Observer Inside Collapsing Shell

[Tomas Vencl]

TL;DR

Inside observer view of a collapsing shell

let's consider spherically symmetrical thin shell of dust, which is collapsing under its own gravity. There are no other forces as pressure or so except gravity, and particles of shell (dust) are in free fall. The shell has total mass M and collapse starts from rest state with diameter of the shell much larger than its Schwarzschild radius. During all collapse the symmetry remains.
Now consider observer (a) with negligible mass in the center of the shell. The observer has a clock and measures its propper time t(a). The observer also has a flashlight, and can send signals and receive reflected signals from inner wall of the shell.

Until the observer is absorbed (collapsed) by the shell, he is still in flat spacetime and he can make some measurements.
If he will measure (or calculate) the "velocity" of inner wall of collapsing shell (meaning dr/dt(a) ,what he will measure/calculate ? Mainly after the shell passes the Schwarzschild radius ? Does the defined "speed" exceeds c, or what are the limits ?
Or generally, what observer (a) can see during collapse ?

 

[PeterDonis]

Until the observer is absorbed (collapsed) by the shell, he is still in flat spacetime and he can make some measurements.

Yes, but the flat spacetime region he is in does not include the shell itself, so he cannot apply the rules of flat spacetime (or the implications those rules have for measurements he makes) to any measurements that involve the shell.

If he will measure (or calculate) the "velocity" of inner wall of collapsing shell (meaning dr/dt(a) ,what he will measure/calculate ?

You would have to specify a particular measurement procedure he is going to use. There is no single well-defined answer.

what observer (a) can see during collapse ?

Probably the best single observation to describe is the frequency shift he observes of light signals coming from the shell. This will be a blueshift, increasing without bound as the shell approaches him.

 

[Tomas Vencl]

You would have to specify a particular measurement procedure he is going to use. There is no single well-defined answer.

Thank you.
I am thinking about next procerure:
At the beginning there is a line of the synchronized buoys in the region inside shell. Line is oriented radially and starts in the centre. Distances between buoys are regular, and matches to radial distance in flat area. Each buoy is continuously emitting signal, and is designed to stop the emitting while in contact with collapsing shell.
Observer receives signals from buoys, and while signal from apropriate buoy stops than he is comparing with his proper time t(a) . So he can calculate required dr/dt(a).
Will he catch the end of the signals from all buoys, before he is hiting by shell ? Or at some time he catch the end of signal from some buoy (not his nearest) and than is crashed by shell ? Or other scenario ?

 

[Ibix]

So the buoys stop transmitting when the inner surface of the shell collapses past them? In that case, an observer at the center will always receive the light from each buoy before the shell reaches him - although the delay will tend to zero as the shell collapses.

 

[PeterDonis]

an observer at the center will always receive the light from each buoy before the shell reaches him - although the delay will tend to zero as the shell collapses.

Also, the time by his clock between signals from successive buoys will tend to zero as the shell collapses (this corresponds to the blueshift that I mentioned before).

 

[timmdeeg]

Interesting gedankenexperiment!
Let's assume that radial thickness of shell and size of observer are almost negligible compared to the Schwarzschildradius of M.
What experiences the observer in the moment and later when the collapsing shell is within it's Schwarzschildradius 2GM (c=1) and a black hole is formed?
Does the spacetime inside change from flat to curved within this moment?

 

[PeterDonis]

What experiences the observer in the center when the collapsing shell is within it's Schwarzschildradius 2GM (c=1) and a black hole is formed?

Nothing.

Does the spacetime inside change from flat to curved within this moment?

No. The entire spacetime region inside the shell is flat. The observer in the center only sees a change in his local spacetime geometry when the shell reaches him.