# Kerr black hole in the formalism of isolated horizons

David Kofroň<EMAIL_ADDRESS>Institute of Theoretical Physics, Faculty of
Mathematics and Physics,
Charles University,
V Holešovičkách 2, 180 00 Prague 8, Czech Republic

###### Abstract

We revise the work of Scholtz, Flandera and Gürlebeck [Kerr-Newman black hole
in the formalism of isolated horizons, Phys. Rev. D 96, 064024 (2017)]. We
cast the Kerr metric explicitly in the form suitable for the framework of
isolated horizons. We proceed in a geometrical fashion and are capable to
provide the results in a compact closed manner, without any unevaluated
integrals. We also discuss the uniqueness and drawbacks of this construction.
We suggest a new vector field to generate the null geodesic foliation.

exact solutions, Kerr black hole, isolated horizons, general relativity

##  I Introduction

The concept of weakly isolated horizons (WIH) has been developed at the
beginning of this millennia by Ashtekar et al. [1, 2, 3, 4] in order to tackle
the notion of a black hole in equilibrium state possibly surrounded by matter
or radiation. Later on this concept was upgraded to notion of dynamical
horizons [5] where even the general dynamical processes are taken into the
account in full general relativity.

This formalism was used in [6] to evolve the solution from the initial data in
a near-horizon expansion using the Newman – Penrose (NP) formalism [7, 8].

The rotating Kerr black hole — one of the most astrophysically relevant
solution of the vacuum Einstein field equations — was discovered in 1963 [9].
Recent historical reviews can be found in [10, 11]. There exists a plethora of
interesting coordinate systems for the Kerr black hole and it is worth to
investigate them on their own. Let us mention some of the most important

  * •

Kerr – Schild form [9, 12] in which it has been found,

  * •

Boyer – Lindquist (BL) coordinates [13] which are one of the most suitable for
interpretation and, simultaneously, the equations governing test fields are
proven to be separable in these coordinates [14],

  * •

Eddington – Finkelstein coordinates [9] which are regular on the horizon,

  * •

Kruskal coordinates which reveals the full structure of the spacetime using
the maximal analytical extension of the coordinates [13],

  * •

Doran coordinates [15] are adapted to family of time-like observers with zero
angular momentum which are freely falling to the black hole,

  * •

Bondi-like coordinates which are propagated from the null infinity [16].

At the same time the Kerr black hole itself is an excellent example of space-
time containing a WIH. An attempt to cast it in this framework has been done
in [17] in the recent years.

In this paper we will revise the work [17] which is difficult to follow and
contains some implicit, unevaluated integrals; and, rather surprisingly, not a
single picture of new coordinates is presented. Using a slightly different
approach, we are able to evaluate all the integrals and derive simple results.
In [17] the authors prescribe what the final vector field generating the null
congruence should be and then they applied 4 successive transformations
(boost, null rotation, spin, null rotation) on the Kinnersley tetrad to align
the tetrad with this vector field. At each step they evaluated the NP spin
coefficients and solve lot of equations in order to make some of he NP spin
coefficients vanishing. Instead of this we simply construct a tetrad based
directly on the final vector field, using the Killing – Yano tensor, and make
just one null rotation so that the tetrad is parallely propagated. We also
discuss the uniqueness of this construction and its drawbacks.

In the end, we reveal so many disadvantages of the null congruence chosen in
[16, 17] that we suggest a completely new congruence which is well behaved.
Alas, this one is much more difficult to treat analytically.

The paper is organized as follows: we briefly introduce the concept of
isolated horizons (IH) in Sec. II and recapitulate the the Kerr solution in
Sec. III. In the Sec. IV we develop the parallely propagated null tetrad with
the help of Killing – Yano tensors. The explicit formulae for the elliptic
integral introduced in this section are postponed to Appendix B. The null
coordinates adapted to this tetrad are then introduced in Sec. V and their
affine parameterization in the following Sec. VI. The final form of the tetrad
in these coordinates is contained in Sec. VII. In Sec. VIII we discuss the
(non)uniqueness of this construction and propose an alternative solution. In
Sec. IX we shortly compare these coordinates with the Bondi coordinates.

##  II Isolated horizons

We follow the construction presented in [6, 17]111We had to adjust a sign on
either $\boldsymbol{l}$ or $\boldsymbol{n}$ since we are using different
signature of the metric., details can be found in [4].

On the manifold $M$ the _non-expanding horizon_ $\mathcal{H}\subset M$ is
defined as a null hypersurface with following properties

  * •

the topology of $\mathcal{H}$ is $R\times S^{2}$

  * •

the expansion of any null normal $\boldsymbol{l}^{a}$ is vanishing

  * •

equations of motion are satisfied on $\mathcal{H}$ and the projection of
stress energy tensor $\boldsymbol{T}_{ab}$ on any future pointing null normal
$\boldsymbol{l}$ is causal (a consequence of dominant energy condition)

It is not possible to define a unique induced covariant derivative on a
general null hypersurface, since the induced metric is degenerate. However,
the existence of additional conditions (namely the vanishing expansion of null
normals) on the non-expanding horizon allows us to define a preferred
connection $\mathcal{D}_{a}$ on the tangent bundle $T\mathcal{H}$ as

$\displaystyle\boldsymbol{X}^{a}\mathcal{D}_{a}\boldsymbol{Y}^{b}$
$\displaystyle\doteq\boldsymbol{X}^{a}\nabla_{a}\boldsymbol{Y}^{b}\,,\quad\text{for
any }\boldsymbol{X}^{a},\,\boldsymbol{Y}^{a}\,\in T\mathcal{H}\,,$ (1)

where $\nabla_{a}$ is connection on $TM$ compatible with the metric
$\boldsymbol{g}_{ab}$ and the relation operator “$\doteq$” means that the
equality holds on $\mathcal{H}$.

This covariant derivative implies the existence of a _rotational 1-form_
$\boldsymbol{\omega}$ which is defined by

$\displaystyle\mathcal{D}_{a}\boldsymbol{l}^{b}$
$\displaystyle\doteq\boldsymbol{\omega}_{a}\boldsymbol{l}^{b}\,.$ (2)

The null normal $\boldsymbol{l}$ can be rescaled by an arbitrary non-vanishing
function. In order to get rid of this ambiguity we define the _weakly isolated
horizon_ as a non-expanding horizon $\mathcal{H}$ together with the
equivalence class of null normals
$[\boldsymbol{l}]=\\{c\boldsymbol{l},\,c\in\mathbb{R},\,c\neq 0\\}$ (the
remaining freedom is scaling by constant) and restrict to the class for which
a representative element $\boldsymbol{l}\in[\boldsymbol{l}]$ satisfies

$\displaystyle\left[\mathcal{L}_{\boldsymbol{l}},\mathcal{D}_{a}\right]\boldsymbol{l}^{b}$
$\displaystyle\doteq 0\,,$ (3)

where $\mathcal{L}$ is the Lie derivative. This condition is equivalent to
$\mathcal{L}_{\boldsymbol{l}}\boldsymbol{\omega}_{a}\doteq 0$.

The weakly isolated horizons are physically identical to non-expanding
horizons. The selection of the proper equivalence class of null normals
$[\boldsymbol{l}^{a}]$, given by Eq. (3) allowed to formulate zeroth law of
thermodynamics (which is equivalent to Eq. (3)) for isolated horizons and give
a physical meaning for the rotational 1-form $\boldsymbol{\omega}$, see [3,
6].

Choosing a particular null generator $\boldsymbol{l}$ of a WIH
$(\mathcal{H},[\boldsymbol{l}])$ we can (non-uniquely) complete the full NP
tetrad
$(\boldsymbol{l},\,\boldsymbol{n},\,\boldsymbol{m},\,\boldsymbol{\bar{m}})$ so
that $\boldsymbol{m},\boldsymbol{\bar{m}}\in T\mathcal{H}$ and
$\boldsymbol{n}$ is transversal to $\mathcal{H}$.

Having the NP tetrad we can demonstrate the significance of rotational 1-form.
It has been proven in [4] that

$\displaystyle\mathrm{d}\boldsymbol{\omega}\doteq\mathop{\rm
Im}\left(\Psi_{2}\right)\boldsymbol{\varepsilon}\,,$ (4)

where $\boldsymbol{\varepsilon}$ is a volume element of cross sections of
$\mathcal{H}$ and $\Psi_{2}$ is the standard projection of Weyl tensor.

Our primary goal is to set up a coordinate system adapted to the isolated
horizon and NP formalism (an analog of Bondi system).

The null generator $\boldsymbol{l}$ gives rise to a preferred foliation
$\mathcal{S}_{v}$ of $\mathcal{H}$ by topological spheres by imposing
$Dv\equiv\boldsymbol{l}^{a}\nabla_{a}v\doteq 1$. On the spherical cut
$\mathcal{S}_{0}$ of $\mathcal{H}$ angular coordinates $\zeta^{2},\zeta^{3}$
are introduced and they are propagated on $\mathcal{H}$ along the null
generators, i.e. we have $D\zeta^{i}\doteq 0$ for $i=(2,3)$. The vectors
$\boldsymbol{m},\,\boldsymbol{\bar{m}}$ are Lie dragged along the horizon as
$\mathcal{L}_{\boldsymbol{l}}\boldsymbol{m}^{a}\doteq\mathcal{L}_{\boldsymbol{l}}\boldsymbol{\bar{m}}^{a}\doteq
0$.

Finally, the null tetrad defined on the horizon is parallely propagated
outward in the transversal direction defined by $\boldsymbol{n}$. This null
geodetic congruence is affinely parameterized by $s$. Starting from a point on
the horizon $(v,\zeta^{2},\zeta^{3})$ every point at least in the vicinity of
$\mathcal{H}$ should be assigned coordinates $(v,s,\zeta^{2},\zeta^{3})$ by
this procedure (for the eternal Kerr black hole we want to construct a global
coordinate system; for more complicated solutions the construction of the
coordinates is not guaranteed globally).

Thus we look for a coordinate system $(v,s,\Theta,\tilde{\Phi})$222We coined a
particular names to $\zeta^{i}$, namely $\zeta^{2}=\Theta$ and
$\zeta^{3}=\tilde{\Phi}$. and a null tetrad such that the metric

$\displaystyle\boldsymbol{g}_{ab}=2\boldsymbol{l}_{(a}\boldsymbol{n}_{b)}-2\boldsymbol{m}_{(a}\boldsymbol{\bar{m}}_{b)}\,,$
(5)

is reconstructed from a NP tetrad in the form

$\displaystyle\boldsymbol{l}$
$\displaystyle=\boldsymbol{\partial_{v}}+U\boldsymbol{\partial_{s}}+X^{\Theta}\boldsymbol{\partial_{\Theta}}+X^{\tilde{\Phi}}\boldsymbol{\partial_{\tilde{\Phi}}}\,,$
(6) $\displaystyle\boldsymbol{n}$ $\displaystyle=\boldsymbol{\partial_{s}}\,,$
(7) $\displaystyle\boldsymbol{m}$
$\displaystyle=\Omega\boldsymbol{\partial_{s}}+\xi^{\Theta}\boldsymbol{\partial_{\Theta}}+\xi^{\tilde{\Phi}}\boldsymbol{\partial_{\tilde{\Phi}}}\,,$
(8)

satisfying following conditions on the horizon: (a) vector $\boldsymbol{l}$ is
generator of the horizon, (b) vectors $\boldsymbol{m},\boldsymbol{\bar{m}}$
are tangent to the horizon, (c) the vector field $\boldsymbol{n}$ is
transversal to the horizon, geodetic, affinely parameterized and twist-free.
The whole tetrad is parallely propagated along $\boldsymbol{n}$.

