Kerr coordinates

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Kerr coordinates describe a rotating black hole. They were published by Roy Kerr in 1963, and represent the first solution of this spacetime. (See also Kerr Cartesian coordinates, given in the same paper). They are a generalisation of Eddington-Finkelstein coordinates for a Schwarzschild-Droste (non-spinning) black hole.

ds^2=-\Big(1-\frac{2Mr}{\rho^2}\Big)(dv-a\sin^2\theta\,d\phi)^2+2(dv-a\sin^2\theta\,d\phi)(dr-a\sin^2\theta\,d\phi)+\rho^2(d\theta^2+\sin^2\theta\,d\phi^2)
(Teukolsky, Kerr tweaked) where \rho^2\equiv r^2+a^2\cos^2\theta.

I have compared numerous sources to ensure this is the standard version, to avoid the confusion and frustration of a minus sign difference between authors, for example.

(These are equivalent to Kerr’s original coordinates but with two minor tweaks of notation. Kerr originally used the opposite sign of a, but this was due to a calculation error, and was corrected by Boyer who compared the angular momentum with the Lense-Thirring results, and this became standard in the subsequent literature (Kerr §2.5 in Wiltshire et al. B-L 1967 has similar sounding, is this what Kerr is quoting?). Also we use v because this is the convention for advanced time; Kerr originally used u but this commonly represents retarded time (Teukolsky 2015, footnote 1). Visser §1.1 in Wiltshire et al uses Kerr’s original a, and this confused me for a long time because of inconsistency with other sources…) Advanced Eddington-Finkelstein form.

B-L give: ds^2=-dt^2+dr^2+2a\sin^2\theta\,dr\,d\phi+\rho^2\,d\theta^2+(r^2+a^2)\sin^2\theta\,d\phi^2+\frac{2Mr}{\rho^2}(dt+dr+a\sin^2\theta\,d\phi)^2  (§2) They call it “(E) frame” since it generalises Eddington-Finkelstein coordinates. Inverse metric in eqn 2.15. Null congruence k^\mu=(-1,0,0,1) in these coords.

B-L “E’ frame” adapted to l vector. ds^2=-dt^2+dr^2-2a\sin^2\theta\,dr\,d\phi+\rho^2\,d\theta^2+(r^2+a^2)\sin^2\theta\,d\phi^2+\frac{2Mr}{\rho^2}(-dt+dr-a\sin^2\theta\,d\phi)^2. d\phi_S=d\phi-\frac{a}{\Delta}dr, dt_S=dt+\frac{2Mr}{\Delta}dr

ds^2=ds_0^2+\frac{2Mr}{\rho^2}k^2
where
ds_0^2=\rho^2(d\theta^2+\sin^2\theta\,d\phi^2)-(2dr+du-a\sin^2\theta\,d\phi)(du+a\sin^2\theta\,d\phi)
and k=du+a\sin^2\theta\,d\phi
u replaced by -u for current convention, and a replaced with -a
(Kerr \S2.5 in Wiltshire et al)

Chandrasekhar electronic p328 – compare metric form!

Ingoing principal null vector -\partial_r
(Teukolsky 2015, footnote 1)

Floyd p.9

Carter [3] integrals of motion
[probably that was a citation in Floyd. And indeed, 3 is: B. Carter; The Physical Review, Vol 174 No.5, p 1559, 1968.]

Kerr \S2.6 in Wiltshire et al: clarifies what he meant in 1963 paper by desiring “interior solution”

Geodesics

Timelike geodesics. See Boyer-Lindquist coordinates, but t and \phi_{BL} are replaced by v and \phi_{Kerr}:

\rho^2\diff{v}{\lambda}=-a(aE\sin^2\theta-L_z)+(r^2+a^2)(\sqrt{R}+P)/\Delta

\rho^2\diff{\phi}{\lambda}=-(aE-L_z/sin^2\theta)+a/\Delta(\sqrt{R}+P)

(MTW §33.5)

Killing tensor

Jezierski & Łukasik (2006, §4) gives the Killing tensor K corresponding to Carter’s constant in raised index form as

K=\begin{pmatrix} -\frac{a^2r^2\sin^2\theta}{\rho^2} & \frac{a^2(r^2+a^2)\cos^2\theta}{\rho^2} & 0 & -\frac{ar^2}{\rho^2} \\ \frac{a^2(r^2+a^2)\cos^2\theta}{\rho^2} & \frac{a^2\Delta\cos^2\theta}{\rho^2} & 0 & \frac{a^3\cos^2\theta}{\rho^2} \\ 0 & 0 & -\frac{r^2}{\rho^2} & 0 \\ -\frac{ar^2}{\rho^2} & \frac{a^3\cos^2\theta}{\rho^2} & 0 & -\frac{r^2}{\rho^2\sin^2\theta} \end{pmatrix}

(check 1/2 coefficients of cross terms…)

They also give its “square root”, a Killing-Yano tensor:

f=r\sin\theta\,d\theta\wedge[(r^2+a^2)d\phi-a\,dv]+a\cos\theta\,dr\wedge(dv-a\sin^2\theta\,d\phi)

which has components

f_{\mu\nu}=\begin{pmatrix} 0 & -a\cos\theta & ar\sin\theta & 0 \\ a\cos\theta & 0 & 0 & -a^2\cos\theta\sin^2\theta \\ -ar\sin\theta & 0 & 0 & r(r^2+a^2)\sin\theta \\ 0 & a^2\cos\theta\sin^2\theta & r(r^2+a^2)\sin\theta & 0 \end{pmatrix}

(check: do I need to halve everything, or double it? Same for corresponding expression in Boyer-Lindquist coordinates)

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