Black holes
Now that all the abstract mathematical tools are in place we can begin with some physical applications. In this chapter we shall see what happens if too much mass is located into a small volume of space --- we shall take a look at the phenomena known as black holes.
The Schwarzschild metric
\begin{quote}
\emph{``It is always pleasant to avail of exact solutions in simple form"}
\begin{flushright}
\textit{- K. Schwarzschild}
\end{flushright}
\end{quote}
Merely months after Einstein's publication of general relativity, Karl Schwarzschild obtained an exact solution to Einstein's vacuum field equations, \cite{tay-whe-00}. The first Schwarzschild had done was to consider vacuum, i.e., $T^{\mu \nu} = 0$. Secondly he made several symmetry assumptions, namely he considered vacuum models that are static and spherically symmetric.
Derivation of the Schwarzschild metric
Since the Schwarzschild solution is one of the simplest and most useful solutions to Einstein's field equations, it is worthwhile to derive this metric in some detail.
Assumptions
The Schwarzschild vacuum solution is spherically symmetric and static, these assumptions give the following simplifications,
\begin{enumerate}
\item A spherically symmetric spacetime implies that all metric components are unchanged under any rotation-reversal $\theta \rightarrow - \theta$ or $\phi \rightarrow - \phi$.
\item A static spacetime is one in which all metric components are independent of the time coordinate and where furthermore the geometry of the spacetime is unchanged under a time-reversal $t \rightarrow -t$.
\item A vacuum solution is one that satisfies the equation $T_{\mu \nu} = 0$. From Einstein's field equations, this implies that $R_{\mu \nu} = 0$.
\end{enumerate}
Diagonalizing the metric
Consider the variable transformation $t = x^0 = -x^{0'} = -t'$, $x^i = x^{i '}$ where, $x^i = \{ r, \theta , \phi \}$; then
\[
g_{i' t'} = \frac{\partial x^{\mu}}{\partial x^{i '}} \frac{\partial x^{\nu}}{\partial x^{t '}} g_{\mu \nu}= -g_{i t},
\]
but the static condition implies that $g_{i ' t '} = g_{i t}$, and hence it follows that $g_{i t} = 0$.
This taken together with the coordinate transformations $(t, r, \theta, \phi) \rightarrow (t, r, \theta, -\phi)$ and $(t, r, \theta, \phi) \rightarrow (t, r, -\theta, \phi )$ similarly implies that the entire metric must be diagonal,
\[
ds^2 = g_{tt} dt ^2 + g_{rr}d r^2 + g_{\theta \theta} d \theta ^2 + g_{\phi \phi} d \phi ^2.
\]
Simplifying the components
On the hypersurfaces of constant $t$ and $r$, it is required that the metric is that of a 2-sphere
\[
d \Omega ^2 = r_{0}^2 \left ( d \theta^2 + \sin^2 \theta d \phi^2 \right ),
\]
where $r_0$ is the radius of the 2-sphere. Comparing the forms of the metric on this hypersurface gives
\[
\tilde{g}_{\theta \theta}\left(d \theta^2 + \frac{\tilde{g}_{\phi \phi}}{\tilde{g}_{\theta \theta}} d \phi^2 \right) = r_{0}^2 \left ( d \theta^2 + \sin^2 \theta d \phi^2 \right ),
\]
which yields
\[
\begin{aligned}
\tilde{g}_{\theta \theta} =& r_{0}^2,\\
\tilde{g}_{\phi \phi} =& r_{0}^2 \sin ^2 \theta.
\end{aligned}
\]
But this is required to hold on each hypersurface and thus
\[
\begin{array}{l}
g_{\theta \theta}= r^2,\\
g_{\phi \phi}= r^2 \sin^2 \theta.
\end{array}
\]
The metric can hence be written on the form
\[
ds^2 = g_{tt} dt^2 + g_{rr} dr^2 + r^2d \theta^2 + r^2 \sin^2 \theta d \phi^2.
\]
So far $g_{tt}$ and $g_{rr}$ are arbitrary. But to preserve spherical symmetry requires that
\[
\begin{aligned}
g_{rr} =& g_{rr} (r,t),\\
g_{tt} =& g_{tt} (r,t).