In NP formalism these conditions translates to the following conditions on the
NP scalars

$\gamma=\tau=\nu=\mu-\bar{\mu}=\pi-\alpha-\bar{\beta}=0\,,$ (9)

which hold everywhere and

$\varrho\doteq\kappa\doteq\sigma\doteq\epsilon-\bar{\epsilon}\doteq 0\,,$ (10)

and

$U\doteq X^{\Theta}\doteq X^{\tilde{\Phi}}\doteq\Omega\doteq 0\,,$ (11)

which hold on the horizon $\mathcal{H}$ only.

##  III Kerr black hole

The standard form of the Kerr metric in BL coordinates [13] reads

$\boldsymbol{\mathrm{d}}s^{2}=-\frac{\Delta}{\Sigma}\left(\boldsymbol{\mathrm{d}}t-a\sin^{2}\theta\,\boldsymbol{\mathrm{d}}\varphi\right)^{2}+\frac{\Sigma}{\Delta}\,\boldsymbol{\mathrm{d}}r^{2}\\\
+\Sigma\,\boldsymbol{\mathrm{d}}\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}\Bigl{(}\left(a^{2}+r^{2}\right)\boldsymbol{\mathrm{d}}\varphi-a\,\boldsymbol{\mathrm{d}}t\Bigr{)}^{2}\,,$
(12)

with the definitions

$\displaystyle\Delta(r)$ $\displaystyle=r^{2}-2Mr+a^{2}=(r-r_{p})(r-r_{m})\,,$
(13) $\displaystyle\rho(r,\theta)$ $\displaystyle=r-ia\cos\theta\,,$ (14)
$\displaystyle\Sigma(r,\theta)$
$\displaystyle=\rho\bar{\rho}=r^{2}+a^{2}\cos^{2}\theta\,.$ (15)

The parameters have the following meaning: $M$ is the mass of the black hole,
$Ma$ is its angular momentum, $r_{p}$ is the position of outer black hole
horizon whereas $r_{m}$ is the position of inner black hole horizon. We will
use all of these parameters ($r_{p},r_{m},M,a$) although only two of them are
independent.

The Kerr metric is endowed by two Killing vector fields, namely, the one
associated with stationarity $\boldsymbol{\xi}=\boldsymbol{\partial_{t}}$ and
the one associated with axial symmetry
$\boldsymbol{\eta}=\boldsymbol{\partial_{\varphi}}$. It is advantageous to
introduce also a quantity

$\Upsilon(r,\theta)=-\frac{\Sigma}{\sin^{2}\theta}\,\boldsymbol{\eta}\cdot\boldsymbol{\eta}=\Delta\Sigma+2Mr\left(r^{2}+a^{2}\right).$
(16)

The Kinnersley NP333Notice the boost given by $\sqrt{2}$ in contrast to the
standard textbook form. The Kinnersley tetrad is denoted by tilde in this
work. tetrad
$(\boldsymbol{\tilde{l}},\,\boldsymbol{\tilde{m}},\,\boldsymbol{\tilde{\bar{m}}},\,\boldsymbol{\tilde{n}})$,
introduced in [18], which is adapted to the principal null directions of the
Weyl tensor reads as follows

$\displaystyle\boldsymbol{\tilde{l}}$
$\displaystyle=\frac{1}{\sqrt{2}\,\Delta}\left[\left(r^{2}+a^{2}\right)\boldsymbol{\partial_{t}}+\Delta\,\boldsymbol{\partial_{r}}+a\,\boldsymbol{\partial_{\varphi}}\right]\,,$
(17) $\displaystyle\boldsymbol{\tilde{n}}$
$\displaystyle=\frac{1}{\sqrt{2}\,\Sigma}\left[\left(r^{2}+a^{2}\right)\boldsymbol{\partial_{t}}-\Delta\,\boldsymbol{\partial_{r}}+a\,\boldsymbol{\partial_{\varphi}}\right]\,,$
$\displaystyle\boldsymbol{\tilde{m}}$
$\displaystyle=\frac{1}{\sqrt{2}\,\bar{\rho}}\bigl{(}ia\sin\theta\,\boldsymbol{\partial_{t}}+\boldsymbol{\partial_{\theta}}+i\csc\theta\,\boldsymbol{\partial_{\varphi}}\bigr{)}\,.$

The appropriate spin coefficients are enlisted in Appendix A.

The Kerr metric posses not only two Killing vector fields but also a principal
Killing – Yano tensor [19, 20] which provides additional hidden symmetry. The
principal Killing – Yano tensor expanded in the Kinnersley tetrad is given by

$\displaystyle\boldsymbol{f}_{ab}$
$\displaystyle=-2ir\,\boldsymbol{\tilde{m}}_{[a}\boldsymbol{\bar{\tilde{m}}}_{b]}+2a\cos\theta\,\boldsymbol{\tilde{l}}_{[a}\boldsymbol{\tilde{n}}_{b]}\,,$
(18)

and its dual is also a Killing – Yano tensor, namely

$\displaystyle\boldsymbol{h}_{ab}$
$\displaystyle=2r\,\boldsymbol{\tilde{l}}_{[a}\boldsymbol{\tilde{n}}_{b]}+2ia\cos\theta\,\boldsymbol{\tilde{m}}_{[a}\boldsymbol{\bar{\tilde{m}}}_{b]}\,.$
(19)

From now on, we use the following shortcuts, to make the formulae visually
more compact

$\displaystyle\Sigma_{0}$ $\displaystyle=\Sigma(r,0)=r^{2}+a^{2}\,,$
$\displaystyle\Upsilon_{\\!\frac{\pi}{2}}$
$\displaystyle=\Upsilon(r,\pi/2)\,.$ (20)

Using the four independent constants of motion we can write a general geodesic
congruence as follows

$\displaystyle\boldsymbol{u}$
$\displaystyle=\frac{1}{\Sigma}\Biggl{[}\left(\frac{\Sigma_{0}\left(\Sigma_{0}E-aL\right)}{\Delta}+a\left(L-a\sin^{2}\theta\,E\right)\right)\boldsymbol{\partial_{t}}$
$\displaystyle\phantom{=}+\epsilon_{r}\sqrt{\left(\Sigma_{0}E-aL\right)^{2}-\left(K-\kappa
r^{2}\right)\Delta}\,\boldsymbol{\partial_{r}}$
$\displaystyle\phantom{=}+\epsilon_{\theta}\sqrt{K+\kappa
a^{2}\cos^{2}\theta-\csc^{2}\theta\left(L-a\sin^{2}\theta
E\right)}\,\boldsymbol{\partial_{\theta}}$
$\displaystyle\phantom{=}+\left(\csc^{2}\theta\left(L-a\sin^{2}\theta
E\right)^{2}+\frac{a\left(\Sigma_{0}E-aL\right)}{\Delta}\right)\boldsymbol{\partial_{\varphi}}\Biggr{]}.$
(21)

The parameters $E$ (energy), $L$ (angular momentum) and $K$ (Carter’s
constant) are constants along a single geodesic. They can be functions of
$(r,\theta)$, however, they have to obey

$\displaystyle\boldsymbol{u}^{a}\nabla_{a}E$ $\displaystyle=0\,,$
$\displaystyle\boldsymbol{u}^{a}\nabla_{a}L$ $\displaystyle=0\,,$
$\displaystyle\boldsymbol{u}^{a}\nabla_{a}K$ $\displaystyle=0\,,$ (22)

for $\boldsymbol{u}$ being the geodesic congruence.

Let us note that Doran coordinates are adapted to the geodetic congruence
$\boldsymbol{u}$ with the constants chosen as
$\kappa=-1,\,L=0,\,E=-1,\epsilon_{r}=1$ and are independent on the value of
$\epsilon_{\theta}$ (since the $\theta$ component vanishes).

##  IV The parallely propagated NP tetrad

In order to obtain a suitable null foliation of the Kerr spacetime we look for
a twist-free null geodesic congruence, which penetrates horizon and can reach
the null infinity. To cover the regions close to the axis, the angular
momentum has to be zero (as can be seen from the $\theta$ component of
$\boldsymbol{u}$). One of the possibilities444The choice $\epsilon_{\theta}=1$
was discussed in [17] and we will cover it in Appendix C. is to demand

$\displaystyle E$ $\displaystyle=-1\,,$ $\displaystyle L$ $\displaystyle=0\,,$
$\displaystyle\kappa$ $\displaystyle=0\,,$ $\displaystyle K$
$\displaystyle=a^{2}\,,$ (23) $\displaystyle\epsilon_{r}$ $\displaystyle=1\,,$
$\displaystyle\epsilon_{\theta}$ $\displaystyle=-1\,,$ (24)

which also ensures the mirror symmetry with respect to the equator.

The vector $\boldsymbol{u}$, given in Eq. (21), with constants of motion given
by Eqs. (23 – 24) then becomes the first vector of our NP tetrad

$\displaystyle\boldsymbol{n}$
$\displaystyle=\frac{-1}{\Sigma}\Bigg{(}\frac{\Upsilon}{\Delta}\,\boldsymbol{\partial_{t}}-\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}\,\boldsymbol{\partial_{r}}+a\cos\theta\,\boldsymbol{\partial_{\theta}}+\frac{2aMr}{\Delta}\,\boldsymbol{\partial_{\varphi}}\Biggr{)},$
(25)

since

$\displaystyle\boldsymbol{n}^{a}\nabla_{a}\boldsymbol{n}^{b}$
$\displaystyle=0\,.$ (26)

Such a geodetic congruence is generated by null rays passing through the
horizon and reaching the null infinity. We may introduce angular coordinates
$\Theta$ and $\Phi$ which will be constant along these null rays, namely

$\displaystyle\Theta$
$\displaystyle=2\arctan\left[\tanh\Gamma^{+}(r,\theta)\right]\,,$ (27)
$\displaystyle\Phi$
$\displaystyle=\varphi+2aM\int_{r_{p}}^{r}\frac{u}{\Delta(u)\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}\,\mathrm{d}u\,,$
(28)

where

$\displaystyle\Gamma^{+}(r,\theta)$ $\displaystyle=\mathop{\rm
arctanh}\left[\tan\left(\frac{\theta}{2}\right)+\frac{a}{2}\mathcal{I}_{0}(r)\right]\,.$
(29)

Here we introduced the integral $\mathcal{I}_{1}$ (and also $\mathcal{I}_{2}$
for further reference) as

$\displaystyle\mathcal{I}_{0}(r)$
$\displaystyle=\int_{r_{p}}^{r}\frac{1}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}\,\mathrm{d}u\,,$
(30) $\displaystyle\mathcal{I}_{2}(r)$
$\displaystyle=\int_{r_{p}}^{r}\frac{u^{2}}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}\,\mathrm{d}u\,.$
(31)

These integrals are explicitly given in terms of elliptic integrals – see
Appendix B for particular formulae.

