if $\ell_{lb}^{+}(\varepsilon)<\ell_{z}<\ell_{ub}^{+}(\varepsilon)$ or
$\ell_{ub}^{-}(\varepsilon)<\ell_{z}<\ell_{lb}^{-}(\varepsilon)$

       * •

If $\tilde{\gamma}$ starts at some point in
$A^{K,trapped}(\varepsilon,\ell_{z})$, then $\tilde{\gamma}$ is either trapped
or spherical with radius $\tilde{r}^{\pm}(\varepsilon,\ell_{z})$, given by
Lemma 33.

       * •

Otherwise, $\tilde{\gamma}$ starts at some point in
$A^{K,abs}(\varepsilon,\ell_{z})$ and reaches the horizon in a finite proper
time.

    4. (d)

if $\ell_{z}=\ell_{ub}^{+}(\varepsilon)$ or
$\ell_{z}=\ell_{ub}^{-}(\varepsilon)$,

       * •

If $\tilde{\gamma}$ starts at
$\displaystyle\left(r^{K}_{2}(\varepsilon,\ell_{z}),\frac{\pi}{2}\right)$,
then $\tilde{\gamma}$ is a circle confined in the equatorial plane.

       * •

Otherwise, $\tilde{\gamma}$ starts at some point in
$A^{K,abs}(\varepsilon,\ell_{z})$ and reaches the horizon in a finite proper
time.

    5. (e)

If $\varepsilon^{2}>1$

      1. i.

If $\ell_{lb}^{-}(\varepsilon)<\ell_{z}<\ell_{lb}^{+}(\varepsilon)$, then

         * •

If $\tilde{\gamma}$ starts at the point
$\displaystyle\left\\{\overline{r}(\varepsilon,\ell_{z})\right\\}\times[\overline{\theta}_{max}^{\geq
1}(\varepsilon,\ell_{z}),\pi-\overline{\theta}_{max}^{\geq
1}(\varepsilon,\ell_{z})]$, then $\tilde{\gamma}$ has a constant radius and
oscillates between $\overline{\theta}_{max}^{\geq 1}(\varepsilon,\ell_{z})$
and $\pi-\overline{\theta}_{max}^{\geq 1}(\varepsilon,\ell_{z})$ in the
$\theta$ direction, where $\displaystyle\overline{\theta}_{max}^{\geq
1}(\varepsilon,\ell_{z})$ is given by (4.115)

         * •

If $\tilde{\gamma}$ starts at some point in the region
$]r_{H},\infty[\times[\theta_{min}(\varepsilon,\ell_{z}),\pi-\theta_{min}(\varepsilon,\ell_{z})]\backslash\left\\{\overline{r}(\varepsilon,\ell_{z})\right\\}\times[\overline{\theta}_{max}^{\geq
1}(\varepsilon,\ell_{z}),\pi-\overline{\theta}_{max}^{\geq
1}(\varepsilon,\ell_{z})]$, then $\tilde{\gamma}$ remains in the region
$A^{K,q\geq 0}$.

           * –

if $\tilde{\gamma}$ starts with a negative radial velocity, then it reaches
the horizon in a finite proper time while oscillating around the equatorial
plane.

           * –

if $\tilde{\gamma}$ starts with a positive radial velocity, then it goes to
infinity while oscillating around the equatorial plane.

         * •

If $\tilde{\gamma}$ starts in the region $A^{K,z>0}(\varepsilon,\ell_{z})$ or
in the region $A^{K,z<0}(\varepsilon,\ell_{z})$. Then,

           * –

if $\tilde{\gamma}$ starts with a negative radial velocity, then it reaches
the horizon in a finite proper time while staying inside
$A^{K,z>0}(\varepsilon,\ell_{z})$ or $A^{K,z<0}(\varepsilon,\ell_{z})$.

           * –

if $\tilde{\gamma}$ starts with a positive radial velocity, then it goes to
infinity while staying in the region $A^{K,z>0}(\varepsilon,\ell_{z})$ or
$A^{K,z<0}(\varepsilon,\ell_{z})$.

      2. ii.

If $\ell_{z}=\ell_{lb}^{+}(\varepsilon)$ or
$\ell_{z}=\ell_{lb}^{-}(\varepsilon)$, then

         * •

If $\tilde{\gamma}$ starts at
$\displaystyle\left(r^{K}_{0}(\varepsilon,\ell_{z}),\frac{\pi}{2}\right)$,
then $\tilde{\gamma}$ is a circle confined in the equatorial plane.

         * •

Otherwise,

           * –

If $\tilde{\gamma}$ starts at some point in $A^{K,abs}(\varepsilon,\ell_{z})$

             * *

If $\gamma$ is confined in the equatorial plane and starts with a positive
radial velocity, then it approaches the circle of radius
$r^{K}_{0}(\varepsilon,\ell_{z})$ in an infinite proper time,

             * *

otherwise, it reaches the horizon in a finite proper time.

           * –

otherwise, i.e if $\tilde{\gamma}$ starts at some point in
$A^{K,scattered}(\varepsilon,\ell_{z})$, then

             * *

If $\gamma$ is confined in the equatorial plane and starts with a negative
radial velocity, then it approaches the circle of radius
$r^{K}_{0}(\varepsilon,\ell_{z})$ in an infinite proper time.

             * *

Otherwise, the orbit (with negative initial radial velocity) hits a potential
barrier and goes back infinity while oscillating around the equatorial plane
or (with positive initial radial velocity) goes to infinity while trapped in
the $\theta-direction$ between $\theta(0)$ and $\pi-\theta(0)$.

approaches the circle of radius $r^{K}_{0}(\varepsilon,\ell_{z},0)$ and
confined in the equatorial plane in an infinite proper time.

      3. iii.

If $\ell_{z}>\ell_{lb}^{+}(\varepsilon)$ or
$\ell_{z}<\ell_{lb}^{-}(\varepsilon)$, then

         * •

If $\tilde{\gamma}$ starts at some point in the region
$A^{K,scattered}(\varepsilon,\ell_{z})$, then the orbit (with negative initial
radial velocity) hits a potential barrier and goes back infinity while
oscillating around the equatorial plane between… or (with positive initial
radial velocity) goes to infinity while trapped in the $\theta-direction$.

         * •

Otherwise, $\tilde{\gamma}$ starts at some point in the region
$A^{K,abs}(\varepsilon,\ell_{z})$. It reaches the horizon in a finite proper
time while oscillating around the equatorial plane.

###### Proof.

Let $\gamma:I\to\mathcal{O}$ be a timelike future-directed geodesic with
initial conditions $(\gamma(0),\dot{\gamma}(0))$. First of all, recall that by
Lemma 5, the nature of $\gamma$ is determined by the set
$(\varepsilon,\ell_{z},q)$ and $\gamma(0)=(t(0),\phi(0),r(0),\theta(0))$.
Moreover, due to spacetime symmetries, it suffices to determine the nature of
its projection in the $(r,\theta)-$plane. We compute
$(\varepsilon,\ell_{z},q)$ from the initial condition by:

$\varepsilon=V_{K}(r(0),\theta(0))v^{t}(0)-W_{K}(r(0),\theta(0))v^{\phi}(0),$
$\ell_{z}=W_{K}(r(0),\theta(0))v^{t}(0)+X_{K}(r(0),\theta(0))v^{\phi}(0),$

and

$q=v_{\theta}(0)^{2}+\cos^{2}\theta(0)\left(d^{2}(1-\varepsilon^{2})+\frac{\ell_{z}^{2}}{\sin^{2}\theta(0)}\right),$

Now, we study the possible maximal solutions of the reduced system (4.29) for
all $(\varepsilon,\ell_{z})\in\mathcal{A}^{admissible}$.
Let $(\tilde{r},\tilde{\theta})\in]r_{H},\infty[\times]0,\pi[$ and let
$(\tilde{v}_{r},\tilde{v}_{\theta})\in\mathbb{R}^{2}$ and consider the
following Cauchy problem

$\left\\{\begin{aligned} \frac{dr}{d\tau}&=\frac{\Delta}{\Sigma^{2}}v_{r},\\\
\frac{dv_{r}}{d\tau}&=\frac{1}{2\Sigma^{2}}\left(-\Delta^{\prime}(r)v_{r}^{2}+\frac{\Delta^{\prime}(r)R(r,\varepsilon,\ell_{z},q)-\Delta(r)\partial_{r}R(r,\varepsilon,\ell_{z},q)}{\Delta(r)^{2}}\right),\\\
\frac{d\theta}{d\tau}&=\frac{1}{\Sigma^{2}}v_{\theta},\\\
\frac{dv_{\theta}}{d\tau}&=\frac{1}{2\Sigma^{2}}\left(\partial_{\theta}\left(\frac{T(\cos\theta,\varepsilon,\ell_{z},q)}{\sin^{2}\theta}\right)\right)\end{aligned}\right.$
(4.135)

with initial conditions

$\displaystyle\tilde{\gamma}(0)$
$\displaystyle=(\overline{r},\overline{\theta}),$
$\displaystyle\dot{\tilde{\gamma}}(0)$
$\displaystyle=(\overline{v}_{r},\overline{v}_{\theta}).$

By Cauchy-Lipschitz theorem, there exists a unique maximal solution for the
above system given by $(\tilde{\gamma},\dot{\tilde{\gamma}},I\ni 0)$. We
decouple the equations for $(r,v_{r})$ and for $(\theta,v_{\theta})$, we use
Mino time, $\Lambda$, defined by (4.6). We have

$\forall\tau\in
I\;:\;\Lambda(\tau)=\int_{0}^{\tau}\frac{1}{\Sigma^{2}(r(s),\theta(s))}\,ds.$

Now set $J:=\Lambda(I)$. Then, the above system becomes: $\forall\lambda\in J$

$\left\\{\begin{aligned}
\frac{dr}{d\lambda}&=\Delta(r(\Lambda^{-1}(\lambda)))v_{r}(\Lambda^{-1}(\lambda)),\\\
\frac{dv_{r}}{d\lambda}&=\frac{1}{2}\left(-\Delta^{\prime}(r(\Lambda^{-1}(\lambda)))v_{r}^{2}(\Lambda^{-1}(\lambda))\right.\\\
&\left.+\frac{\Delta^{\prime}(r(\Lambda^{-1}(\lambda)))R(r(\Lambda^{-1}(\lambda)),\varepsilon,\ell_{z},q)-\Delta(r(\Lambda^{-1}(\lambda)))\partial_{r}R(r(\Lambda^{-1}(\lambda)),\varepsilon,\ell_{z},q)}{\Delta(r(\Lambda^{-1}(\lambda)))^{2}}\right),\\\
\frac{d\theta}{d\lambda}&=v_{\theta}(\Lambda^{-1}(\lambda)),\\\
\frac{dv_{\theta}}{d\lambda}&=\frac{1}{2}\left(\partial_{\theta}\left(\frac{T(\cos(\theta(\Lambda^{-1}(\lambda))),\varepsilon,\ell_{z},q)}{\sin^{2}(\theta(\Lambda^{-1}(\lambda)))}\right)\right)\end{aligned}\right.$

In the following, we identify $r\circ\Lambda^{-1}$, $\theta\circ\Lambda^{-1}$,
$v_{r}\circ\Lambda^{-1}$ and $v_{\theta}\circ\Lambda^{-1}$with $r$, $\theta$,
$v_{r}$ and $v_{\theta}$ respectively. We consider the two Cauchy problems:

$\left\\{\begin{aligned}
\frac{dr}{d\lambda}&=\Delta(r(\lambda))v_{r}(\lambda),\\\
\frac{dv_{r}}{d\lambda}&=\frac{1}{2}\left(-\Delta^{\prime}(r(\lambda))v_{r}^{2}(\lambda)+\frac{\Delta^{\prime}(r(\lambda))R(r(\lambda),\varepsilon,\ell_{z},q)-\Delta(r(\lambda))\partial_{r}R(r(\lambda),\varepsilon,\ell_{z},q)}{\Delta(r(\lambda))^{2}}\right),\\\
r(0)&=\overline{r}\;\;,\;\;v_{r}(0)=\overline{v}_{r}\end{aligned}\right.$
(4.136)

and

$\left\\{\begin{aligned} \frac{d\theta}{d\lambda}&=v_{\theta}(\lambda),\\\
\frac{dv_{\theta}}{d\lambda}&=\frac{1}{2}\left(\partial_{\theta}\left(\frac{T(\cos(\theta(\lambda)),\varepsilon,\ell_{z},q)}{\sin^{2}(\theta(\lambda))}\right)\right)\\\
\theta(0)&=\overline{\theta}\;\;,\;\;v_{\theta}(0)=\overline{v}_{\theta}.\end{aligned}\right.$
(4.137)

It is easy to see that at a given $(\varepsilon,\ell_{z},q)$, if
$\left(r,\theta,v_{r},v_{\theta},I\right)$ is the maximal solution of (4.135),
then $(r,v_{r},J)$ and $(\theta,v_{\theta},J)$ are the maximal solutions of
(4.136) and (4.137) respectively. Reciprocally, if $(r,v_{r},J_{r})$ and
$(\theta,v_{\theta},J_{\theta})$ are the maximal solutions of (4.136) and
(4.137) respectively, then
$(r,\theta,v_{r},v_{\theta},I=\Lambda^{-1}(J_{r}\cap J_{\theta}))$.
Consequently, we classify the maximal solutions of (4.136) and (4.137) in
order to obtain the general classification.
In the following, we tackle in details the case when
$(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$. The remaining cases are
similar.

  1. 1.

If $(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, then $\forall\lambda\in
J:=J_{r}\cap J_{\theta},\;(r,\theta)(\lambda)\in
A^{K,abs}(\varepsilon,\ell_{z})\sqcup A^{K,trapped}(\varepsilon,\ell_{z})$ and
more precisely, $\forall\lambda\in J$

$r(\lambda)\in]r_{H},r_{0}^{K}(\varepsilon,\ell_{z},q)]\sqcup[r_{1}^{K}(\varepsilon,\ell_{z},q),r_{2}^{K}(\varepsilon,\ell_{z},q)]\quad\text{and}\quad\theta(\lambda)\in[\theta_{min}(\varepsilon,\ell_{z},q,d),\pi-\theta_{min}(\varepsilon,\ell_{z},q,d)]$

where $r_{0}^{K}(\varepsilon,\ell_{z},q),r_{1}^{K}(\varepsilon,\ell_{z},q)$
and $r_{2}^{K}(\varepsilon,\ell_{z},q)$ are the roots of the equation (4.109)
and $\displaystyle\theta_{min}(\varepsilon,\ell_{z},q,d)$ is the unique angle
in $\left]0,\frac{\pi}{2}\right]$ that solves (4.33).

    1. (a)

Let $(r,v_{r},J_{r})$ be the maximal solution of (4.136). Then,
$\forall\lambda\in J_{r},$ we have

$2\frac{dv_{r}}{d\lambda}=-\Delta^{\prime}(r(\lambda))v_{r}^{2}(\lambda)+\frac{\Delta^{\prime}(r(\lambda))R(r(\lambda),\varepsilon,\ell_{z},q)-\Delta(r(\lambda))\partial_{r}R(r(\lambda),\varepsilon,\ell_{z},q)}{\Delta(r(\lambda))^{2}}.$

We multiply the latter by $v_{r}$ to obtain

$\frac{d(v_{r})^{2}}{d\lambda}=-\Delta^{\prime}(r(\lambda))v_{r}^{3}(\lambda)+\frac{\Delta^{\prime}(r(\lambda))R(r(\lambda),\varepsilon,\ell_{z},q)-\Delta(r(\lambda))\partial_{r}R(r(\lambda),\varepsilon,\ell_{z},q)}{\Delta(r(\lambda))^{2}}v_{r}.$

By the first equation of (4.136), we have

$\frac{d(v_{r})^{2}}{d\lambda}=-\Delta^{\prime}(r(\lambda))v_{r}^{2}(\lambda)\dot{r}\frac{1}{\Delta}+\frac{\Delta^{\prime}(r(\lambda))R(r(\lambda),\varepsilon,\ell_{z},q)-\Delta(r(\lambda))\partial_{r}R(r(\lambda),\varepsilon,\ell_{z},q)}{\Delta(r(\lambda))^{2}}\frac{\dot{r}}{\Delta}.$

Therefore,

$\frac{d}{d\lambda}(\Delta
v_{r}^{2})=\frac{d}{d\lambda}\left(\frac{R(r,\varepsilon,\ell_{z},q)}{\Delta}\right).$

Finally we integrate between $0$ and some $\lambda\in J_{r}$ to obtain

$\Delta^{2}v_{r}^{2}=R(r,\varepsilon,\ell_{z},\varepsilon,\ell_{z})+(\Delta^{2}(\overline{r})\overline{v}_{r}-R(\overline{r},\varepsilon,\ell_{z},\varepsilon,q))$

Now, we have

$(\Delta^{2}(\overline{r})\overline{v}_{r}-R(\overline{r},\varepsilon,\ell_{z},\varepsilon,q))=0.$

Therefore,

$\Delta^{2}v_{r}^{2}=R(r,\varepsilon,\ell_{z},q).$

    2. (b)

     * •

If $(\tilde{r},\tilde{\theta})\in A^{K,abs}(\varepsilon,\ell_{z})$. Then,
$\forall\lambda\in J_{r}\cap J_{\theta}$

$r(\lambda)\in]r_{H},r_{0}^{K}(\varepsilon,\ell_{z},q)]\quad\text{and}\quad\theta(\lambda)\in[\theta_{min}(\varepsilon,\ell_{z},q),\pi-\theta_{min}(\varepsilon,\ell_{z},q)].$

Let $J_{r}=]T_{min},T^{max}[$. We claim that

$-\infty<T_{min}\quad,\quad T_{max}<+\infty$

and

$\lim_{\lambda\to T_{max}}\,r(\lambda)=\lim_{\lambda\to
T_{min}}\,r(\lambda)=r_{H}.$

       1. (a)

If $(\overline{r},\overline{\theta})\in\partial
A^{K,abs}(\varepsilon,\ell_{z})$, then
$(\overline{v}_{r},\overline{v}_{\theta})=(0,0)$ and

$\displaystyle\overline{r}$ $\displaystyle=r^{K}_{0}(\varepsilon,\ell_{z},q),$
$\displaystyle\overline{\theta}$
$\displaystyle\in\left\\{\theta_{min}(\varepsilon,\ell_{z},q),\pi-\theta_{min}(\varepsilon,\ell_{z},q)\right\\}.$

We claim that $\forall\lambda\in J_{r}\backslash\left\\{0\right\\}$,

$R(r(\lambda))>0.$

In fact, suppose that there exists
$\tilde{\lambda}\in]T^{min},0[\cup]0,T^{max}[$ such that $\displaystyle
R(r(\tilde{\lambda}))=0.$ Then,