\end{aligned}
\]
Furthermore, due to the static requirement $g_{tt}$ and $g_{rr}$ cannot depend on $t$, and thus the diagonalized line element takes the form
\[
ds^2 = g_{tt}(r) dt^2 + g_{rr}(r) dr^2 + r^2d \theta^2 + r^2 \sin^2 \theta d \phi^2.
\]
Using the field equations to find $g_{tt}$ and $g_{rr}$
To determine $g_{tt}$ and $g_{rr}$, the vacuum field equations will be used, $R_{\mu \nu} = 0$. Only three of these vacuum equations are nontrivial, and they can be written as
\[
\begin{aligned}
4 g_{rr,r} g_{tt}^2 - 2 r g_{tt,r} g_{rr} g_{tt} + r g_{rr,r} g_{tt,r} g_{tt} + r g_{tt,r} ^2 g_{rr} & = 0,\\
r g_{rr,r}g_{tt} + 2 g_{rr}^2 g_{tt} - 2g_{rr}g_{tt} - r g_{tt,r} g_{rr} & = 0,\\
- 2 r g_{tt,r} g_{rr}g_{tt} + r g_{rr,r} g_{tt,r}g_{tt} + r g_{tt,r}^2 g_{rr} - 4 g_{tt,r} g_{rr}g_{tt} & = 0;
\end{aligned}
\]
subtracting the first and third equations yields
\[
\begin{aligned}
g_{rr,r}g_{tt} + g_{rr} g_{tt,r} = 0, \\
\Rightarrow \left ( g_{rr} g_{tt} \right ) _{,r} = 0,\\
\Rightarrow g_{rr} g_{tt} = K,
\end{aligned}
\]
where $K$ is a non-zero real constant. Substituting $g_{rr} g_{tt} = K$ into the second equation and tidying up gives
\[
r g_{rr,r} = g_{rr} \left ( 1 - g_{rr} \right ),
\]
which has the general solution
\[
g_{rr} = \left ( 1+ \frac{1}{Sr} \right ) ^{-1}
\]
for some real constant $S$. Hence, the static, spherically symmetric vacuum metric takes the form
\[
ds^2 = K \left ( 1 + \frac{1}{S r} \right ) dt^2 + \left ( 1 + \frac{1}{S r} \right ) ^{-1} dr^2 + r^2 \left ( d \theta^2 + \sin^2 \theta d \phi^2 \right ).
\]
Using the weak field approximation to find $K$ and $S$
Using the weak field approximation\footnote{where the metric approaches the Minkowski metric.} to identify the constants $K$ and $S$, yields
\[
g_{tt} = K \left ( 1 + \frac{1}{Sr} \right) \approx - 1 + \frac{2M}{r} = - \left( 1 - \frac{2M}{r} \right )
\]
where $M$ is the mass of the gravitational source; it follows that
\[
K= -1 \quad \textrm{and} \quad \frac{1}{S}= - 2M,
\]
and hence
\[
g_{rr} = \left ( 1 - \frac{2M}{r} \right ) ^{-1} \quad \textrm{and} \quad g_{tt} = - \left ( 1 - \frac{2M}{r} \right ),
\]
which yields the line element
\begin{equation}\label{Schwarzschild}
ds^2 = - \left ( 1 - \frac{2M}{r} \right ) dt^2 + \left ( 1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2 \left (d \theta^2 +\sin^2 \theta d \phi^2 \right ).
\end{equation}
This expression can be written in terms of the so-called Schwarzschild radius, $2M = R_S$, and $d \theta^2 +\sin^2 \theta d \phi^2 = d \Omega ^2$:
\[
ds^2 = - \left ( 1 - \frac{R_S}{r} \right ) dt^2 + \left ( 1 - \frac{R_S}{r}\right)^{-1} dr^2 + r^2 d \Omega ^2.
\]
Here $t$ can be interpreted as ``far away time", the time that an observer far away from the mass, i.e. in Minkowski space-time, is measuring; $r$ is the ``reduced circumference" and is determined by
\[
r = \frac{C}{2 \pi},
\]
where $C$ is the circumference of a circle; this is needed due to the impossibility of measuring the radius when $r \leq R_S$ in the same manor that it is measured on a flat paper.
