Thin shell velocity is greater than speed of light?

[mersecske]

Spherically symmetric infinitesimally thin shells
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?

 

[George Jones]

If this is just a coordinate velocity,
how can i express a velocity formula,
which has to satisfy the casuality condition |v| < 1?

At the boundary, calculate the physical speed between a static Schwarzschild observer and an observer comoving with the shell.

 

[mersecske]

Why is the static Schwaryschild observer what we need?

You mean dr/dt is the velocity formula,
where t is the Schwarzschild time?

 

[George Jones]

Why is the static Schwaryschild observer what we need?

A static observer isn't necessarily needed. Because spatial distance, in general, is not defined in general relativity, physical speeds between separated observers are not defined. If two observers are next to each other (coincident), the observers can use local clocks and measuring sticks (or clocks, mirrors, and light signals) to measure the relative speed between. This relative speed always satisfies

how can i express a velocity formula, which has to satisfy the casuality condition |v| < 1?

Here is one nice way to calculate relative physical (not coordinate) speed $v$ between two observers who are coincident at an event. Suppose the 4-velocities of the two observers are are $u$ and $u'$. Then,

$$\gamma = \left( 1 - v^2 \right)^{-\frac{1}{2}} = g \left( u , u' \right) = g_{\alpha \beta}u^\alpha u'^\beta.$$

This is an invariant quantity, and, consequently, can be calculated using any coordinate system/basis, i.e., this works in all coordinate systems in both special and general relativity.

You mean dr/dt is the velocity formula, where t is the Schwarzschild time?

No, dr/dt is a coordinate speed. Let's work through, in the context of a thin shell, an example of a physical velocity. To find find a physical speed for the shell, find the relative speed between an observer who is comoving with the shell and some other observer who is coincident with the comoving observer. A static Schwarzschild observer is a natural choice for this second observer, but it is not the only possible choice.

For concreteness, let's use section 16.4 Spherical shell of dust in vacuum from

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.118.2817&rep=rep1&type=pdf .

Then (equation 16.59), in terms of the exterior Schwarzschild coordinates,

$$u = \left( \dot{t}, \dot{R}, 0, 0 \right)$$

with (equation 16.62)

$$\dot{t} = \frac{\sqrt{1 - \frac{2M}{R} + \dot{R}^2}}{1 - \frac{2M}{R}}$$

is the 4-velocity of the comoving observer,

$$u' = \left( \left( 1 - \frac{2M}{R} \right)^{-\frac{1}{2}}, 0, 0, 0 \right)$$

is the 4-velocity of the static observer, and $g_{\alpha \beta}$ are the standard Schwarzschild coordinates of the metric. This gives

$$\begin{equation*}<br /> \begin{split}<br /> \gamma &= g \left( u , u' \right) = g_{\alpha \beta} u^\alpha u'^\beta \\<br /> \left( 1 - v^2 \right)^{-\frac{1}{2}} &= g_{00} u^0 u'^0 \\<br /> &= \left( 1 - \frac{2M}{R} \right) \dot{t} \left( 1 - \frac{2M}{R} \right)^{-\frac{1}{2}} \\<br /> &= \sqrt{\frac{1 - \frac{2M}{R} + \dot{R}^2}{1 - \frac{2M}{R}}}.<br /> \end{split}<br /> \end{equation*}$$

Rearranging results in (taking speed to be positve)

$$v = \sqrt{\frac{\dot{R}^2}{1 - \frac{2M}{R} + \dot{R}^2}},$$

which, is the speed, as measured with clocks and rulers, with which the collapsing shell recedes from a hovering observer. Above the Schwarzschild radius, $v$ is always less than the speed of light, and, as $R$ approaches the Schwarzschild radius, $v$ approaches the speed of light, which makes sense.

The physical speeds of the shell with respect to other observers also could be calculated, and these speeds will all be less than the speed of light.

 

[mersecske]

Thank you very much the detailed answer!