$(\Delta^{-1}(r(\cdot))\dot{r})(\tilde{\lambda})=0.$

Besides, we have

$(\Delta^{-1}(r(\cdot))\dot{r})(0)=0.$

By Rolle’s theorem, there exists $\lambda_{0}\in]0,\tilde{\lambda}[$ such that

$\frac{d}{d\lambda}(\Delta^{-1}(r(\cdot))\dot{r})(\lambda_{0})=0.$

Therefore, by the second equation of (4.136),

$\partial_{r}R(r(\lambda_{0}),\varepsilon,\ell_{z},q)=0$

. Contradiction since the $r-$derivative of $R$ is negative on
$]r_{H},r^{K}_{0}(\varepsilon,\ell_{z},q)]$.
This implies

$\forall\lambda\in
J_{r}\backslash\left\\{0\right\\}\;,\;\frac{dr}{d\lambda}=\pm\sqrt{R(r(\lambda),\varepsilon,\ell_{z},q)}.$

More precisely,

$\displaystyle\forall\lambda\in]0,T^{max}[\;,\;\dot{r}=-\sqrt{R(r(\lambda),\varepsilon,\ell_{z},q)},\;$
$\displaystyle\forall\lambda\in]T^{min},0[\;,\;\dot{r}=\sqrt{R(r(\lambda),\varepsilon,\ell_{z},q)}.$

Now, we claim that

$T^{max}=\int_{r_{H}}^{r_{0}^{K}(\varepsilon,\ell_{z},q)}\;\frac{1}{\sqrt{R(r(s),\varepsilon,\ell_{z},q)}}\,ds$

and

$T^{min}=-\int_{r_{H}}^{r_{0}^{K}(\varepsilon,\ell_{z},q)}\;\frac{1}{\sqrt{R(r(s),\varepsilon,\ell_{z},q)}}\,ds.$

Let $\lambda\in]0,T^{max}[$. We have

$1=-\frac{\dot{r}(\lambda)}{{\sqrt{R(r(\lambda),\varepsilon,\ell_{z},q)}}}$

We integrate between $0$ and $\lambda$ to obtain

$\lambda=-\int_{0}^{\lambda}\;\frac{\dot{r}(s)}{{\sqrt{R(r(s),\varepsilon,\ell_{z},q)}}}\,ds.$

Now we make the change of variable $\displaystyle u=r(s)$ in the right hand
side. We obtain

$\lambda=\int_{r_{H}}^{r_{0}^{K}(\varepsilon,\ell_{z},q)}\;\frac{du}{{\sqrt{R(u,\varepsilon,\ell_{z},q)}}}.$

We introduce the function $G(\cdot,\varepsilon,\ell_{z},q)$ defined on
$]r_{H},r^{K}_{0}(\varepsilon,\ell_{z},q)]$ by

$G(s,\varepsilon,\ell_{z},q):=\int_{s}^{r^{K}_{0}(\varepsilon,\ell_{z},q)}\;\frac{du}{{\sqrt{R(u,\varepsilon,\ell_{z},q)}}}.$

Since $r^{K}_{0}(\varepsilon,\ell_{z},q)$ is a simple root of $R$, $G$ is well
defined. Moreover $G(\cdot,\varepsilon,\ell_{z},q)$ is monotonically
decreasing on $]r_{H},r^{K}_{0}(\varepsilon,\ell_{z},q)]$. Therefore, it is
bijective from $]r_{H},r^{K}_{0}(\varepsilon,\ell_{z},q)]$ to $[0,T^{max}[$
where

$T^{max}:=\lim_{r\to
r_{H}}G(s)=\int_{r_{H}}^{r_{0}^{K}(\varepsilon,\ell_{z},q)}\;\frac{1}{\sqrt{R(r(s),\varepsilon,\ell_{z},q)}}\,ds<+\infty.$

Hence,

$\forall\lambda\in[0,T^{max}[\;,\;r(\lambda)=G^{-1}(\lambda).$

In the same manner, we obtain

$T^{min}:=\lim_{r\to
r_{H}}H(s)=-\int_{r_{H}}^{r_{0}^{K}(\varepsilon,\ell_{z},q)}\;\frac{1}{\sqrt{R(r(s),\varepsilon,\ell_{z},q)}}\,ds>-\infty$

and

$\forall\lambda\in]T^{min},0]\;,\;r(\lambda)=H^{-1}(\lambda)$

where $H(\cdot,\varepsilon,\ell_{z},q)$ is the function defined on
$]r_{H},r^{K}_{0}(\varepsilon,\ell_{z},q)]$ by

$H(s,\varepsilon,\ell_{z},q):=-\int_{s}^{r^{K}_{0}(\varepsilon,\ell_{z},q)}\;\frac{du}{{\sqrt{R(u,\varepsilon,\ell_{z},q)}}}.$

It remains to analyse the motion in the $\theta$-direction in order to
determine $J_{\theta}$: $\forall\lambda\in
J_{\theta}\;,\;\theta(\lambda)\in\tilde{\theta}\in[\theta_{min}(\varepsilon,\ell_{z},q),\pi-\theta_{min}(\varepsilon,\ell_{z},q)]$.
By compactness, $J_{\theta}=\mathbb{R}$. Moreover, $\lambda\to\theta(\lambda)$
is periodic with period

$T_{\theta}:=2\int_{\theta_{min}(\varepsilon,\ell_{z},q)}^{\pi-\theta_{min}(\varepsilon,\ell_{z},q)}\;\frac{\sin\sigma}{\sqrt{T(\cos\sigma,\varepsilon,\ell_{z},q)}}\,d\sigma.$
(4.138)

In fact, let $(\theta,v_{\theta},J_{\theta})$ be the maximal solution of
(4.137). Introduce the function $\tilde{G}_{1}(\cdot,\varepsilon,\ell_{z},q)$
defined on
$[\theta_{min}(\varepsilon,\ell_{z},q),\pi-\theta_{min}(\varepsilon,\ell_{z},q)]$
by

$\tilde{G}_{1}(s,\varepsilon,\ell_{z},q):=\int_{\theta_{min}(\varepsilon,\ell_{z},q)}^{s}\frac{\sin\sigma}{\sqrt{T(\cos\sigma,\varepsilon,\ell_{z},q)}}\,d\sigma$

Since $\cos(\theta_{min}(\varepsilon,\ell_{z},q))$ is a simple root of $T$,
$\tilde{G}(\cdot,\varepsilon,\ell_{z},q)$ is well-defined. Moreover, it is
monotonically increasing on its domain so that it defines a bijection from
$[\theta_{min}(\varepsilon,\ell_{z},q),\pi-\theta_{min}(\varepsilon,\ell_{z},q)]$
to $\left[0,\frac{T_{\theta}}{2}\right]$. Now we denote its inverse by
$\tilde{G}^{-1}_{1}(\cdot,\varepsilon,\ell_{z},q)$. In the same way, we define
the bijective function $\tilde{G}_{2}(\cdot,\varepsilon,\ell_{z},q)$ defined
from
$[\theta_{min}(\varepsilon,\ell_{z},q),\pi-\theta_{min}(\varepsilon,\ell_{z},q)]$
to $\left[\frac{T_{\theta}}{2},T_{\theta}\right]$ by

$\tilde{G}_{2}(s,\varepsilon,\ell_{z},q):=\frac{T_{\theta}}{2}+\int^{\pi-\theta_{min}(\varepsilon,\ell_{z},q)}_{s}\frac{\sin\sigma}{\sqrt{T(\cos\sigma,\varepsilon,\ell_{z},q)}}\,d\sigma$

and we denote its inverse by
$\tilde{G}_{2}^{-1}(\cdot,\varepsilon,\ell_{z},q)$. Now we define
$\tilde{\theta}$ on $[0,T_{\theta}]$ by

$\tilde{\theta}(\lambda):=\left\\{\begin{aligned}
&\tilde{G}^{-1}_{1}(\lambda,\varepsilon,\ell_{z},q)\;\text{if}\;\lambda\in\left[0,\frac{T_{\theta}}{2}\right]\\\
&\tilde{G}^{-1}_{2}(\lambda,\varepsilon,\ell_{z},q)\;\text{if}\;\lambda\in\left[\frac{T_{\theta}}{2},T_{\theta}\right].\end{aligned}\right.$

and $\tilde{v}_{\theta}$ on $[0,T_{\theta}]$ by

$\tilde{v}_{\theta}:=\theta^{\prime}(\lambda).$

Now, we can extend $(\tilde{\theta},\tilde{v}_{\theta})$ to a periodic
solution defined on $\mathbb{R}$. Moreover, It easy to see that
$(\tilde{\theta},\tilde{v}_{\theta})$ satisfies(4.137). By uniqueness,
$\theta$ is periodic with period $T_{\theta}$.

       2. (b)

If $(\overline{r},\overline{\theta})\in
A^{K,abs}(\varepsilon,\ell_{z})\backslash(\partial
A^{K,abs}(\varepsilon,\ell_{z}))$. Then, we proceed as above to obtain the
same result.

     * •

If $(\tilde{r},\tilde{\theta})\in A^{K,trapped}(\varepsilon,\ell_{z})$.
Suppose that $(\tilde{r},\tilde{\theta})\in\partial
A^{K,trapped}(\varepsilon,\ell_{z})$. The other cases are studied in the same
manner.

$\tilde{r}\in[r_{1}^{K}(\varepsilon,\ell_{z},q),r_{2}^{K}(\varepsilon,\ell_{z},q)]\quad\text{and}\quad\tilde{\theta}\in[\theta_{min}(\varepsilon,\ell_{z},q),\pi-\theta_{min}(\varepsilon,\ell_{z},q)]$

Therefore, $\displaystyle\forall\lambda\in
J\;,\;(r,\theta)(\lambda)\in[r_{1}^{K}(\varepsilon,\ell_{z},q),r_{2}^{K}(\varepsilon,\ell_{z},q)]\times[\theta_{min}(\varepsilon,\ell_{z},q,d),\pi-\theta_{min}(\varepsilon,\ell_{z},q,d)]$
for all $\tau\in I$. By compactness, $J=\mathbb{R}$ and $\tilde{\gamma}$ is
trapped. Moreover, $r$ is periodic with period:

$T_{r}:=\int_{r_{1}^{K}(\varepsilon,\ell_{z},q)}^{r_{2}^{K}(\varepsilon,\ell_{z},q)}\;\frac{dr}{\sqrt{R(r,\varepsilon,\ell_{z},q)}}.$
(4.139)

and $\theta$ is periodic with period $T_{\theta}$ defined by (4.138). For the
periodicity of $\theta$, it has already been tackled in the first case. As for
the periodicity of $r$, we proceed in the same way: let $(r,v_{r},J_{r})$ be
the maximal solution of (4.136). Introduce the function
$\overline{G}_{1}(\cdot,\varepsilon,\ell_{z},q)$ defined on
$[r^{K}_{1}(\varepsilon,\ell_{z},q),r^{K}_{2}(\varepsilon,\ell_{z},q)]$ by

$\overline{G}_{1}(s,\varepsilon,\ell_{z},q):=\int_{r^{K}_{1}(\varepsilon,\ell_{z},q)}^{s}\frac{d\sigma}{\sqrt{R(\sigma,\varepsilon,\ell_{z},q)}}\,d\sigma$

Since $r^{K}_{1}(\varepsilon,\ell_{z},q)$ and
$r^{K}_{2}(\varepsilon,\ell_{z},q)$ are simple roots of $R$,
$\overline{G}(\cdot,\varepsilon,\ell_{z},q)$ is well-defined. Moreover, it is
monotonically increasing on its domain so that it defines a bijection from
$[r^{K}_{1}(\varepsilon,\ell_{z},q),r^{K}_{2}(\varepsilon,\ell_{z},q)]$ to
$\left[0,\frac{T_{r}}{2}\right]$. Now, we denote its inverse by
$\overline{G}^{-1}_{1}(\cdot,\varepsilon,\ell_{z},q)$. In the same way, we
define the bijective function $\overline{G}_{2}(\cdot,\varepsilon,\ell_{z},q)$
defined from
$[r^{K}_{1}(\varepsilon,\ell_{z},q),r^{K}_{2}(\varepsilon,\ell_{z},q)]$ to
$\left[\frac{T_{r}}{2},T_{r}\right]$ by

$\overline{G}_{2}(s,\varepsilon,\ell_{z},q):=\frac{T_{r}}{2}+\int^{r^{K}_{2}(\varepsilon,\ell_{z},q)}_{s}\frac{d\sigma}{\sqrt{R(\sigma,\varepsilon,\ell_{z},q)}}\,d\sigma$

and we denote its inverse by
$\overline{G}_{2}^{-1}(\cdot,\varepsilon,\ell_{z},q)$. Now we define
$\tilde{r}$ on $[0,T_{r}]$ by

$\tilde{r}(\lambda):=\left\\{\begin{aligned}
&\overline{G}^{-1}_{1}(\lambda,\varepsilon,\ell_{z},q)\;\text{if}\;\lambda\in\left[0,\frac{T_{r}}{2}\right]\\\
&\overline{G}^{-1}_{2}(\lambda,\varepsilon,\ell_{z},q)\;\text{if}\;\lambda\in\left[\frac{T_{r}}{2},T_{r}\right].\end{aligned}\right.$

and $\tilde{v}_{r}$ on $[0,T_{r}]$ by

$\tilde{v}_{r}:=\Delta^{-1}(r(\lambda))r^{\prime}(\lambda).$

Now, we can extend $(\tilde{r},\tilde{v}_{r})$ to a periodic solution defined
on $\mathbb{R}$. Moreover, It easy to see that $(\tilde{r},\tilde{v}_{r})$
satisfies(4.136). By uniqueness, $r$ is periodic with period $T_{r}$.

  2. 2.

The remaining cases follow using similar arguments.

∎

###  4.2 Study of the geodesic motion in Weyl coordinates

The aim of this section is to analyse the geodesic motion in the Weyl
coordinates defined in Section 3.4. We will focus on trapped non-spherical
geodesics. Weyl coordinates are well adapted to the axisymmetric problem
especially when it comes to the resolution of the reduced Einstein Vlasov
system. In a Kerr exterior, we have already seen that the geodesic motion
forms an integrable system in BL coordinates. In particular, the $r-$motion
decouples from the $\theta-$motion. In a general stationary and axisymmetric
spacetime, this is not necessarily true. Therefore, it is useful to study the
Kerr geodesic motion in Weyl coordinates without relying on the decoupling of
the $r-$motion and the $\theta$-motion, i.e without relying on the existence
of $q$.
Let $x=(t,\phi,\rho,z)\in\mathcal{M}$, let $v=(v^{t},v^{\phi},v^{\rho},v^{z})$
be the conjugate coordinates to the spacetime coordinates. In view of Section
3.2.5, the geodesics equation reduces to the following two degree of freedom
problem:

$\left\\{\begin{aligned} \frac{d\rho}{d\tau}&=v^{\rho},\\\
\frac{dz}{d\tau}&=v^{z},\\\
\frac{dv^{\rho}}{d\tau}&=-\frac{1}{2}e^{-2\lambda}\partial_{\rho}J^{K}(\rho,z,\varepsilon,\ell_{z},d)-\tensor{\Gamma}{{}^{\rho}_{ij}}v^{i}v^{j},\quad
i,j\in\left\\{\rho,z\right\\}\\\
\frac{dv^{z}}{d\tau}&=-\frac{1}{2}e^{-2\lambda}\partial_{z}J^{K}(\rho,z,\varepsilon,\ell_{z},d)-\tensor{\Gamma}{{}^{z}_{ij}}v^{i}v^{j}.\end{aligned}\right.$
(4.140)

where $J^{K}:\mathscr{B}\times\mathbb{R}\times\mathbb{R}$ is defined by

$J^{K}(\rho,z,\varepsilon,\ell_{z}):=-1+\frac{X_{K}}{\sigma_{K}^{2}}\varepsilon^{2}+\frac{2W_{K}}{\sigma_{K}^{2}}\varepsilon\ell_{z}-\frac{V_{K}}{\sigma_{K}^{2}}\ell_{z}^{2}$
(4.141)

We introduce the effective potential energy
$E_{\ell_{z}}^{K}:\mathscr{B}\to\mathbb{R}$ relative to a timelike future
directed geodesic, $(\gamma,I),$ with angular momentum $\ell_{z}$ and energy
$\varepsilon$:

$E^{K}_{\ell_{z}}(\rho,z):=\frac{-W_{K}(\rho,z)}{X_{K}(\rho,z)}\ell_{z}+\frac{\sigma_{K}}{X_{K}(\rho,z)}\sqrt{\ell_{z}^{2}+X_{K}(\rho,z)}.$
(4.142)

We refer to Figure 10 for the shape of $E^{K}_{\ell_{z}}$ and we recall that
the allowed region for $\gamma$ is given by

$\displaystyle A^{K}(\varepsilon,\ell_{z})$
$\displaystyle=\left\\{(\rho,z)\in\mathscr{B}\;:\;J^{K}(\rho,z,\varepsilon,\ell_{z})\geq
0\right\\}$
$\displaystyle=\left\\{(\rho,z)\in\mathscr{B}\;:\;E_{\ell_{z}}^{K}(\rho,z)\leq\varepsilon\right\\}$
$\displaystyle=\left\\{(r,\theta)\in]r_{H}(d),\infty[\times]0,\pi[\;:\;R(r,\varepsilon,\ell_{z},q)\geq
0\quad\text{and}\quad T(\cos\theta,\varepsilon,\ell_{z},q)\geq 0\right\\}$

We also recall that the boundary of $A^{K}(\varepsilon,\ell_{z})$ is the zero
velocity curve $Z^{K}(\varepsilon,\ell_{z})$ given by Definition 12.

######  Remark 19.

$Z^{K}(\varepsilon,\ell_{z})$ can also be seen as the level sets of the
effective potential energy $E^{K}_{\ell_{z}}$ at $\varepsilon$.