Thus, we have

$\displaystyle\boldsymbol{n}^{a}\nabla_{a}\Theta(r,\theta)$
$\displaystyle=0\,,$ $\displaystyle\boldsymbol{n}^{a}\nabla_{a}\Phi(r,\theta)$
$\displaystyle=0\,,$ (32)

The inverse of the relations (27) – (28), i.e. how $\theta$ and $\varphi$
change along the null ray defined by $\Theta=\mathrm{const}$ and
$\Phi=\mathrm{const}$, read

$\displaystyle\theta$
$\displaystyle=2\arctan\left[\tanh\Gamma^{-}(r,\Theta)\right]\,,$ (33)
$\displaystyle\varphi$
$\displaystyle=\Phi-2aM\int_{r_{p}}^{r}\frac{u}{\Delta(u)\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}\,\mathrm{d}u\,,$
(34)

where

$\displaystyle\Gamma^{-}(r,\Theta)$ $\displaystyle=\mathop{\rm
arctanh}\left(\tan\frac{\Theta}{2}\right)-\frac{a}{2}\,\mathcal{I}_{0}(r)\,.$
(35)

The program of the construction of the tetrad and coordinate system is now
straightforward

  1. 1.

Employing the Killing – Yano tensors we complete the NP tetrad555The dots over
the vectors does not represent derivatives. We choose them instead of indices
$(1)$ and $(2)$ which would spoil the equations. to get
$\left(\boldsymbol{\ddot{n}}=\boldsymbol{n},\,\boldsymbol{\ddot{l}},\,\boldsymbol{\ddot{m}},\,\boldsymbol{\ddot{\bar{m}}}\right)$.
See the explicit construction below in Eqs. (36 – 39).

  2. 2.

We perform null rotation around $\boldsymbol{n}$, so that the all the vectors
are parallely propagated along $\boldsymbol{n}$ and, moreover, on the horizon
$\boldsymbol{l}$ coincides with its generator, to obtain
$\left(\boldsymbol{n},\,\boldsymbol{l},\,\boldsymbol{\dot{m}},\,\boldsymbol{\dot{\bar{m}}}\right)$.
This is done in Eqs. (57) – (62).

  3. 3.

If desired, we may perform final rotation in
$\boldsymbol{\dot{m}}-\boldsymbol{\dot{\bar{m}}}$ plane (using functions of
$\Theta$ coordinate, such that the parallel transport of the new vector is not
violated) and thus obtain
$\left(\boldsymbol{n},\,\boldsymbol{l},\,\boldsymbol{m},\,\boldsymbol{\bar{m}}\right)$.

  4. 4.

Finally, we perform the coordinate transformation to the coordinates of the
isolated horizon formalism, see Sections V,VI.

A well known fact is that the principal Killing – Yano tensor can be used to
produce a parallely transported vector along the geodesic congruence given by
$\boldsymbol{n}$

$\displaystyle\boldsymbol{e_{(f)}}^{b}$
$\displaystyle=\frac{1}{a}\,\boldsymbol{f}^{b}_{\phantom{b}c}\boldsymbol{n}^{c}\,,$
(36)

which is space-like in our case. We can complete the tetrad by the following
two vectors:

$\displaystyle\boldsymbol{e_{(h)}}^{b}$
$\displaystyle=\frac{1}{a}\,\boldsymbol{h}^{b}_{\phantom{b}c}\boldsymbol{n}^{c}\,,$
(37) $\displaystyle\boldsymbol{{e_{(3)}}}^{b}$
$\displaystyle=\frac{1}{a^{2}}\left(\boldsymbol{f}^{b}_{\phantom{b}c}\boldsymbol{f}^{c}_{\phantom{c}d}\boldsymbol{n}^{d}+\boldsymbol{f}_{jk}\boldsymbol{f}^{jk}\boldsymbol{n}^{b}\right).$
(38)

From these we can construct a null tetrad as

$\left(\boldsymbol{n},\,\boldsymbol{\ddot{l}},\,\boldsymbol{\ddot{m}},\,\boldsymbol{\ddot{\bar{m}}}\right)\\\
=\left(\boldsymbol{n},\,\boldsymbol{e_{(3)}},\,\frac{1}{\sqrt{2}}\left(\boldsymbol{e_{(f)}}+i\boldsymbol{e_{(h)}}\right),\frac{1}{\sqrt{2}}\left(\boldsymbol{e_{(f)}}-i\boldsymbol{e_{(h)}}\right)\right),$
(39)

which explicitly reads as follows

$\displaystyle\boldsymbol{n}$ $\displaystyle=\mathrm{Eq.\ (\ref{eq:n})}\,,$
(40) $\displaystyle\boldsymbol{\ddot{l}}$
$\displaystyle=-\frac{1}{2}\Biggl{(}\frac{\Sigma_{0}^{2}+a^{2}\sin^{2}\theta\,\Delta}{a^{2}\Delta}\,\boldsymbol{\partial_{t}}-\frac{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}}{a^{2}}\,\boldsymbol{\partial_{r}}$
$\displaystyle\qquad\qquad-\frac{\cos\theta}{a}\,\boldsymbol{\partial_{\theta}}+\frac{\Sigma_{0}+\Delta}{a\Delta}\,\boldsymbol{\partial_{\varphi}}\Biggr{)},$
(41) $\displaystyle\boldsymbol{\ddot{m}}$
$\displaystyle=-\frac{1}{\sqrt{2}\bar{\rho}}\Biggl{(}\frac{a^{2}\sin
2\theta\,\Delta+i\Sigma_{0}\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}}{2a\Delta}\,\boldsymbol{\partial_{t}}-\frac{i\Sigma_{0}}{a}\,\boldsymbol{\partial_{r}}$
$\displaystyle\qquad\qquad+\sin\theta\,\boldsymbol{\partial_{\theta}}+\frac{\cot\theta\,\Delta+i\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}}{\Delta}\,\boldsymbol{\partial_{\varphi}}\Biggr{)}.$
(42)

The Weyl scalars in this tetrad have the following form

$\displaystyle\left(\ddot{\psi}_{0},\ddot{\psi}_{1},\ddot{\psi}_{2},\ddot{\psi}_{3},\ddot{\psi}_{4}\right)=-\left(\frac{3}{4}\frac{\rho^{2}}{a^{2}}\,,0\,,\frac{1}{2}\,,0\,,\frac{3}{4}\,\frac{a^{2}}{\rho^{2}}\right)\tilde{\psi}_{2}\,,$
(43)

where $\tilde{\psi}_{2}$ is the only non-vanishing Weyl projection in the
Kinnersley tetrad (17), which is adapted to principal null directions of the
Weyl tensor.

The appropriate spin coefficients are as follows

$\displaystyle\ddot{\kappa}$
$\displaystyle=\frac{-\Sigma_{0}\cos\theta+iar\sin^{2}\theta}{\sqrt{2}a^{2}\bar{\rho}}\,,$
(44) $\displaystyle\ddot{\nu}$ $\displaystyle=0\,,$ (45)
$\displaystyle\ddot{\sigma}$
$\displaystyle=\frac{1}{16a^{2}\bar{\rho}}\left(4a\frac{ia\cos\theta-r\cos
2\theta}{\sin\theta}+\frac{\ddot{\upsilon}}{\bar{\rho}\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}}\right)\,,$
(46) $\displaystyle\ddot{\lambda}$
$\displaystyle=\frac{1}{8\rho^{3}}\left(4a\frac{ia\cos\theta-r\cos
2\theta}{\sin\theta}-\frac{\ddot{\upsilon}}{\bar{\rho}\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}}\right)\,,$
(47) $\displaystyle\ddot{\varrho}$
$\displaystyle=\frac{1}{8a^{2}}\left(-2a\frac{\cos
2\theta}{\sin\theta}-\frac{\partial_{r}\Upsilon_{\\!\frac{\pi}{2}}}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}}\right),$
(48) $\displaystyle\ddot{\mu}$
$\displaystyle=\frac{1}{4\Sigma}\left(-2a\frac{\cos
2\theta}{\sin\theta}+\frac{\partial_{r}\Upsilon_{\\!\frac{\pi}{2}}}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}}\right)\,,$
(49) $\displaystyle\ddot{\tau}$ $\displaystyle=-\frac{i}{\sqrt{2}a}\,,$ (50)
$\displaystyle\ddot{\pi}$ $\displaystyle=-\frac{i}{\sqrt{2}a}\,,$ (51)
$\displaystyle\ddot{\beta}$ $\displaystyle=-\frac{i}{\sqrt{2}a}\,,$ (52)
$\displaystyle\ddot{\alpha}$ $\displaystyle=0\,,$ (53)
$\displaystyle\ddot{\epsilon}$
$\displaystyle=\frac{ia^{2}\cos\theta\,\sin\theta+\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}}{2a^{2}\rho}\,,$
(54) $\displaystyle\ddot{\gamma}$ $\displaystyle=0\,,$ (55)

where we defined a shortcut

$\displaystyle\ddot{\upsilon}$
$\displaystyle=-8ia\Upsilon_{\\!\frac{\pi}{2}}\,\cos\theta$ (56)
$\displaystyle\qquad+a^{2}\left(2Ma^{2}+2a^{2}r-12Mr^{2}+\cos
2\theta\,\partial_{r}\Upsilon_{\\!\frac{\pi}{2}}\right).$

Notice that, at this stage, the limit $a\rightarrow 0$ does not exist for the
tetrad
$\left(\boldsymbol{n},\,\boldsymbol{\ddot{l}},\,\boldsymbol{\ddot{m}},\,\boldsymbol{\ddot{\bar{m}}}\right)$.
However, the limit will be recovered once the final tetrad is constructed.

