Spherically symmetric infinitesimally thin shells(adsbygoogle = window.adsbygoogle || []).push({});

can be described via the well known junction formalism of Israel.

The equation of motion of thin shells is (G=c=1):

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

This is an energy balance equation,

where r is the circumferential radius,

tau is the proper time, measured by co-moving observer,

mc > 0 is the central mass (in case of Schwarzschild spacetime,

this is the mass parameter of the inner Schwarzschild spacetime)

mr > 0 is the rest mas of the shell = 4*pi*r^2*sigma,

where sigma is the surface energy density,

mg is the gravitational mass of the shell,

mg = M - mc by definition, where M is the total mass of the system,

if the outer spacetime is Schwarzschild vacuum,

then M is the outer Schwarzschild mass parameter.

mc and M has to be non-negative.

There are lots of papers studying for example the dust case,

when mr is constant during the motion.

My question:

Usually (dr/dtau) > 1 (speed of light), how is it possible?

If this is just a coordinate velocity,

how can i express a velocity formula,

which has to satisfy the casuality condition |v| < 1?

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# Thin shell velocity is greater than speed of light?

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