The Schwarzschild metric has some features that are easily checked.
\begin{itemize}
\item When $r \rightarrow \infty$ the metric reduces to the flat Minkowski metric.
\item When $M \rightarrow 0$ the metric reduces to the flat Minkowski metric.
\item By setting $dt = d\Omega = 0$, one considers the distance between two events on a radial line at the same time, as described by (\ref{Schwarzschild}), which leads to
\[
ds = dr_{shell} = \frac{dr}{\left ( 1 - \dfrac{2M}{r} \right )^{1/2}}\, .
\]
From this it is obvious that $dr_{shell} \geq dr$, a difference that indicates that curvature is present.
\item By setting $dr = d\Omega = 0$, i.e., by placing a clock at a fixed point and measuring the time between two ticks, yields
\[
ds = -dt_{shell} = - \left ( 1 - \frac{2M}{r} \right )^{1/2} dt.
\]
From this it is obvious that $dt_{shell} \leq dt$; this is an effect known as gravitational redshift/time dilation. In fact, as it turns out, any massive object slows down time.
\end{itemize}
Singularities
When analyzing (\ref{Schwarzschild}) one finds singularities when solving the equation $g_{tt} = 0$ (the same result is obtained by setting $g_{rr} = 0$). This leads to the solutions $r = 0$ and $r = R_S$. For normal stars these radii are never any problem, since typical values are $R_{\odot} \simeq 7 \cdot 10 ^8 m$ and $2 M_{\odot} \simeq 3 \cdot 10^3 m$, which means that $r = 0$ as well as $R_S$ are within the star where the Schwarzschild vacuum solution is invalid. But let us not trifle with a trivial thing like a star, let us instead look at what would happen if something was compacted enough so that $r \leq R_S$.
At $r = R_S$ the Schwarzschild solution still makes perfect sense; nothing special happens with the curvature and one does not have any singularity there, other than a bad choice of coordinates, and thus it is only a so-called coordinate singularity, \cite{wal-84}, \cite{car-03}, \cite{sch-85}. There are more suitable sets of coordinates available such as Kruskal-Szekeres coordinates, which allows one to obtain a maximally extended Schwarzschild solution.
Even though $r = R_S$ is not a singularity, there are some bizarre effects that occur when $r$ passes through $R_S$; in particular the radial coordinate $r$ becomes timelike and the time coordinate $t$ becomes spacelike when $r < R_S$. Thus there is no possible way to keep a particle at a constant $r$ when $r < R_S$, it will, as time does, inexorably move forward, i.e. into the singularity $r = 0$, even a photon will inevitably move towards the singularity. Thus the surface associated with $r=R_S$ can never be seen from the outside as it will neither emit any radiation of any kind to $r > R_S$ nor will anything else leave there, according to general relativity it is completely black.
At $r = 0$ there really is a space-time singularity where the curvature becomes infinite, and thus the metric and the space-time cease to be well defined. This was for a long time discarded as a non-physical feature associated with too strong symmetry assumptions, but later it was realized that it is a generic feature of general relativity and not some rare special case.
We shall regard the solutions we consider as describing real physical phenomena and call them by the name given to them by J. A. Wheeler: black holes, for rather obvious reasons. The Schwarzschild radius is called a horizon, due to the fact that it it is impossible to see beyond it, moreover, the surface with $r = R_S$ is also known as the surface of the Schwarzschild black hole.
The Reissner-Nordstr\"{o}m metric
Instead of just looking for vacuum solutions one can also consider matter solutions \cite{gravitation}. An electric charge yields that one has to solve $G_{\mu \nu} = 8\pi T_{\mu \nu}$ for a source
\[
T_{\mu \nu} = F_\mu{}^\alpha F_{\alpha \nu} + \frac{1}{4} \eta_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}.
\]
A static spherically symmetric electric field is described by
\[
\begin{aligned}
E_r &= E(r),\\
E_\theta = E_\phi &= 0,\\
\vec B &= 0,\\
\end{aligned}
\]
which yields,
\[
F^{\mu \nu} =
\left( {\begin{array}{*{20}c}
0 & -E_r & 0 & 0 \\
E_r & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}} \right).