The vector $\boldsymbol{n}$ is already fixed, we thus perform a null rotation
about it to get a new tetrad, namely

$\left(\boldsymbol{n},\,\boldsymbol{l},\,\boldsymbol{\dot{m}},\,\boldsymbol{\dot{\bar{m}}}\right)\\\
=\left(\boldsymbol{n},\,\boldsymbol{\ddot{l}}+E\boldsymbol{{\ddot{\bar{m}}}}+\bar{E}\boldsymbol{\ddot{m}}+E\bar{E}\boldsymbol{n},\,\boldsymbol{\ddot{m}}+E\boldsymbol{n},\,\boldsymbol{\ddot{\bar{m}}}+\bar{E}\boldsymbol{n}\right).$
(57)

Imposing the condition of parallel transport of vector $\boldsymbol{\dot{m}}$
along $\boldsymbol{n}$, i.e. demanding

$\displaystyle\boldsymbol{n}^{a}\nabla_{a}\boldsymbol{\dot{m}}$
$\displaystyle=0\,,$ (58)

leads to a first order partial differential equation for $E$. Its real and
imaginary parts are as follows

$\displaystyle\left(-\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}\,\partial_{r}+a\cos\theta\,\partial_{\theta}\right)\mathop{\rm
Re}(E)$ $\displaystyle=0\,,$ (59)
$\displaystyle\left(\Sigma-\sqrt{2}a\sqrt{\Upsilon_{\\!\frac{\pi}{2}}}\,\partial_{r}+\sqrt{2}a^{2}\cos\theta\,\partial_{\theta}\right)\mathop{\rm
Im}(E)$ $\displaystyle=0\,.$ (60)

We may try to solve for $E$ in a separated form
$E(r,\theta)=E_{r}(r)+E_{\theta}(\theta)$. The general separable solution
reads

$\displaystyle E$
$\displaystyle=i\frac{r_{p}-a\sin\theta+\mathcal{I}_{2}}{\sqrt{2}a}+F(\Gamma^{+})\,,$
(61)

where $F$ is an arbitrary complex function of $\Gamma^{+}(r,\theta)$ which is
defined in Eq. (29).

If $\boldsymbol{n}$ and $\boldsymbol{\dot{m}}$ (and thus also
$\boldsymbol{\dot{\bar{m}}}$) are parallely transported along $\boldsymbol{n}$
then, inevitably, also the last vector $\boldsymbol{\dot{l}}$ is parallely
transported.

This freedom in the choice of $F$ allows us to fix $\boldsymbol{l}$ in such a
way that on the horizon it coincides with the generator of the horizon. The
parallely transported tetrad is given by the function

$\displaystyle E$
$\displaystyle=\frac{1}{\sqrt{2}}\left(i\frac{r_{p}-a\sin\theta+\mathcal{I}_{2}}{a}-\sec
2\Gamma^{+}+i\tanh 2\Gamma^{+}\right).$ (62)

The Weyl scalars in this tetrad are related to the only non-vanishing
projection $\tilde{\psi}_{2}$ of the Weyl tensor onto the Kinnersley tetrad as
follows

$\displaystyle\dot{\psi}_{0}$
$\displaystyle=-\frac{3}{4}\left(\frac{2a^{2}\bar{E}^{2}+\rho^{2}}{a\rho}\right)^{2}\tilde{\psi}_{2}\,,$
(63) $\displaystyle\dot{\psi}_{1}$
$\displaystyle=-\frac{3}{2}\,\frac{2a^{2}\bar{E}^{2}+\rho^{2}\bar{E}}{\rho^{2}}\,\tilde{\psi}_{2}\,,$
(64) $\displaystyle\dot{\psi}_{2}$
$\displaystyle=-\frac{1}{2}\frac{6a^{2}\bar{E}^{2}+\rho^{2}}{\rho^{2}}\,\tilde{\psi}_{2}\,,$
(65) $\displaystyle\dot{\psi}_{3}$
$\displaystyle=-\frac{3a^{2}\bar{E}}{\rho^{2}}\,\tilde{\psi}_{2}\,,$ (66)
$\displaystyle\dot{\psi}_{4}$
$\displaystyle=-\frac{3a^{2}}{\rho^{2}}\,\tilde{\psi}_{2}\,.$ (67)

At this point we need to investigate the behaviour of the Weyl scalars (and of
the null tetrad itself) in the limit $a\rightarrow 0$. Careful calculations
show that

$\bar{E}\approx-i\frac{r}{\sqrt{2}}\frac{1}{a}-\frac{\cos\theta}{\sqrt{2}}+\mathcal{O}(a)\,,$
(68)

and thus

$\displaystyle\dot{\psi}_{0}$ $\displaystyle=\mathcal{O}(a^{2})\,,$ (69)
$\displaystyle\dot{\psi}_{1}$ $\displaystyle=\mathcal{O}(a)\,,$ (70)
$\displaystyle\dot{\psi}_{2}$
$\displaystyle=\bigl{.}\tilde{\psi}_{2}\bigr{|}_{a=0}+\mathcal{O}(a)\,,$ (71)
$\displaystyle\dot{\psi}_{3}$ $\displaystyle=\mathcal{O}(a)\,,$ (72)
$\displaystyle\dot{\psi}_{4}$ $\displaystyle=\mathcal{O}(a^{2})\,.$ (73)

The NP spin coefficients of the tetrad
$(\boldsymbol{n},\boldsymbol{l},\boldsymbol{\dot{m}},\boldsymbol{\dot{\bar{m}}})$
are given by standard transformation rules (see [7, 8] for details).

The last, optional, step is to perform a rotation in
$\boldsymbol{\dot{m}}-\boldsymbol{\dot{\bar{m}}}$ plane

$\left(\boldsymbol{n},\,\boldsymbol{l},\,\boldsymbol{m},\,\boldsymbol{\bar{m}}\right)=\left(\boldsymbol{n},\,\boldsymbol{l},\,e^{i\alpha}\boldsymbol{\dot{m}},\,e^{-i\alpha}\boldsymbol{\dot{\bar{m}}}\right),$
(74)

so that
$\boldsymbol{e}_{\varphi}=-i(\boldsymbol{m}-\boldsymbol{\bar{m}})/\sqrt{2}$
has only $t$ and $\varphi$ components on the horizon. This can be done with

$\displaystyle\alpha$
$\displaystyle=\arctan\left(\frac{a\cos\Theta}{r_{p}}\right)-\Theta-\frac{\pi}{2}\,.$
(75)

This way we also reconstruct the non-rotating limit

$\displaystyle\lim_{a\rightarrow 0}\boldsymbol{m}=\lim_{a\rightarrow
0}\boldsymbol{\tilde{m}}\,,$ (76)

and the Weyl scalars are affected in a trivial way.

##  V Null coordinates

Following [16, 17], we define ingoing null coordinates $(v,r,\Theta,\Phi)$ by
the relations

$\displaystyle t$
$\displaystyle=-v-\int_{r_{p}}^{r}\frac{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}{\Delta(u)}\,\mathrm{d}u+a\sin\theta(r,\Theta)\,,$
(77) $\displaystyle r$ $\displaystyle=r\,,$ (78) $\displaystyle\theta$
$\displaystyle=2\arctan\left[\tanh\Gamma^{-}(r,\Theta)\right]\,,$ (79)
$\displaystyle\varphi$
$\displaystyle=\Phi-2aM\int_{r_{p}}^{r}\frac{u}{\Delta(u)\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}\,\mathrm{d}u\,,$
(80)

where $\theta(r,\Theta)$ entering (77) is defined in Eq. (79). In this
coordinates the vector $\boldsymbol{n}$ takes the form666We did not want to
rename all the coordinates and we thus write the vanishing components
explicitly in gray color. Compare the form of $\boldsymbol{n}$ given in Eq.
(25) and Eq. (81).

$\displaystyle\boldsymbol{n}$
$\displaystyle={\color[rgb]{.75,.75,.75}0\,\boldsymbol{\partial_{v}}+}\frac{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}{\Sigma(r,\theta(r,\Theta))}\,\boldsymbol{\partial_{r}}{\color[rgb]{.75,.75,.75}+0\,\boldsymbol{\partial_{\Theta}}+0\,\boldsymbol{\partial_{\Phi}}}\,.$
(81)

##  VI Affinely parameterized null geodesics

The geodesic on Kerr background has been shown to be separable in Mino time
[21]. Yet, we need an affine parameter, not a Mino time. It may seem hopeless
to evaluate the integral which enters the definition of an affine parameter
for the null ray given by $\boldsymbol{n}$

$s(r,\Theta)=\int_{r_{p}}^{r}\frac{\Sigma(u,\theta(u,\Theta))}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}\,\mathrm{d}u\,,$
(82)

but we found that the integral can be analytically calculated and the result
is

$\displaystyle s(r,\Theta)$
$\displaystyle=a\left[\sin\Theta-\tanh\Gamma^{-}(r,\Theta)\right]+\mathcal{I}_{2}(r)$
$\displaystyle=a\left[\sin\Theta-\sin\theta(r,\Theta)\right]+\mathcal{I}_{2}(r)\,,$
(83)

where we use the function $\mathcal{I}_{2}$ which has been defined in Eq.
(31).

The very last step would have been to invert this relation to get
$r=r(s,\Theta)$ in order to be able to explicitly express the final
expressions in terms of $v,s,\Theta,\tilde{\Phi}$ (let us remind that along a
particular given ray $\Theta$ is constant). Alas, we were unable to proceed in
this direction.

In order to eliminate the axial component of the vector $\boldsymbol{\dot{l}}$
on the horizon, we also perform a final coordinate transformation to
coordinates $(v,s,\Theta,\tilde{\Phi})$ defined by

$\displaystyle v$ $\displaystyle=v\,,$ (84) $\displaystyle s$
$\displaystyle=s(r,\Theta)\,,$ (85) $\displaystyle\Theta$
$\displaystyle=\Theta\,,$ (86) $\displaystyle\tilde{\Phi}$
$\displaystyle=\Phi+\frac{a}{a^{2}+r_{\\!p}^{2}}\,v\,,$ (87)

where $a/(a^{2}+r_{\\!p}^{2})$ is the angular velocity of the horizon. The
coordinate $s$ is now the affine parameter along the null geodesics given by
$\boldsymbol{n}$.

\begin{overpic}[keepaspectratio,width=86.72267pt,percent]{coords.png}
\put(55.0,42.0){$R$} \put(-2.0,95.0){$z$} \end{overpic} Figure 1: Red lines
represent projection of integral lines of $\boldsymbol{n}$ (i.e. lines of
constant $\Theta$) in BL coordinates meanwhile the light blue lines are
equipotentials of constant affine parameter $s(r,\theta)$. The thick blue
half-circle is black hole horizon $r=r_{\\!p}$, axis of rotation is vertical.
The black line denotes vanishing expansion of the congruence $\boldsymbol{n}$.
Calculations are done in BL coordinates, which are later identified with polar
coordinates $(R=r\sin\theta,\,z=r\cos\theta$).

The newly constructed coordinates $(s,\Theta)$ are visualised in Fig. 1. The
coordinates $\Theta$ and $\theta$ coincide on the outer horizon.

The null geodesic vector field $\boldsymbol{n}$ is singular near the axis (has
non-vanishing $\theta$ component). This has some unpleasant consequences, as
the presence of caustics.

