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

  1. Jun 22, 2006 #1
    John Baez wrote:

    >So one thing you're saying
    >is that a product of rotations in two different rest frames can be
    >a boost??


    One way to see why the products of some rotations will be boosts is as
    follows.

    Suppose that for a pair of linearly independent vectors u and v in R^3 we
    can find an element of O(2,1) with the following properties:

    * it preserves both u and v
    * its determinant is -1
    * its square is the identity

    If we can find such an element, we'll call it ref(u,v).

    Now, if we pick three linearly independent vectors u, v and w so that
    ref(u,v), ref(u,w) and ref(v,w) all exist, we can construct the following
    elements of SO(2,1):

    g1 = ref(u,v) ref(v,w) which preserves v
    g2 = ref(v,w) ref(u,w) which preserves w
    g3 = ref(u,v) ref(u,w) which preserves u

    and we have g3 = g1.g2 because ref(v,w)^2 = I.

    So if we pick a spacelike u and a timelike v and w, we'll have a boost
    that's equal to a product of rotations.

    The one question that remains is, when can we find ref(u,v)? This is
    trivial in O(3), but O(2,1) is trickier. Generically we can construct a
    normal n to the plane spanned by u and v:

    n^a = g^{ab} eps_{bcd} u^c v^d

    and if it's not a null vector we can construct ref(u,v) as the projector
    into the plane minus the projector onto n. But if n is null, I don't
    think ref(u,v) exists.
     
  2. jcsd
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