 #1
JD_PM
 1,128
 158
 TL;DR Summary

I would like to gain more insight and discuss rotating black holes.
I am particularly interested in understanding the isometries related to the metric (i.e. Killing vectors), the conserved quantities associated to them and in what cases we have event horizons.
I will summarize what I know so far about it and then make some questions to get started. Please feel free to correct me if any of my statements is wrong.
I study from Carroll's book and the beautiful Tong's lecture notes on GR.
Before explicitly stating the Kerr metric let us discuss a bit what to expect, comparing it to the easiest solution to (invacuum) Einstein's equations that I know: the Schwarzschild metric.
I studied that the Schwarzschild metric is derived under the following assumptions: the metric must be spherically symmetric and static (the latter word means that the metric has a Killing vector associated to the time coordinate ##K=\partial_t## and that the metric is invariant under time reversal, ##t \rightarrow t##).
For the Kerr metric I see we do not make such assumptions. It makes sense to me to not assume spherical symmetry, as a black hole (I think) cannot be always approximated to be a sphere. Besides, (I think) its shape can change over time so indeed it makes sense to not assume that the metric is timeindependent.
Let's now state the Kerr metric
\begin{align*}
(ds)^2 &= \left(1\frac{2GMr}{\rho^2} \right)(dt)^2  \frac{2GMar \sin^2 \theta}{\rho^2}(dt d\phi + d\phi dt) \\
&+ \frac{\rho^2}{\Delta}(dr)^2 + \rho^2 (d \theta)^2 + \frac{\sin^2 \theta}{\rho^2} \left[ (r^2+a^2)^2a^2 \Delta \sin^2 \theta \right] (d \phi)^2
\end{align*}
Where
\begin{equation*}
\Delta = r^2 2GMr+a^2
\end{equation*}
And
\begin{equation*}
\rho^2 = r^2+a^2 \cos^2 \theta, \ \ \ \ a=\mathcal{J}/M
\end{equation*}
Where ##\mathcal{J}## is the (Komar) angular momentum.
Let us now study the Killing vectors associated to such metric. We notice that no coefficient of the coordinates depends neither on time nor ##\phi## so we indeed have ##K=\partial_t## and ##K=\partial_{\phi}##. Incidentally we notice that indeed the metric is static, as ##t \rightarrow t## does not hold due to ##(dt d\phi + d\phi dt)##.
We have ##K=\partial_t## and ##K=\partial_{\phi}## as Killing vectors so we expect to have 2 conserved quantities associated to this metric: energy and angular momentum. Regarding the latter: my guess is that what is conserved is the magnitude of the angular momentum, am I right?
Let me make the Schwarzschild metric comparison again: that one has 4 Killing vectors: one associated to energy conservation and the other three to conservation of (3D) angular momentum, which indeed makes sense as the Schwarzschild metric is spherically symmetric. However, in Kerr's metric, I did not expect the presence of a Killing vector associated to angular momentum as we made no assumption regarding shape/motion.
To find at what values of ##r## we have an event horizon, we search for values which blow up our metric i.e. ##\Delta = 0##.
Thus we obtain two physical event horizons out of solving
\begin{equation*}
\Delta = r^2 2GMr+a^2=0
\end{equation*}
Those are
\begin{equation*}
r_{\pm} = GM \pm \sqrt{G^2 M^2  a^2}
\end{equation*}
My questions are
1) Is ##K=\partial_{\phi}## associated to angular momentum's magnitude?
2) Why are not ##\rho = 0 \Rightarrow r=\pm a \cos \theta## valid event horizon locations?
Please feel free to add extra insight, I really enjoy learning in PF!
Thank you!
I studied that the Schwarzschild metric is derived under the following assumptions: the metric must be spherically symmetric and static (the latter word means that the metric has a Killing vector associated to the time coordinate ##K=\partial_t## and that the metric is invariant under time reversal, ##t \rightarrow t##).
For the Kerr metric I see we do not make such assumptions. It makes sense to me to not assume spherical symmetry, as a black hole (I think) cannot be always approximated to be a sphere. Besides, (I think) its shape can change over time so indeed it makes sense to not assume that the metric is timeindependent.
Let's now state the Kerr metric
\begin{align*}
(ds)^2 &= \left(1\frac{2GMr}{\rho^2} \right)(dt)^2  \frac{2GMar \sin^2 \theta}{\rho^2}(dt d\phi + d\phi dt) \\
&+ \frac{\rho^2}{\Delta}(dr)^2 + \rho^2 (d \theta)^2 + \frac{\sin^2 \theta}{\rho^2} \left[ (r^2+a^2)^2a^2 \Delta \sin^2 \theta \right] (d \phi)^2
\end{align*}
Where
\begin{equation*}
\Delta = r^2 2GMr+a^2
\end{equation*}
And
\begin{equation*}
\rho^2 = r^2+a^2 \cos^2 \theta, \ \ \ \ a=\mathcal{J}/M
\end{equation*}
Where ##\mathcal{J}## is the (Komar) angular momentum.
Let us now study the Killing vectors associated to such metric. We notice that no coefficient of the coordinates depends neither on time nor ##\phi## so we indeed have ##K=\partial_t## and ##K=\partial_{\phi}##. Incidentally we notice that indeed the metric is static, as ##t \rightarrow t## does not hold due to ##(dt d\phi + d\phi dt)##.
We have ##K=\partial_t## and ##K=\partial_{\phi}## as Killing vectors so we expect to have 2 conserved quantities associated to this metric: energy and angular momentum. Regarding the latter: my guess is that what is conserved is the magnitude of the angular momentum, am I right?
Let me make the Schwarzschild metric comparison again: that one has 4 Killing vectors: one associated to energy conservation and the other three to conservation of (3D) angular momentum, which indeed makes sense as the Schwarzschild metric is spherically symmetric. However, in Kerr's metric, I did not expect the presence of a Killing vector associated to angular momentum as we made no assumption regarding shape/motion.
To find at what values of ##r## we have an event horizon, we search for values which blow up our metric i.e. ##\Delta = 0##.
Thus we obtain two physical event horizons out of solving
\begin{equation*}
\Delta = r^2 2GMr+a^2=0
\end{equation*}
Those are
\begin{equation*}
r_{\pm} = GM \pm \sqrt{G^2 M^2  a^2}
\end{equation*}
My questions are
1) Is ##K=\partial_{\phi}## associated to angular momentum's magnitude?
2) Why are not ##\rho = 0 \Rightarrow r=\pm a \cos \theta## valid event horizon locations?
Please feel free to add extra insight, I really enjoy learning in PF!
Thank you!