All Questions
Tagged with black-holes vector-fields
24
questions
9
votes
1
answer
819
views
Defining Surface gravity of a black hole
For a Killing horizon associated with a Killing vector $K$, the surface gravity $\kappa$ can be computed by various methods, like
$$
\kappa^2 = - \frac{1}{2} \nabla^\mu K^\nu \nabla_\mu K_\nu \ .
$$
...
- 669
8
votes
2
answers
792
views
Puzzle concerning the Divergence Theorem
Something is puzzling me concerning the divergence theorem. Usually, one writes the divergence theorem as
\begin{equation}
\int_\mathcal{M} d^4x \sqrt{-g} \nabla_\mu v^\mu=\int_{\partial \mathcal{M}} ...
- 521
5
votes
2
answers
4k
views
Surface gravity of a Killing horizon
I have two questions about this:
Surface gravity is defined on the Killing horizon by $\xi^\mu \nabla_\nu \xi^\nu = \kappa \xi^\nu$ for the Killing vector $\xi$. Why can we interpret this as the ...
- 759
4
votes
1
answer
1k
views
I'm confused about the number of Killing vectors in Schwarzschild metric
I'm trying to perform a calculation to derive the Killing vectors of a spherically symmetric metric (so I use the Schwarzschild metric without loss of generality because the Birkhoff theorem tells me ...
- 180
4
votes
3
answers
672
views
What is the proof that the Schwarzschild metric is not static inside the horizon?
In Lecture Notes on General Relativity, Sean M. Carroll shows that the Schwarzschild metric is not only stationary but also static (Chapter 7, page 169, Eq. 7.20 and following interpretation). On the ...
- 1,948
4
votes
2
answers
1k
views
A simple calculation about surface gravity in classical GR
I am reading An Introduction to General Relativity Spacetime and Geometry by Sean Carroll, but simple calculations stop me.
At page 245, a formula for the surface gravity is given $$\kappa^2=-\frac{1}...
- 41
4
votes
1
answer
680
views
Conformal Killing fields on Schwarzschild
I am trying to understand which are the conformal Killing Fields on the Schwarzschild spacetime.
I say that $X$ is a conformal Killing field on $S$ ($S$ is Schwarzschild) if there exists a function $f:...
- 61
3
votes
1
answer
279
views
Why is $\chi_{[\mu}\nabla_\nu \chi_{\sigma ]} = 0$ at the Killing horizon?
Let $\chi$ be a Killing vector field that is null along a Killing horizon $\Sigma$
Why is $\chi_{[\mu}\nabla_\nu \chi_{\sigma ]} = 0$ at $\Sigma$?
- 669
3
votes
1
answer
2k
views
What's the significance of a Killing horizon?
A Killing horizon is defined as a null hypersurface generated by a Killing vector, which is then null at that surface. Some often cited examples come from the Kerr spacetime, where the Killing vector $...
- 28.3k
2
votes
1
answer
318
views
Killing vectors in General Relativity?
I'm looking to derive the surface area of the event horizon of a Schwarzschild black hole. I was just wondering if it were possible for someone to explain to me this:
$$
\sqrt{g_{\theta\theta}g_{\phi\...
- 1,634
2
votes
1
answer
139
views
Energy conservation around a black hole
In the Schwarzschild black hole, the Killing vector "time translation" $k^a$, so that the following quantity is conserved along a geodesic:
$$E = -g_{ab}k^au^b = (1 - \frac{2GM}{r})\frac{dt}{d\tau}.$$...
- 415
2
votes
1
answer
646
views
Killing Horizon for Kerr Black Hole
I have some confusion about Killing Horizon of BH.
Since a Killing Horizon (KH) is a null hyper-surface at which killing vector $k^{\mu}$ is null; $k^{\mu}k_{\mu}=0.$ For time translation symmetry $...
- 1,125
2
votes
1
answer
824
views
Killing Vectors of BTZ black hole and their calculation in general
I was wondering what are the Killing vectors of BTZ black hole and how to guess them easily? Will it be the same as of AdS? What then will be Killing vectors for AdS-Schwarzschild e.g.?
- 2,099
1
vote
1
answer
121
views
Calculating divergence and flux of geodesic word lines
Given a family of neighbouring geodesic word lines, is there a way of calculating properties such as their divergence or flux? maybe by converting the tangent vectors of the world lines to a vector ...
- 633
1
vote
1
answer
337
views
Killing vector $\xi_\alpha$ at event horizon of Kerr black hole
I am calculating surface gravity of Kerr Black Hole following 'A Relativist's toolkit' which uses the definition
$$ \left(-\xi^\beta \xi_\beta\right);_\alpha=2\kappa \xi_\alpha$$
where $\kappa$ is ...
- 687