From the definition of the transformation $\theta=\theta(r,\Theta)$ we can see
that the range of variable $\Theta$ is $r$-dependent

$\displaystyle\Theta$
$\displaystyle\in\langle\Theta_{0}(r),\,\pi-\Theta_{0}(r)\rangle\,,$ (88)
$\displaystyle\Theta_{0}(r)$
$\displaystyle=2\arctan\tanh\left(\frac{1}{2}\,a\,\mathcal{I}_{0}(r)\right)\,,$
(89)

and some regions below the horizon are not covered at all.

It is peculiar that the expansion, i.e., an invariant quantity, of null
geodetic congruence $\boldsymbol{n}$,

$\Theta_{\boldsymbol{n}}=\nabla^{a}\boldsymbol{n}_{a}=\frac{1}{\Sigma}\left(\frac{\mathrm{d}\Upsilon_{\\!\frac{\pi}{2}}(r)}{\mathrm{d}r}-a\,\frac{\cos
2\theta}{\cos\theta}\right),$ (90)

is divergent close to the axis (as visualised in Fig. 1).

##  VII Final tetrad and coordinate system

In the coordinates constructed in the last section we can write down the final
tetrad

$\displaystyle\boldsymbol{l}$
$\displaystyle=\boldsymbol{\partial_{v}}+U\boldsymbol{\partial_{s}}+X^{\Theta}\boldsymbol{\partial_{\Theta}}+X^{\tilde{\Phi}}\boldsymbol{\partial_{\tilde{\Phi}}}\,,$
(91) $\displaystyle\boldsymbol{n}$
$\displaystyle=\boldsymbol{\partial_{s}}\,,$ (92)
$\displaystyle\boldsymbol{\dot{m}}$
$\displaystyle=\Omega\boldsymbol{\partial_{s}}+\xi^{\Theta}\boldsymbol{\partial_{\Theta}}+\xi^{\tilde{\Phi}}\boldsymbol{\partial_{\tilde{\Phi}}}\,,$
(93)

where the functions
$U,X^{\Theta},X^{\tilde{\Phi}},\Omega,\xi^{\theta},\xi^{\tilde{\Phi}}$ are
given as follows

$\displaystyle U(r,\Theta)$
$\displaystyle=E(r,\theta)\bar{E}(r,\theta)+\cos^{2}\Theta+\frac{\Sigma^{-}(r,\theta)}{2a^{2}}+\left[\frac{iE(r,\theta)}{\sqrt{2}\bar{\rho}(r,\theta)}\left(\frac{i\Sigma_{0}(r)\Sigma(r,\Theta)}{a\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}-a\left(\cos^{2}\Theta-\cos^{2}\theta\right)\tan\theta\right)+\mathrm{c.c.}\right],$
(94) $\displaystyle X^{\Theta}(r,\Theta)$
$\displaystyle=\left\\{\frac{1}{a}+\left[\frac{E(r,\theta)}{\sqrt{2}\bar{\rho}(r,\theta)}\left(\frac{\Sigma_{0}(r)}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}+i\tan\theta\right)+\mathrm{c.c.}\right]\right\\}\cos\Theta,$
(95) $\displaystyle X^{\tilde{\Phi}}(r,\Theta)$
$\displaystyle=-\frac{r_{\\!p}}{2Ma}-\left[\frac{E(r,\theta)}{\sqrt{2}\bar{\rho}(r,\theta)}\left(\frac{r^{2}}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}+\cot\theta\right)+\mathrm{c.c.}\right],$
(96) $\displaystyle\Omega(r,\Theta)$
$\displaystyle=\bar{E}(r,\theta)+\frac{1}{\sqrt{2}\bar{\rho}(r,\theta)}\left(\frac{i\Sigma_{0}(r)\Sigma(r,\Theta)}{a\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}-a\left(\cos^{2}\Theta-\cos^{2}\theta\right)\tan\theta\right),$
(97) $\displaystyle\xi^{\Theta}(r,\Theta)$
$\displaystyle=\frac{i\cos\Theta}{\sqrt{2}\bar{\rho}(r,\theta)}\left(\frac{\Sigma_{0}(r)}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}+i\tan\theta\right),$
(98) $\displaystyle\xi^{\tilde{\Phi}}(r,\Theta)$
$\displaystyle=\frac{-1}{\sqrt{2}\bar{\rho}(r,\theta)}\left(\frac{i\,r^{2}}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}+\cot\theta\right),$
(99)

where we implicitly assume $r=r(s,\Theta)$ and also
$\theta=\theta(r,\Theta)=\theta(r(s,\Theta),\Theta)$. We also introduced

$\Sigma^{-}(r,\theta)=\frac{1}{2}\,\boldsymbol{f}_{ab}\boldsymbol{f}^{ab}=r^{2}-a^{2}\cos^{2}\theta\,.$
(100)

The following identities were used during the simplification process

$\displaystyle\frac{\partial}{\partial s}\,r(s,\Theta)$
$\displaystyle=\frac{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}{\Sigma(r,\theta(r,\Theta))}\,,$
(101) $\displaystyle\frac{\partial}{\partial r}\,s(r,\Theta)$
$\displaystyle=\frac{\Sigma(r,\theta(r,\Theta))}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}\,,$
(102) $\displaystyle\frac{\partial}{\partial\Theta}\,s(r,\Theta)$
$\displaystyle=a\,\frac{\cos^{2}\Theta-\cos^{2}\theta(r,\Theta)}{\cos\Theta}\,,$
(103) $\displaystyle\frac{\partial}{\partial\Theta}\,r(s,\Theta)$
$\displaystyle=-a\,\frac{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(r)}}{\Sigma(r,\theta(r,\Theta))}\frac{\cos^{2}\Theta-\cos^{2}\theta(r,\Theta)}{\cos\Theta}\,,$
(104) $\displaystyle\frac{\partial}{\partial\Theta}\,\theta(r,\Theta)$
$\displaystyle=\frac{\cos\theta(r,\Theta)}{\cos\Theta}\,.$ (105)

Also, we can simplify the form of the null rotation $E$, given in Eq. (62), as

$E(r,\theta)=\frac{1}{\sqrt{2}}\left(-e^{-i\Theta}+i\,\frac{\mathcal{I}_{2}(r)+r_{\\!p}-a\sin\theta}{a}\right)\,.$
(106)

The null rotation parameter $E$ has the following relation to $s$

$\displaystyle E(r,\Theta)$
$\displaystyle=i\,\frac{s(r,\Theta)+r_{\\!p}+ia\cos\Theta}{\sqrt{2}\,a}\,.$
(107)

In the coordinates $(v,s,\Theta,\tilde{\Phi})$ the metric takes the following
form

$\boldsymbol{g}_{\mu\nu}=\begin{pmatrix}g_{vv}&1&g_{v\Theta}&g_{v\tilde{\Phi}}\\\
1&0&0&0\\\ g_{\Theta v}&0&g_{\Theta\Theta}&g_{\Theta\tilde{\Phi}}\\\ g_{\Phi
v}&0&g_{\tilde{\Phi}\Theta}&g_{\tilde{\Phi}\tilde{\Phi}}\end{pmatrix},$ (108)

where

$\displaystyle g_{vv}$ $\displaystyle=\frac{4M^{2}r_{\\!p}\,\Delta-
r_{m}\sin^{2}\theta\left(4M^{2}r_{\\!p}(r_{\\!p}-2r)+\Upsilon(r,\theta)\right)}{4M^{2}r_{\\!p}\,\Sigma(r,\theta)}\,,$
(109) $\displaystyle g_{v\Theta}$
$\displaystyle=a\,\frac{\cos^{2}\theta}{\cos\Theta}\left(\frac{r}{r_{\\!p}}\frac{\Sigma(r_{\\!p},\theta)}{\Sigma(r,\theta)}-\frac{\cos^{2}\Theta}{\cos^{2}\theta}\right)\,,$
(110) $\displaystyle g_{v\tilde{\Phi}}$
$\displaystyle=-\frac{r_{m}}{2Ma}\frac{4M^{2}r_{\\!p}\,r-\Upsilon(r,\theta)}{\Sigma(r,\theta)}\,\sin^{2}\theta\,,$
(111) $\displaystyle g_{\Theta\Theta}$
$\displaystyle=-\frac{\cos^{2}\theta}{\cos^{2}\Theta}\left(r^{2}+\frac{2Ma^{2}r\cos^{2}\theta}{\Sigma(r,\theta)}\right)\,,$
(112) $\displaystyle g_{\Theta\tilde{\Phi}}$
$\displaystyle=\frac{2Ma^{2}r}{\Sigma(r,\theta)}\frac{\cos^{2}\theta}{\cos\Theta}\,\sin^{2}\theta\,,$
(113) $\displaystyle g_{\tilde{\Phi}\tilde{\Phi}}$
$\displaystyle=-\frac{\Upsilon(r,\theta)}{\Sigma(r,\theta)}\,\sin^{2}\theta\,.$
(114)

As has been discussed in a paragraph after the Eq. (83), the inversion of the
explicit relation $s=s(r,\Theta)$ is not analytically possible. Therefore,
also the above stated metric functions contains implicitly defined
coordinates.

##  VIII Uniqueness

First of all we notice that for an arbitrary choice of constants
$E,K,\epsilon_{r},\epsilon_{\theta}$ (the defining parameters of the vector
field $\boldsymbol{n}$), we can complete the tetrad in such a way, that
$\boldsymbol{l},\boldsymbol{m},\boldsymbol{\bar{m}}$ are tangent to the
horizon (on the horizon) and start evolving from the horizon. But these
coordinate systems suffer the same disadvantages (and, actually, are even
“worse”: not defined everywhere, not having mirror symmetry with respect to
the equatorial).

In order to find a field which is defined everywhere and is regular around the
axis we need to investigate the twist-free geodesic congruence in a more
general setting.

The vector field $\boldsymbol{u}$ given by Eq. (21) is a geodesic congruence
if

$\displaystyle\boldsymbol{u}^{a}\nabla_{a}E(r,\theta)$ $\displaystyle=0\,,$
$\displaystyle\boldsymbol{u}^{a}\nabla_{a}L(r,\theta)$ $\displaystyle=0\,,$
$\displaystyle\boldsymbol{u}^{a}\nabla_{a}K(r,\theta)$ $\displaystyle=0\,.$
(115)

Specializing for null geodesic, the twist-free condition

$\displaystyle\boldsymbol{n}_{[a}\nabla_{b}\boldsymbol{n}_{c]}=0$ (116)

leads to

$\displaystyle L=\mathrm{const}\times E\,,$ (117)

and subsequently thus vanishing angular momentum $L=0$, since we need
$\boldsymbol{n}$ to be defined close to the axis.