####  4.2.1 Properties of the effective potential energy $E^{K}_{\ell_{z}}$

The classification of timelike geodesics is based on the topology of the
$Z^{K}(\varepsilon,\ell_{z})$ curves whose shapes (depending on
$(\varepsilon,\ell_{z})$) were already determined in Section 4.1.7,
Proposition 12. In this section, we will rewrite the latter proposition in
terms of the level sets of $E_{\ell_{z}}^{K}$. From this perspective, the
shape of Zero velocity curves associated to timelike trapped future directed
geodesics will not depend on the Carter constant. This is key to the
identification trapped geodesics in stationary and axisymmetric spacetimes
close to Kerr.
Since we are interested in the level sets of $E_{\ell_{z}}^{K}$, we will first
study its critical points. We make the difference between direct critical
points and retrograde critical points defined by

######  Definition 17.

Let $\ell_{z}\in\mathbb{R}$. A point $(\rho^{+}_{c},z^{+}_{c})$ is a direct
critical point of $E_{\ell_{z}}$ if

$\nabla_{(\rho,z)}E_{\ell_{z}}(\rho^{+}_{c},z^{+}_{c})=0\quad\text{with}\quad-W\ell_{z}>0.$

A point $(\rho^{-}_{c},z^{-}_{c})$ is a retrograde critical point of
$E_{\ell_{z}}$ if

$\nabla_{(\rho,z)}E_{\ell_{z}}(\rho^{-}_{c},z^{-}_{c})=0\quad\text{with}\quad-W\ell_{z}<0.$

We begin the analysis of $E_{\ell_{z}}$ with the study of critical lemma

######  Proposition 14 (Existence of critical points for $E^{K}_{\ell_{z}}$).

Let $\ell_{z}\in\mathbb{R}$. Then,

  * •

$E_{\ell_{z}}^{K}$ admits direct critical point if and only if
$\ell_{z}\in]\\-\infty,\ell_{min}^{-}]$,

  * •

$E_{\ell_{z}}^{K}$ admits retrograde critical point if and only if
$\ell_{z}\in[\ell_{min}^{+},\infty[$,

where $\ell_{min}^{\pm}$ is given by

$\ell_{min}^{\pm}=\Psi_{\pm}(r_{ms}^{\pm}),$

and $r_{ms}^{\pm}$161616See Lemma 18. is given by

${r_{ms}^{\pm}}(d)=3+Z_{2}(d)\mp\sqrt{(3-Z_{1})(3+Z_{1}(d)+2Z_{2}(d))}$

where

$Z_{1}(d)=1+(1-d^{2})^{\frac{1}{3}}((1+d)^{\frac{1}{3}}+(1-d)^{\frac{1}{3}})\;,\;Z_{2}(d)=\sqrt{3d^{2}+Z_{1}^{2}}.$

Moreover,

  * •

if $\displaystyle\ell_{z}\geq\ell_{min}^{+}$, then the critical points are
given by

$(\rho^{+}_{s},z^{+}_{s}):=(\sqrt{\Delta(r_{max}^{+}(\ell_{z}))},0)\quad\text{and}\quad(\rho^{+}_{min},z^{+}_{min}):=(\Delta(\sqrt{r_{min}^{+}(\ell_{z})}),0).$
(4.143)

  * •

if $\displaystyle\ell_{z}\leq\ell_{min}^{-}$, then the critical points are
given by

$(\rho^{-}_{s},z^{-}_{s}):=(\sqrt{\Delta(r_{max}^{-}(\ell_{z})}),0)\quad\text{and}\quad(\rho^{-}_{min},z^{-}_{min}):=(\Delta(\sqrt{r_{min}^{-}(\ell_{z})}),0),$
(4.144)

where $r_{max}^{\pm}(\ell_{z})$ and $r_{min}^{\pm}(\ell_{z})$ are given by

$r^{\pm}_{max}(\ell_{z}):=(\Psi^{1})^{-1}_{\pm}(\ell_{z})\quad\text{and}\quad
r^{\pm}_{max}(\ell_{z}):=(\Psi^{2})^{-1}_{\pm}(\ell_{z}),$

where $(\Psi^{1})_{\pm}^{-1}$ and $(\Psi^{2})_{\pm}^{-1}$ are the inverse of
the restriction of $(\Psi)_{\pm}$ on $]r_{ph},r_{ms}[$ and $]r_{ms},\infty[$
respectively.

###### Proof.

We only consider the case of direct critical points. The remaining case
follows in the same manner. We henceforth drop the $\pm$ symbol from all the
quantities and assume that $\ell_{z}\in[0,\infty[$.

  1. 1.

By Lemma 3, $(\rho_{c},z_{c})$ is a critical point of $E^{K}_{\ell_{z}}$ if
and only $(\rho_{c},z_{c},0,0)$ is a stationary solution of the reduced system

$\left\\{\begin{aligned} \frac{d\rho}{d\tau}&=v^{\rho},\\\
\frac{dz}{d\tau}&=v^{z},\\\
\frac{dv^{\rho}}{d\tau}&=-\frac{1}{2}e^{-2\lambda}\partial_{\rho}J^{K}(\rho,z,\varepsilon,\ell_{z},d)-\tensor{\Gamma}{{}^{\rho}_{ij}}v^{i}v^{j},\quad
i,j\in\left\\{\rho,z\right\\}\\\
\frac{dv^{z}}{d\tau}&=-\frac{1}{2}e^{-2\lambda}\partial_{z}J^{K}(\rho,z,\varepsilon,\ell_{z},d)-\tensor{\Gamma}{{}^{z}_{ij}}v^{i}v^{j}.\end{aligned}\right.$
(4.145)

with parameters
$(\varepsilon_{c},\ell_{z})=(E_{\ell_{z}}^{K}(\rho_{c},z_{c}),\ell_{z})$.

  2. 2.

Now, we claim $(\rho_{c},z_{c},0,0)$ is a stationary solution of if and only
if $(r_{c},\theta_{c},0,0)$ is a stationary solution of the reduced system

$\left\\{\begin{aligned} \frac{dr}{d\tau}&=\frac{\Delta}{\Sigma^{2}}v_{r},\\\
\frac{dv_{r}}{d\tau}&=\frac{1}{2\Sigma^{2}}\left(-\Delta^{\prime}(r)v_{r}^{2}+\frac{\Delta^{\prime}(r)R(r,\varepsilon,\ell_{z},q)-\Delta(r)\partial_{r}R(r,\varepsilon,\ell_{z},q)}{\Delta(r)^{2}}\right),\\\
\frac{d\theta}{d\tau}&=\frac{1}{\Sigma^{2}}v_{\theta},\\\
\frac{dv_{\theta}}{d\tau}&=\frac{1}{2\Sigma^{2}}\partial_{\theta}\left(\frac{T(\cos\theta,\varepsilon,\ell_{z},q)}{\sin^{2}\theta}\right).\end{aligned}\right.$
(4.146)

with parameters $(\varepsilon_{c},\ell_{z})$. To prove the latter, it suffices
to note that the system (4.146) is the system (4.145) written in the
isothermal coordinates $(\rho,z)$.

  3. 3.

By Lemma 34, stationary solutions of (4.146) exist if and only if
$\ell_{z}\in[\ell_{min},\infty[$, where $\ell_{min}$ is defined by (4.48).
They are given by

$\left(r_{max}(\ell_{z}),\frac{\pi}{2}\right)\quad\text{and}\quad\left(r_{min}(\ell_{z}),\frac{\pi}{2}\right).$

  4. 4.

Therefore $(\rho_{c},z_{c})$ are given by

$\left(\rho_{s}^{\pm}(\ell_{z}),0\right)\quad\text{and}\quad(\rho_{min}^{\pm}(\ell_{z}),0),$

where $\displaystyle\rho_{s}^{\pm}(\ell_{z})$ and
$\displaystyle\rho_{min}^{\pm}(\ell_{z})$ are defined by (4.143) and (4.144).
Moreover, the corresponding $\varepsilon_{c}$ satisfy

$\varepsilon_{c}^{s}(\ell_{z}):=E^{K}_{\ell_{z}}(\rho_{s}(\ell_{z}),0)=\varepsilon_{s}(\ell_{z})$

and

$\varepsilon_{c}^{m}(\ell_{z}):=E^{K}_{\ell_{z}}(\rho_{min}(\ell_{z}),0)=\varepsilon_{m}(\ell_{z})$

where $\varepsilon_{s}(\ell_{z})$ and $\varepsilon_{m}(\ell_{z})$ are given by
(4.61) and (4.62) respectively.

∎

######  Proposition 15 (Study of critical points).

Let $\ell_{z}\geq\ell^{+}_{min}$ or $\ell_{z}\leq\ell^{-}_{min}$. Then,

  * •

$\displaystyle(\rho_{s}^{\pm}(\ell_{z}),0)$ corresponds to a saddle point.

  * •

$\displaystyle(\rho_{min}^{\pm}(\ell_{z}),0)$ corresponds to a local minimum.

###### Proof.

Let $\ell_{z}\in]-\infty,\ell^{-}_{min}]\cup[\ell_{min}^{+},\infty[.$ We will
study the critical points of $E_{\ell_{z}}^{K}$ in the BL coordinates. We
recall that

$\tilde{E}^{K}_{\ell_{z}}(r,\theta):=E^{K}_{\ell_{z}}(\rho(r,\theta),z(r,\theta)).$

We have, $\forall(r,\theta)\in]r_{H},\infty[\times]0,\pi[$,

$\tilde{J}^{K}(r,\theta,\tilde{E}^{K}_{\ell_{z}}(r,\theta),\ell_{z})=0,$
(4.147)

where $\tilde{J}^{K}$ is defined by (4.17). Now, we differentiate twice the
expression (4.147) in order to obtain

$\displaystyle\partial_{\varepsilon}^{2}\tilde{J}^{K}(r,\theta,\tilde{E}^{K}_{\ell_{z}}(r,\theta),\ell_{z})\nabla_{(r,\theta)}\tilde{E}^{K}_{\ell_{z}}(r,\theta)\left(\nabla_{(r,\theta)}\tilde{E}^{K}_{\ell_{z}}(r,\theta)\right)^{t}+\partial_{\varepsilon}\tilde{J}^{K}(r,\theta,\tilde{E}^{K}_{\ell_{z}}(r,\theta),\ell_{z})\nabla^{2}_{(r,\theta)}\tilde{E}^{K}_{\ell_{z}}(r,\theta)$
$\displaystyle+\nabla^{2}_{(r,\theta)}\tilde{J}^{K}(r,\theta,\tilde{E}^{K}_{\ell_{z}}(r,\theta),\ell_{z})+\nabla_{(r,\theta)}\tilde{E}_{\ell_{z}}(r,\theta)\left(\nabla_{(r,\theta)}\partial_{\varepsilon}\tilde{J}^{K}(r,\theta,\tilde{E}^{K}_{\ell_{z}}(r,\theta),\ell_{z})\right)^{t}$
$\displaystyle=0.$

In particular, if $(r,\theta)$ is a critical point, then the latter expression
reduces to

$\displaystyle\partial_{\varepsilon}\tilde{J}^{K}(r_{c},\theta_{c},\tilde{E}^{K}_{\ell_{z}}(r_{c},\theta_{c}),\ell_{z})\nabla^{2}_{(r,\theta)}\tilde{E}^{K}_{\ell_{z}}(r_{c},\theta_{c})+\nabla^{2}_{(r,\theta)}\tilde{J}^{K}(r_{c},\theta_{c},\tilde{E}^{K}_{\ell_{z}}(r_{c},\theta_{c}),\ell_{z})=0.$

Now, we recall from Section 3.2.5, that the term
$\displaystyle\partial_{\varepsilon}\tilde{J}^{K}(r_{c},\theta_{c},\tilde{E}^{K}_{\ell_{z}}(r_{c},\theta_{c}),\ell_{z})$
does not vanish. Moreover, it is positive on its domain. Therefore,

$\displaystyle\nabla^{2}_{(r,\theta)}\tilde{E}^{K}_{\ell_{z}}(r_{c},\theta_{c})=-\frac{\nabla^{2}_{(r,\theta)}\tilde{J}^{K}(r_{c},\theta_{c},\tilde{E}^{K}_{\ell_{z}}(r_{c},\theta_{c}),\ell_{z})}{\partial_{\varepsilon}\tilde{J}^{K}(r_{c},\theta_{c},\tilde{E}^{K}_{\ell_{z}}(r_{c},\theta_{c}),\ell_{z})}.$

Now, we compute the Hessian of $\tilde{J}^{K}$ with respect to $(r,\theta)$ at
the points
$(\varepsilon_{c}(\ell_{z})=\tilde{E}^{K}_{\ell_{z}}(r_{c},\theta_{c}),\ell_{z},r_{c}(\ell_{z}),\theta_{c}(\ell_{z}))$.
We find that

$\displaystyle\partial_{rr}\tilde{J}^{K}\left(\varepsilon_{c}(\ell_{z}),\ell_{z},r_{c}(\ell_{z}),\frac{\pi}{2}\right)$
$\displaystyle=\Sigma^{-2}\left(r_{c}(\ell_{z}),\frac{\pi}{2}\right)\partial_{rr}R(r_{c}(\ell_{z}),\varepsilon_{c}(\ell_{z}),\ell_{z},0),$
$\displaystyle\partial_{\theta\theta}\tilde{J}^{K}\left(\varepsilon_{c}(\ell_{z}),\ell_{z},r_{c}(\ell_{z}),\frac{\pi}{2}\right)$
$\displaystyle=\Sigma^{-2}\left(r_{c}(\ell_{z}),\frac{\pi}{2}\right)\partial_{\theta\theta}{T(0,\varepsilon_{c}(\ell_{z}),\ell_{z},0)},$
$\displaystyle\partial_{r\theta}\tilde{J}^{K}\left(\varepsilon_{c}(\ell_{z}),\ell_{z},r_{c}(\ell_{z}),\frac{\pi}{2}\right)$
$\displaystyle=0.$

Here, we used that

  * •

$\displaystyle\cos\theta_{c}(\ell_{z})=0$ is a double root of the polynomial
$T(\cdot,\varepsilon_{c}(\ell_{z}),\ell_{z},q)$ so that $q=0$, see Lemma 13,

  * •

$r_{c}(\ell_{z})$ is a double root of the polynomial
$R(\cdot,\varepsilon_{c}(\ell_{z}),\ell_{z},0)$.

Moreover,

$\partial_{\theta\theta}T(0,\varepsilon_{c}(\ell_{z}),\ell_{z},0)>0$

and

$\partial_{rr}R(r_{c}(\ell_{z}),\varepsilon_{c}(\ell_{z}),\ell_{z},0)=\left\\{\begin{aligned}
&>0\quad\text{if}\quad r_{c}(\ell_{z})=r^{\pm}_{min}(\ell_{z}),\\\
&<0\quad\text{if}\quad r_{c}(\ell_{z})=r^{\pm}_{max}(\ell_{z}).\\\
\end{aligned}\right.$

Hence,

  * •

$\displaystyle\left(r_{max}^{\pm}(\ell_{z}),\frac{\pi}{2}\right)$ corresponds
to a saddle point for $\tilde{E}^{K}_{\ell_{z}}$.

  * •

$\displaystyle\left(r_{min}^{\pm}(\ell_{z}),\frac{\pi}{2}\right)$ corresponds
to a local minimum $\tilde{E}^{K}_{\ell_{z}}$.

∎

Now, we study the basic properties of $E_{\ell_{z}}^{K}$:

######  Lemma 36 (Properties of $E^{K}_{\ell_{z}}$).

We have

  1. 1.

Let $\tilde{\mathscr{A}}$ be a neighbourhood of the axis. Then

$\displaystyle\lim_{||(\rho,z)||\to\infty,(\rho,z)\in(\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}})\backslash\tilde{\mathscr{A}}}\;E^{K}_{\ell_{z}}(\rho,z)=1.$

  2. 2.

$\forall(\rho,z)\in\mathscr{B},$

$E^{K}_{\ell_{z}}(\rho,z)=E^{K}_{\ell_{z}}(\rho,-z).$

  3. 3.

For $d\ell_{z}<0$, $E^{K}_{\ell_{z}}$ is negative in a neighbourhood of the
horizon 171717In this region, particles with positive energies in the local
observer’s frame can have negative energy with respect to infinity..

###### Proof.

  1. 1.

We have $\forall(\rho,z)\in\mathscr{B}\,,\,\forall\ell_{z}\in\mathbb{R}$

$-\frac{W_{K}(\rho,z)}{X_{K}(\rho,z)}\ell_{z}=\frac{2dr(\rho,z)}{\tilde{\Pi}(\rho,z)}\ell_{z},$

where $r(\rho,z)$ is given (3.36) and

$\tilde{\Pi}(\rho,z):=(r^{2}(\rho,z)+d^{2})^{2}-\rho^{2}d^{2}.$

Therefore,

$\lim_{||(\rho,z)||\to\infty}\;-\frac{W_{K}(\rho,z)}{X_{K}(\rho,z)}\ell_{z}=\lim_{||(\rho,z)||\to\infty}\frac{2dr(\rho,z)}{r^{4}(\rho,z)}\ell_{z}=0.$

Now, we compute
$\displaystyle\lim_{||(\rho,z)||\to\infty}\;\frac{\rho}{X_{K}(\rho,z))}\sqrt{\ell_{z}^{2}+X_{K}(\rho,z)}.$
We have $\forall(\rho,z)\in\mathscr{B}\,,\,\forall\ell_{z}\in\mathbb{R}$

$\frac{\rho}{X_{K}(\rho,z)}=\frac{1}{\rho}\frac{\Delta(r(\rho,z))\tilde{\Sigma}^{2}(\rho,z)}{\tilde{\Pi}(\rho,z)}$

where

$\tilde{\Sigma}^{2}(\rho,z):=r^{2}(\rho,z)+\frac{d^{2}z^{2}}{(r(\rho,z)-1)^{2}}$

We have

$\lim_{||(\rho,z)||\to\infty}\frac{\Delta(r(\rho,z))\tilde{\Sigma}^{2}(\rho,z)}{\tilde{\Pi}(\rho,z)}=1.$

  2. 2.

By (3.36), $\forall(\rho,z)\in\mathscr{B}$,

$r(\rho,z)=r(\rho,-z).$

Therefore, $\forall\rho>0$, $W_{K}(\rho,\cdot)$ and $X_{K}(\rho,\cdot)$ are
even.

  3. 3.

