Exercise 5.3 Inside the event horizon

Question

Consider a particle (not necessarily on a geodesic) that has fallen inside the event horizon, ##r<2GM##. Use the ordinary Schwarzschild coordinates ##\left\{t,r,\theta,\phi\right\}##.

Observable Universe

a) Show that the radial coordinate must decrease at a minimum rate given by $$
\left|\frac{dr}{d\tau}\right|\geq\sqrt{\frac{2GM}{r}-1}
$$b) Calculate the maximum lifetime for a particle along a trajectory from ##r=2GM## to ##r=0##.
c) Express this in seconds for a black hole with mass measured in solar masses.
d) Show that this maximum proper time is achieved by falling freely with ##E\rightarrow0##.

Answers

The most interesting exercise so far, especially parts b and c. If the sun were a black hole then the maximum lifetime for our particle would be $$
{\Delta\tau}_\bigodot=1.55\times\ {10}^{-5}\  \rm{s}
$$That's a pretty short time, but if the Sun were a black hole it would have ##2GM=3\ \text{km}## so our test particle would have an average speed of about ##2\ \times\ {10}^8\ \text{m s}^{-1}## which is just below the speed of light and a hundred times faster than the Parker Solar Probe launched in 2018 which should only reach 0.064% the speed of light.

However big the black hole is, the average minimum speed for the fall for the centre is constant at$$
v_\rm{AvMin}=\frac{2c}{\pi}
$$M87* the black hole at the centre of our galaxy is about ##6.5\times\ {10}^9## solar masses so we get$$
{\Delta\tau}_\rm{M87\ast}=1.55\times\ {10}^{-5}\times6.5\times\ {10}^9={10}^5\ s=28\ \rm{hours}
$$There is a short time to prepare in M87*.

We can do the same for a black hole with the mass of the Universe: The observable Universe contains ordinary matter equivalent to ##{10}^{23}## solar masses. So$$
{\Delta\tau}_{Universe}=1.55\times\ {10}^{-5}\times{10}^{23}\approx{10}^{18}s=300\ \text{billion years}
$$The estimated 'age' of the Universe is 14 billion years, so there is plenty of time inside the big black hole - we have hardly started the journey.

See proof and calculations at Ex 5.3 Inside the event horizon.pdf (4 pages). Also contains speculations on what happens to a photon and links to other answers.
My document on Constants and conversion factors also came in very handy.