Finding Event Horizon & Ergosphere: Derivations & Formulas

[user1139]

Homework Statement:: See below.
Relevant Equations:: See below.

I am trying to calculate the event horizon and ergosphere of the Kerr metric. However, I could not seem to find a proper derivation or formula to calculate the event horizon and ergosphere. Could someone point me to the appropriate derivations or formulas?

 

[hutchphd]

Calling @ergospherical

 

[George Jones]

Homework Statement:: See below.
Relevant Equations:: See below.

I am trying to calculate the event horizon and ergosphere of the Kerr metric. However, I could not seem to find a proper derivation or formula to calculate the event horizon and ergosphere. Could someone point me to the appropriate derivations or formulas?

What texts have you consulted?

 

[ergospherical]

Hey, sorry for the late reply, I've been a bit pre-occupied today and I'm currently fairly drunk.

It's most tractable to use the Boyer-Lindquist form of the metric, for which $g_{rr}$ is seen to diverge whenever $1-2m/r + a^2/r^2 = 0$. The roots $r_{\pm}$ of this equation correspond to the inner and outer event horizons. One can show that the 3-metric $\gamma_{ab}$ induced on the 3-surfaces $\Sigma_{\pm}$ (defined by $r = r_{\pm}$ respectively, i.e. just put $dr=0$ and $r=r_{\pm}$ in the 4-metric) has vanishing determinant, thereby implying that for any $p \in \Sigma_{\pm}$ there exists a non-zero vector $v^a$ such that $\gamma_{ab} v^b = 0 \implies \gamma_{ab} v^a v^b = 0 \implies ||v|| = 0$, i.e. a tangent vector to null curves (photon orbits) lying entirely within $\Sigma_{\pm}$; it should then be clear that null trajectories starting at less than $r_{+}$ do not ever breach $\Sigma_+$, for example.

As for the ergospheres, they are defined as the region within which it's not possible to remain at fixed spatial $(r,\theta,\phi)$, for which a necessary condition is that the "still" trajectory (with tangent vector $u = \partial/\partial t$) is spacelike (and therefore unattainable), $g_{ab} u^a u^b = g_{tt} u^t u^t > 0 \implies g_{tt} > 0$, i.e.\begin{align*}
\frac{2mr}{r^2 + a^2 \cos^2{\theta}} - 1 > 0
\end{align*}the roots ($r_{\pm}^{\mathrm{e}}$, say) of which define the inner and outer surfaces of the ergosphere.

 

[user1139]

@ergospherical could you list a reference or two on what you have written? I would like to read more about it.

 

[ergospherical]

Any book on black hole solutions ought to do really; the last chapter of Poisson's book, "A Relativist's Toolkit", is good, for instance.

 

[user1139]

@ergospherical I see. However, I’m trying to understand specifically your argument. Hence, I’m hoping there’s a reference that details the arguments you laid out.

 

[PeroK]

@ergospherical I see. However, I’m trying to understand specifically your argument. Hence, I’m hoping there’s a reference that details the arguments you laid out.

https://arxiv.org/pdf/0706.0622.pdf

It took me less than a minute to find that!

 

[ergospherical]

@ergospherical I see. However, I’m trying to understand specifically your argument. Hence, I’m hoping there’s a reference that details the arguments you laid out.

Which bits are bothering you? (Then we can try and explain.)

 

[PeterDonis]

@ergospherical could you list a reference or two on what you have written? I would like to read more about it.

You have already been asked what texts you have already consulted. Pretty much any GR textbook will give the formulas you are asking for. So will many published papers, such as the one @PeroK has referenced (which is specifically intended for pedagogy). So if you are seriously saying you can't find the formulas you are looking for, you need to look harder. Once you have done that, if you have questions about something specific in whatever reference you find that you decide to use, you can start a new thread asking those questions and giving the specific reference. But expecting others here to hold your hand and lead you to something that can be found in any GR textbook, or in a minute or two of online search as @PeroK did, is not reasonable.

This thread is closed.