Let $(\rho,z)\in\mathscr{B}$. Then,

$\displaystyle E^{K}_{\ell_{z}}(\rho,z)$
$\displaystyle=-\frac{W_{K}(\rho,z)}{X_{K}(\rho,z)}\ell_{z}+\frac{\rho}{X_{K}(\rho,z))}\sqrt{\ell_{z}^{2}+X_{K}(\rho,z)}$
$\displaystyle=\frac{2d\ell_{z}r(\rho,z)}{\tilde{\Pi}(\rho,z)}+\frac{\rho}{X_{K}(\rho,z))}\sqrt{\ell_{z}^{2}+X_{K}(\rho,z)}.$

We show that there exists a neighbourhood of $\mathscr{H}$, say
$\tilde{\mathscr{H}}\subset\overline{\mathscr{B}_{H}}$, such that $\forall
d\ell_{z}\leq 0$, $E_{\ell_{z}}^{K}$ is negative on $\tilde{\mathscr{H}}$. By
Lemma 4, we have the following asymptotics for $X_{K}$ near the horizon
$X_{\mathscr{H}}:\tilde{\mathscr{H}}\to\mathbb{R}$ with
$X_{\mathscr{H}}(0,z)>0$ such that

$\left.X(\rho,z)\right|_{\tilde{\mathscr{H}}}=X_{\mathscr{H}}(\rho^{2},z).$

Moreover the function
$\displaystyle(\rho,z)\to\frac{r(\rho,z)}{\tilde{\Pi}(\rho,z)}$ is smooth and
positive on $\overline{\mathscr{B}_{H}}$. Therefore, $E^{K}_{\ell_{z}}$
extends smoothly to $\overline{\mathscr{B}_{H}}$. Now we make a first order
Taylor expansion for $E_{\ell_{z}}^{K}$ around $(0,z)$ with $|z|<\gamma$:
$\forall\rho\geq 0$ small,

$E^{K}_{\ell_{z}}(\rho,z)=E^{K}_{\ell_{z}}(0,z)+\rho\partial_{\rho}E^{K}_{\ell_{z}}(0,z)+O(\rho^{2}).$

We compute:

$E^{K}_{\ell_{z}}(0,z)=2d\ell_{z}\frac{r_{H}}{\tilde{\Pi}(0,z)}=\frac{d\ell_{z}}{2r_{H}}$

and

$\partial_{\rho}E^{K}_{\ell_{z}}(0,z)=\frac{1}{X_{\mathscr{H}}(0,z))}\sqrt{\ell_{z}^{2}+X_{\mathscr{H}}(0,z)}+2d\ell_{z}\frac{\partial_{\rho}r(0,z)\tilde{\Pi}(0,z)-r_{H}\partial_{\rho}\tilde{\Pi}(0,z)}{16r_{H}^{4}}.$

We have

$\partial_{\rho}r(0,z)=0\quad\text{and}\quad\partial_{\rho}\tilde{\Pi}(0,z)=8r_{H}^{2}.$

Hence

$\partial_{\rho}r(0,z)\tilde{\Pi}(0,z)-r_{H}\partial_{\rho}\tilde{\Pi}(0,z)=-8r_{H}^{2}$

and

$\partial_{\rho}E^{K}_{\ell_{z}}(0,z)>0.$

Now, we choose $\rho>0$ such that

$\frac{d\ell_{z}}{2r_{H}}+\rho\partial_{\rho}E^{K}_{\ell_{z}}(0,z)<0.$

∎

Figure 10: Shape of $E^{K}_{\ell_{z}}$ with $d=0.8$. On the left:
$\ell_{z}=1$. On the right: $\ell_{z}=4$

####  4.2.2 Trapped timelike future-directed geodesics

A full classification of timelike future-directed geodesics in Kerr can be
achieved using the integrability of the geodesics equation in BL coordinates
(see Proposition 13). A priori, the nature of the orbits then depends also on
the Carter constant $Q$. In stationary and axisymmetric spacetimes, the
geodesic motion does not necessarily form an integrable Hamiltonian system
since there is no generalisation of $Q$ and there are only three constants of
motion $E$, $L_{z}$ and $m$.
However, since only an open set of trapped non-spherical orbits is relevant to
our work, we will determine sufficient conditions on $(\varepsilon,\ell_{z})$
which are independent of $Q$ and on the precise initial position
$(\rho(0),z(0))$ for a trajectory to be trapped. This will allow us to
construct an open subset of parameters $(\varepsilon,\ell_{z})$ on which the
distribution function $f$ will be supported.
We begin by recalling from Proposition 13 the necessary and sufficient
conditions for a timelike future-directed orbit to be trapped and non-
spherical.
Let $\gamma:I\ni 0\to\mathcal{O}$ be a timelike future-directed geodesic with
constants of motion $(\varepsilon,\ell_{z})$ and let $\tilde{\gamma}$ its
projection in $\mathscr{B}$. Then $\gamma$ is trapped if and only if one of
the following cases occur:

  1. 1.

$\displaystyle\varepsilon\in]\varepsilon^{+}_{min},\varepsilon^{-}_{min}]$,
$\displaystyle\ell_{z}\in]\tilde{\ell}_{min}(\varepsilon),\ell^{+}_{lb}(\varepsilon)[$
and

$\displaystyle\tilde{\gamma}(0)$
$\displaystyle\in\left(A^{K,abs}(\varepsilon,\ell_{z})\cap(]\tilde{\rho}^{+}(\varepsilon,\ell_{z}),\rho^{K}_{0}(\varepsilon,\ell_{z})[\times]\gamma-\tilde{z}^{+}(\varepsilon,\ell_{z}),\gamma[)\right)$
$\displaystyle\cup\left(A^{K,abs}(\varepsilon,\ell_{z})\cap(]\tilde{\rho}^{+}(\varepsilon,\ell_{z}),\rho^{K}_{0}(\varepsilon,\ell_{z})[\times]-\gamma,\gamma+\tilde{z}^{+}(\varepsilon,\ell_{z})[)\right),$

  2. 2.

$\displaystyle\varepsilon\in]\varepsilon^{+}_{min},\varepsilon^{-}_{min}]$,
$\displaystyle\ell_{z}\in]\ell^{+}_{lb}(\varepsilon),\ell^{+}_{ub}(\varepsilon)[$
and $\displaystyle\tilde{\gamma}(0)\in A^{K,trapped}(\varepsilon,\ell_{z})$,

  3. 3.

$\displaystyle\varepsilon\in]\varepsilon^{-}_{min},1[$,
$\displaystyle\ell_{z}\in]\ell^{-}_{ub}(\varepsilon),\ell^{-}_{lb}(\varepsilon)\cup]\ell^{+}_{lb}(\varepsilon),\ell^{+}_{ub}(\varepsilon)[$
and $\displaystyle\tilde{\gamma}(0)\in A^{K,trapped}(\varepsilon,\ell_{z})$,

  4. 4.

$\displaystyle\varepsilon\in]\varepsilon^{-}_{min},1[$,
$\displaystyle\ell_{z}\in]\ell^{-}_{lb}(\varepsilon),\ell^{+}_{lb}(\varepsilon)[$
and

$\displaystyle\tilde{\gamma}(0)$
$\displaystyle\in\left(A^{K,abs}(\varepsilon,\ell_{z})\cap(]\tilde{\rho}^{1}(\varepsilon,\ell_{z}),\rho^{K}_{0}(\varepsilon,\ell_{z})[\times]\gamma-\tilde{z}^{1}(\varepsilon,\ell_{z}),\gamma[)\right)$
$\displaystyle\cup\left(A^{K,abs}(\varepsilon,\ell_{z})\cap(]\tilde{\rho}^{1}(\varepsilon,\ell_{z}),\rho^{K}_{0}(\varepsilon,\ell_{z})[\times]-\gamma,-\gamma+\tilde{z}^{1}(\varepsilon,\ell_{z}),\pi[)\right),$

where

$(\tilde{\rho}^{+}(\varepsilon,\ell_{z}),\tilde{z}^{+}(\varepsilon,\ell_{z})):=(\sqrt{\Delta(\tilde{r}^{+}(\varepsilon,\ell_{z}))}\sin(\theta_{1}(\varepsilon,\ell_{z})),(\tilde{r}^{+}(\varepsilon,\ell_{z})-1)\cos(\theta_{1}(\varepsilon,\ell_{z})))$

and

$(\tilde{\rho}^{1}(\varepsilon,\ell_{z}),\tilde{z}^{1}(\varepsilon,\ell_{z})):=(\sqrt{\Delta(\tilde{r}^{1}(\varepsilon,\ell_{z}))}\sin(\tilde{\theta}_{1}(\varepsilon,\ell_{z})),(\tilde{r}^{1}(\varepsilon,\ell_{z})-1)\cos(\tilde{\theta}_{1}(\varepsilon,\ell_{z})))$

In particular, if $\gamma$ is a timelike future-directed geodesic with
constants of motion $(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, where
$\mathcal{A}_{bound}$ is defined by (4.125) (see Figure 11), then
$\tilde{\gamma}$ is either trapped or plunging. Moreover, the zero velocity
curve associated to $\gamma$, $Z^{K}(\varepsilon,\ell_{z})$, has two connected
components. Indeed, by Lemma 12, we have

$Z^{K}(\varepsilon,\ell_{z})=Z^{K,abs}(\varepsilon,\ell_{z})\cup
Z^{K,trapped}(\varepsilon,\ell_{z})$

where $Z^{K,abs}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{R}$ and
$Z^{K,trapped}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{S}^{1}$.

In the following section, we will reparameterize181818For the sole purpose of
the main theorem, we could have constructed an atlas only for
$Z^{K,trapped}(\varepsilon,\ell_{z})$. We also do the analysis of
$Z^{K,abs}(\varepsilon,\ell_{z})$ $Z^{K}(\varepsilon,\ell_{z})$ for
$(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$.

####  4.2.3 Reparameterization of the zero velocity curves associated to
trapped geodesics

Let $\gamma$ be a timelike future-directed geodesic with constants of motion
$(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$ and let $\tilde{\gamma}$ be its
projection in $\mathscr{B}$. We recall that the allowed region for
$\tilde{\gamma}$ has two connected components
$A^{K,abs}(\varepsilon,\ell_{z})$ and $A^{K,trapped}(\varepsilon,\ell_{z})$
bounded respectively by $Z^{K,abs}(\varepsilon,\ell_{z})$ and
$Z^{K,trapped}(\varepsilon,\ell_{z})$.
Since $Z^{K,abs}(\varepsilon,\ell_{z})$ and
$Z^{K,trapped}(\varepsilon,\ell_{z})$ are one-dimensional manifolds, we will
construct atlases
$(\Phi^{K,abs}_{(\varepsilon,\ell_{z})},I_{(\varepsilon,\ell_{z})}\subset\mathbb{R})$
and
$(\Phi^{K,trapped}_{(\varepsilon,\ell_{z}),i},I_{(\varepsilon,\ell_{z}),i})_{i=1\cdots
i_{0}}$ for $Z^{K,abs}(\varepsilon,\ell_{z})$ and
$Z^{K,trapped}(\varepsilon,\ell_{z})$ respectively. This will allow us to see
locally the solutions of the equation

$\varepsilon=E^{K}_{\ell_{z}}(\rho,z).$ (4.148)

as the graph of smooth functions. As a consequence, the problem of finding
solutions to Equation (4.148) on $\mathscr{B}$ is equivalent to writing
locally $\rho$ as a function of $z$ or $z$ as a function of $\rho$. This
representation of the solutions will help us to formulate the stability result
for trapped non-spherical timelike future-directed geodesics in the following
section.

Figure 11: The set $\mathcal{A}_{bound}$ with $d=0.8$ as the union of the two
set bounded by the red curve, the blue curve and the black curve. On the left:
$\mathcal{A}_{bound}^{+}$. On the right: $\mathcal{A}_{bound}^{-}$.

To this end, we begin with the following lemma

######  Lemma 37.

$\forall(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, we have

$Z^{K,trapped}(\varepsilon,\ell_{z})\subset[\rho_{1}^{K}(\varepsilon,\ell_{z}),\rho_{2}^{K}(\varepsilon,\ell_{z})]\times[-z_{max}^{K}(\varepsilon,\ell_{z}),z_{max}^{K}(\varepsilon,\ell_{z})]$

where $\rho_{i}^{K}(\varepsilon,\ell_{z})$ are the two largest roots of the
equation (4.148) with $z=0$ and $z_{max}^{K}(\varepsilon,\ell_{z})$ is defined
by

$z_{max}^{K}(\varepsilon,\ell_{z}):=\left(\tilde{r}(\varepsilon,\ell_{z})-1\right)\cos\overline{\theta}^{<1}_{max}(\varepsilon,\ell_{z})$
(4.149)

where $\tilde{r}(\varepsilon,\ell_{z})$ is defined in Lemma 33 and
$\cos\overline{\theta}^{<1}_{max}(\varepsilon,\ell_{z})$ is given by (4.114).
Moreover, $\forall(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, the equation
(4.148) with $z=\pm z_{max}^{K}(\varepsilon,\ell_{z})$ admits a unique
solution $\rho_{max}(\varepsilon,\ell_{z})$ in the region
$[\rho_{1}^{K}(\varepsilon,\ell_{z}),\rho_{2}^{K}(\varepsilon,\ell_{z})]$,
given by

$\rho_{max}(\varepsilon,\ell_{z})=\sqrt{\Delta(\tilde{r}(\varepsilon,\ell_{z}))}\sin\overline{\theta}^{<1}_{max}(\varepsilon,\ell_{z}).$
(4.150)

###### Proof.

We recall from Proposition 12 that

$\displaystyle Z^{K,trapped}(\varepsilon,\ell_{z})$
$\displaystyle=Graph\left(r^{1}_{tr}(\varepsilon,\ell_{z},q(\cdot))\right)\cup
Graph\left(r^{2}_{tr}(\varepsilon,\ell_{z},q(\cdot))\right)$
$\displaystyle\cup
Graph\left(\theta_{tr}(\varepsilon,\ell_{z},q(\cdot))\right)\cup
Graph\left(\pi-\theta_{tr}(\varepsilon,\ell_{z},q(\cdot))\right),$

where $r^{1}_{tr}$, $r^{2}_{tr}$ and $\theta_{tr}$ are given by (4.120),
(4.121) and (4.122) respectively. Moreover, the latter functions verify

  * •

$r^{1}_{tr}(\varepsilon,\ell_{z},q(\cdot))$ has a global minimum at
$r_{1}^{K}(\varepsilon,\ell_{z})$ and

$r^{1}_{tr}(\varepsilon,\ell_{z},q(\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})))=r^{1}_{tr}(\varepsilon,\ell_{z},q(\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})))=\tilde{r}(\varepsilon,\ell_{z}).$

Therefore,

$Graph\left(r^{1}_{tr}(\varepsilon,\ell_{z},q(\cdot))\right)\subset\left[r_{1}^{K}(\varepsilon,\ell_{z}),\tilde{r}^{+}(\varepsilon,\ell_{z})\right]\times\left[\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z}),\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\right].$

  * •

$r^{2}_{tr}(\varepsilon,\ell_{z},q(\cdot))$ has a global maximum at
$r_{2}^{K}(\varepsilon,\ell_{z})$ and

$r^{2}_{tr}(\varepsilon,\ell_{z},q(\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})))=r^{2}_{tr}(\varepsilon,\ell_{z},q(\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})))=\tilde{r}(\varepsilon,\ell_{z}),$

Thus,

$Graph\left(r^{2}_{tr}(\varepsilon,\ell_{z},q(\cdot))\right)\subset\left[\tilde{r}(\varepsilon,\ell_{z}),r_{2}^{K}(\varepsilon,\ell_{z})\right]\times\left[\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z}),\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\right].$

  * •

$\displaystyle\theta_{tr}(\varepsilon,\ell_{z},q(\cdot))$ has a global maximum
at $\displaystyle\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})$ and

$Graph\left(\theta_{tr}(\varepsilon,\ell_{z},q(\cdot))\right)\subset\left[r_{1}^{K}(\varepsilon,\ell_{z}),r_{2}^{K}(\varepsilon,\ell_{z})\right]\times\left[0,\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\right].$

  * •

$\displaystyle\pi-\theta_{tr}(\varepsilon,\ell_{z},q(\cdot))$ has a global
minimum at
$\displaystyle\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})$ and

$Graph\left(\theta_{tr}(\varepsilon,\ell_{z},q(\cdot))\right)\subset\left[r_{1}^{K}(\varepsilon,\ell_{z}),r_{2}^{K}(\varepsilon,\ell_{z})\right]\times\left[0,\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\right].$

Therefore,

$\displaystyle Z^{K,trapped}(\varepsilon,\ell_{z})$
$\displaystyle\subset[r_{1}^{K}(\varepsilon,\ell_{z}),r_{2}^{K}(\varepsilon,\ell_{z})]\times[\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z}),\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})],$

or if we use the coordinates $(\rho,z)$, we have

$Z^{K,trapped}(\varepsilon,\ell_{z})\subset[\rho_{1}^{K}(\varepsilon,\ell_{z}),\rho_{2}^{K}(\varepsilon,\ell_{z})]\times[-z_{max}^{K}(\varepsilon,\ell_{z}),z_{max}^{K}(\varepsilon,\ell_{z})].$

∎

Now, let $\mathcal{B}_{bound}\subset\subset\mathcal{A}_{bound}$. We claim that
$\mathcal{B}_{bound}$ can be included in a finite union of products of closed
intervals. More precisely, we have

######  Lemma 38.

Let $\mathcal{B}_{bound}\subset\subset\mathcal{A}_{bound}$. Then, there exists
a finite number $N$ of products of closed intervals
$[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}]\times[\ell^{i,\pm}_{1},\ell^{i,\pm}_{2}]$
such that

$[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}]\times[\ell^{i,\pm}_{1},\ell^{i,\pm}_{2}]\subset\mathcal{A}_{bound}^{\pm},$
$\mathcal{B}_{bound}^{\pm}\subset\bigcup_{i=1}^{N}[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}]\times[\ell^{i,\pm}_{1},\ell^{i,\pm}_{2}]$

and $\displaystyle\varepsilon^{i,\pm}_{j}$ and $\displaystyle\ell^{i,\pm}_{j}$
satisfy

$\varepsilon^{\pm}_{min}<\varepsilon^{i,\pm}_{1}<\varepsilon^{i,\pm}_{2}<1\quad\text{,}\quad\ell_{lb}(\varepsilon^{+}_{2})<\ell^{i,+}_{1}<\ell^{i,+}_{2}<\ell_{ub}(\varepsilon^{+}_{1})\quad\text{and}\quad\ell_{ub}(\varepsilon^{-}_{1})<\ell^{i,-}_{1}<\ell^{i,-}_{2}<\ell_{lb}(\varepsilon^{-}_{2}).$
(4.151)

###### Proof.

By compactness of $\mathcal{B}_{bound}$, there exists a finite number $N$ of
product closed intervals
$[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}]\times[\ell^{i,\pm}_{1},\ell^{i,\pm}_{2}]$
such that

$[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}]\times[\ell^{i,\pm}_{1},\ell^{i,\pm}_{2}]\subset\mathcal{A}_{bound}^{\pm},$
(4.152)
$\mathcal{B}_{bound}^{\pm}\subset\bigcup_{i=1}^{N}[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}]\times[\ell^{i,\pm}_{1},\ell^{i,\pm}_{2}].$

We need to check that $\displaystyle\varepsilon^{i,\pm}_{j}$ and
$\displaystyle\ell^{i,\pm}_{j}$ satisfy (5.3). Let $i\in\left\\{1,\dots
N\right\\}$. Then,

$[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}]\subset]\varepsilon^{+}_{min},1[$

and

$[\ell^{i,\pm}_{1},\ell^{i,\pm}_{2}]\subset]\ell_{lb}^{+}(\varepsilon),\ell_{ub}^{+}(\varepsilon)[\quad\text{for
all}\;\varepsilon\in[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}].$

By monotonicity properties of $\ell_{ub}^{+}$ and $\ell_{lb}^{+}$ (see Lemma
23), we have

$\displaystyle\ell_{lb}^{+}(\varepsilon^{i,+}_{1})<\ell_{lb}^{+}(\varepsilon)<\ell_{ub}^{+}(\varepsilon^{i,+}_{2})$
$\displaystyle\ell_{ub}^{+}(\varepsilon^{i,+}_{1})<\ell_{ub}^{+}(\varepsilon)<\ell_{ub}^{+}(\varepsilon^{i,+}_{2}).$

Therefore, $\ell_{1}^{i,+}$ and $\ell_{2}^{i,+}$ verify

$\ell_{lb}^{+}(\varepsilon^{i,+}_{1})<\ell_{lb}^{+}(\varepsilon_{2})<\ell_{1}^{i,+}<\ell_{2}^{i,+}<\ell_{lb}^{+}(\varepsilon)<\ell_{ub}^{+}(\varepsilon^{i,+}_{1}).$

∎

######  Remark 20.

In the remaining of our work, we will suppose that $\mathcal{B}_{bound}$ is
included in one of the product intervals
$\displaystyle[\varepsilon^{i,\pm}_{1},\varepsilon^{i,\pm}_{2}]\times[\ell^{i,\pm}_{1},\ell^{i,\pm}_{2}]$.
The general case can be dealt by a partition of unity argument.