Then the energy $E$ can be scaled to a constant and we need to solve the
equation for the Carter’s constant

$\displaystyle\frac{\epsilon_{r}K_{,r}}{\sqrt{E\Sigma_{0}^{2}-K\Delta}}+\frac{\epsilon_{\theta}K_{,\theta}}{\sqrt{K-a^{2}E^{2}\sin^{2}\theta}}=0\,.$
(118)

If this equation is satisfied, $\boldsymbol{n}$ is geodesic and twist-free. So
far the only explicitly known solutions to this partial differential equation
are $K=\mathrm{const}$ and we need to overcome this complication. Notice that

$K(r,\theta)=a^{2}E^{2}\sin^{2}\theta$

everywhere does not lead to a geodetic vector field.

Let us define the initial values for $\boldsymbol{n}$ on the horizon and then
evolve the geodesic equations. A natural choice for the Carter’s constant on
the horizon is

$\displaystyle K$ $\displaystyle\doteq a^{2}E^{2}\sin\theta^{2}\,,$ (119)

since it makes the $\theta$ component of the field $\boldsymbol{n}$ vanishing.

In the following we need to work in null coordinates given by

$\displaystyle t$
$\displaystyle=u-\int\frac{a^{2}+r^{2}}{\Delta(r)}\,\mathrm{d}r\,,$ (120)
$\displaystyle\varphi$
$\displaystyle=\tilde{\varphi}-\int\frac{a}{\Delta(r)}\,\mathrm{d}r\,.$ (121)

Introducing

$\displaystyle R(r,\theta)$
$\displaystyle=E^{2}\Sigma_{0}^{2}(r)-K(r,\theta)\Delta(r)\,,$ (122)
$\displaystyle S(r,\theta)$
$\displaystyle=K(r,\theta)-a^{2}E^{2}\sin^{2}\theta\,,$ (123)

we thus look for an integral lines of the field

$\displaystyle\boldsymbol{n}$
$\displaystyle=\frac{1}{\Sigma}\Biggl{(}\frac{\Sigma_{0}\sqrt{R}+E\left(\Sigma_{0}^{2}-a^{2}\sin^{2}\theta\,\Delta\right)}{\Delta}\,\boldsymbol{\partial_{u}}$
$\displaystyle\qquad+\epsilon_{r}\sqrt{R}\,\boldsymbol{\partial_{r}}+\epsilon_{\theta}\sqrt{S}\,\boldsymbol{\partial_{\theta}}+a\frac{2MEr+\epsilon_{r}\sqrt{R}}{\Delta}\,\boldsymbol{\partial_{\tilde{\varphi}}}\Biggr{)}$
(124)

with the initial values for the vector field $\boldsymbol{n}$ on the horizon

$\displaystyle\boldsymbol{n}$
$\displaystyle\doteq\frac{1}{2\Sigma(r_{\\!p},\theta_{0})}\Biggl{(}a^{2}\sin^{2}\theta_{0}\,\boldsymbol{\partial_{u}}+2Mr_{\\!p}\,\boldsymbol{\partial_{r}}$
$\displaystyle\qquad{+\;\color[rgb]{.75,.75,.75}0\,\boldsymbol{\partial_{\theta}}}+\left(a+\frac{r_{m}\Sigma(r_{\\!p},\theta_{0})}{2aM}\right)\boldsymbol{\partial_{\tilde{\varphi}}}\Biggr{)},$
(125)

where we already set $E=-1$ and $\epsilon_{r}=1$. We shall see later that
there is no dependence on $\epsilon_{\theta}$.

Using the recent developments in analytical treating of geodesics within the
Kerr spacetime [22, 23] (and following the notation of [23]) we can write the
explicit solution in terms of the Weierstrass $\wp$ function as

$\displaystyle r(z)$
$\displaystyle=r_{0}+\frac{-\varepsilon_{r}\sqrt{R(r_{0})}\,\wp_{r}^{\prime}(z)+\frac{1}{2}R^{\prime}(r_{0})\left(\wp_{r}(z)-\frac{1}{24}\,R^{\prime\prime}(r_{0})\right)+\frac{1}{24}R(r_{0})R^{\prime\prime\prime}(r_{0})}{2\left(\wp_{r}(z)-\frac{1}{24}R^{\prime\prime}(r_{0})\right)^{2}-\frac{1}{48}R(r_{0})R^{(4)}(r_{0})}\,,$
(126) $\displaystyle\mu(z)$
$\displaystyle=\mu_{0}+\frac{\varepsilon_{\theta}\sqrt{G(\mu_{0})}\,\wp_{\theta}^{\prime}(z)+\frac{1}{2}G^{\prime}(\mu_{0})\left(\wp_{\theta}(z)-\frac{1}{24}\,G^{\prime\prime}(\mu_{0})\right)+\frac{1}{24}G(\mu_{0})G^{\prime\prime\prime}(\mu_{0})}{2\left(\wp_{\theta}(z)-\frac{1}{24}G^{\prime\prime}(\mu_{0})\right)^{2}-\frac{1}{48}G(\mu_{0})G^{(4)}(\mu_{0})}\,,$
(127)

where $z$ is a Mino time and the angular variable has been re-parameterize
such that $\mu=\cos\theta$. Therefore, also $G$ is re-parameterized $S$ from
Eq.(123) and is of the form

$\displaystyle G(\mu)$
$\displaystyle=(1-\mu^{2})\left[K-a^{2}(1-\mu^{2})\right]\,.$ (128)

The complete form of the Weierstrass functions $\wp_{r}(z)$ and
$\wp_{\theta}(z)$ where the Weierstrass invariants $g_{2}$, $g_{3}$ (which are
given in terms of roots of $R$ or $G$) is as follows

$\displaystyle\wp_{r}(z)$ $\displaystyle\equiv\wp(z;g^{r}_{2},g^{r}_{3})\,,$
(129) $\displaystyle\wp_{\theta}(z)$
$\displaystyle\equiv\wp(z;g^{\theta}_{2},g^{\theta}_{3})\,.$ (130)

The Weierstrass invariants expressed in terms of black hole parameters and the
Carter constant read

$\displaystyle g^{\theta}_{2}$
$\displaystyle=\frac{1}{12}\left(K^{2}-16aK+16a^{2}\right),$ (131)
$\displaystyle g_{3}^{\theta}$
$\displaystyle=\frac{1}{216}\left(K-2a\right)\left(K^{2}+32aK-32a^{2}\right)\,,$
(132) $\displaystyle g_{2}^{r}$ $\displaystyle=g^{\theta}_{2}\,,$ (133)
$\displaystyle g^{r}_{3}$
$\displaystyle=g_{3}^{\theta}-\frac{K^{2}M^{2}}{4}\,,$ (134)

Let us emphasize, that $K(r,\theta)$ varies as a field over the manifold, but
it is constant along every single geodesic.

In the Eqs. (126) – (127) we will start from the horizon, thus
$r_{0}=r_{\\!p}$, and we will consider $\theta_{0}$, being constant along the
geodesic as the new coordinate $\Theta$. Second new coordinate will be the
Mino time $z$. Taking the initial conditions into the account, the expression
for $\mu(z)$ greatly simplifies and we can explicitly write
$\theta(z)=\arccos\mu(z)$. Thus we can turn a solution to the geodesic
equation into the coordinate transformation as follows

$\displaystyle r(z,\Theta)$ $\displaystyle=\text{Eq.~{}(\ref{eq:geoa}) with
}r_{r}=r_{\\!p}\,,$ (135) $\displaystyle\theta(z,\Theta)$
$\displaystyle=\arccos\left(\cos\Theta+\frac{6a^{2}\sin\Theta\sin
2\Theta}{a^{2}(3+5\cos^{2}\Theta)+24\wp_{\theta}(z)}\right)\,.$ (136)

\begin{overpic}[keepaspectratio,width=86.72267pt,percent]{Kcoords.png}
\put(55.0,42.0){$R$} \put(-2.0,95.0){$z$} \end{overpic} Figure 2: The integral
lines of $\boldsymbol{n}$ and equipotentials of affine parameter in BL
coordinates for $\epsilon_{r}=1$ emanating from the horizon with initial
conditions given by Eq. (125). The geodesics, i.e. lines of constant $\Theta$,
are slightly bent which can be difficult to observe by naked eye, therefore we
included lines of constant $\theta$ as green dashed lines (with half density
of lines). The explanation of structures is the same as in Fig. 1.

Now it is straightforward to perform the change of variables given by
coordinate transformation (135) – (136) in the partial differential equation
for the Carter’s constant $K$ given by Eq. (118) and check directly that

$\displaystyle K(z,\Theta)$ $\displaystyle=a^{2}\sin^{2}\Theta\,,$ (137)

is a valid solution. The value of $K$ differs just slightly from the
$a^{2}\sin^{2}\theta$. In the Fig. 3 we plot the value of expression

$K(r(z,\Theta),\theta(z,\theta))-a^{2}\sin^{2}\theta$

to visualize these small differences. We can see that there is no difference
neither on the axes nor the equatorial.

\begin{overpic}[keepaspectratio,width=195.12767pt,percent]{Kval.png}
\put(2.5,93.0){$z$} \put(90.0,54.0){$R$} \put(2.0,62.0){$\Delta K$}
\end{overpic} Figure 3: The value of $\Delta K\equiv
K(r(z,\Theta),\theta(z,\theta))-a^{2}\sin^{2}\theta$ above the horizon (small
half-circle in the middle) in BL coordinates. The red line is a rotational
axis ($z$), the black line depicts the equatorial plane, in the green
direction the function value is plotted. The exact values have no particular
meaning. The yellow line is one particular geodesic with $\Theta=\pi/4$ for
which the function is in greater detail in Fig. 4. The blue line is a black
hole horizon.
\begin{overpic}[keepaspectratio,width=195.12767pt,percent]{Kpi.png}
\put(96.0,7.0){$r$} \put(-2.0,25.0){$\Delta K$} \end{overpic} Figure 4: The
vertical axis represents the value of $\Delta K\equiv
K(r(z,\Theta),\theta(z,\theta))-a^{2}\sin^{2}\theta$ for one particular
geodesic from Fig. 3. The values of the black hole are $r_{\\!p}=2,\,r_{m}=1$
and $\Theta=\pi/4$. On the horizontal axis is the BL coordinate $r$.

The coordinates are constructed such that

  * •

on the horizon we have $\Theta\doteq\theta$,

  * •

the axes and equatorial plane are generated by geodesics,

  * •

the expansion is positive and bounded (the explicit formula is exceedingly
long for presentation),

  * •

the vector field $\boldsymbol{n}$ is smooth.

