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

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In summary: Ex 21.25. After struggling for some time and receiving guidance from TSny, TerryW offers some thoughts on how to approach the problem. They mention that there are three metrics to be used and explain the concept of a "world tube" created by the evolving shell. They also suggest looking at a specific page in the Problem Book in Relativity and Gravitation for further help. Overall, the conversation revolves around finding a solution to the problem with the assistance of TSny. In summary, TerryW and TSny discuss hints for solving a problem using the formalism of Ex 21.25, including the use of three metrics and the concept of a "world
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TerryW
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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

1659109760540.png
 

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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
 

1. What is the concept of mass/energy in a collapsing gas shell?

The concept of mass/energy in a collapsing gas shell refers to the total amount of matter and energy present in the shell as it collapses under its own gravity. This includes both the rest mass of the particles making up the shell and the energy associated with their motion.

2. How is the mass/energy of a collapsing gas shell calculated?

The mass/energy of a collapsing gas shell can be calculated using the equation E=mc², where E is the total energy, m is the rest mass, and c is the speed of light. This equation takes into account both the rest mass and the kinetic energy of the particles in the shell.

3. What is the significance of the mass/energy of a collapsing gas shell?

The mass/energy of a collapsing gas shell is significant because it determines the strength of the gravitational pull of the shell. As the shell collapses, the mass/energy increases, resulting in a stronger gravitational force that can cause further collapse or even lead to the formation of a black hole.

4. How does the mass/energy of a collapsing gas shell affect its collapse?

The mass/energy of a collapsing gas shell plays a crucial role in determining the outcome of the collapse. If the mass/energy is not enough to overcome the internal pressure of the shell, it will stop collapsing and reach a stable state. However, if the mass/energy is sufficient, the shell will continue to collapse until it reaches a point of infinite density known as the singularity.

5. Can the mass/energy of a collapsing gas shell be converted into other forms?

Yes, according to Einstein's famous equation E=mc², mass and energy are interchangeable. This means that the mass/energy of a collapsing gas shell can be converted into other forms, such as radiation, as the collapse progresses. This process is a crucial factor in understanding the evolution of stars and the formation of black holes.

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