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

  1. Jun 13, 2006 #1
    In article <20060610152605.CF2D11310F5@mail.netspace.net.au>, Greg Egan
    <gregegan@netspace.net.au> wrote:

    > Ralph Hartley wrote:


    [snip]

    > >An *arbitrary* spacelike surface can have curvature anywhere, not just
    > >at the particles.

    >
    > Good point! Your comment made me think of the hyperbolic plane, and the
    > opportunity it gives for the sums of angles in triangles to be less than
    > 2pi.
    >

    [snip description of a 2-particle model with spacetime foliated by
    hyperboloids]

    > [W]e can't
    > extend things back into the indefinite past, though it might be possible
    > to get around that with some further tricks.


    I can't see how to extend this solution to infinite -ve time, but why not
    make a virtue out of necessity and consider a family of Big Bang style
    solutions which start from a singularity?

    If we pick an origin in Minkowski spacetime, we can take the topological
    interior of the forward light cone of the origin and declare that this
    set, minus some wedges excluded along particle world lines, will be our
    entire solution.

    If we have a collection of world lines starting from the origin (which
    itself is excluded), we then cut out wedges around them and identify the
    opposite faces to give these particles various masses. If we were using
    all of Minkowski spacetime for our solution, we would need to be sure
    that the wedge cut out around one worldline never overlapped with that of
    any other particle, but in this case instead we only have to be sure that
    there's no overlap *inside the light cone*.

    That allows the particle masses to be greater than they otherwise would
    be, and this is what enabled the extra mass in the construction I
    described in my previous post, where two particles each with deficit
    angles approaching 2pi are possible.

    The same construction extends naturally to a symmetrical configuration of
    N particles, each of mass m. If we arrange their worldlines
    symmetrically around the origin of our chosen coordinate system, the
    maximum possible value of m will occur when two wedge planes associated
    with two neighbouring particles meet precisely on the light cone. Take
    one particle's world line to lie on the xt plane, and to be generated by
    the timelike vector:

    u = (1,v,0)

    The two wedge planes for this world line will pass through null vectors
    that are halfway to this particle's clockwise and counterclockwise
    neighbours. These null vectors are:

    q1 = (1, cos pi/N, sin pi/N)
    q2 = (1, cos pi/N, -sin pi/N)

    Normals to these planes are:

    n1 = (v, 1, (v-cos(pi/N))/sin(pi/N) )
    n2 = (v, 1, -(v-cos(pi/N))/sin(pi/N) )

    c = n1.n2 / |n1||n2| =

    (cos(pi/N)^2 - 2) v^2 + 2 cos(pi/N) v - cos(2pi/N)
    --------------------------------------------------
    (cos(pi/N) v - 1)^2

    The deficit angle associated with each particle is then:

    A = pi - arccos(c) v < cos(pi/N)
    pi + arccos(c) v >= cos(pi/N)

    where 0 <= arccos <= pi.

    Between v=0 and v=1 (where these endpoints should be taken only as
    limiting cases, not values that can actually be achieved):

    v c A
    ---------- ------------ ------------
    0 -cos(2pi/N) 2pi/N
    cos(pi/N) 1 pi
    1 -1 2pi

    So in the high-velocity limit v->1 each particle has a deficit angle
    approaching 2pi.

    For 0 < v < cos(pi/N), in the many-particle limit N->infinity we have a
    (maximum) total deficit angle of:

    N A -> 2pi sqrt((1+v)/(1-v))

    This means that if we want a system with a scalar sum of rest mass equal
    to M (which, with a certain choice of units, means a total deficit angle
    of 2M), there will be a minimum velocity required in these models of:

    v_min = (M^2 - pi^2) / (M^2 + pi^2)

    For, say, M=4.5pi, v_min = 0.905. Interestingly, this comes at a point
    where the total holonomy has ceased to be cyclic in v; see the plots at
    the end of:

    http://gregegan.customer.netspace.net.au/GR2plus1/GR2plus1.htm
     
  2. jcsd
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