Black holes

Despite the very wide variety of topics and weird phenomenon in general relativity, to this day the very most common questions on it remain about black holes, with the big bang as a distant second. It's a topic fraught with minsinterpretations, half-truths and simplifications, so I thought an article on the topic wouldn't go amiss.

What is a black hole

Quite possibly one of the most confusing aspect of the whole topic of black holes is that the term is used to refer to a wide variety of spacetimes, with fairly different properties. These range from the abstract black holes defined from symmetric or algebraic properties to the stellar black holes supposed to represent real astrophysical objects.

Despite not covering properly every possible case that might be called a black hole, the most common definition involves the presence of a horizon.


For an asymptotically flat spacetime, the horizon can be defined as

\begin{equation} \text{Hor}(M) = M \setminus I^-(\mathscr{I}) \end{equation}

The Schwarzschild solution

The archetypal and oldest black hole solution remains the Schwarzschild metric. Typically in four dimensions, it is defined by a spherically symmetric vacuum solution, ie it possesses a continuous group of transformations of three parameters $G_3$ with orbits homeomorphic to $S^2$.

As the orbits $S^2$ are of dimension $2$ with a $3$-dimensional isometry group, they have a $2 (2+1) / 2 = 3$ isometry group, making them maximally symmetric. The induced metric therefore has constant curvature on that sphere, and as it is a sphere, it has the standard sphere metric on it,

\begin{equation} d\Omega^2 = R^2 (d\theta^2 + \sin(\theta) d\varphi^2) \end{equation}

Thm 8.16 Stefani : If a group $G_r$ of $r = d(d+1)/2$, $d>1$ has orbits of dimension $d$ the orbits admit orthogonal surfaces

A spherically symmetric metric can therefore be put in the form

\begin{equation} ds^2= -e^{2f(r,t)} dt^2 + e^{g(r,t)} dr^2 + R^2(r,t) (d\theta^2 + \sin(\theta) d\varphi^2) \end{equation}

We don't know yet what role $r$ and $t$ will play, but everything has been put in rather suggestive form.


The Schwarzschild metric is then

\begin{equation} ds^2 = -(1 - \frac{r_s}{r}) dt^2 + (1 - \frac{r_s}{r}) dr^2 + r^2 (d\theta^2 + \sin(\theta) d\varphi^2) \end{equation}

Outside of the usual and not terribly menacing singular behaviour for the angular components (which vanish for $r = 0$), we have two potentially

Last updated : 2019-08-24 20:20:45
Tags : physics , general-relativity , black-hole