The Mino time $z$ is given by

$\displaystyle z(r)$
$\displaystyle=\epsilon_{r}\int_{r_{\\!p}}^{r}\frac{1}{\sqrt{R(r^{\prime})}}\,\mathrm{d}r^{\prime}.$
(138)

The geodesic leaving horizon at $z=0$ reach infinity in finite Mino time
$z_{\infty}$ which can be evaluated in terms of elliptic integrals as follows

$\displaystyle z_{\infty}$
$\displaystyle=\left[\frac{2F\left(\varphi_{\infty},\frac{(r_{2}-r_{3})(r_{1}-r_{4})}{(r_{1}-r_{3})(r_{2}-4_{4})}\right)}{\sqrt{(r_{1}-r_{3})(r_{2}-r_{4})}}\right]_{r=r_{\\!p}}^{r=\infty}\,,$
(139) $\displaystyle\varphi_{\infty}$
$\displaystyle=\arcsin\left(\frac{\sqrt{r-r_{1}}\sqrt{r_{2}-r_{4}}}{\sqrt{r-r_{2}}\sqrt{r_{2}-r_{4}}}\right)\,.$
(140)

The detailed factorization of polynomial $R(r)$ is given in Appendix D.

In principle it may be feasible to write down the complete solution of the
geodesic equations

$\displaystyle t$ $\displaystyle=t(t_{0},z,\Theta)\,,$ (141) $\displaystyle r$
$\displaystyle=r(z,\Theta)\,,$ (142) $\displaystyle\theta$
$\displaystyle=\theta(z,\Theta)\,,$ (143) $\displaystyle\varphi$
$\displaystyle=\varphi(\varphi_{0},z,\Theta)\,,$ (144)

in terms of the Weierstrass $\wp$ functions, see [23]. Alas, to proceed
further with the explicit coordinate transformation to coordinates
$t_{0},z,\Theta,\varphi_{0}$ and solving the parallel transport equations
seems to be too complicated.

But we advocate that this vector field $\boldsymbol{n}$ is the best one for
construction of a null foliation of the Kerr spacetime generated by non-
twisting congruence. It is, in some sense, unique, since it does not depend on
the choice of $\epsilon_{\theta}$.

##  IX Comparison with Bondi coordinates

The coordinates $(v,s,\Theta,\Phi)$ were called “Bond-like” in [17]. It’s
difficult to judge whether such a designation is correct or not since the
rigorous definition of the “likeness” has not been provided.

In the original Bondi – Sachs formalism the coordinate system is based upon a
congruence of null geodesics emanating from a spatial spherical cross section
of a chosen world-tube. The null coordinate labels the cross sections and the
angular coordinates are the introduced on such a cross section (and stay
constant along the rays). The radial distance is then a parameter along a
particular ray. The metric is then of the form

$\mathrm{d}s^{2}=-\frac{V}{r}\,e^{2\beta}\mathrm{d}u^{2}-2e^{2\beta}\,\mathrm{d}u\,\mathrm{d}r\\\
+r^{2}h_{AB}\left(\mathrm{d}x^{A}-U^{A}\mathrm{d}u\right)\left(\mathrm{d}x^{B}-U^{B}\mathrm{d}u\right),$
(145)

with $h_{AB}$ being the metric on topological spheres, such that $\det
h_{AB}=\det q_{AB}$, where $q_{AB}$ is metric on a sphere. The coordinate $r$
is called a luminosity distance since a surface $u=\mathrm{const}$ and
$r=\mathrm{const}$ has a surface $4\pi r^{2}$.

A version of Bondi – Sachs approach, which fits more appropriately to NP
formalism, is affine-null metric formulation provided by Sachs and Winicour
[24], where the affine parameter along the null rays is used instead of the
luminosity distance. A simple transformation

$\partial_{r}\lambda(u,r,x^{A})=e^{2\beta}\,,$ (146)

leads to the metric in the form

$\mathrm{d}s^{2}=-\left(\mathcal{V}-g_{AB}W^{A}W^{B}\right)\mathrm{d}u^{2}-2\mathrm{d}u\mathrm{d}\lambda\\\
-2g_{AB}W^{A}\,\mathrm{d}u\,\mathrm{d}x^{B}+g_{AB}\mathrm{d}x^{A}\mathrm{d}x^{B}\,,$
(147)

with $g_{AB}=r^{2}h_{AB}$ and $\det h_{AB}=\det q_{AB}$, where $q_{AB}$ is
metric on a round sphere. Now, $r,\,\mathcal{V},\,W^{A}$ as well as $h_{AB}$
are functions of $(u,\lambda,x^{A})$.

The null geodetic congruence is given by

$\displaystyle\boldsymbol{n}_{W}$
$\displaystyle=\boldsymbol{\partial_{\lambda}}\,,$ (148)

and the expansion of this vector field

$\displaystyle\Theta_{\boldsymbol{n}_{W}}$
$\displaystyle=\nabla_{a}\boldsymbol{n}_{W}^{a}=-\frac{1}{2}\frac{\partial\det
g_{AB}}{\partial\lambda}=4r^{3}r_{,\lambda}\,.$ (149)

Clearly, the coordinate system of formalism of isolated horizon is a member of
affine-null metric formulation. It is moreover endowed with a particular
parallely propagated null tetrad.

##  X Conclusions

We explicitly constructed a parallely propagated null tetrad and an
appropriate coordinate system of Kerr metric, following the existing work on
the formulation of Kerr solution in the formalism of isolated horizons. We
provided a physical interpretation and showed that these coordinate systems
are not well behaved (not covering the whole space-time, the vector field
$\boldsymbol{n}$ being irregular on the axis with diverging expansion).

Demanding the regularity of the vector field $\boldsymbol{n}$ we found a well
behaved geodetic congruence in terms of the Weierstrass $\wp$ functions. We
suggest that this is the proper non-twisting null geodetic field. Alas, the
coordinate transformation is extremely difficult to treat analytically and we
were able neither to cast the Kerr metric in coordinates adapted to this
congruence neither to construct a parallely propagated tetrad in the sense of
isolated horizons.

###### Acknowledgements.

D.K. acknowledges the support from the Czech Science Foundation, Grant
21-11268S. The calculations were performed using _Wolfram Mathematica_ ® and
the powerful package _xAct_.

##  Appendix A NP quantities of Kerr black hole

The nonzero NP spin coefficients corresponding to the tetrad (17) are

$\displaystyle\tilde{\pi}$
$\displaystyle=\frac{i}{\sqrt{2}}\,\frac{a\sin\theta}{\rho^{2}}\,,\qquad$
$\displaystyle\tilde{\mu}$
$\displaystyle=\frac{-1}{\sqrt{2}}\,\frac{\Delta}{\Sigma\rho}\,,$ (150)
$\displaystyle\tilde{\tau}$
$\displaystyle=\frac{-i}{\sqrt{2}}\,\frac{a\sin\theta}{\Sigma}\,,$
$\displaystyle\tilde{\varrho}$
$\displaystyle=\frac{-1}{\sqrt{2}}\,\frac{1}{\rho}\,,$
$\displaystyle\tilde{\gamma}$
$\displaystyle=\tilde{\mu}+\frac{1}{\sqrt{2}}\,\frac{r-M}{\Sigma}\,,\qquad$
$\displaystyle\tilde{\beta}$
$\displaystyle=\frac{1}{\sqrt{2}}\,\frac{\cot\theta}{\bar{\rho}}\,,$
$\displaystyle\tilde{\alpha}$
$\displaystyle=\tilde{\pi}-\bar{\tilde{\beta}}\,,$

and the only nonzero Weyl scalar reads

$\tilde{\psi}_{2}=-\frac{M}{\rho^{3}}\,.$ (151)

##  Appendix B The explicit form of integral $\mathcal{I}_{1}$ and
$\mathcal{I}_{2}$

The integrals $\mathcal{I}_{0}$ and $\mathcal{I}_{2}$ have been introduced in
Eqs. (30) – (31) as

$\displaystyle\mathcal{I}_{0}(r)$
$\displaystyle=\int_{r_{p}}^{r}\frac{1}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}\,\mathrm{d}u\,,$
(152) $\displaystyle\mathcal{I}_{2}(r)$
$\displaystyle=\int_{r_{p}}^{r}\frac{u^{2}}{\sqrt{\Upsilon_{\\!\frac{\pi}{2}}(u)}}\,\mathrm{d}u\,.$
(153)

In order to evaluate these (in terms of elliptic integral) we need to
factorize the function $\Upsilon_{\\!\frac{\pi}{2}}(u)$ as

$\displaystyle\Upsilon_{\\!\frac{\pi}{2}}(u)$
$\displaystyle=u(u^{3}+r_{\\!p}r_{m}u+r_{\\!p}r_{m}(r_{\\!p}+r_{m}))$
$\displaystyle=u(u^{3}+a^{2}u+2Ma^{2})$
$\displaystyle=u(u-u_{r})(u-u_{c})(u-\bar{u_{c}})$
$\displaystyle=u(u+2Z)(u^{2}-2Zu+R)$ (154)

which has two real roots $0$, $u_{r}$ and two mutually complex conjugated
roots $u_{c}$ and $\bar{u_{c}}$, namely

$\displaystyle u_{r}$
$\displaystyle=-\frac{2^{1/3}a^{2}}{3^{2/3}w}+\frac{w}{2^{1/3}3^{2/3}}\,,$
(155) $\displaystyle u_{c}$
$\displaystyle=\frac{(1+i\sqrt{3})a^{2}}{2^{2/3}3^{1/3}w}-\frac{(1-i\sqrt{3})w}{2^{4/3}3^{1/3}w}\,,$
(156)

where

$\displaystyle w$ $\displaystyle=2a^{2}(\sqrt{3}\sqrt{27M^{2}+a^{2}}-9M)\,,$
(157) $\displaystyle Z$ $\displaystyle=(u_{c}+\bar{u_{c}})/2=-u_{r}/2\,,$
(158) $\displaystyle R$ $\displaystyle=\sqrt{u_{c}\bar{u_{c}}}\,.$ (159)

Then we can write the integrals $\mathcal{I}_{0}$ and $\mathcal{I}_{2}$ easily
in terms of primitive functions

$\displaystyle\tilde{\mathcal{I}}_{0}(u)$
$\displaystyle=2i\frac{1}{\sqrt{q}}F\left(\varphi_{0}|m_{0}\right)\,,$ (160)
$\displaystyle\varphi_{0}$
$\displaystyle=\arcsin\left(\frac{1}{\sqrt{2}}\sqrt{1+\frac{Z-R^{2}/u}{p}}\right)\,,$
(161) $\displaystyle m_{0}$ $\displaystyle=\frac{4Zp}{q}\,,$ (162)

and

$\displaystyle\tilde{\mathcal{I}}_{2}(u)$
$\displaystyle=\sqrt{-Q}E\left(\varphi_{2}|m_{2}\right)+\sqrt{-\frac{R^{4}}{Q}}F\left(\varphi_{2}|m_{2}\right)$
$\displaystyle\quad+\sqrt{\frac{u(u^{2}-2uZ+R^{2})}{r+2Z}}\,,$ (163)
$\displaystyle\varphi_{2}$
$\displaystyle=\arcsin\left(\frac{1}{\sqrt{2}}\sqrt{1-\frac{Z+\frac{R^{2}-4Zu}{u+2Z}}{p}}\right),$
(164) $\displaystyle m_{2}$ $\displaystyle=-\frac{4Zp}{Q}\,,$ (165)

with constants

$\displaystyle p$ $\displaystyle=\sqrt{Z^{2}-R^{2}}\,,$ (166) $\displaystyle
q$ $\displaystyle=R^{2}+2Zp+2Z^{2}\,,$ (167) $\displaystyle Q$
$\displaystyle=R^{2}-2Zp+2Z^{2}\,.$ (168)