\]
Let us still assume that one has a static spherically symmetric space-time, but that the body with mass $M$ now also carries an electrical charge $Q$. This yields a derivation that is very much the same as for the Schwarzschild case, and gives a metric
\[
\begin{aligned}
ds^2 = - \left( 1 - \frac{R_S}{r} + \frac{R_Q^2}{r^2} \right) dt^2 + \left (1 - \frac{R_S}{r} + \frac{R_Q ^2}{r^2} \right )^{-1} dr^2 + r^2 d\Omega^2,
\end{aligned}
\]
where $R_Q$ is a length-scale corresponding to the electric charge $Q$ given by
\[
R_Q^2 = \frac{Q^2}{4\pi},
\]
where $\frac{1}{4\pi}$ is the Coulomb force constant in suitable units.
This so-called Reissner-Nordstr\"{o}m metric has some features that are worthwhile to point out:
\begin{itemize}
\item In the limit where the charge $Q \rightarrow 0$, the Reissner-Nordstr\"{o}m metric reduces to the Schwarzschild metric.
\item $R_Q \leq R_S$, otherwise one would have a naked singularity, i.e., one without horizon; actually one needs $R_Q \ll R_S$, or otherwise atoms would be ripped apart by the immense electromagnetic field.
\end{itemize}
Singularities
As for the Schwarzschild metric the horizons are located by analyzing the equation $g_{tt} = 0$; this yields
\[
g_{tt} = 1 - \frac{R_S}{r} + \frac{R_Q^2}{r^2} = 0,
\]
which has the solutions
\[
r_\pm = \frac{1}{2} \left ( R_S \pm \sqrt{R_S^2 - 4R_Q^2} \right ).
\]
The radius
\[
r_+ = \frac{1}{2} \left ( R_S + \sqrt{R_S^2 - 4R_Q^2} \right )
\]
is very similar to the horizon in the Schwarzschild case; here space and time change meaning.
At the radius
\[
r_- = \frac{1}{2}\left ( R_S - \sqrt{R_S^2 - 4R_Q^2} \right )
\]
time and space change meaning again, making the space-time static. So instead of a singularity that is in the future of all objects passing through the horizon of $r_+$, it is now possible to avoid the singularity and get away from the anomaly (though not back into the same part of space-time one came from).
The Cosmic censorship hypothesis
In the case $2R_Q = R_S$ one can see that
\[
r_+ = r_- = \frac{R_S}{2},
\]
and thus the horizons become degenerate; this is a so-called extremal Reissner-Nordstr\"{o}m metric. When $2R_Q > R_S$ the solutions are unphysical, and thus a naked singularity is created, which makes the universe unpredictable, something that must be avoided at all costs, which is expressed as follows in the cosmic censorship hypothesis \cite{haw-94}.
\begin{quote}
\emph{``Nature abhors a naked singularity"}
\begin{flushright}
\textit{- R. Penrose}
\end{flushright}
\end{quote}
Hence one assumes $2 R_Q \leq R_S$.
The Kerr metric
Instead of an electric charge, let us now consider the case where a body with mass $M$ rotates in a stationary way with an angular momentum $J$. This implies that the metric is not spherically symmetric, but at least it is axially symmetric, \cite{ker-63} \cite{tay-whe-00}. The associated solution to Einstein's field equations is known as the Kerr solution and can be written in the form
\[
\begin{aligned}
ds^2 = -& \left( 1 - \frac{R_S r}{\rho^2} \right) dt^2 + \frac{\rho^2}{\Lambda^2} dr^2 + \rho^2 d\theta^2\\
+&\left( r^2 + \alpha^2 + \frac{R_S r \alpha^2}{\rho^2} \sin^2 \theta \right) \sin^2 \theta \ d\phi ^2 - \frac{2R_S r\alpha \sin^2 \theta }{\rho^2} dt d\phi,
\end{aligned}
\]
where $\alpha$, $\rho$, $\Lambda$ have been introduced for brevity, and are given by
\[
\begin{aligned}
\alpha =& \frac{J}{M},\\
\rho^2 =& r^2 + \alpha^2 \cos^2 \theta,\\
\Lambda^2 =& r^2 - R_S r + \alpha^2.
