# A Hollow Black Hole

After I commented on the question I started wondering what an observer inside a collapsing shell would experience.

If you construct a spherical shell then an observer inside it feels no gravity. This is true in Newtonian gravity, and is also true in General Relativity as a consequence of Birkhoff's theorem i.e. the metric inside the shell is the Minkowski metric.

In principle we can take the shell and compress it until it's external radius falls below the Schwarzschild radius $r = 2GM/c^2$, at which point the shell will start collapsing inwards and form a singularity in a finite time. In fact it's a very short time indeed. Calculating the lapsed time to fall from the horizon to the singularity of an existing black hole is a standard exercise in GR, and the result is:

$$ \tau \approx 6.57 \frac{M}{M_{Sun}} \mu s $$

That is, for a black hole of 10 solar masses the fall takes 65.7 microseconds! I would have to indulge in some head scratching to work out if the same time would be measured by an observer riding on the collapsing shell, but if the time isn't the same it will be of a similar order of magnitude. This means much of the question doesn't apply, since the shell cannot be stable long enough for the black hole to evaporate. However it leaves open the interesting question of what the observer inside the shell experiences.

Curious as it seems, Birkhoff's theorem implies the observer experiences absolutely nothing until the collapsing shell hits them and sweeps them, along with the shell, to an untimely end (a few microseconds later!).

**Response to comment: time dilation**

The infall time I calculated above is the proper time, that is the time measured by the freely falling observer on their wristwatch. You need to tread carefully when talking about time in relativity, but the proper time is usually easy to understand.

Re time dilation: again we need to be careful to define exactly what we mean. In the context of black holes we usually take an observer far from the black hole (strictly speaking at an infinite distance) as a reference and compare their clock to a clock near the black hole. By *time dilation* we mean that the observer at infinity sees the clock near the black hole running slowly.

A clock in a gravitational potential well runs slowly compared to the clock at infinity. This was discussed in the higher you go the slower is ageing (and also in Gravitational time dilation at the earth's center). It's important to understand that it's the potential that matters, not the gravitational acceleration, so even though the observer inside the shell feels no gravitational acceleration they are still time dilated compared to the observer at infinity.

Note that the time dilation relative to the observer at infinity goes to infinity at the event horizon, so it makes no sense to compare times inside the event horizon to anything outside.

Take a shell with outer radius $R$ and inner radius $r$. If the density of the ring material is $d$, The formula to calculate the mass of the shell is $$\frac{4}{3}d\pi(R^3-r^3)$$ The Schwarzschild radius of this shell is $$\frac{\frac{8}{3}dG\pi(R^3-r^3)}{c^2}$$ It follows that if this equals $R$, the shell should collapse into a black hole. Solving for $d$: $$d=\frac{3c^2R}{8G\pi (R^3-r^3)}$$ A sphere of density $d$ and radius $R-r$ has a Schwazschild radius of $$\frac{8\pi dG(R-r)^3}{3c^2}$$ Substituting $d$ gives us: $$\frac{R(R-r)^3}{R^3-r^3}$$ If the Schwarzschild radius of a sphere inside the ring of the shell is less than the radius of the shell at collapse, the shell breaks apart into infinite black holes: $$\frac{R(R-r)^3}{R^3-r^3}<\frac{\frac{8}{3}\frac{3c^2R}{8G\pi (R^3-r^3)}G\pi(R^3-r^3)}{c^2}$$ The radius of the shell at collapse is R anyway, so $$\frac{R(R-r)^3}{R^3-r^3}<R$$ This inequality holds true only if both $R$ and $r$ are positive.

This solution seems to suggest, that any shell with a thin radius will collapse into local black holes at $r + \frac{(R-r)}{2}$ before the shell reaches a density high enough to collapse into the black hole at a smaller radius.

I am unsure whether any of this is true. It may only work for very small values of $R-r$. Even so, I don't know how black holes would form in that region, or if they dissapear due to hawking radiation. If the latter is the case, this would be a way of converting mass into energy in the form of gravitational waves 100% efficiently.