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 and trapped surfaces. This itself is not that helpful, as there are many different types of both
The most commonly used definition of a horizon, as the one commonly or 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. For now let's call $r$ the radial distance, $t$ the time and $\theta, \varphi$ the angular coordinates.
The Schwarzschild metric can be found in a few different ways from this, either by looking for a spherically symmetric vacuum solution or looking for a stationary spherically symmetric vacuum solution (or, since we do not know the role of $t$ yet, a solution where $\partial_t$ is a Killing vector). We will later see the equivalence of the two conditions, but for now let's assume that we have a stationary spherically symmetric vacuum solution.
Killing vectors : $\xi$, $S^{n-2}$
In general, given a spacetime with a single Killing vector $\xi$,
\begin{equation} ds^2= -e^{2f(r)} dt^2 + e^{g(r)} dr^2 + R^2(r) (d\theta^2 + \sin(\theta) d\varphi^2) \end{equation}In those coordinates, the Christoffel symbols are then
\begin{eqnarray} {\Gamma^t}_{tt} &=& \end{eqnarray}Riemann tensor
Ricci tensor
Ricci scalar
...
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 problematic points in these coordinates : the event horizon $r = r_s$, which is a submanifold of topology $\mathbb{R} \times S^2$, and the singularity $r = 0$.
There are a few different definitions and tests we can do on a singularity. First it is to be kept in mind that divergence of metric components are not necessarily pathological, as can be seen from the coordinate patch of Minkowski space under the coordinate change $t' = t^{-1}$ for the region $t > 0$ :
\begin{equation} ds^2 = -\frac{1}{t'^2} dt'^2 + dx^2 + dy^2 + dz^2 \end{equation}The simplest way to check for singularities is to check for divergences of some polynomial expression of the curvature.
The Nordstrom solution
Tomimatsu and Kerr solution
Observers and black holes
Let's consider a variety of observers in the Schwarzschild metric.
Our first observer $A$ is at some distance considered particularly large from the event horizon, with no angular momentum.
Bibliography
Last updated : 2023-04-17 16:03:30