We shall henceforth assume that $\mathcal{B}_{bound}$ has the following form

$\mathcal{B}_{bound}:=\mathcal{B}_{bound}^{-}\cup\mathcal{B}_{bound}^{+}$
(4.153)

where $\mathcal{B}_{bound}^{-}$ and $\mathcal{B}_{bound}^{+}$ are defined by

$\mathcal{B}_{bound}^{\pm}:=[\varepsilon_{1}^{\pm},\varepsilon_{2}^{\pm}]\times[\ell_{1}^{\pm},\ell_{2}^{\pm}]$

where $\displaystyle\varepsilon^{i,\pm}_{j}$ and
$\displaystyle\ell^{i,\pm}_{j}$ satisfy

$\varepsilon^{\pm}_{min}(d)<\varepsilon^{\pm}_{1}<\varepsilon^{\pm}_{2}<1\quad\text{,}\quad\ell_{lb}(\varepsilon^{+}_{2})<\ell^{+}_{1}<\ell^{+}_{2}<\ell_{ub}(\varepsilon^{+}_{1})\quad\text{and}\quad\ell_{ub}(\varepsilon^{-}_{1})<\ell^{-}_{1}<\ell^{-}_{2}<\ell_{lb}(\varepsilon^{-}_{2}).$
(4.154)

In the remaining of this section, we omit the $\pm$ in order to lighten the
expressions. We state the following lemma

######  Lemma 39.

Let $(\varepsilon,\ell_{z})\in\mathcal{B}_{bound}$ and let
$\rho^{K}_{i}(\varepsilon,\ell_{z}),i\in\left\\{0,1,2\right\\}$ be the
solutions of the equation

$E^{K}_{\ell_{z}}(\rho,0)=\varepsilon$

such that

$\rho^{K}_{0}(\varepsilon,\ell_{z})<\rho^{K}_{1}(\varepsilon,\ell_{z})<\rho^{K}_{2}(\varepsilon,\ell_{z}).$

Then,

  1. 1.

$\forall\varepsilon\in[\varepsilon_{1},\varepsilon_{2}]$,

     * •

$\displaystyle\rho^{K}_{0}(\varepsilon,\cdot)$ decreases monotonically on
$[\ell_{1},\ell_{2}]$.

     * •

$\displaystyle\rho^{K}_{1}(\varepsilon,\cdot)$ increases monotonically on
$[\ell_{1},\ell_{2}]$.

     * •

$\displaystyle\rho^{K}_{2}(\varepsilon,\cdot)$ decreases monotonically on
$[\ell_{1},\ell_{2}]$.

  2. 2.

$\forall\ell_{z}\in[\ell_{1},\ell_{2}]$,

     * •

$\displaystyle\rho^{K}_{0}(\cdot,\ell_{z})$ decreases monotonically on
$[\varepsilon_{1},\varepsilon_{2}]$.

     * •

$\displaystyle\rho^{K}_{1}(\cdot,\ell_{z})$ increases monotonically on
$[\varepsilon_{1},\varepsilon_{2}]$.

     * •

$\displaystyle\rho^{K}_{2}(\cdot,\ell_{z})$ decreases monotonically on
$[\varepsilon_{1},\varepsilon_{2}]$.

###### Proof.

First of all, we recall from Proposition 15 that $E^{K}_{\ell_{z}}(\cdot,0)$
admits two critical points: a maximum at $\rho_{s}(\ell_{z})$ and a minimum at
$\rho_{s}(\ell_{z})$. It follows that

  * •

$E^{K}_{\ell_{z}}(\cdot,0)$ is monotonically increasing on
$]0,\rho_{s}(\ell_{z})[$,

  * •

$E^{K}_{\ell_{z}}(\cdot,0)$ is monotonically decreasing on
$]\rho_{s}(\ell_{z}),\rho_{m}(\ell_{z})[$,

  * •

$E^{K}_{\ell_{z}}(\cdot,0)$ is monotonically increasing on
$]\rho_{m}(\ell_{z}),\infty[$.

Moreover, $\forall\rho>0$, $E^{K}_{\cdot}(\rho,0)$ is monotonically increasing
on $]\ell_{1}^{+},\ell_{2}^{+}[$. Furthermore, we have

$\rho^{K}_{0}(\varepsilon,\cdot)<\rho_{s}(\ell_{z})<\rho^{K}_{1}(\varepsilon,\ell_{z})<\rho_{m}(\ell_{z})<\rho^{K}_{2}(\varepsilon,\ell_{z})$

and

$\frac{\partial\rho^{K}_{i}}{\partial\varepsilon}(\varepsilon,\ell_{z})=\frac{1}{\frac{\partial
E^{K}_{\ell_{z}}}{\partial\rho}(\rho^{K}_{i}(\varepsilon,\ell_{z}),0)}\quad\text{and}\quad\frac{\partial\rho^{K}_{i}}{\partial\ell_{z}}(\varepsilon,\ell_{z})=-\frac{\frac{\partial
E^{K}_{\ell_{z}}}{\partial\ell_{z}}(\rho^{K}_{i}(\varepsilon,\ell_{z}),0)}{\frac{\partial
E^{K}_{\ell_{z}}}{\partial\rho}(\rho^{K}_{i}(\varepsilon,\ell_{z}),0)}.$

This yields the result. ∎

Now, we state the following lemma

######  Lemma 40.

There exists $\eta>0$ such that $\forall(\varepsilon,\ell_{z})\in B_{bound}$,
we have

$\rho_{1}^{K}(\varepsilon,\ell_{z})-\rho_{0}^{K}(\varepsilon,\ell_{z})>2\eta.$

###### Proof.

Let $(\varepsilon,\ell_{z})\in\mathcal{B}_{bound}$. Then,
$\varepsilon\in[\varepsilon_{1},\varepsilon_{2}]$ and
$\ell_{z}\in[\ell_{1}(\varepsilon),\ell_{2}(\varepsilon)]$ where
$\varepsilon_{min}(d)<\varepsilon_{1}<\varepsilon_{2}<1$ and
$\ell_{lb}(\varepsilon)<\ell_{1}(\varepsilon)<\ell_{2}(\varepsilon)<\ell_{ub}(\varepsilon)$.
By monotonicity properties of $\rho_{i}^{K}$, we have
$\forall\varepsilon\in[\varepsilon_{1},\varepsilon_{2}],\forall\ell_{z}\in[\ell_{1}(\varepsilon),\ell_{2}(\varepsilon)]$

$\rho_{1}^{K}(\varepsilon,\ell_{z})-\rho_{0}^{K}(\varepsilon,\ell_{z})\geq\rho_{1}^{K}(\varepsilon,\ell_{1}(\varepsilon))-\rho_{0}^{K}(\varepsilon,\ell_{1}(\varepsilon))\geq\min_{[\varepsilon_{1},\varepsilon_{2}]}\rho_{1}^{K}(\varepsilon,\ell_{1}(\varepsilon))-\rho_{0}^{K}(\varepsilon,\ell_{1}(\varepsilon))=:2\eta.$

By compactness and regularity of $\rho_{1}^{K}$ and $\rho_{0}^{K}$,

$\eta:=\frac{1}{2}\left(\rho_{1}^{K}(\varepsilon_{0},\ell_{1}(\varepsilon_{0}))-\rho_{0}^{K}(\varepsilon_{0},\ell_{1}(\varepsilon_{0}))\right)\quad\text{for
some $\varepsilon_{0}\in[\varepsilon_{1},\varepsilon_{1}]$}.$

In order to prove that $\eta>0$, it suffices to note that
$\displaystyle\ell_{1}(\varepsilon_{0})>\ell_{lb}(\varepsilon_{0})$ so that
$\rho^{K}_{0}$ and $\rho^{K}_{1}$ do not coincides (recall that this case only
occurs when
$\displaystyle\ell_{1}(\varepsilon_{0})=\ell_{lb}(\varepsilon_{0})$). In
particular, we have:

$\rho_{1}^{K}(\varepsilon_{0},\ell_{1}(\varepsilon_{0}))>\rho_{0}^{K}(\varepsilon_{0},\ell_{1}(\varepsilon_{0})).$

∎

Consequently, the quantities

$\rho^{K}_{0,max}:=\max_{\mathcal{B}_{bound}}\rho^{K}_{0}(\varepsilon,\ell_{z})\quad,\quad\rho^{K}_{1,min}:=\min_{\mathcal{B}_{bound}}\rho^{K}_{1}(\varepsilon,\ell_{z})\quad\text{and}\quad\rho^{K}_{2,max}:=\max_{\mathcal{B}_{bound}}\rho^{K}_{2}(\varepsilon,\ell_{z})$
(4.155)

are well-defined. and we have

$\rho^{K}_{1,min}-\rho^{K}_{0,max}>2\eta.$ (4.156)

Now, we claim that

######  Lemma 41.

$\forall z\in[0,z_{max}^{K}(\varepsilon,\ell_{z})]$, there exist
$\rho_{1}(z,\varepsilon,\ell_{z}),\rho_{2}(z,\varepsilon,\ell_{z})\in[\rho_{1}^{K}(\varepsilon,\ell_{z}),\rho_{2}^{K}(\varepsilon,\ell_{z})]$
solutions of the equation

$E_{\ell_{z}}^{K}(\rho,z)=\varepsilon$

which satisfy

$\rho_{1}(z,\varepsilon,\ell_{z})\leq\rho_{max}(\varepsilon,\ell_{z})\leq\rho_{2}(z,\varepsilon,\ell_{z})$

Let $(\varepsilon,\ell_{z})\in\mathcal{B}_{bound}$.

  * •

If $z_{max}(\varepsilon,\ell_{z})=0$, then by Lemma 37,
$Z^{K}(\varepsilon,\ell_{z})$ is confined in the equatorial plane:

$\displaystyle Z^{K,abs}=\left\\{\rho_{0}^{K}(\varepsilon,\ell_{z})\right\\},$
$\displaystyle
Z^{K,trapped}(\varepsilon,\ell_{z})=\left\\{\rho_{1}^{K}(\varepsilon,\ell_{z}),\rho_{2}^{K}(\varepsilon,\ell_{z})\right\\}.$

  * •

Otherwise, we can choose $\overline{z}_{max}(\varepsilon,\ell_{z})$ and
$\tilde{z}_{max}(\varepsilon,\ell_{z})$ such that

$0<\overline{z}_{max}(\varepsilon,\ell_{z})<\tilde{z}_{max}(\varepsilon,\ell_{z})<z_{max}(\varepsilon,\ell_{z}).$

By Lemma 41, there exist
$\overline{\rho}_{i}(\varepsilon,\ell_{z}),\tilde{\rho}_{i}(\varepsilon,\ell_{z})\in]\rho_{1}^{K}(\varepsilon,\ell_{z}),\rho_{2}^{K}(\varepsilon,\ell_{z})[,i\in\left\\{1,2\right\\}$
which solve the equations

$E_{\ell_{z}}^{K}(\rho,\overline{z}_{max}(\varepsilon,\ell_{z}))=\varepsilon\quad\text{and}\quad
E_{\ell_{z}}^{K}(\rho,\tilde{z}_{max}(\varepsilon,\ell_{z}))=\varepsilon$

respectively and which satisfy

$\rho_{1}^{K}(\varepsilon,\ell_{z})<\overline{\rho}_{1}(z,\varepsilon,\ell_{z})<\tilde{\rho}_{1}(z,\varepsilon,\ell_{z})<\rho_{max}(\varepsilon,\ell_{z})<\tilde{\rho}_{2}(z,\varepsilon,\ell_{z})<\overline{\rho}_{2}(z,\varepsilon,\ell_{z})<\rho_{2}^{K}(\varepsilon,\ell_{z}).$

We introduce the following open subsets of $\mathscr{B}$:

$\displaystyle\mathscr{B}^{abs}$
$\displaystyle:=\left]0,\rho^{K}_{0,max}+\frac{3\eta}{2}\right[\times\mathbb{R},$
(4.157) $\displaystyle\mathscr{B}_{1}$
$\displaystyle:=\left]\rho^{K}_{0,max}+\frac{\eta}{2},\infty\right[\times\left]\overline{z}_{max}(\varepsilon,\ell_{z}),+\infty\right[,$
(4.158) $\displaystyle\mathscr{B}_{2}$
$\displaystyle:=\left]\rho^{K}_{0,max}+\frac{\eta}{2},\infty\right[\times\left]-\infty,-\overline{z}_{max}(\varepsilon,\ell_{z})\right[,$
(4.159) $\displaystyle\mathscr{B}_{3}$
$\displaystyle:=\left]\rho^{K}_{0,max}+\frac{\eta}{2},\tilde{\rho}_{1}(\varepsilon,\ell_{z})\right[\times\mathbb{R},$
(4.160) $\displaystyle\mathscr{B}_{4}$
$\displaystyle:=\left]\tilde{\rho}_{2}(\varepsilon,\ell_{z}),\infty\right[\times\mathbb{R}.$
(4.161)

In order to define $\mathscr{B}_{5}$, we choose
$z_{r}(\varepsilon,\ell_{z})\in]\overline{z}_{max}(\varepsilon,\ell_{z}),\tilde{z}_{max}(\varepsilon,\ell_{z})[$,
$\rho^{1}_{r}(\varepsilon,\ell_{z})\in]\overline{\rho}_{1}(z,\varepsilon,\ell_{z}),\tilde{\rho}_{1}(z,\varepsilon,\ell_{z})[$
and
$\rho^{2}_{r}(\varepsilon,\ell_{z})\in]\overline{\rho}_{2}(z,\varepsilon,\ell_{z}),\tilde{\rho}_{2}(z,\varepsilon,\ell_{z})[$
and we set

$\mathscr{B}_{5}:=]\rho_{r}^{1}(\varepsilon,\ell_{z}),\rho_{r}^{2}(\varepsilon,\ell_{z})[\times]-z_{r}(\varepsilon,\ell_{z}),z_{r}(\varepsilon,\ell_{z})[.$
(4.162)

######  Remark 21.

If $z_{max}(\varepsilon,\ell_{z})=0$, then the geodesic is confined in the
equatorial plane and $Z^{K}(\varepsilon,\ell_{z})$ is reduced to the set of
points $\rho_{i}^{K}(\varepsilon,\ell_{z})$. In this case, there is no work to
be done here. Indeed, the classification of the equatorial orbits is the same
as that of Schwarzschild geodesics.
In our work, we are interested in the general case (not necessarily equatorial
orbits.). Therefore, we will assume that $\displaystyle
z_{max}(\varepsilon,\ell_{z})>0$.

By construction, the above subsets cover $\mathscr{B}$ (See Figure 12):

######  Lemma 42.

We have

$\mathscr{B}=\mathscr{B}^{abs}\cup\left(\cup_{i=1\cdots
5}\mathscr{B}_{i}\right).$

Now, we introduce the following functions

$\displaystyle\rho_{abs}^{0}(\varepsilon,\ell_{z},\theta)$
$\displaystyle:=\sqrt{\Delta(r_{abs}^{0}(\varepsilon,\ell_{z},\theta))}\sin\theta,$
(4.163) $\displaystyle z_{abs}^{0}(\varepsilon,\ell_{z},\theta)$
$\displaystyle:=(r_{abs}^{0}(\varepsilon,\ell_{z},\theta)-1)\cos\theta,$
$\displaystyle\rho_{tr}^{1}(\varepsilon,\ell_{z},\theta)$
$\displaystyle:=\sqrt{\Delta(r_{tr}^{1}(\varepsilon,\ell_{z},\theta))}\sin\theta,$
(4.164) $\displaystyle z_{tr}^{1}(\varepsilon,\ell_{z},\theta)$
$\displaystyle:=(r_{tr}^{1}(\varepsilon,\ell_{z},\theta)-1)\cos\theta,$

$\displaystyle\rho_{tr}^{2}(\varepsilon,\ell_{z},\theta)$
$\displaystyle:=\sqrt{\Delta(r_{tr}^{2}(\varepsilon,\ell_{z},\theta))}\sin\theta,$
(4.165) $\displaystyle z_{tr}^{2}(\varepsilon,\ell_{z},\theta)$
$\displaystyle:=(r_{tr}^{2}(\varepsilon,\ell_{z},\theta)-1)\cos\theta,$

$\displaystyle\rho_{tr}(\varepsilon,\ell_{z},r)$
$\displaystyle:=\sqrt{\Delta(r)}\sin\theta_{tr}(\varepsilon,\ell_{z},r),$
(4.166) $\displaystyle z_{tr}(\varepsilon,\ell_{z},r)$
$\displaystyle:=(r-1)\cos\theta_{tr}(\varepsilon,\ell_{z},r),$

where $r_{abs}^{0}$, $r_{abs}^{1}$, $r_{abs}^{2}$ and (4.166) are defined by
(4.119), (4.120), (4.121), (4.122) respectively. We state the following lemma

######  Lemma 43.

$\forall(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, we have

  * •

$z_{abs}^{0}(\varepsilon,\ell_{z},\cdot)$,
$z_{tr}^{1}(\varepsilon,\ell_{z},\cdot)$ and
$z_{tr}^{2}(\varepsilon,\ell_{z},\cdot)$ are monotonically decreasing on
$]0,\pi[$,

  * •

$\rho_{tr}(\varepsilon,\ell_{z},\cdot)$ is monotonically increasing on
$]r^{K}_{1}(\varepsilon,\ell_{z}),r^{K}_{2}(\varepsilon,\ell_{z})[$.

###### Proof.

  1. 1.

First of all, it is easy to see that $\displaystyle
z_{abs}^{0}(\varepsilon,\ell_{z},\cdot)$ is smooth on $]0,\pi[$ and
$\forall\theta\in]0,\pi[,$ we have

$\frac{\partial
z_{abs}^{0}}{\partial\theta}(\varepsilon,\ell_{z},\theta)=\frac{\partial
r_{abs}^{0}}{\partial\theta}(\varepsilon,\ell_{z},\theta)\cos\theta-(r_{abs}^{0}-1)\sin\theta.$

By the monotonicity properties of $r_{abs}^{0}(\varepsilon,\ell_{z},\cdot)$
(decreasing on $\displaystyle\left]0,\frac{\pi}{2}\right[$ and increasing on
$\displaystyle\left]\frac{\pi}{2},\pi\right[$), the first term of the right
hand side is always negative. Moreover, the second term is always negative on
$]0,\pi[$. Therefore, $z_{abs}^{0}(\varepsilon,\ell_{z},\cdot)$ is
monotonically decreasing on $]0,\pi[$ and we can write write $\theta$ in terms
of $z$. The same analysis can be made for
$z_{tr}^{1}(\varepsilon,\ell_{z},\cdot)$ and
$z_{tr}^{2}(\varepsilon,\ell_{z},\cdot)$.

  2. 2.

