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posted by CoolHand on Tuesday April 07 2015, @03:58PM   Printer-friendly
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.

 
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  • (Score: 3, Interesting) by wonkey_monkey on Tuesday April 07 2015, @05:58PM

    by wonkey_monkey (279) on Tuesday April 07 2015, @05:58PM (#167511) Homepage

    Here's how I think of relativity, most of which comes (IIRC) from Brian Greene's Fabrice of the Cosmos, and might be subject to my faulty memory.

    Imagine a field, and imagine you want to find a way to be able to specify the locations of objects in a field (let's say there's a tree and a rock). A grid system is the obvious solution - two perpendicular axes, X and Y, which will let you do what you want.

    The axes are arbitrary, though - someone else could come along and place them differently, but both are equally valid ways of dividing up the field and specificying the locations of objects. Also note that there is no "base" coordinate system - no grid that is an intrinsically better way of dividing up the field than any other.

    So, under one grid system, the rock and the tree might have equal X coordinates, but different Y coordinates. Under another, they might have equal Y coordinates but different X coordinates. And under another, both coordinates may differ. In a sense, by moving from one grid to another, you're "converting" some of the distance along one axis into distance along the other, and vice versa.

    Spacetime (if you reduce it to one space dimension to go easy on your imagination) is a bit like that field, with a couple of important differences. Firstly, we don't talk about objects, but events - events which occur at a single specific time and a single specific location - say, a flashbulb firing. Secondly, you can't arbitrarily choose where the grid goes. From your now/here point in spacetime, one axis will go out and join together all the events which are happening (as far as you determine) at the same point in time. The other axis, perpendicular to the first, will go out and join together all the events that will happen, or did happen (again, as far as you can determine) at the same location in space.

    So why isn't that grid choice arbitrary? Because, if you do/did nothing to change your position, you must have been and will continue to be at that same location in the past and future. Space (by definition?) is the dimension along which your position will not change (in your grid system) if nothing is done to alter your trajectory. Time, being perpendicular, is... well, the other one. You might object and say, "but what about someone in a car? They're moving," - but that's the crux of relativity. They're not moving, not along their own space axis. They're only moving when you consider their coordinate along your space axis.

    So, back to the grid overlaying the field. If you compare two people's arbitrary grids, you'll see you can turn one into the other simply by rotating it. From your grid, someone's else grid will look like a rotated version of yours.

    But instead of thinking of it as rotating the whole grid, think of it as rotating both axes in the same direction. The funny thing about doing the same thing in spacetime, though, is that the axes rotate in opposite directions. So instead of looking at someone else's grid and seeing a rotation of yours, what you instead see is a diamond-shape distortion of your own grid, with two opposite corners pulled "outwards" along one diagonal, and the other two corners pushed inwards along another. From now on I'll call this "rotation," because it's not the rotation we're all used to (though the maths of it is very similar).

    So, imagine grid A, your grid, a nice square grid. Someone else, moving relative to you, will be in grid B - let's say his axes have "rotated" 10 degrees, in opposite directions, compared to yours. If you then switched to his grid (by getting up to his velocity), it'd be like pinching the corners of his grid as drawn on a stretchy sheet and pulling them back into a square. Your grid would then be distorted as his was before, but the opposite way. It's hard to imagine this next bit, but try this: if there's another traveller in grid C, and the "rotation" from grid B to grid C is the same as the "rotation" from grid A to grid B, you'll find that you can't just add up the rotations as you could with normal space. A "rotation" by forty degrees will not "rotate" an already-rotated-by-40-degrees grid to 80 degrees - instead it will only "rotate" it to something like... well, I'm not really sure how the maths goes here, exactly, but something like 44 degrees.

    But, no matter what you do, you'll never be able to alter your grid (by changing your velocity) so much than B's grid flattens down to a line, with both axes "rotated" by 45 degrees to lie on top of each other. You can get arbitrarily close, but you'll never quite get there. And the same goes for Mr. B; no matter how much he tries to, he can never get anyone else's grid to flatten down to a line either. And that, I think, is directly analogous to the speed of light. Reaching the speed of light relative to someone else would mean that you'd collapsed their grid down to a line, and their space and time would become the same thing (and the same would go for you, from their perspective). You would no longer have a sensible grid for any of the events they could experience.

