliate" spacetime with a load of spatial sheets along a time coordinate. My usual metaphor is a loaf of sliced bread. The theory of GR then breaks down into a theory based on the intrinsic curvature, describing what's happening on the sheet itself, and the extrinsic curvature, describing how the sheet distorts as one moves along the time coordinate.
(Score: 3, Insightful) by melikamp on Wednesday April 08 2015, @06:06PM
Mmm I think a cylinder would be a better example (I figured that's what you meant). A torus is rather complicated. A familiar donut shape has non-zero gaussian curvature, which one can guess after trying to actually fold a piece of paper into a donut. In fact, if we put three points on any of the circular cross-sections, they will form a triangle with angles summing up to 3π, so the curvature appears to be positive.
I am not a geometer, so I can't say offhand whether there is a way to embed a flat torus into a flat euclidean space (looks like 4 dimensions may be enough), but when embedding is not an issue, we can certainly flatten it by either identifying opposing sides of a unit square, or the circular edges of a cylindrical tube. And that flat torus is, of course, good enough for your example.
(Score: 2) by boristhespider on Wednesday April 08 2015, @06:22PM
Very good point - I conflated topologically flat (in that non-tearing distortions can transform a uniform surface into it, which *is* true for a torus) with geometrically flat. A cylinder is a much better example.
(Score: 2) by hendrikboom on Tuesday April 07 2015, @09:27PM
Well, isn't there a theorem that any curved space can be faithfully embedded i a space f sufficiently higher dimension, at least locally?
I found it hard to get that it was a 3+1 metric, where some of the directions effectively contribute negatively to distance.
-- hendrik
(Score: 2) by HiThere on Wednesday April 08 2015, @06:24PM
Yes, but Differential Geometry is supposed to be about how that n+1 dimensional space isn't necessary. But talking about bending or curving is just the wrong way to explain that (to me). Instead the analogy should be to compression and rarefaction, as in sound waves. THAT I'd have had (I think) little difficulty in grasping.
Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.
(Score: 3, Interesting) by hendrikboom on Thursday April 09 2015, @08:46PM
I am looking forward to the lectures on differential geometry. When I took it it ws all tensors and Christoffel symbols and the like. You know, indexed clusters of numbers that transform in a particular way.
Nowadays it's differential forms, connections, fiber bundles and the like -- something for those numbers to be coordinates of. A reason for those numbers to behave the way they do.
At least, I hope they do it in the modern way. It might make sense now.
-- hendrik