\end{aligned}
\]
It is of interest to note some things:
\begin{enumerate}
\item When the body stops to rotate one gets that
\[
J \to 0 \Rightarrow
\left\{
\begin{aligned}
\alpha \to & 0 \\
\rho ^2 \to & r^2 \\
\Lambda ^2 \to & r^2 - R_S r \\
\end{aligned}
\right . ;
\]
and thus the metric reduces to the Schwarzschild metric.
\item The so-called extremal Kerr metric is obtained when $J = M^2$, which gives
\[
J = M^2 \Rightarrow
\left\{
\begin{aligned}
\alpha = & M \\
\rho ^2 = & r^2 + M^2 \cos ^2 \theta \\
\Lambda ^2 = & r^2 - R_S r + M^2 \\
\end{aligned}
\right . .
\]
\item One new feature that the previous metrics do not share is the mixed $dt d\phi$ term; the effects of this term will be discussed below.
\end{enumerate}
Singularities
Previously it sufficed to consider only $g_{tt} = 0$ since we could use that $g_{tt} = g_{rr} ^{-1}$; this is no longer the case. Now different surfaces come from $g_{rr} \rightarrow \infty$ and the usual $g_{tt} = 0$; it will be appropriate to start from the inside and move outwards.
By setting $g_{rr} \rightarrow \infty$ one really means that $\Lambda ^2 = 0$, and thus
\[
r^2 - R_S r + \alpha^2 = 0,
\]
with the solutions
\[
r_\pm = \frac{1}{2} \left ( R_S \pm \sqrt{R_S^2 - 4\alpha^2} \right ).
\]
Here we have that
\[
r_+ = \frac{1}{2} \left ( R_S + \sqrt{R_S^2 - 4\alpha^2} \right ) = R_K
\]
is a sphere that is also a horizon of the metric; inside $R_K$ the present form of the Kerr metric is invalid and thus $r_-$ is unphysical and will hence be discarded.
The solutions to $g_{tt} = 0$ are
\[
r_\pm = \frac{1}{2} \left ( R_S \pm \sqrt{R_S^2 - 4\alpha^2 \cos^2\theta}\right ) .
\]
By the same argument as before the solution $r_-$ is unphysical, and will thus be discarded. The only physical solution will henceforth be known as $r(\theta)$,
\[
r(\theta) = \frac{1}{2} \left ( R_S + \sqrt{R_S^2 - 4\alpha^2 \cos^2\theta} \right ),
\]
which describes an oblate spheroid, due to the $\cos^2 \theta$ factor; at the poles of the axis of rotation $\left ( \theta = 0 \textrm{ or } \pi \right )$ one obtains
\[
r (0) = \frac{1}{2} \left ( R_S + \sqrt{R_S^2 - 4\alpha^2} \right ),
\]
which coincides with the horizon associated with $R_K$. In the plane of rotation $\left ( \theta = \frac{\pi}{2} \right )$ one gets
\[
r \left ({\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2}\right ) = \frac{1}{2} \left ( R_S + \sqrt{R_S^2} \right ) = R_S.
\]
Both $R_K$ and $r(\theta)$ are apparent singularities created from a bad choice of coordinates, they are just, as in the Schwarzschild case, coordinate singularities.
The cross term $dt d\phi$
\begin{quote}
\emph{``Space-time itself on the move"}
\begin{flushright}
\textit{- Taylor \& Wheeler}
\end{flushright}
\end{quote}
From the metric one can see the new feature of a cross term $dt d\phi$, implying that the coordinates $t$ and $\phi$ are intertwined, which yields a prediction of a phenomenon called frame dragging (or the Lense-Thirring effect), \cite{tay-whe-00}, \cite{wal-84}.
Near a non-rotating gravitating mass thrust is needed to stay at a fixed radius, but if the gravitating mass is rotating an additional tangential thrust is required to stay at a fixed position, i.e., a position where the fixed stars stay in fixed positions. Thus the space-time is swept along with the rotation of the mass.
