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.

Eddington and Finkelstein take us into a black hole

In the first part of section 5.2 (two posts ago) it seemed to be impossible to get inside the Schwarzschild radius. In the second part we look at other coordinate systems and find the Eddington-Finkelstein coordinates which show us how. The metric then takes a different form (which says something about Birkhoff's theorem) and it does not have an infinity at the Schwarzschild radius (##r=2GM##). 

The properties of the function ##1-2GM/r## (which causes the trouble) frequently amaze. It keeps eating itself up which is very satisfactory.

Eddington-Finkelstein coordinates eliminate the time coordinate ##t## and introduce

$$v=t+r+2GM\ln{\left(\frac{r}{2GM}-1\right)}$$

so the relationship between ##v## and ##t## is complicated.

The image shows a light cone at various distances out from the centre at ##r=0## . The ingoing light beam always heads for the centre the outgoing beam can get away when ##r>2GM## but flips towards the centre once it originates at a distance less than the Schwarzschild radius (aka the event horizon). The closer the starting point is to the centre, the less room there is for manoeuvre. 

I am slightly dubious about the direction of the ingoing side of a light cone inside the Schwarzschild radius.

See why and check my maths: Commentary 5.6#2 Schwarzschild Black Holes.pdf (6 pages)

Geodesics of Schwarzschild

In section 5.4 Carroll explores the geodesics of Schwarzschild. These turn out to be almost useless as far as I can see. What are very useful are the Killing vector fields in Schwarzschild. In particular the energy Killing vector field which will eventually enable us to find an equation fro the path of a radially free-falling test particle. I learnt why Killing vectors are so important!

In this section we also write out the useless geodesic equations and work out some potentials. Most of it is pretty dry stuff. So dry that I forgot to post it until I actually used the equations with some success in late June.

All at Commentary 5.4 Geodesics of Schwarzschild.pdf 

2 years on

Two years on and I am about half way through Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll. It's probably the best value for money book I have read (er... studied) in my life. When I started I could barely remember how to differentiate. Now I can use the chain rule almost without thinking and the tensor things which I had never met before are a doddle. I am now on the Eddington-Finkelstein metric and discovering how to get into a black hole (which seems to be impossible in the obvious ##t,r,\theta,\phi## coordinates) and why you never get out. I can't praise the book and Sean Carroll highly enough!

Schwarzschild Black Holes

We're just looking at the first two pages of section 5.2 here where Carroll shows closing up light cones as one approaches an event horizon and then a beacon on a radial geodesic which purports to show that clock ticks on the beacon appear to get slower and slower according to a stationary observer who maintains a safe distance. 

I slightly improve on the former and show the world lines of in- and out-going photons starting at a given radius. I then try to reproduce the latter, first with an invented geodesic, and then with a properly calculated one. The invented geodesic (shown above) produced the result Carroll suggests but the 'properly calculated ones' did not. The first method produced correct geodesics, but not the desired ones. I believe that the second method failed due to my mathematical inexperience. Possibly there is no exact solution.

The diagram is like Carroll's Fig 5.8. It shows the world lines of a photon and a beacon falling directly towards the centre of the black hole. There is an observer hovering above them at a safe distance. Using the Schwarzschild metric the photon never seems to cross the event horizon at ##r=R_S##! The beacon, also falling directly in, sends signals (flashes of light) back out at intervals ##\Delta\tau_1##. They arrive at the observer separated by longer and longer times. The beacon also appears to take forever to get to the event horizon!

The maths didn't really work. Witness my struggles at Commentary 5.6 Schwarzschild Black Holes.pdf (8 pages)

Gravitational redshift

The second piece of evidence for general relativity we examine is gravitational redshift in section 5.5. That's when the wavelength (or frequency) of light changes as it moves to stronger or weaker parts of a gravitational field.

Apparently Pound and Rebka were the first to measure it using gamma rays going up 72 feet (that's 22m in new money). They did it in the Jefferson laboratory (pictured)  at Harvard in 1959. That's about 40 years after Einstein predicted it. The change in the wavelength was 2 parts in a thousand trillion (##2## in ##10^{15}##). They measured it by wiggling the source of the gamma rays about in a speaker cone and seeing when the Doppler shift cancelled the gravitational shift!

The calculations are quite simple (and I got the right answer without cheating!) but I really needed to understand a few other things which sent me right back to chapters 3 and then 1. It all concerns the energy-momentum vector for a massive particle and then a massless e.g. gamma ray) particle and how the energy and therefore frequency fits into that. Light dawned.

The basic calculations: Commentary 5.5 Gravitational Redshift.pdf (2 pages)
On four-momentum and energy: Commentary 3.4 Particle energy.pdf (6 pages)

Precession of perihelion of Mercury

In 5.5, we continue our adventures from the previous section 5.4 where we calculated a few things from the Killing vectors and geodesics of the Schwarzschild metric. I learnt a good lesson about Killing vectors then that went over my head previously.

There are several amazing things in this section one group being mathematical dexterity and the other being the feats of astronomers. The latter have measured the precession of the perihelion of Mercury at ##{44}^{\prime\prime}  \text{per century}##. ##{44}^{\prime\prime}## is 44 seconds of an arc or 44 times 1/3600 degrees. That's about the angle that a soccer ball would subtend if it was one kilometer away and if you could see it. How did they do that?

Apparently we are following d'Inverno 1992, not Einstein, and we start with a differential equation for the orbit in terms of radial distance ##r## and affine parameter ##\lambda## and turn it into a simpler differential equation in terms of ##x\propto1/r## and azimuth ##\phi##. A crafty differentiation then makes the equation further collapse into the equation of an ellipse with a small perturbation, as witnessed by astronomers. Using more trigonometrical tricks we then solve the GR perturbation part and find a term in that which must correspond to the precession. A further trig trick gets us to "the equation for an ellipse with an angular period that is not quite ##2\pi##" and from that we extract the precession of the perihelion of Mercury. How did anybody think up all that?

The image shows the path of a Mercury that precesses about a million times faster than our own Mercury. I plotted it to check the  "the equation for an ellipse with an angular period that is not quite ##2\pi##". It is surprisingly accurate for such a large precession.

Read my attempts to follow Carroll at
Commentary 5.5 Precession of perihelia.pdf (7 pages)
Commentary 5.4 Geodesics of Schwarzschild.pdf (8 pages)