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Re: This Week's Finds in Mathematical Physics (Week 232)

  1. Nov 4, 2006 #1
    I wrote:

    >I just figured out how to construct the complete spacetime for a general
    >2-particle collision in 2+1 gravity. It's mind-bogglingly simple.

    D'oh! It *is* simple, but I messed up a crucial detail. And the reasons
    stretch back a long time, to when I stupidly calculated "g1.g2" for the
    holonomy of going around loop 1 and then loop 2, when of course that
    should be "g2.g1".

    The upshot is that when two particles collide, observers on either side
    of the "plane" of the collision will think that the outgoing particle is
    coming *towards* them, not moving away from them.

    So you really do need to get these ordering conventions right in 2+1
    gravity, or you won't know when a particle is coming to smack you in the

    >Take one copy of Minkowski spacetime. Draw in two world lines W1 and W2
    >for the two incoming particles, coming in from the past and meeting and
    >terminating at some point C. Then draw in the world line W3 for the
    >outgoing particle, starting at C and heading off in to the future. Don't
    >make these lines all coplanar (unless you're interested in a trivial
    >limiting case).

    The above is all OK.

    >Throw away all of the spacetime that lies *inside* the
    >"infinite tetrahedron" with its vertex at C and the three world lines as
    >its edges, leaving behind a kind of "concave infinite tetrahedron".
    >Now take a mirror image of that "concave infinite tetrahedron", and glue
    >it to the original along all three congruent faces.

    Don't use the concave tetrahedron defined by the three world lines, use
    the convex one. You still glue it face-to-face to its mirror image.

    Everything works now. This recipe gives angular deficits around the
    incoming particles' world lines (the concave one didn't, it gave angular
    excesses) and it gives what I now realise is the correct behaviour for
    the geodesics of observers on either side of the collision: it makes
    them collide!

    A slice that cuts through both tetrahedra along the outgoing particle's
    world line will look like this, where G and G' are geodesics through
    observers at B and B'.

    ........ ........
    | \ / |
    | \ / |
    G' | \ W3 W3 / | G
    | \ / |
    | \ / |
    | | | |
    B' + | | + B
    | | L L | |
    | | | |
    | | | |
    ............... ...............
  2. jcsd
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