The static limit
The static limit is where this rotation of space-time becomes so fast that the thrust required to stay in a fixed position is equal to the speed of light. To determine where the static limit is, one sends in a beacon in the plane of rotation $\left ( \theta = \frac{\pi}{2}, \ d \theta = 0, \ \rho ^2 = r^2 \right )$ that sends light flashes $\left ( ds^2 = 0 \right )$ in the $\phi$ direction $\left ( dr = 0 \right )$. Inserting all this into the metric and multiplying with $\left (dt \right ) ^{-2}$ gives, after some tidying up,
\[
\begin{aligned}
ds ^2 = & g_{tt} dt^2 + g_{t\phi} dtd\phi + g_{\phi \phi} d\phi ^2,\\
\Rightarrow & \quad \frac{d\phi}{dt} = -\frac{g_{t\phi}}{g_{\phi \phi}} \pm \sqrt{\left ( \frac{g_{t\phi}}{g_{\phi \phi}} \right ) ^2 - \frac{g_{tt}}{g_{\phi \phi}}}.
\end{aligned}
\]
At the surface where $g_{tt} = 0$ one gets the two solutions
\[
\frac{d\phi}{dt} = 0 \quad \textrm{and} \quad \frac{d\phi}{dt} = -2\frac{g_{t\phi}}{g_{\phi \phi}}.
\]
The first solution tells us that a photon that is sent in the opposite direction to the rotation will stand still, the dragging has become so strong that the photon cannot move in the opposite direction to the rotation, which clearly means that any massive particle will be forced to move along with the rotation, even if it has an arbitrary large angular momentum opposite to the rotation.
\begin{figure}[h]
\begin{minipage}{7cm}
\begin{tabular}{c}
\includegraphics[width=1\columnwidth]{ergoplane} \\
(a)
\end{tabular}
\end{minipage}
\hfill
\begin{minipage}{7cm}
\begin{tabular}{c}
\includegraphics[width=1\columnwidth]{ergoaxel}\\
(b)
\end{tabular}
\end{minipage}
\caption{A simple schematic overview of a rotating black hole, showed (a) from the 'side', and (b) from the 'top'}
\label{fig:ergo}
\end{figure}
The ergosphere and the Penrose process
The region inside $r(\theta)$ but outside $R_K$ is known as the ergosphere. Inside this sphere it is possible to have a negative total energy, which R. Penrose took an interest in and devised a method that shows that it is theoretically possible to extract energy from the rotational energy \cite{Pen-69}, \cite{tay-whe-00}. The Penrose process can be described with three (in theory) rather simple steps.
\begin{enumerate}
\item Two particles with energy, $\mathcal{E} = \mathcal{E}_1 + \mathcal{E}_2$, start from a far away position and descend to the inside of the ergosphere.
\item The two particles are then separated in such a way that one particle gives more than its total energy to the other particle, and hence acquires negative energy, $\mathcal{E}_1 < 0$.
\item The negative energy particle falls through the horizon, while the other return to its faraway position with more energy than the two particles started with $\mathcal{E'} > \mathcal{E}_1 + \mathcal{E}_2$.
Thus one has transferred energy from the black hole to the escaping particle. This process decreases the black hole's rotational energy and mass, and, if sufficiently repeated, the Kerr black hole will finally become a Schwarzschild black hole.
\end{enumerate}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.5\columnwidth]{penroseprocess}\\
\caption{The Penrose process for energy extraction, see the text for details.}
\label{fig:penroseprocess}
\end{center}
\end{figure}
To find he maximum amount of energy extracted from this, one needs the irreducible mass, $M_{irr}$, described by
\[
M_{irr}^2 = \frac{1}{2} \left ( M^2 + M \sqrt {M^2 - \alpha ^2 } \right );
\]
using that the maximal rotation is determined by, $\alpha = M$, it is found that
\[
M_{irr} = \frac{\sqrt 2}{2} M.
\]
From this it follows that $M_{irr}$ contains $\sim 70.7\%$ of the total energy, which means that $\sim 29.3\%$ of the total energy of the black hole can be extracted \cite{chr-ruf-71}, \cite{Chr-70}.
