1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Jun 22, 2006 #1
    I wrote:

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

    I think the only way the normal to the plane can be null is if the plane
    is tangent to the light cone, in which case it will contain a single null
    ray (which is itself normal to the plane), and all the other vectors in
    it will be spacelike.

    So if (but not only if) u and v are timelike, ref(u,v) will exist.

    What's the significance of this construction failing if two of the
    vectors lie on a tangent plane to the light cone? Well, if u and v are
    *not* on such a plane, then you can choose w to be almost anywhere:
    anywhere such that (u,w) and (v,w) also don't lie on tangent planes to
    the light cone. In other words, given such generic u and v, you can find
    SO(2,1) elements g_u and g_v that preserve them, *and* such that the
    eigenvector of g_u g_v lies *almost* anywhere in R^3.

    However, if u and v lie on a tangent plane to the light cone, then I
    think the eigenvector of (g_u g_v) will always lie on that same tangent
    plane, for all g_u and g_v that preserve u and v respectively.

    The upshot of this for collisions in 2+1 gravity would be that certain
    tachyon-tachyon and tachyon-luxon collisions (which yield either tachyons
    or luxons as the outgoing particle, never tardyons) involve *coplanar*
    vector-valued momenta.

    What I'm claiming amounts to the statement that for each null ray n there
    is a subgroup H(n) of SO(2,1), consisting of those elements whose
    eigenvectors are normal to n.

    So the simplest way to check this would be to look for a subalgebra of
    so(2,1) consisting of elements whose null space lies in the plane normal
    to n.

    If we take the example of n being the null ray generated by (1,1,0), then
    so(2,1) elements of the form:

    | 0 a b |
    | a 0 b |
    | b -b 0 |

    have their null space generated by (b,b,-a), which is always normal to
    (1,1,0). And this set turns out to be closed under the Lie bracket, with
    [e_a,e_b]=e_b (where by e_a I mean the element above with a=1, b=0, and
    by e_b vice versa).

    So it looks like those subgroups of H(n) do exist. But they won't be
    Abelian! So those weird coplanar collisions in 2+1 gravity still won't
    obey the old vector addition law.
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted

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