Since $\theta_{tr}(\varepsilon,\ell_{z},\cdot)$ is smooth,
$\rho_{tr}(\varepsilon,\ell_{z},\cdot)$ is smooth on
$]r^{K}_{1}(\varepsilon,\ell_{z}),r^{K}_{2}(\varepsilon,\ell_{z})[$ and we
have

$\partial_{r}\rho_{tr}(\varepsilon,\ell_{z},r)=\frac{\Delta^{\prime}(r)}{2\sqrt{\Delta(r)}}\sin\theta_{tr}(\varepsilon,\ell_{z},r)+\sqrt{\Delta(r)}\partial_{r}\theta_{tr}(\varepsilon,\ell_{z},r)\cos\theta_{tr}(\varepsilon,\ell_{z},r).$

Since $\theta_{tr}(\varepsilon,\ell_{z},\cdot)$ is monotonically increasing on
$]r^{K}_{1}(\varepsilon,\ell_{z}),\tilde{r}(\varepsilon,\ell_{z})[$ and
$\theta_{tr}(\varepsilon,\ell_{z},r)\in\left]\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z}),\frac{\pi}{2}\right[$,
the second term is always positive. Therefore,
$\rho_{tr}(\varepsilon,\ell_{z},\cdot)$ is monotonically increasing on
$]r^{K}_{1}(\varepsilon,\ell_{z}),r^{K}_{2}(\varepsilon,\ell_{z})[$ and we can
write $r$ in terms of $\rho$.

∎

Now, we introduce the following functions

######  Definition 18.

Let $(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$.

  1. 1.

$\displaystyle\Phi^{K,abs}_{(\varepsilon,\ell_{z})}:]-\gamma,\gamma[\mapsto]0,\rho_{0}^{K}(\varepsilon,\ell_{z})]$
is defined by

$\Phi^{K,abs}_{(\varepsilon,\ell_{z})}(z):=\rho^{0}_{abs}(\varepsilon,\ell_{z},(z_{abs}^{0}(\varepsilon,\ell_{z},\cdot))^{-1}(z)),$
(4.167)

  2. 2.

$\displaystyle\Phi^{K,i}_{(\varepsilon,\ell_{z})}$ are defined in the
following way

$\displaystyle\Phi^{K,1}_{(\varepsilon,\ell_{z})}$
$\displaystyle:I^{1}_{(\varepsilon,\ell_{z})}:=]\overline{\rho}_{1}(\varepsilon,\ell_{z}),\overline{\rho}_{2}(\varepsilon,\ell_{z})[\mapsto]\overline{z}_{max}(\varepsilon,\ell_{z}),z_{max}(\varepsilon,\ell_{z})],$
(4.168) $\displaystyle\Phi^{K,2}_{(\varepsilon,\ell_{z})}$
$\displaystyle:I^{2}_{(\varepsilon,\ell_{z})}:=]\overline{\rho}_{1}(\varepsilon,\ell_{z}),\overline{\rho}_{2}(\varepsilon,\ell_{z})[\mapsto[-z_{max}(\varepsilon,\ell_{z}),-\overline{z}_{max}(\varepsilon,\ell_{z})[$
$\displaystyle\Phi^{K,3}_{(\varepsilon,\ell_{z})}$
$\displaystyle:I^{3}_{(\varepsilon,\ell_{z})}:=]-\tilde{z}_{max}(\varepsilon,\ell_{z}),\tilde{z}_{max}(\varepsilon,\ell_{z})[\mapsto]\tilde{\rho}_{1}(\varepsilon,\ell_{z}),\rho_{1}^{K}(\varepsilon,\ell_{z})],$
$\displaystyle\Phi^{K,4}_{(\varepsilon,\ell_{z})}$
$\displaystyle:I^{4}_{(\varepsilon,\ell_{z})}:=]-\tilde{z}_{max}(\varepsilon,\ell_{z}),\tilde{z}_{max}(\varepsilon,\ell_{z})[\mapsto]\tilde{\rho}_{2}(\varepsilon,\ell_{z}),\rho_{2}^{K}(\varepsilon,\ell_{z})],$

where

$\displaystyle\Phi^{K,1}_{(\varepsilon,\ell_{z})}(\rho):=z_{tr}(\varepsilon,\ell_{z},(\rho_{tr}(\varepsilon,\ell_{z},\cdot))^{-1}(\rho)),$
(4.169)
$\displaystyle\Phi^{K,2}_{(\varepsilon,\ell_{z})}(\rho):=-\Phi^{K,1}_{(\varepsilon,\ell_{z})}(\rho),$
$\displaystyle\Phi^{K,3}_{(\varepsilon,\ell_{z})}(z):=\rho^{1}_{tr}(\varepsilon,\ell_{z},(z_{tr}^{1}(\varepsilon,\ell_{z},\cdot))^{-1}(z)),$
$\displaystyle\Phi^{K,4}_{(\varepsilon,\ell_{z})}(z):=\rho^{2}_{tr}(\varepsilon,\ell_{z},(z_{tr}^{2}(\varepsilon,\ell_{z},\cdot))^{-1}(z)).$

Following Lemma 43, we obtain

######  Lemma 44.

$\forall(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, the functions
$\Phi^{K,abs}_{\varepsilon,\ell_{z}}$ and $\Phi^{K,i}_{\varepsilon,\ell_{z}}$
are well-defined and smooth on $]-\gamma,\gamma[$ and
$I^{i}_{(\varepsilon,\ell_{z})}$ respectively. Moreover,

  1. 1.

$\Phi^{K,abs}_{(\varepsilon,\ell_{z})}$ admits a unique critical point (a
global maximum) on $]-\gamma,\gamma[$ given by
$\rho^{0}_{abs}(\varepsilon,\ell_{z})$, reached at the point $z=0$.

  2. 2.

$\Phi^{K,1}_{(\varepsilon,\ell_{z})}$ admits a unique critical point (a global
maximum) on $I^{1}_{(\varepsilon,\ell_{z})}$ given by
$z_{max}(\varepsilon,\ell_{z})$, reached at the point
$\rho=\rho_{max}(\varepsilon,\ell_{z})$.

  3. 3.

$\Phi^{K,2}_{(\varepsilon,\ell_{z})}$ admits a unique critical point (a global
minimum) on $I^{2}_{(\varepsilon,\ell_{z})}$ given by
$-z_{max}(\varepsilon,\ell_{z})$, reached at the point
$\rho=\rho_{max}(\varepsilon,\ell_{z})$.

  4. 4.

$\Phi^{K,3}_{(\varepsilon,\ell_{z})}$ admits a unique critical point (a global
minimum) on $I^{3}_{(\varepsilon,\ell_{z})}$ given by
$\rho^{1}_{abs}(\varepsilon,\ell_{z})$, reached at the point $z=0$.

  5. 5.

$\Phi^{K,4}_{(\varepsilon,\ell_{z})}$ admits a unique critical point (a global
maximum) on $I^{4}_{(\varepsilon,\ell_{z})}$ given by
$\rho^{2}_{abs}(\varepsilon,\ell_{z})$, reached at the point $z=0$.

Now, we use the previous results to obtain

######  Proposition 16.

$\forall(\varepsilon,\ell_{z})\in\mathcal{B}_{bound}$ ,

  1. 1.

there exists
$\displaystyle\Phi^{K,abs}_{(\varepsilon,\ell_{z})}:]-\gamma,\gamma[\mapsto]0,\rho_{0}^{K}(\varepsilon,\ell_{z})]$
such that

$Z^{K,abs}(\varepsilon,\ell_{z})=\text{Gr}\left(\Phi^{K,abs}_{(\varepsilon,\ell_{z})}\right),$

  2. 2.

there exist
$\displaystyle\Phi^{K,i}_{(\varepsilon,\ell_{z})}:I^{i}_{(\varepsilon,\ell_{z})}\mapsto\mathbb{R},\;i=1\cdots
4$ such that

$Z^{K,trapped}(\varepsilon,\ell_{z})=\bigcup_{i=1\cdots
4}\text{Gr}\left(\Phi^{K,i}_{(\varepsilon,\ell_{z})}\right),$

where $\displaystyle\Phi^{K,abs}_{(\varepsilon,\ell_{z})}$ and
$\displaystyle\Phi^{K,i}_{(\varepsilon,\ell_{z})}$ are given by Definition 18.

To summarise, instead of solving the equation on $\mathscr{B}$, we solve it on
$\mathscr{B}_{i}$ and $\mathscr{B}^{abs}$ and on each of these region, the
solutions are either functions of $\rho$ or $z$.
Therefore, the problem of finding solutions on $\mathscr{B}$ to the equation

$E_{\ell_{z}}^{K}(\rho,z)=\varepsilon$ (4.170)

is equivalent to the problem of finding solutions on $\mathscr{B}_{i}$ and on
$\mathscr{B}^{abs}$. Given $(\varepsilon,\ell_{z})\in\mathcal{B}_{bound}$, the
latter is equivalent to the problem of finding a function defined on
$I^{i}_{(\varepsilon,\ell_{z})}$ or $I^{abs}_{(\varepsilon,\ell_{z})}$. Hence,
by the above proposition, the solutions are given by
$\Phi^{K,abs}_{(\varepsilon,\ell_{z})}$ in $\mathscr{B}^{abs}$ and by
$\Phi^{K,i}_{(\varepsilon,\ell_{z})}$ in $\mathscr{B}_{i}$.

(a) First component $Z^{K,abs}(\varepsilon,\ell_{z})$

(b) Second component $Z^{K,trapped}(\varepsilon,\ell_{z})$

Figure 12: The zero velocity curve (in blue) with $d=0.5$ and
$(\varepsilon,\ell_{z})=(0.98,4)\in\mathcal{A}_{bound}$. The dashed lines
separate $\mathscr{B}$ into six regions: $\mathscr{B}_{0}$ is the region that
contains $Z^{K,abs}$ and it is delimited by the red line (Figure (a)). The
region $\mathscr{B}_{5}$ is the region bounded by the cyan rectangle in Figure
(b). $\mathscr{B}_{3}$ is the region delimited by the two green lines in
Figure (b). $\mathscr{B}_{4}$ is the region delimited by the cyan vertical
line from the left. The horizontal black line above the equatorial plane
delimits the region $\mathscr{B}_{1}$ from below and the horizontal black line
below the equatorial plane delimits the region $\mathscr{B}_{2}$ from above.

####  4.2.4 Domain of trapped geodesics in a sub-extremal Kerr spacetime

The aim of this section is to prove that the region
$\mathscr{B}^{trapped}\subset\overline{\mathscr{B}}$ where geodesic motion
occurs stay away from $\partial\overline{\mathscr{B}}$. From the previous
section, we recall that trapped timelike future-directed orbits occur when the
associated zero velocity curve has a compact connected component,
$Z^{K,trapped}\subset\mathscr{B}$. In this case, the orbit is confined in the
region bounded by the latter curve. We note that the zero velocity curves are
only defined in $\mathscr{B}$. More precisely, the effective potential energy
is only defined in the exterior region of the spacetime minus the axis of
symmetry. However, since we are interested in metrics which are at least
$C^{2}-$extendable to the horizon and the axis of symmetry, the boundaries of
$\mathscr{B}$, we need to consider the extension of
$Z^{K}(\varepsilon,\ell_{z})$ in $\overline{\mathscr{B}}$ associated to
timelike future-directed geodesics in order to determine
$\mathscr{B}^{trapped}\subset\overline{\mathscr{B}}$. Given
$(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, $Z^{K}(\varepsilon,\ell_{z})$
may have accumulation points on the boundary, namely on the poles. However, in
the case of the second connected component of $Z^{K}(\varepsilon,\ell_{z})$,
that is $Z^{K,trapped}(\varepsilon,\ell_{z})$, the accumulation points are
distincts from $\partial\overline{\mathscr{B}}$ and thus from the poles.
Before we state the precise result of this section, we recall the definition
of $\overline{\mathscr{B}}$

$\overline{\mathscr{B}}:=\overline{\mathscr{B}_{\mathcal{A}}}\cup\overline{\mathscr{B}_{\mathcal{H}}}\cup\overline{\mathscr{B}_{N}}\cup\overline{\mathscr{B}_{S}}.$

where

${\overline{\mathscr{B}_{H}}}:=\left\\{(\rho,z)\in\overline{\mathscr{B}},\rho^{2}+(z\pm\beta)^{2}>\frac{\beta}{a},|z|+|\rho|<\left(1+\frac{1}{b}\right)\beta\right\\},$

${\overline{\mathscr{B}_{A}}}:=\left\\{(\rho,z)\in\overline{\mathscr{B}},\rho^{2}+(z\pm\beta)^{2}>\frac{\beta}{a},|z|+|\rho|>\left(1-\frac{1}{b}\right)\beta\right\\}.$

${\overline{\mathscr{B}_{N}}}:=\left\\{(\rho,z)\in\overline{\mathscr{B}},\;z\neq\beta,\;\rho^{2}+(z-\beta)^{2}<\frac{\beta}{c}\right\\}\cup\left\\{(s,\chi)\in\overline{\mathscr{B}}\quad/0\leq
s,\chi<\left(\frac{\beta}{e}\right)^{\frac{1}{4}}\right\\}.$

${\overline{\mathscr{B}_{N}}}:=\left\\{(\rho,z)\in\overline{\mathscr{B}},\;z\neq-\beta,\;\rho^{2}+(z+\beta)^{2}<\frac{\beta}{c}\right\\}\cup\left\\{(s^{\prime},\chi^{\prime})\in\overline{\mathscr{B}}\quad/0\leq
s^{\prime},\chi^{\prime}<\left(\frac{\beta}{e}\right)^{\frac{1}{4}}\right\\}.$

for $0<e<c<a<b$.

We state the following result

######  Lemma 45.

Let $\mathcal{B}_{bound}\subset\subset\mathcal{A}_{bound}$. Then, we can
choose $0<e<c$ uniform in $(\varepsilon,\ell_{z})$ such that
$\forall(\varepsilon,\ell_{z})\in\mathcal{B}_{bound}$,

$\overline{\mathscr{B}_{N}},\overline{\mathscr{B}_{S}}\cap
Z^{K,trapped}(\varepsilon,\ell_{z})=\emptyset.$ (4.171)

Moreover, $\forall(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, the
accumulation points for $Z^{K,abs}(\varepsilon,\ell_{z})$ are
$p_{N}:=(0,\beta)$ and $p_{S}:=(0,-\beta)$.

###### Proof.

∎

Now, we define the domain of trapped timelike geodesics,
$\mathscr{B}^{\pm,trapped}(a,M)$, which depends only on $(a,M)$ by the
following proposition

######  Proposition 17 (Domain of trapped geodesics in a sub-extremal Kerr
spacetime).

Let $(a,M)$ be such that $0<|a|<M$. Then, there exists $\rho^{mb,\pm}(a,M)>0$

$\forall(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}\quad\;\quad\rho^{K,\pm}_{1}(\varepsilon,\ell_{z},a,M)>\rho^{mb,\pm}(a,M)>0,$

where $\rho_{1}^{K,\pm}(\varepsilon,\ell_{z},a,M)$ is the second largest root
of the equation

$E_{\ell_{z}}(\rho,0)=\varepsilon$

and

$\rho^{mb,\pm}(a,M):=\sqrt{\Delta\left(r^{\pm}_{mb}(a,M)\right)},\quad
r^{\pm}_{mb}(a,M):=2M\mp a+2\sqrt{M^{2}\mp aM}.$

Hence, we set

$\mathscr{B}^{\pm,trapped}(a,M):=\left]\rho^{mb,\pm}(a,M),\infty\right[\times\mathbb{R}$

######  Remark 22.

We recall that $r^{\pm}_{mb}(a,M)$ is defined to be the unique solution of the
equation

$\Phi_{\pm}(r)=1,$

where $\Phi_{\pm}$ is defined by (4.41).

###### Proof.

Let $(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$ and recall that
$\displaystyle d=\frac{a}{M}$. Then, by monotonicity properties of
$\tilde{\rho}_{1}^{K,\pm}$191919Recall that $\tilde{\rho}$ is defined by
$\rho=M\tilde{\rho}$., proved in Lemma 39, we have

$\tilde{\rho}_{1}^{K,\pm}(\varepsilon,\ell_{z},d)>\tilde{\rho}_{1}^{K,\pm}(\varepsilon,\ell_{ub}(\varepsilon,d),d)=\tilde{\rho}^{K,\pm}_{s}(\ell_{ub}^{\pm}(\varepsilon,d),d)>\tilde{\rho}^{K,\pm}_{s}(\ell_{ub}^{\pm}(1,d),d)=\sqrt{\Delta\left(\tilde{r}^{\pm}_{mb}(d)\right)}.$

It remains to show that $\displaystyle\rho^{mb,\pm}(a,M)>0$ for all
$\displaystyle(a,M)$ such that $\displaystyle 0<|a|<M$.
By the definition of $\displaystyle\rho^{mb,\pm}(a,M)$, it is easy to see that
$\displaystyle\rho^{mb,-}(a,M)$ never vanishes and
$\displaystyle\rho^{mb,+}(a,M)$ vanishes if and only if $|a|=M$. Indeed,

  1. 1.

$\tilde{\rho}^{mb,+}:[0,1[\to\mathbb{R}_{+}$ is monotonically decreasing on
$]0,1[$ and we have

$\lim_{d\to 1}\,\tilde{\rho}^{mb,+}(d)=0.$

  2. 2.

$\tilde{\rho}^{mb,-}:[0,1[\to\mathbb{R}_{+}$ is monotonically increasing on
$]0,1[$ and we have

$\lim_{d\to 1}\,\tilde{\rho}^{mb,-}(d)\approx 4.83.$

∎

For later purposes, in particular the study of regularity of the matter terms,
It will be convenient to adjust the definition of $\overline{\mathscr{B}_{H}}$
and $\overline{\mathscr{B}_{A}}$ so that the support of Vlasov matter remain
in the region $\overline{\mathscr{B}_{A}}$. More precisely, we will show the
following

######  Lemma 46.

There exists $b>0$ such that
$\forall(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$, we have

$\overline{\mathscr{B}_{H}}\cap
Z^{K,trapped}(\varepsilon,\ell_{z})=\emptyset\quad\text{and}\quad
Z^{K,trapped}(\varepsilon,\ell_{z})\subset\overline{\mathscr{B}_{A}}.$

###### Proof.

First of all, we claim that $\forall d\in]0,1[$

$\tilde{\rho}^{mb,-}(d)>\sqrt{1-d^{2}}\quad\text{and}\quad\tilde{\rho}^{mb,+}(d)>\sqrt{1-d^{2}}.$

Therefore,

$\forall(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}\;,\;\tilde{\rho}_{1}^{K,\pm}(\varepsilon,\ell_{z})>\tilde{\rho}^{mb,\pm}(d)>\left(1+\frac{1}{b}\right)\sqrt{1-d^{2}}.$

∎

######  Remark 23.

We note that, by Proposition 17, in a sub-extremal Kerr black hole, all
trapped timelike geodesics lie in $\mathscr{B}^{\pm,trapped}(a,M)$. However,
in the limiting case (extremal Kerr), $\mathscr{B}^{+,trapped}(a,M)$ coincides
with the whole exterior region, $\mathscr{B}$. As for,
$\mathscr{B}^{-,trapped}(a,M)$, it is located at approximatively $4.83M$ away
from the horizon (See proof of Proposition 17).
In this work, we use the fact that trapped timelike geodesics lie away from
the horizon and thus we consider only the sub extremal case. In fact, we will
need a non-trivial lower bound on the inner boundary of the support of the
Vlasov matter.
In the extremal case, one possibility could be to consider only orbits which
are retrograde.

