How do we determine the global geometry of thin shells in GR?

[mersecske]

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

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!

 

[mersecske]

"Gauss-Codazzi" or "Gauss-Kodazzi"?

 

[George Jones]

"Gauss-Codazzi" or "Gauss-Kodazzi"?

I just checked in four of my books, and they all write "Codazzi."

 

[mersecske]

Kodazzi is used also lots of place in the literature, why?

 

[mersecske]

The master equation of thin shells is:

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

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.