    And... now I've run out of steam and need to go and lie down. Did that help? Did I get anything outrageously wrong?

    --
    systemd is Roko's Basilisk
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  • (Score: 3, Informative) by maxwell demon on Tuesday April 07 2015, @06:47PM

    by maxwell demon (1608) on Tuesday April 07 2015, @06:47PM (#167537) Journal

    Well, the main problem of your post is that you described special relativity, while this story is about general relativity. In particular, in general relativity, spacetime is curved, therefore you'll no longer be able to get away with a simple axes grid. Think of the surface of the earth: You still can describe it with two coordinates (longitude and latitude) which form a grid, but you cannot get them by simply choosing two axes and saying "those are the directions which I'll use everywhere". Indeed, you get even pathological points (north pole and south pole) where the coordinates fail (if you are on the north pole, all directions are south). Note that you can change your coordinates (for example, you could decide to choose two "poles" on the equator, or use the angles the connection lines from you to New York and San Francisco form with the connection line between those cities, or use the distances from sun and moon, etc.), but you'll not get rid of the fact that you'll always have such pathological points.

    Moreover, if you look at the path you make on the surface of earth when just going straight on (which on earth means going along a great circle), in the coordinates it will look like you're going an extremely curved way. For example, you'll find that the equator has a strange attractive force, since no matter where and in which direction you start, sooner or later you'll find yourself at the equator; indeed unless you're staying at the equator to begin with, you'll find that your latitude oscillates around the equator.

    This is basically what goes on in GR spacetime as well: All free-falling objects just go straight on in spacetime (technically: the follow spacetime geodesics), but since spacetime (not just space!) is curved (with the curvature determined by energy and momentum of the matter and fields inside), it looks as if objects were attracted by massive objects.

    --
    The Tao of math: The numbers you can count are not the real numbers.
    • (Score: 2) by wonkey_monkey on Tuesday April 07 2015, @07:34PM

      by wonkey_monkey (279) on Tuesday April 07 2015, @07:34PM (#167558) Homepage

      Dammit. Rookie mistake! For some reason I was thinking of GR being the general, broad one, and SR being the special one because it included gravity...

      For example, you'll find that the equator has a strange attractive force, since no matter where and in which direction you start, sooner or later you'll find yourself at the equator

      But then I'd also notice that every great circle has the same "strange attractive force," wouldn't I? Then it stops being strange...

      --
      systemd is Roko's Basilisk
      • (Score: 2) by boristhespider on Tuesday April 07 2015, @07:39PM

        by boristhespider (4048) on Tuesday April 07 2015, @07:39PM (#167562)

        Other way round, Special Relativity is specialised to the case where there is no gravity, and General Relativity applies in the general case where there *is* gravity...

      • (Score: 2) by wonkey_monkey on Tuesday April 07 2015, @07:40PM

        by wonkey_monkey (279) on Tuesday April 07 2015, @07:40PM (#167564) Homepage

        Broad as in undetailed.

        --
        systemd is Roko's Basilisk
      • (Score: 2) by maxwell demon on Tuesday April 07 2015, @07:53PM

        by maxwell demon (1608) on Tuesday April 07 2015, @07:53PM (#167572) Journal

        Well, the potential of the equator obviously is that of a harmonic oscillator: all orbits have the same period.

        --
        The Tao of math: The numbers you can count are not the real numbers.
        • (Score: 2) by wonkey_monkey on Tuesday April 07 2015, @09:29PM

          by wonkey_monkey (279) on Tuesday April 07 2015, @09:29PM (#167598) Homepage

          Still not getting what can be special about the equator instead of any other great circle, if we're just talking about the Earth's surface as an example of curved 2d space. Only now you've mentioned orbits...

          --
          systemd is Roko's Basilisk