One can wonder what is the location of trapped non-spherical orbits with
respect to the ergoregion $\mathscr{E}$. Recall that the ergosurface, the
boundary of $\mathscr{E}$, denoted by $\mathscr{S}$, is defined to be the set
of points $(\rho,z)\in\mathscr{B}$ such that

$g(T,T)=0.$

In a sub-extremal Kerr exterior $0<d<1$, the latter is equivalent to the set
of points $(\rho,z)\in\overline{\mathscr{B}}$ such that

$V_{K}(\rho,z)=0.$

Direct computations lead to

$\mathscr{S}=\left\\{(\rho,z)\in\overline{\mathscr{B}}\;:\;z^{2}=\left(1-\frac{\rho}{d}\right)\left(1-d^{2}\left(1-\frac{\rho}{d}\right)\right)\right\\}.$

We refer to Figure 14 for the shape of $\mathscr{S}$. On the other hand, we
have

  * •

$\displaystyle\forall d\in]0,1[,$

$\tilde{\rho}^{mb,-}(d)>d.$

  * •

$\displaystyle d_{0}:=2(\sqrt{2}-1)$ is the unique solution in $]0,1[$ to

$\tilde{\rho}^{mb,+}(d)=d.$

Note that $\mathscr{S}$ intersect the equatorial plane uniquely at the point
$\rho_{eq}=d$. Therefore, it is straightforward to see that

$\mathscr{B}^{-,trapped}(a,M)\cap\mathscr{E}=\emptyset.$ (4.172)

and

$\mathscr{B}^{+,trapped}(a,M)\cap\mathscr{E}=\emptyset\quad\text{if and only
if}\quad d<d_{0}.$ (4.173)

We refer to Figure 13 to visualise the intersection when $d>d_{0}$ .

Figure 13: Intersection of $\mathscr{E}(d)$ (blue curve) and
$Z^{K,trapped}(\varepsilon,\ell_{z})$ (red curve) when $d=0.95$. The remaining
parameters are set to :$\varepsilon=0.99$ and $\ell_{z}=2.5$.  Figure 14:
Shape of $S$ in the following cases from the left to the right: $d=0.1$,
$d=\sqrt{0.5}$ and $d=0.99$

######  Remark 24.

  * -

Recall that a necessary condition for energy extraction from the black hole
and thus a negative energy $\varepsilon$ is that $d\ell_{z}<0$ and $g_{tt}>0$,
which means that retrograde orbits that lie inside the ergoregion have
negative energies. This gives another explanation for (4.172).

  * -

Finally, we note that when computing the matter terms in 9.2.1, we will have
to take into account the ergoregion. In this case, we will have to split the
matter sources into a term supported in the ergoregion and another which is
supported in the remaining region of the spacetime exterior.

##  5 Reduced Einstein-Vlasov system

In this section, we compute the components of the energy momentum tensor and
we reduce the Einstein equations with a source arising from a Vlasov field to
a system of elliptic equations.

###  5.1 Assumptions and General Framework

Recall from Section 3.2.1 the metric ansatz considered throughout this work:

$g=-Vdt^{2}+2Wdtd\phi+Xd\phi^{2}+e^{2\lambda}\left(d\rho^{2}+dz^{2}\right),$
(5.1)

where $V,W,X,\lambda:\mathscr{B}\to\mathbb{R}$ and
$\mathscr{B}=\left\\{(\rho,z)\;:\;\rho>0\;,\;z\in\mathbb{R}\right\\}.$
Following the work [16], we replace the metric components $V,W,X,\lambda$ by
$(X,W,\theta,\sigma,\lambda)$, called "metric data", which reduces under
symmetries in a nice manner and where $\theta$ and $\sigma$ are given by
(3.12) and (3.13) respectively.

###  5.2 Ansatz for the distribution function

We are interested in stationary and axisymmetric distribution functions.
Therefore, we assume that $f:\Gamma_{1}\to\mathbb{R}_{+}$ takes the form

$f(t,\phi,\rho,z,\phi,v^{\rho},v^{\phi},v^{z})=\Phi(\varepsilon,\ell_{z})\Psi_{\eta}(\rho,(\varepsilon,\ell_{z}),(X,W,\sigma))$
(5.2)

where

  * •

$\displaystyle\Phi:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{+}$ is a $C^{2}$
function and is supported on some compact set
$\displaystyle\mathcal{B}_{bound}$ of $\mathcal{A}_{bound}$. Without loss of
generality (See Lemma 38), we assume that $\mathcal{B}_{bound}$ has the form

$\mathcal{B}_{bound}:=\mathcal{B}_{bound}^{-}\cup\mathcal{B}_{bound}^{+}$

where $\mathcal{B}_{bound}^{-}$ and $\mathcal{B}_{bound}^{+}$ are defined by

$\mathcal{B}_{bound}^{\pm}:=[\varepsilon_{1}^{\pm},\varepsilon_{2}^{\pm}]\times[\ell_{1}^{\pm},\ell_{2}^{\pm}]$

where $\displaystyle\varepsilon^{i,\pm}_{j}$ and
$\displaystyle\ell^{i,\pm}_{j}$ satisfy

$\varepsilon^{\pm}_{min}(d)<\varepsilon^{\pm}_{1}<\varepsilon^{\pm}_{2}<1\quad\text{,}\quad\ell_{lb}(\varepsilon^{+}_{2})<\ell^{+}_{1}<\ell^{+}_{2}<\ell_{ub}(\varepsilon^{+}_{1})\quad\text{and}\quad\ell_{ub}(\varepsilon^{-}_{1})<\ell^{-}_{1}<\ell^{-}_{2}<\ell_{lb}(\varepsilon^{-}_{2}).$
(5.3)

  * •

$\eta>0$ is a constant that will be specified later (see Section 7),
$\Psi_{\eta}(\cdot,\cdot,h)\in
C^{\infty}(]0,\infty[\times\mathcal{A}^{admissible},\mathbb{R}_{+})$ is a cut-
off function depending on the metric data $h:=(X,W,\sigma)$, such that

$\Psi_{\eta}(\cdot,(\varepsilon,\ell),h):=\left\\{\begin{aligned}
&\mbox{\Large$\chi$}_{\eta}(\cdot-\rho_{1}(h,(\varepsilon,\ell_{z}))),\quad(\varepsilon,\ell_{z})\in\mathcal{B}_{bound},\\\
&0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(\varepsilon,\ell_{z})\notin\mathcal{A}_{bound},\end{aligned}\right.$
(5.4)

where $\rho_{1}$ is a positive function of $(h,\varepsilon,\ell_{z})$ which
will be defined later 202020$\rho_{1}$ is defined by (7.14). It can be seen as
the perturbation of $\rho^{1}_{K}(\varepsilon,\ell_{z})$, the second largest
root of the equation $\displaystyle E^{K}_{\ell_{z}}(\rho,0)=0$. and
$\mbox{\Large$\chi$}_{\eta}\in C^{\infty}(\mathbb{R},\mathbb{R}_{+})$ is a
cut-off function such that

$\mbox{\Large$\chi$}_{\eta}(s)=\left\\{\begin{aligned} &1\quad s\geq 0,\\\
&\leq 1\quad s\in[-\eta,0],\\\ &0\quad s<-\eta.\end{aligned}\right.$ (5.5)

  * •

$\varepsilon$ and $\ell_{z}$ are defined by (3.18) and (3.19).

###  5.3 Reparametrization of the fibre $\Gamma_{x}$ and the components of the
energy-momentum tensor

In this section, we will compute the components of the energy momentum tensor
$\tensor{{T}}{{}_{\alpha\beta}}$ provided $g$ has the form (3.11) and $f$ has
the form (5.2). First of all, we recall the definition of $\Omega$:

$\Omega=\frac{\partial}{\partial
t}+\omega\frac{\partial}{\partial\phi}\quad\text{where}\quad\omega=-\frac{W}{X},$

the timelike vector field defined on $\mathscr{B}$ which was fixed in Section
3.2.5 for the time orientation.
Recall that by definition, if $c_{\alpha}$ is the coordinate basis associated
to the spacetime coordinates, then any tangent vector can be written as

$v=v^{\alpha}c_{\alpha}.$

The $v^{\alpha}$ are then called the conjugate coordinates to the spacetime
coordinates.
Now let $(v^{t},v^{\phi},v^{\rho},v^{z})$ be the conjugate coordinates to the
spacetime coordinates $(t,\phi,\rho,z)$. Let $x=(t,\phi,\rho,z)\in\mathscr{B}$
and denote by

$c_{0}=\frac{\partial}{\partial
t}\;,\;c_{1}=\frac{\partial}{\partial\phi}\;,\;c_{2}=\frac{\partial}{\partial\rho}\;,\;c_{3}=\frac{\partial}{\partial
z}$

the canonical basis determined by the local coordinates. We consider the
orthonormal frame defined by

$e_{0}:=\sqrt{\frac{X}{\sigma^{2}}}\Omega\;,\;e_{1}:=\sqrt{\frac{1}{X}}\frac{\partial}{\partial\phi}\;,\;e_{2}:=e^{-\lambda}\frac{\partial}{\partial\rho}\;,\;e_{3}:=e^{-\lambda}\frac{\partial}{\partial
z}.$

Let $p^{\alpha}$ be the corresponding coordinates. Therefore:

$\displaystyle(v^{\alpha})=G(p^{\alpha})\quad\text{where}\quad\quad
G=\begin{pmatrix}\frac{\sqrt{X}}{\sigma}&0&0&0\\\
\frac{\omega\sqrt{X}}{\sqrt{\sigma}}&\frac{1}{\sqrt{X}}&0&0\\\
0&0&e^{-\lambda}&0\\\ 0&0&0&e^{-\lambda}\end{pmatrix}$

In the orthonormal frame, the mass shell condition is given by

$-(p^{0})^{2}+(p^{1})^{2}+(p^{2})^{2}+(p^{3})^{2}=-1.$

Now, we can express the $p^{0}$ in terms of the remaining $p^{i}$:

$p^{0}=\sqrt{1+|p|^{2}}.$

We recall the energy-momentum tensor associated to the metric $g$ and the
distribution function $f$,

$\forall(t,\phi,\rho,z)\in\mathcal{M}\;\quad\tensor{{T}}{{}_{\alpha\beta}}(t,\phi,\rho,z)=\int_{\Gamma_{x}}v_{\alpha}v_{\beta}f(t,\phi,\rho,z,v^{\rho},v^{\phi},v^{z})\;d\text{vol}_{x}(v),$

where $\Gamma_{x}$ is given by

$\Gamma_{x}:=\left\\{v^{\alpha}\in\mathbb{T}_{x}\mathcal{M}\;:\;\tensor{g}{{}^{\alpha\beta}}v_{\alpha}v_{\beta}=-1,\;v^{0}>0\right\\},$

where we recall that the condition $v^{0}>0$ is equivalent to $v$ is future-
pointed. Thus, after a first change of variables in the fibre $\Gamma_{x}$,
the energy momentum tensor is given by

$\tensor{{T}}{{}_{\alpha\beta}}[f]=\int_{\mathbb{R}^{3}}\;v_{\alpha}(p)v_{\beta}(p)f(x,v(p))\frac{dp^{1}dp^{2}dp^{3}}{\sqrt{1+|p|^{2}}}.$

We will make a second change of variables in order to simplify the expressions
of the energy momentum tensor components. To this end, we compute
$\varepsilon$ and $\ell_{z}$ in terms of the $p^{i}$s:

$\displaystyle\varepsilon=-v_{0}=\frac{\sigma}{\sqrt{X}}\left(1+|p|^{2}\right)^{\frac{1}{2}}+\omega\ell_{z}$

and

$\ell_{z}=\sqrt{X}p^{1}.$

From the mass shell condition, we have

$e^{2\lambda}\left((v^{\rho})^{2}+(v^{z})^{2}\right)=J(\rho,z,\varepsilon,\ell_{z}),$
(5.6)

where $J$ is given by

$J(\rho,z,\varepsilon,\ell_{z})=-1+\frac{X}{\sigma^{2}}\varepsilon^{2}+\frac{2W}{\sigma^{2}}\varepsilon\ell_{z}-\frac{V}{\sigma^{2}}\ell_{z}^{2}.$

We recall that $\sigma^{2}=XV+W^{2}$. We introduce the polar variables
$(\sqrt{J},\vartheta)\in[0,\infty[\times[0,2\pi]$ in the following way:

$\displaystyle v^{2}$ $\displaystyle=e^{-\lambda}\sqrt{J}\sin\vartheta,$
$\displaystyle v^{3}$ $\displaystyle=e^{-\lambda}\sqrt{J}\cos\vartheta.$

Therefore,

$\displaystyle p^{2}$ $\displaystyle=\sqrt{J}\sin\vartheta,$ $\displaystyle
p^{3}$ $\displaystyle=\sqrt{J}\cos\vartheta.$

Now we make the change of variables from $(p^{1},p^{2},p^{3})$ to
$(p^{1},\sqrt{J},\vartheta)$. We have

$\frac{dp^{1}dp^{2}dp^{3}}{\sqrt{1+|p|^{2}}}=\frac{\sqrt{J}}{\sqrt{1+J+(p^{1})^{2}}}d\sqrt{J}d\vartheta
dp^{\phi}.$

Therefore,

$\tensor{{T}}{{}_{\alpha\beta}}[f]=\int_{\mathbb{R}_{+}}\int_{0}^{2\pi}\int_{\mathbb{R}}\;v_{\alpha}(p)v_{\beta}(p)f(x,v(p))\frac{\sqrt{J}}{\sqrt{1+J+(p^{1})^{2}}}d\sqrt{J}d\vartheta
dp^{1}.$ (5.7)

We make a last change of variables
$H(\rho,z):(p^{1},\sqrt{J},\vartheta)\mapsto(E,L,\vartheta)$ defined by

$\displaystyle E$
$\displaystyle:=\varepsilon-\omega\ell_{z}=\frac{\sigma}{\sqrt{X}}\left(1+J+(p^{1})^{2}\right)^{\frac{1}{2}},$
(5.8) $\displaystyle L$ $\displaystyle:=\frac{\sqrt{X}}{\rho}p^{1}.$

It is easy to see that $H(\rho,z)$ is a smooth diffeomorphism on its image and
we have

$\frac{\sqrt{J}}{\sqrt{1+J+(p^{1})^{2}}}dp^{1}d\sqrt{J}d\vartheta=\frac{\rho}{\sigma}dEdLd\vartheta.$

It remains to compute the domain of the variables $E$ and $L$, which is the
image of above change of variables. From (5.8), $\forall J\geq 0,\forall
p^{1}\in\mathbb{R}$, we have

$E\geq\frac{\sigma}{\sqrt{X}}.$

By straightforward computations, we obtain

$J=\frac{X}{\sigma^{2}}E^{2}-1-\frac{\rho^{2}}{X}L^{2}.$

Since $J$ is positive, $L$ satisfies

$|L|\leq\frac{\sqrt{X}}{\rho}\left(-1+\frac{X}{\sigma^{2}}E^{2}\right)^{\frac{1}{2}}=:\tilde{L}(E,X,\sigma,\rho,z).$
(5.9)

Now we define $D(\rho,z)$ to be the set:

$D(\rho,z):=\left\\{(E,L)\;:\;E\geq\frac{\sigma}{\sqrt{X}}\quad\text{and
}\quad|L|\leq\tilde{L}(E,X,\sigma,\rho,z)\right\\}.$ (5.10)

Therefore,

$\text{Im}\,H(\rho,z)=D(\rho,z).$

By symmetry considerations, we have

$\tensor{{T}}{{}_{\rho\rho}}[f](\rho,z)=\tensor{{T}}{{}_{zz}}[f](\rho,z).$

Moreover, the only non-vanishing components of $\mathbb{T}_{\alpha\beta}$ are
$\tensor{{T}}{{}_{tt}},\tensor{{T}}{{}_{t\phi}},\tensor{{T}}{{}_{\phi\phi}},\tensor{{T}}{{}_{\rho\rho}}$
and $\tensor{{T}}{{}_{zz}}$. Now we compute

$\displaystyle\tensor{{T}}{{}_{tt}}[f](\rho,z)$
$\displaystyle=\int_{\mathbb{R}_{+}}\int_{0}^{2\pi}\int_{\mathbb{R}}\;(v_{0}(p))^{2}f(x,v(p))\frac{\sqrt{J}}{\sqrt{1+J+(p^{1})^{2}}}d\sqrt{J}d\chi
dp^{1}$ $\displaystyle=\int_{D(\rho,z)}\int_{0}^{2\pi}\;(E+\rho\omega
L)^{2}\Phi(E+\rho\omega,L)\frac{\rho}{\sigma}\Psi_{\eta}(\rho,(E+\rho\omega
L,\rho L),(X,W,\sigma))dEdLd\vartheta$
$\displaystyle=\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\left(E+\rho\omega
L\right)^{2}\Phi(E+\rho\omega L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL.$

Here, we used that $v_{0}(p)=-\varepsilon=-(E+\rho\omega L)$, $\ell_{z}=\rho
L$ and the independence of the integrand on $\vartheta$. The components
$\tensor{{T}}{{}_{tt}}[f](\rho,z)$, $\tensor{{T}}{{}_{t\phi}}[f](\rho,z)$ and
$\tensor{{T}}{{}_{\phi\phi}}[f](\rho,z)$ are computed in the same way. As for
$\tensor{{T}}{{}_{\rho\rho}}[f](\rho,z)$, we have

$\displaystyle\tensor{{T}}{{}_{\rho\rho}}[f](\rho,z)$
$\displaystyle=\int_{\mathbb{R}_{+}}\int_{0}^{2\pi}\int_{\mathbb{R}}\;(v_{\rho}(p))^{2}f(x,v(p))\frac{\sqrt{J}}{\sqrt{1+J+(p^{1})^{2}}}d\sqrt{J}d\chi
dp^{1}$
$\displaystyle=e^{2\lambda}\int_{D(\rho,z)}\int_{0}^{2\pi}\;J(E,L)\sin^{2}\vartheta\frac{\rho}{\sigma}\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))dEdLd\vartheta$
$\displaystyle=\frac{\pi\rho}{\sigma}e^{2\lambda}\int_{D(\rho,z)}\left(\frac{X}{\sigma^{2}}E^{2}-1-\frac{\rho^{2}}{X}L^{2}\right)\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL$
$\displaystyle=\frac{\pi\rho}{\sigma}e^{2\lambda}\int_{D(\rho,z)}\left(\tilde{L}(E,X,\sigma,\rho,z)^{2}-L^{2}\right)\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL.$

The final expression is obtained by (5.9). Hence,

$\tensor{{T}}{{}_{tt}}[f](\rho,z)=\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\left(E+\rho\omega
L\right)^{2}\Phi(E+\rho\omega L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL,$ (5.11)
$\tensor{{T}}{{}_{t\phi}}[f](\rho,z)=-\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\rho
L\left(E+\rho\omega L\right)\Phi(E+\rho\omega L,\rho
L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL,$ (5.12)
$\tensor{{T}}{{}_{\phi\phi}}[f](\rho,z)=\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}(\rho
L)^{2}\Phi(E+\rho\omega L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL,$ (5.13)
$\tensor{{T}}{{}_{\rho\rho}}[f](\rho,z)=\frac{\pi\rho}{\sigma}e^{2\lambda}\int_{D(\rho,z)}\left(\tilde{L}(E,X,\sigma,\rho,z)^{2}-L^{2}\right)\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL,$ (5.14)
$\tensor{{T}}{{}_{zz}}[f](\rho,z)=\frac{\pi\rho}{\sigma}e^{2\lambda}\int_{D(\rho,z)}\left(\tilde{L}(E,X,\sigma,\rho,z)^{2}-L^{2}\right)\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL,$ (5.15)

where $\tilde{L}$ was defined in (5.9). It remains to compute the intersection
of $D(\rho,z)$ and the support of $\Phi$ and $\Psi_{\eta}$. This will be done
in Section 9.2 concerning the regularity of the matter terms.

