40
votes
Accepted
Do black holes have a moment of inertia?
The angular velocity of a Kerr black hole with mass $M$ and angular momentum $J$ is
$$ \Omega = \frac{J/M}{2M^2 + 2M \sqrt{M^2 - J^2/M^2}} $$
The moment of inertia of an object can be thought of as ...
- 6,335
35
votes
Accepted
Wouldn't Miller's planet be fried by blueshifted radiation?
Miller's world would be fried by a strong flux of extreme ultraviolet (EUV) radiation. The cosmic microwave background (CMB) would be blueshifted by gravitational time dilation and then would be very ...
- 128k
32
votes
Accepted
What do black holes spin relative to?
But if you start running around it, it will move faster or slower relative to you.
In this case, the disc has a ground speed, 60 rpm, because it has something to spin in relation to, in this case, the ...
- 94.4k
30
votes
Accepted
Massless Kerr black hole
It's simply flat space in Boyer-Lindquist coordinates. By writing
$\begin{cases}
x=\sqrt{r^2+a^2}\sin\theta\cos\phi\\
y=\sqrt{r^2+a^2}\sin\theta\sin\phi\\
z=r\cos\theta
\end{cases}$
you'll get good ol'...
- 5,728
15
votes
Massless Kerr black hole
This is presumably a flat spacetime described in funny coordinates. You can check this by calculating the Riemann tensor to see if it's zero. If I was going to do this, I would code it in the open-...
- 151
11
votes
Does the twist of spacetime by a black hole increase over time?
The spacetime around a rotating black hole (Kerr metric) is stationary, this means that you can choose a coordinate system where the metric doesn't depend on the time coordinate. In layman's terms ...
- 2,839
10
votes
Accepted
How does the Penrose diagram for a spinning black hole differ in realistic scenarios (formed by stellar collapse)?
Your question basically boils down to a recognition of the following fact:
The Schwarzschild metric, with spacelike $r=0$, admits an "eternal" BH to form by stellar collapse, like the one you've ...
- 1,361
9
votes
Accepted
Does the twist of spacetime by a black hole increase over time?
Yes, you are taking the metaphor too literally. When we visualize spacetime like a flexible rubber sheet there is an accidental and erroneous impression that a twist will lead to spacetime getting ...
- 30.9k
9
votes
Accepted
Are rotating black holes producing a frame dragging effect inside the event horizon?
Krešimir Bradvica asked: "Are rotating black holes producing a frame dragging effect inside the event horizon? Is that effect moving space inside the event horizon at speeds far greater than the ...
- 10.9k
8
votes
Do black holes have a moment of inertia?
Moments of inertia are defined about a given axis of rotation.
Moment of inertia is the name given to rotational inertia, the rotational analog of mass for linear motion. It appears in the ...
- 232k
8
votes
Killing tensor in the Kerr metric
The Killing tensor is defined as a symmetric tensor $K_{\alpha \beta}$ whose total symmetrization of the covariant gradient vanishes
$$K_{(\alpha \beta;\gamma)} = K_{\alpha \beta;\gamma} + K_{ \beta\...
- 19.8k
7
votes
Accepted
Kerr spacetime and Carter time machine
Suppose a spacetime has a global time function: a globally defined difference between past and future. Then we can divide timelike curves into two classes. When the proper time along a "future-...
- 4,191
7
votes
Accepted
What is exactly rotating in a rotating black hole?
So let us first ask ourselves: What makes us believe that anything is rotating in the Kerr space-time?
The answer is that we go very far away from the black hole and look at the asymptotics of the ...
- 19.8k
7
votes
What do black holes spin relative to?
This is just Newton's bucket in modern garb. The best explanation of this effect that I have seen is in Carlo Rovelli's book Quantum Gravity, which explains it as rotation with respect to the ...
- 411
7
votes
Accepted
Event horizon of a rotating black hole
The event horizon' radius of black holes is an infinite-redshift surface (a one-way surface where particles can never
escape to infinity). It can be computed analytically (or at least numerically) by ...