To clarify the notation we use the incomplete elliptic integral of the first
kind $F$ and of the second kind $E$ which are defined as

$\displaystyle F(\varphi|m)$
$\displaystyle=\int_{0}^{\varphi}\frac{1}{\sqrt{1-m\sin^{2}\theta}}\,\mathrm{d}\theta\,,$
(169) $\displaystyle E(\varphi|m)$
$\displaystyle=\int_{0}^{\varphi}\sqrt{1-m\sin^{2}\theta}\,\mathrm{d}\theta\,.$
(170)

##  Appendix C The choice $\epsilon_{\theta}=1$

The other choice of $\epsilon_{r}=\epsilon_{\theta}=1$ was investigated in
[17]. The main disadvantage of this choice is that the appropriate coordinates
does not cover a huge portion of space-time outside the horizon as can be seen
in Fig. 5. On the other hand the expansion of this congruence newer changes
the sign, yet it is still divergent in the vicinity of the axis. The explicit
value is

$\Theta_{\boldsymbol{n}}=\nabla^{a}\boldsymbol{n}_{a}=\frac{1}{\Sigma}\left(\frac{\mathrm{d}\Upsilon_{\\!\frac{\pi}{2}}(r)}{\mathrm{d}r}+a\,\frac{\cos
2\theta}{\cos\theta}\right),$ (171)

\begin{overpic}[keepaspectratio,width=86.72267pt,percent]{c2.png}
\put(55.0,42.0){$R$} \put(-2.0,95.0){$z$} \end{overpic} Figure 5: The integral
lines of $\boldsymbol{n}$ and equipotentials of affine parameter in BL
coordinates for $\epsilon_{r}=\epsilon_{\theta}=1$ emanating from the horizon.
The detailed explanation as in Fig. 1.

The coordinate transformations given by Eqs. (27 – 28) and (33 – 34) differ
just by different signs in relations for $\varphi$ and $\Phi$ and
interchanging $\Gamma^{-}\longleftrightarrow\Gamma^{+}$.

In [16] they were propagating the null geodesics from null infinity, thus
covering the whole space-time.

##  Appendix D Factorization of polynomial $R(r)$

In this Appendix we provide the factorization of polynomial $R(r)$ which has
been introduced in Eq. (122). Taking into the account that we set $E=-1$ we
need to factorize

$\displaystyle R$ $\displaystyle=\Sigma_{0}^{2}-K\Delta(r)$
$\displaystyle=(r^{2}+a^{2})^{2}-a^{2}(r-r_{\\!p})(r-r_{m})\sin^{2}\Theta$
$\displaystyle=(r-r_{1})(r-r_{2})(r-r_{3})(r-r_{4})\,.$ (172)

This can be done in terms of real variables $r_{r},r_{i}^{\pm}$ as follows

$\displaystyle r_{1}$ $\displaystyle=r_{r}+ir_{i}^{+}\,,$ $\displaystyle
r_{3}$ $\displaystyle=-r_{r}+ir_{i}^{-}\,,$ (173) $\displaystyle r_{2}$
$\displaystyle=r_{r}-ir_{i}^{+}\,,$ $\displaystyle r_{3}$
$\displaystyle=-r_{r}-ir_{i}^{-}\,,$ (174)

where

$\displaystyle r_{r}$
$\displaystyle=\frac{1}{2\sqrt{6}}\Biggl{[}2^{2/3}(q+\sqrt{w})^{1/3}+24g_{2}^{\theta}\left(\frac{2}{q+\sqrt{w}}\right)^{1/3}$
$\displaystyle\qquad\qquad-2a^{2}(3+\cos 2\Theta)\Biggr{]}^{1/3}\,,$ (175)
$\displaystyle r_{i}^{\pm}$ $\displaystyle=\frac{\mp
1}{2}\sqrt{4r_{r}^{2}+a^{2}(3+2\cos
2\Theta)\pm\frac{2Ma^{2}\,\sin^{2}\Theta}{r_{r}}}\,,$ (176) $\displaystyle
g_{2}^{\theta}$ $\displaystyle=\frac{1}{96}a^{4}\left(67+60\cos 2\Theta+\cos
4\Theta\right)\,,$ (177) $\displaystyle q$
$\displaystyle=a^{2}\biggl{[}-128a^{2}+192a^{2}\sin^{2}\Theta$
$\displaystyle\qquad\qquad+12(9M^{2}-5a^{2})-2a^{2}\sin^{6}\Theta\biggr{]}\,,$
(178) $\displaystyle w$ $\displaystyle=q^{2}-6912(g_{2}^{\theta})^{2}\,.$
(179)

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$\bar{\mathcal{K}},w\Vdash\beta(x)$ we get some $w^{\prime}\in\widehat{W}$
such that $\bar{\mathcal{K}},w\Vdash Q_{w^{\prime}}$ and
$\widehat{\mathcal{K}},w^{\prime}\Vdash x$. From $\bar{\mathcal{K}},w\Vdash
Q_{w^{\prime}}$ we have $\bar{\mathcal{K}},w\Vdash q_{w^{\prime}}$ and hence
$w^{\prime}\sqsubseteq v\preccurlyeq w$ for some $v\in W$ with
$\mathcal{K},w^{\prime}\Vdash x$. If $w^{\prime}=v$ then
$w^{\prime}\preccurlyeq w$ and thus $\mathcal{K},w\Vdash x$. Otherwise we have
$w^{\prime}\sqsubset v$ and hence $w^{\prime}\mathrel{\widehat{\sqsubset}}v$
which implies $\bar{\mathcal{K}},w\Vdash\neg q_{v}$ and thus
$\bar{\mathcal{K}},w\nVdash q_{v}$. On the other hand since $v\preccurlyeq w$
we have $v\mathrel{\widetilde{\sqsubseteq}}w$ and hence
$\bar{\mathcal{K}},w\Vdash q_{v}$, a contradiction. ∎

######  Lemma .

$\beta(X^{A}_{w})\mathrel{\begin{picture}(2.0,2.0)\put(0.0,0.0){$\vdash$}
\put(0.1,1.2){ $$} \put(0.1,-0.4){ ${{\sf iGLC_{a}}}$} \end{picture}}\ \
\beta(E)$ implies $X^{A}_{w}\vdash E$ for every $\Box E\in{\sf sub}(A)$.

###### Proof.

We reason contrapositively. Assume that $X^{A}_{w}\nvdash E$ seeking to show
$\beta(X^{A}_{w})\mathrel{\begin{picture}(2.0,2.0)\put(0.0,0.0){$\nvdash$}
\put(0.1,1.2){ $$} \put(0.1,-0.4){ ${{\sf iGLC_{a}}}$} \end{picture}}\ \
\beta(E)$. From $X^{A}_{w}\nvdash E$ we have $\Phi_{w}\nvdash E$ and hence
$\mathcal{K}^{w}_{E},w_{E}\Vdash X^{A}_{w}$ and
$\mathcal{K}^{w}_{E},w_{E}\nVdash E$. Section 5.2 imples
$\bar{\mathcal{K}}^{w}_{E},w_{E}\Vdash\beta(X^{A}_{w})$ and
$\bar{\mathcal{K}}^{w}_{E},w_{E}\nVdash E$ and thus
$\beta(X^{A}_{w})\mathrel{\begin{picture}(2.0,2.0)\put(0.0,0.0){$\nvdash$}
\put(0.1,1.2){ $$} \put(0.1,-0.4){ ${{\sf iGLC_{a}}}$} \end{picture}}\ \
\beta(E)$. ∎

######  Lemma .

For every $u\in W^{w}_{E}$ and $B\in\mathcal{L}_{\Box}$ with ${\sf sub^{{\sf
par}}}(B)\subseteq{\sf sub^{{\sf par}}}(A)$ we have
$\mathcal{K}^{w}_{E},u\Vdash B$ iff
$\bar{\mathcal{K}}^{w}_{E},u\Vdash\beta(B)$.

###### Proof.

We prove this lemma by induction on complexity of $B$. All inductive steps are
derived easily by induction hypothesis, except for basic steps:

  * •

$B=x$ is an atomic variable. Section 5.2.

  * •

$B=p$ is a parameter. By assumption of this lemma we must have $p\in{\sf
sub}(A)$ and hence by definition of $\bar{\mathcal{K}}^{w}_{E}$ we have
$\mathcal{K}^{w}_{E},u\Vdash p$ iff $\bar{\mathcal{K}}^{w}_{E},u\Vdash p$. ∎

######  Lemma .

Let $\vdash F\to\beta(E)$ for some $F\in{{\sf S}{\sf N}(\Box)}$ and $E\in
Y\cup Y^{\prime\prime}$. Then there is some $G\in Y^{\prime\prime}$ such that
$\vdash F\to\beta(G)$ and $\vdash\beta(G)\to\beta(E)$.

###### Proof.

If $E\in Y^{\prime\prime}$ we are trivially done by $G:=E$. So assume that
$E\in Y$. Assume that $\vdash F\to\beta(E)$ seeking some $G\in
Y^{\prime\prime}$ with $\vdash F\to\beta(G)$ and $\vdash\beta(G\to E)$. Since
$F\in{{\sf S}{\sf N}(\Box)}$, sections 3.7 and 3.19 imply
$\vdash(\beta(E))^{\star}\to\beta(E)$ and $\vdash F\to(\beta(E))^{\star}$ with
$(\beta(E))^{\star}\in{{\sf N}(\Box)}$ and ${\sf
sub_{o}}((\beta(E))^{\star})\subseteq{\sf sub_{o}}(\beta(E))$. Hence there is
some $G_{0}\in{{\sf N}(\Box)}$ such that $\beta(G_{0})=(\beta(E))^{\star}$ and
${\sf sub_{o}}(G_{0})\subseteq{\sf sub_{o}}(E)$ and thus $G_{0}\in Y$. Let
$G:=G_{0}^{\Box}$. Obviously we have $G\in Y^{\prime\prime}$. Since $\vdash
F\to\beta(G_{0})$, theorem 2.2 implies $\vdash F\to\beta(G)$. Moreover since
$\vdash\beta(G)\to\beta(G_{0})$ and $\vdash\beta(G_{0})\to\beta(E)$ we get
$\vdash\beta(G\to E)$. ∎

## Acknowledgement

Special acknowledgement heartily granted to Amirhossein Akbar Tabatabai,
Mohammad Ardeshir, Lev Beklemishev and Albert Visser for stimulating
discussions or communications we had on provability logic.

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