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Dynamics of thin shells in GR

  1. Aug 7, 2010 #1
    There is a great resume on dynamics of thin shells in GR:
    Berezin87: Dynamics of bubbles in GR (Phys. Rev. D 36, 2919)

    In section III/A above the equation (3.1) there is the following statement:

    "Thus, for given inner and outer metrics sigma determine the global geometry
    (i.e. how the inner geometry is stuck together to the outer one)"

    Sigma is a sign +1 or -1, see for example in the master equation
    (Schwarzschild-Schwarzschild thin shell):

    sigma_in*sqrt(1-2mc/r+v^2) - sigma_out*sqrt(1-2(mc+mg)/r+v^2) = mr/r

    where r is the circumferential radius;
    v = dr/dtau, and tau is the proper time of the shell;
    mc is the central Schwarzschild mass parameter;
    mg is the gravitational mass of the shell, this means
    that the outer Schwarzschild mass parameter is mc+mg;
    and mr is the rest mass of the shell, mr > 0;

    It can be shown for all four possibilities of the signs that
    the following equation of motion
    can be derived independent of the signs:

    (dr/dtau)^2 = (mg/mr)^2 - 1 + (2mc+mg)/r + (mr/2r)^2

    This coincide with the above statement,
    because he signs do not influence the local motion.

    But in another paper of Goldwirth & Katz:
    http://arxiv.org/abs/gr-qc/9408034
    they have a nice illustration of gluing manifolds together:
    http://arxiv.org/PS_cache/gr-qc/ps/9408/9408034v3.fig1-1.png [Broken]

    They suggest that the signs comes from the four possibilities:
    witch half of the manifolds is chosen.
    But if we have already chosen the half, we also fix the metric.
    I think Berezin's statement means that the signs comes from
    how to join chosen metrics together, and not how to chose the metrics!
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 31, 2010 #2
    "Gauss-Codazzi" or "Gauss-Kodazzi"?
     
  4. Aug 31, 2010 #3

    George Jones

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    I just checked in four of my books, and they all write "Codazzi."
     
  5. Sep 1, 2010 #4
    Kodazzi is used also lots of place in the literature, why?
     
  6. Oct 25, 2010 #5
    The master equation of thin shells is:

    [tex]s_{-}\sqrt(1-2m_c/r+v^2) - s_{+}\sqrt(1-2(m_c+m_g)/r+v^2) = m_s/r[/tex]

    Where mc is the central mass (Schwarzschild mass parameter of the inner '-' spacetime),
    mc+mg is the total mass (Schwarzschild mass parameter of the outer '+' spacetime),
    therefore mg can be interpreted as gravitational mass of the shell.
    ms=4*pi*sigma*r^2, where sigma is the surface energy density, ms>0.
    r is the area radius, v is dr/dtau, where tau is the proper time on the shell.

    s_{-} and s_{+} are sign factors.
    If we are talking about conventional shells
    (not wormhole solution, and not Univerze with two centers)
    than s is nothing else, but sgn(f*Tdot),
    where f=1-2M/r is the metric function, and Tdot=dt/dtau,
    where t is the Schwarzschild time.

    If we give mc,ms=m0,r=r0,v=v0, as initial data set,
    we can calculate the gravitational mass mg.
    But we get two solutions:
    mg=m0(-m0/(2*r0)+sqrt(1-2*mc/r0+v0^2))
    or
    mg=m0(-m0/(2*r0)-sqrt(1-2*mc/r0+v0^2))
    If r0>2*mc, than only the first solution is possible.
    But below the horizon I cannot choose.
    In genereal both solutions can describe a valid system, I think.
    If we check mc+mg>0, do not help to choose.
     
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