- 4,560
7
votes
Wouldn't Miller's planet be fried by blueshifted radiation?
For a 2.7 K blackbody (like the CMB) the calculator at
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/radfrac.html
gives me 3 $\mu$W/m$^2$. This is how much Earth is heated by the CMB — not a lot!
...
- 943
6
votes
Accepted
Torsion in kerr black holes
Torsion is not frame dragging. Torsion is having an anti-symmetric spacetime connection. As you do parallel transport in general relativity (GR) you drag frames the frames roll as they move. With ...
- 13.9k
6
votes
Area of the event horizon of a rotating black hole
If you write the element surface as:
$d\sigma$ = $dl_{\theta}$ $dl_{\phi}$
you should have:
$dl_{\theta}$ = $\sqrt{g_{\theta\theta}}$ $d\theta$
$dl_{\phi}$ = $\sqrt{g_{\phi\phi}}$ $d\phi$
Hence the ...
- 3,731
6
votes
Accepted
How can a black hole rotate if time dilation stops time at the event horizon?
The statement "time is dilated to infinity at the horizon" is a (very imprecise) way of saying that the event horizon is a null/lightlike surface. However, as is clear from light-rays, being ...
- 10.3k
6
votes
Accepted
What's the inner ergosphere in a Kerr black hole?
If you are learning about the Kerr geometry I strongly recommend Matt Visser's paper The Kerr spacetime: A brief introduction as he manages to pack in all the essential information and still keep it ...
- 351k
6
votes
Accepted
Exceeding the Kerr black hole spin limit
This is an interesting question that could really be formulated as follows:
When collapsing an object that has the ratio $a^* = cJ/(GM^2)$ larger than one, can an over-extremal black hole form?
...
- 19.8k
6
votes
Accepted
Black holes in general relativity: angular velocity of the horizon
As a warm-up, consider the non-rotating case $\Omega_H=0$. From the perspective of a distant observer, any pointlike test object that falls into a non-rotating black hole will appear to freeze at the ...
- 53.6k
5
votes
Accepted
Particle falling into a Kerr black hole
No the angular momentum of the particle would remain zero (it is still a constant of motion). The angular velocity of the particle, on the other hand, would steadily increase as the particle ...
- 10.3k
5
votes
Accepted
Anatomy of a black hole: which are its layers?
I'm not sure calling the things on your list "layers" is the most accurate way of putting it. It'd rather say features. Your list is mostly complete. You could also add the innermost bound ...
- 10.3k
5
votes
What do black holes spin relative to?
What do black holes spin relative to?
Relative to an inertial reference frame infinitely far from the hole, in which the hole has no translational motion.
And what happens if you move around it?
A ...
- 51.3k
5
votes
Accepted
Wouldn't Millers planet be ripped to shreds by extreme tidal forces?
According to Fishbone, 1973, ApJ, 185, 43 the minimum density defining the Roche limit for something in the innermost stable, equatorial, circular orbit around a Kerr black hole is the same as that ...
- 128k
4
votes
Time independent Kerr metric
The metric is telling you how to calculate the proper time along a path of your choosing. If you select a path where the time is everywhere constant then as you integrate along that path $dt = 0$ and ...
- 351k
4
votes
What happens to a particle in the exact center of a Kerr black hole?
A maximally extended Kerr solution has multiple horizon. The outer horizon is an event horizon and the inner horizon is a Cauchy horizon.
Outside both is a normal type region of spacetime. In between ...
- 25.4k
4
votes
Closed timelike curves in the Kerr metric
Let me write down the metric in the equatorial plane ($\vartheta = \pi/2$) of the Kerr space-time in Boyer-Lindquist coordinates:
$$
d s^2 = -\left(1 - \frac{2M r}{r^2 + a^2}\right) d t^2 + \frac{r^2+...
- 19.8k
4
votes
Accepted
The physical meaning of the cross-term of Kerr metric
Obviously, you can transform away individual terms of any tensor, but here's the idea:
The Kerr metric is asymptotically flat. An observer stationary with respect to the asymptotically flat boundary ...
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