Mass/Energy of a collapsing gas shell (MTW 21.27)

[TerryW]

Homework Statement

Derive equation 21.176e for a collapsing gas shell

Relevant Equations

See attached extract from MTW

Hi Everyone.

Can anyone give me some hints which will point out how to solve this problem, particularly using 'the formalism of Ex 21.25'.

I've kicked this around for a couple of weeks now and I haven't been able to come up with anything.

Regards

TerryW

 

[TerryW]

Hi to any reader passing this way.

If you too are stuck on this problem, I can now offer a few thoughts on how to get to the solution after being given some very helpful guidance by TSny.

1. There are three metrics to be used - The metric outside the shell (Schwartzschild), the metric inside the shell (flat space-time) and the metric on the shell itself (which isn't given but is
$ds^2 = -d\tau^2 + R^2(\tau)(d\theta^2 + sin^2\theta d\phi^2$)

2. As the shell evolves with time ($\tau$), it creates a 'world tube', the normal to which is space-like. The consequence of this is that in ADM co-ordinate speak, the three co-ordinates of the shell are $\tau, \theta and \phi$ which correspond to i,j, and k.

(It took a good bit of time and some valuable help from TSny to get comfortable with the idea that n doesn't have to be time-like).

3. The quick way is to locate a copy of Problem Book in Relativity and Gravitation by Lightman, Press, Price and Teukolsky and turn to page 586.

So thanks to TSnyTerryW