We had a midterm in the undergraduate General Relativity class this week. Midterm II. We’d just come off a batch of lectures on black holes (focussing on Schwarzschild and Kerr), and so it seemed very natural to focus on that as the topic. Schwarzschild is the most basic (vanilla) black hole and Kerr is the case of having rotation for a bit of extra flavour. They are both hugely important in real physics – the former for the basic phenomena and then the latter since most objects out there in the astrophysical realm are actually going to be rotating to some degree (and after gravitational collapse, probably a high degree). So I focussed on those in class.

For the midterm therefore, I got the opportunity to have them discover properties of the (less astrophysically important perhaps) charged black hole – the Reissner-Nordstrøm solution*:

[tex] ds^2=-\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)dt^2+\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)^{-1}dr^2+r^2\left(d\theta^2+\sin^2\theta d\phi^2\right)\ ,[/tex]

…and do a few computations with it analogous to what they did for the other cases in class and in homework. I hope they had fun (like discovering that for a charged particle probing the geometry there’s a new (unstable) stationary solution corresponding to the result of balance between and attraction and repulsion).

For the last part of the exam I got them to sprinkle in some aspects that connect directly to contemporary research in quantum gravity. They had to deduce the temperature of Hawking radiation for the charged hole and for the rotating hole, by assuming the Bekenstein-Hawking area-entropy relation, [tex]S=(k_{\rm B}c^3/\hbar G)\cdot (A/4)[/tex], is true and then trying to deduce a First Law of Thermodynamics for each type of black hole. It is a fun exercise, and a nice taste of things we do in research in this area, mixing quantum and gravity in an interesting and (it turns out) remarkably natural way. There’s a snapshot of an extract in the photo (click to enlarge). ([Update: I like to point out to them (and did) how interesting and exciting it is that fundamental constants from at least three or four different classes all end up in the same formulae in this subject. Look a few lines up. There’s Newton’s gravitational constant, Boltzman’s constant that’s usually sitting in a thermo class, the speed of light from electromagnetism, and Planck’s constant from quantum mechanics. Wow!])

I hope *they* had fun.

-cvj

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*Written in geometrical units here where [tex]G=c=1[/tex].

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Exam Heat