The Penrose process is probably at least partly responsible for some of the most energetic processes in the universe, i.e. Quasars, gamma-ray bursts (GRB) and active galactic nuclei (AGN).
The Kerr-Newman metric
The final, and most general, stationary asymptotically flat metric we will consider is the so-called Kerr-Newman metric \cite{gravitation}. It is a stationary axisymmetric solution with mass $M$, electric charge $Q$, and angular momentum $J$. The line element can be written in the form:
\[
\begin{aligned}
ds^2 & = - \left( 1 - \frac{R_S r - R_Q^2}{\rho^2} \right) dt^2 + \frac{\rho^2}{\Lambda^2} d r^2 + \rho^2 d\theta^2 \\
& + \left( r^2 + \alpha^2 + \left( R_S r - R_Q^2 \right) \frac{\alpha^2}{\rho^2}\sin^2\theta \right) \sin^2 \theta \ d\phi^2 \\
& - \left( R_S r - R_Q^2 \right) \frac{2\alpha\sin^2\theta}{\rho^2} dt d\phi,
\end{aligned}
\]
where $\alpha$ and $\rho ^2$ are the same as in the Kerr case, but where there is a slight change in $\Lambda ^2$:
\[
\Lambda ^2 = r^2 - R_S r + \alpha ^2 + R_Q ^2.
\]
There are some aspects that deserve notice:
\begin{itemize}
\item $Q \rightarrow 0$ implies that the metric reduces to the Kerr metric.
\item $J \rightarrow 0$ yields that the metric reduces to the Reissner-Nordstr\"{o}m metric.
\item $Q,\ J \rightarrow 0$ leads to that the metric reduces to the Schwarzschild metric.
\end{itemize}
Singularities
Just as for the Kerr metric one has two different cases, $g_{tt} = 0$ and $g_{rr} = 0$.
By setting $g_{rr} \rightarrow \infty$ one really means that $\Lambda ^2 = 0$, and thus
\[
r^2 - R_S r + \alpha^2 + R_Q = 0,
\]
with the solutions
\[
r_\pm = \frac{1}{2} \left ( R_S \pm \sqrt{R_S^2 - 4 \left (\alpha^2 + R_Q ^2 \right )} \right ).
\]
Here we have that
\[
r_+ = \frac{1}{2} \left ( R_S + \sqrt{R_S^2 - 4\left (\alpha^2 + R_Q ^2 \right )} \right ) = R_{KN}
\]
is a sphere, which also is a horizon of the metric; inside $R_{KN}$ the Kerr-Newman metric is invalid, and thus $r_-$ is unphysical and is hence discarded.
The solutions to $g_{tt} = 0$ are
\[
r_\pm = \frac{1}{2} \left ( R_S \pm \sqrt{R_S^2 - 4 \left ( \alpha^2 \cos^2\theta + R_Q ^2 \right )} \right ).
\]
By the same argument as in the Kerr case the solution $r_-$ is unphysical, and is hence discarded. As there is only one physically valid solution it will henceforth be known as $r(\theta)$,
\[
r(\theta) = \frac{1}{2} \left ( R_S + \sqrt{R_S^2 - 4\left ( \alpha^2 \cos^2\theta + R_Q ^2 \right )} \right ),
\]
which describes an oblate spheroid due to the $\cos^2 \theta$ term; at the poles of the axis of rotation $\left ( \theta = 0 \textrm{ or } \pi \right )$ one gets
\[
r (0) = \frac{1}{2} \left ( R_S + \sqrt{R_S^2 - 4\left (\alpha^2 + R_Q ^2 \right )} \right ),
\]
which one can see is the same as the horizon $R_{KN}$. In the plane of rotation $\left ( \theta = \frac{\pi}{2} \right )$ one finds
\[
r \left ({\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2}\right ) = \frac{1}{2} \left ( R_S + \sqrt{R_S^2 - 4 R_Q ^2} \right ).
\]
Both $R_{KN}$ and $r(\theta)$ are apparent singularities created from a bad choice of coordinates, they are just as in the Schwarzschild (and Kerr) case coordinate singularities.