###  5.4 Static and axisymmetric black holes with matter

In this section, we apply Theorem $1.1$ in [16] concerning the modified
Carter-Robinson theory. First of all, we recall the following theorem

######  Theorem 2 (O.CHODOSH, Y.SHLAPENTOKH-ROTHMAN).

Suppose that $(\mathcal{M},g)$ solves the Einstein equations for some energy-
momentum tensor $\mathbb{T}$ satisfying

$\mathbb{T}(T,\partial_{\rho})=\mathbb{T}(T,\partial_{z})=\mathbb{T}(\Phi,\partial_{\rho})=\mathbb{T}(\Phi,\partial_{z})=0.$
(5.16)

Then, the metric data $(X,W,\theta,\sigma,\lambda)$ satisfies the following
equations on
$\mathscr{B}=\left\\{(\rho,z)\in\mathbb{R}^{2}\;:\;\rho>0\right\\}$:

  1. 1.

X satisfies

$\sigma^{-1}\partial_{\rho}(\sigma\partial_{\rho}X)+\sigma^{-1}\partial_{z}(\sigma\partial_{z}X)=e^{2\lambda}\left(-2\mathbb{T}(\Phi,\Phi)+{\text{Tr}(\mathbb{T})}X\right)+\frac{(\partial_{\rho}X)^{2}+(\partial_{z}X)^{2}-\theta_{\rho}^{2}-\theta_{z}^{2}}{X},$
(5.17)

  2. 2.

W satisfies

$\partial_{\rho}(X^{-1}W)d\rho+\partial_{z}(X^{-1}W)dz=\frac{\sigma}{X^{2}}(\theta_{\rho}dz-\theta_{z}d\rho),$
(5.18)

  3. 3.

$\theta$ satisfies

$d\theta=2\sigma^{-1}e^{2\lambda}\left(\mathbb{T}(\Phi,\Phi)W-\mathbb{T}(\Phi,T)X\right)d\rho\wedge
dz,$ (5.19)

as well as

$\sigma^{-1}\partial_{\rho}(\sigma\partial_{\rho}\theta)+\sigma^{-1}\partial_{z}(\sigma\partial_{z}\theta)=\frac{2\theta_{\rho}\partial_{\rho}X+2\theta_{z}\partial_{z}X}{X}$
(5.20)

  4. 4.

$\sigma$ satisfies

$X^{-1}\exp{-2\lambda}\sigma(\partial^{2}_{\rho}\sigma+\partial^{2}_{z}\sigma)=\mathbb{T}\left(T-X^{-1}W\Phi,T-X^{-1}W\Phi\right)-X^{-2}\sigma^{2}\mathbb{T}(\Phi,\Phi)+X^{-1}\sigma^{2}\text{Tr}(\mathbb{T}),$
(5.21)

  5. 5.

$\lambda$ satisfies the following equations at the points where
$|\partial\sigma|\neq 0$

$\partial_{\rho}\lambda=\alpha_{\rho}-\frac{1}{2}\partial_{\rho}\log
X,\quad\partial_{z}\lambda=\alpha_{z}-\frac{1}{2}\partial_{z}\log X,$ (5.22)

where

$\displaystyle((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})\alpha_{\rho}$
$\displaystyle=\frac{1}{2}(\partial_{\rho}\sigma)\sigma\left(\mathbb{T}(\partial_{\rho},\partial_{\rho})-\mathbb{T}(\partial_{z},\partial_{z})+\frac{1}{2}\frac{(\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}\right)$
$\displaystyle+(\partial_{\rho}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)+(\partial_{z}\sigma)(\partial^{2}_{\rho,z}\sigma)$
$\displaystyle+(\partial_{z}\sigma)\sigma\left(\mathbb{T}(\partial_{\rho},\partial_{z})+\frac{1}{2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z}))\right),$
$\displaystyle((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})\alpha_{z}$
$\displaystyle=-\frac{1}{2}(\partial_{z}\sigma)\sigma\left(\mathbb{T}(\partial_{\rho},\partial_{\rho})-\mathbb{T}(\partial_{z},\partial_{z})+\frac{1}{2}\frac{(\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}\right)$
$\displaystyle+(\partial_{\rho}\sigma)((\partial^{2}_{\rho,z}\sigma))-(\partial_{z}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)+(\partial_{\rho}\sigma)((\partial^{2}_{\rho,z}\sigma))$
$\displaystyle+(\partial_{\rho}\sigma)\sigma\left(\mathbb{T}(\partial_{\rho},\partial_{z})+\frac{1}{2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z}))\right),$

Independent of the behaviour of $\sigma$, $\lambda$ satisfies

$\displaystyle 2\partial^{2}_{\rho}\lambda+2\partial^{2}_{z}\lambda$
$\displaystyle=-\partial^{2}_{\rho}\log{X}-\partial^{2}_{z}\log{X}+\sigma^{-1}((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})$
$\displaystyle+e^{2\lambda}\text{Tr}(\mathbb{T})-\mathbb{T}(\partial_{\rho},\partial_{\rho})-\mathbb{T}(\partial_{z},\partial_{z})$
$\displaystyle-2X^{-1}\left(\mathbb{T}(\Phi,\Phi)-\frac{1}{2}\text{Tr}(\mathbb{T})X\right)e^{2\lambda}$
$\displaystyle-\frac{1}{2}X^{-2}((\partial_{\rho}X)^{2}+(\partial_{z}X)^{2}+(\theta_{\rho})^{2}+(\theta_{z})^{2}).$

Conversely, if the metric data solves each of these equations, and
$|\partial\sigma|\neq 0$ on $\mathscr{B}$, then we may recover the metric $g$
defined on $\mathcal{M}=\mathbb{R}\times(0,2\pi)\times\mathscr{B}$ such that
$(\mathcal{M},g)$ solves the Einstein equations with energy-momentum tensor
$\mathbb{T}$.

In our case, all the assumptions of Theorem 2 are satisfied. Therefore, we
apply the latter with the components of the energy momentum tensor given by
(5.11), (5.12), (5.13), (5.14) and (5.15) and obtain

######  Proposition 18.

Suppose that $(\mathcal{M},g,f)$ solves the Einstein-Vlasov equations where
$g$ is given by

$g=-Vdt^{2}+2Wdtd\phi+Xd\phi^{2}+e^{2\lambda}\left(d\rho^{2}+dz^{2}\right)$

and $f$ is given by

$f(x,v)=\Phi(\varepsilon,\ell_{z})\Psi_{\eta}\left(\rho,(\varepsilon,\ell_{z}),h:=(X,W,\sigma)\right).$

Then the metric data $(X,W,\theta,\sigma,\lambda)$ satisfies the following
equations on $\mathscr{B}$

  1. 1.

X satisfies

$\sigma^{-1}\partial_{\rho}(\sigma\partial_{\rho}X)+\sigma^{-1}\partial_{z}(\sigma\partial_{z}X)=F_{1}(W,X,\sigma)(\rho,z)+\frac{(\partial_{\rho}X)^{2}+(\partial_{z}X)^{2}-\theta_{\rho}^{2}-\theta_{z}^{2}}{X},$
(5.23)

  2. 2.

W satisfies

$\partial_{\rho}(X^{-1}W)d\rho+\partial_{z}(X^{-1}W)dz=\frac{\sigma}{X^{2}}(\theta_{\rho}dz-\theta_{z}d\rho),$
(5.24)

  3. 3.

$\theta$ satisfies

$d\theta=2\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(\rho,z)d\rho\wedge dz,$
(5.25)

as well as

$\sigma^{-1}\partial_{\rho}(\sigma\partial_{\rho}\theta)+\sigma^{-1}\partial_{z}(\sigma\partial_{z}\theta)=\frac{2\theta_{\rho}\partial_{\rho}X+2\theta_{z}\partial_{z}X}{X}.$
(5.26)

  4. 4.

$\sigma$ satisfies

$X^{-1}\exp{(-2\lambda)}\sigma(\partial^{2}_{\rho}\sigma+\partial^{2}_{z}\sigma)=F_{3}(W,X,\sigma)(\rho,z),$
(5.27)

  5. 5.

$\lambda$ satisfies the following equations at the points where
$|\partial\sigma|\neq 0$

$\partial_{\rho}\lambda=\alpha_{\rho}-\frac{1}{2}\partial_{\rho}\log
X,\quad\partial_{z}\lambda=\alpha_{z}-\frac{1}{2}\partial_{z}\log X,$ (5.28)

where

$\displaystyle((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})\alpha_{\rho}$
$\displaystyle=\frac{1}{4}(\partial_{\rho}\sigma)\sigma\frac{(\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}+(\partial_{\rho}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)$
$\displaystyle+(\partial_{z}\sigma)(\partial^{2}_{\rho,z}\sigma)+\frac{1}{2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z}))),$
$\displaystyle((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})\alpha_{z}$
$\displaystyle=-\frac{1}{4}(\partial_{z}\sigma)\sigma\frac{(\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}-(\partial_{z}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)$
$\displaystyle+(\partial_{\rho}\sigma)(\partial^{2}_{\rho,z}\sigma)+\frac{1}{2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z}))),$

Independent of the behaviour of $\sigma$, $\lambda$ satisfies

$\displaystyle 2\partial^{2}_{\rho}\lambda+2\partial^{2}_{z}\lambda$
$\displaystyle=-\partial^{2}_{\rho}\log{X}-\partial^{2}_{z}\log{X}+\sigma^{-1}((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})+F_{4}(W,X,\sigma)(\rho,z)$
$\displaystyle-\frac{1}{2}X^{-2}((\partial_{\rho}X)^{2}+(\partial_{z}X)^{2}+(\theta_{\rho})^{2}+(\theta_{z})^{2}),$

where $F_{1},F_{2},F_{3},F_{4}$ are given by:

$F_{1}(W,X,\sigma)(\rho,z):=-e^{2\lambda}\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}(X+2(\rho
L)^{2})\Phi(E+\rho\omega L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL,$ (5.29)
$F_{2}(W,X,\sigma)(\rho,z):=\frac{2\pi\rho}{\sigma}X\int_{D(\rho,z)}\rho
LE\Phi(E+\rho\omega L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL,$ (5.30) $\displaystyle F_{3}(W,X,\sigma)(\rho,z)$
$\displaystyle:=\frac{2\pi\rho^{3}\sigma}{X^{2}}\int_{D(\rho,z)}\left(\tilde{L}^{2}-L^{2}\right)\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL,$ (5.31)
$\displaystyle F_{4}(W,X,\sigma,\lambda)(\rho,z):=-\frac{4\pi
e^{2\lambda}\rho}{\sigma}$
$\displaystyle\int_{D(\rho,z)}\left(\frac{X^{2}}{\rho^{2}\sigma^{2}}E^{2}+\left(1-\frac{X}{\rho^{2}}\right)\left(1+\frac{\rho^{2}}{X}L^{2}\right)\right)\Phi(E+\rho\omega
L,\rho L)$ (5.32) $\displaystyle\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL,$

Conversely, if the metric data solves each of these equations, and
$|\partial\sigma|\neq 0$ on $\mathscr{B}$, then we may recover the metric and
the distribution function $(\mathcal{M},g,f)$, solving the Einstein-Vlasov
equations.

###### Proof.

We apply Theorem 2 and we compute the components of the energy-momentum
tensor.
First of all, we have

$\displaystyle\tensor{{T}}{{}_{\rho\rho}}$
$\displaystyle=\tensor{{T}}{{}_{zz}},$ $\displaystyle\tensor{{T}}{{}_{\rho
z}}$ $\displaystyle=0.$

Therefore, $\alpha_{\rho}$ and $\alpha_{z}$ do not depend on the the matter
terms. Now, we compute

$\displaystyle\text{Tr}(\mathbb{T})$
$\displaystyle=\tensor{g}{{}^{\alpha\beta}}\tensor{{T}}{{}_{\alpha\beta}}$
$\displaystyle=\tensor{g}{{}^{tt}}\tensor{{T}}{{}_{tt}}+2\tensor{g}{{}^{t\phi}}\tensor{{T}}{{}_{t\phi}}+\tensor{g}{{}^{\phi\phi}}\tensor{{T}}{{}_{\phi\phi}}+2e^{-2\lambda}\tensor{{T}}{{}_{\rho\rho}}$
$\displaystyle=\frac{1}{\sigma^{2}}\left(-X\tensor{{T}}{{}_{tt}}+2W\tensor{{T}}{{}_{t\phi}}+V\tensor{{T}}{{}_{\phi\phi}}\right)+2e^{-2\lambda}\tensor{{T}}{{}_{\rho\rho}}$
$\displaystyle=\frac{2\pi\rho}{\sigma^{3}}\int_{D(\rho,z)}\left(-X(E+\rho\omega
L)^{2}-2W\rho L(E+\rho\omega L)+V(\rho
L)^{2}+\sigma^{2}\left(\tilde{L}(E,X,\sigma,\rho)^{2}-L^{2}\right)\right)$
$\displaystyle\Phi(E+\rho\omega L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL,$

Now we recall that $\sigma^{2}=XV+W^{2}$ and
$\displaystyle\tilde{L}(E,X,\sigma,\rho)=\frac{\sqrt{X}}{\rho}\left(-1+\frac{X}{\sigma^{2}}E^{2}\right)^{\frac{1}{2}}$
. We use the latter to obtain

$\displaystyle-X(E+\rho\omega L)^{2}-2W\rho L(E+\rho\omega L)+V(\rho
L)^{2}+\sigma^{2}\left(\tilde{L}(E,X,\sigma,\rho)^{2}-L^{2}\right)=-\sigma^{2}.$

Therefore,

$\text{Tr}(\mathbb{T})=-\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\,\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL.$

Now, we compute

$\displaystyle F_{1}(W,X,\sigma)(\rho,z)$
$\displaystyle=e^{2\lambda}\left(-2\tensor{{T}}{{}_{\phi\phi}}+{\text{Tr}(\mathbb{T})}X\right)$
$\displaystyle=-e^{2\lambda}\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\,(X+2(\rho
L)^{2})\Phi(E+\rho\omega L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL.$ $\displaystyle
F_{2}(W,X,\sigma)(\rho,z)=\tensor{{T}}{{}_{\phi\phi}}W-\tensor{{T}}{{}_{\phi
t}}X$ $\displaystyle=\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\,\left((\rho
L)^{2}W+X\rho L(E+\rho\omega L)\right)\Phi(E+\rho\omega L,\rho
L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL$
$\displaystyle=\frac{2\pi\rho}{\sigma}X\int_{D(\rho,z)}\,\rho
LE\Phi(E+\rho\omega L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL.$

We have used $\omega:=-WX^{-1}$ to obtain the latter expression. Now we
compute

$\displaystyle\mathbb{T}\left(T-X^{-1}W\Phi,T-X^{-1}W\Phi\right)-X^{-2}\sigma^{2}\mathbb{T}(\Phi,\Phi)+X^{-1}\sigma^{2}\text{Tr}(\mathbb{T})$
$\displaystyle=\tensor{{T}}{{}_{tt}}+\omega^{2}\tensor{{T}}{{}_{\phi\phi}}+2\omega\tensor{{T}}{{}_{t\phi}}$
$\displaystyle=\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\,\left((E+\rho\omega
L)^{2}+(\rho\omega L)^{2}-2\rho\omega L(E+\rho\omega
L)\right)\Phi(E+\rho\omega L,\rho L)$
$\displaystyle\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL.$
$\displaystyle=\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\,E^{2}\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL.$

Hence,

$\displaystyle
F_{3}(W,X,\sigma)(\rho,z)=\mathbb{T}\left(T-X^{-1}W\Phi,T-X^{-1}W\Phi\right)-X^{-2}\sigma^{2}\mathbb{T}(\Phi,\Phi)+X^{-1}\sigma^{2}\text{Tr}(\mathbb{T})$
$\displaystyle=\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\,\left(E^{2}-X^{-2}\sigma^{2}(\rho
L)^{2}-X^{-1}\sigma^{2}\right)\Phi(E+\rho\omega L,\rho
L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL$
$\displaystyle=\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\,X^{-2}\sigma^{2}\rho^{2}\left(\frac{X^{2}}{\rho^{2}\sigma^{2}}E^{2}-L^{2}-\frac{X}{\rho^{2}}\right)\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL$
$\displaystyle=\frac{2\pi\rho^{3}\sigma}{X^{2}}\int_{D(\rho,z)}\,\left(\frac{X}{\rho^{2}}\left(\frac{X}{\sigma^{2}}E^{2}-1\right)-L^{2}\right)\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL$
$\displaystyle=\frac{2\pi\rho^{3}\sigma}{X^{2}}\int_{D(\rho,z)}\,\left(\tilde{L}^{2}(E,X,\sigma,\rho,z)-L^{2}\right)\Phi(E+\rho\omega
L,\rho L)\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL.$

Finally, we compute

$\displaystyle F_{4}(W,X,\sigma,\lambda)(\rho,z)$
$\displaystyle=e^{2\lambda}\text{Tr}(\mathbb{T})-\mathbb{T}(\partial_{\rho},\partial_{\rho})-\mathbb{T}(\partial_{z},\partial_{z})-2X^{-1}\left(\mathbb{T}(\Phi,\Phi)-\frac{1}{2}\text{Tr}(\mathbb{T})X\right)e^{2\lambda}$
$\displaystyle=2e^{2\lambda}\left(\text{Tr}(\mathbb{T})-e^{-2\lambda}\tensor{{T}}{{}_{\rho\rho}}-X^{-1}\tensor{{T}}{{}_{\phi\phi}}\right)$
$\displaystyle=-2e^{2\lambda}\frac{2\pi\rho}{\sigma}\int_{D(\rho,z)}\,\left(1+\frac{1}{2}\left(\tilde{L}^{2}(E,X,\sigma,\rho,z)-L^{2}\right)+X^{-1}(\rho
L)^{2}\right)\Phi(E+\rho\omega L,\rho L)$
$\displaystyle\Psi_{\eta}(\rho,(E+\rho\omega L,\rho L),(X,W,\sigma))\,dEdL$
$\displaystyle=-\frac{4\pi
e^{2\lambda}\rho}{\sigma}\int_{D(\rho,z)}\left(\frac{X^{2}}{\rho^{2}\sigma^{2}}E^{2}+\left(1-\frac{X}{\rho^{2}}\right)\left(1+\frac{\rho^{2}}{X}L^{2}\right)\right)\Phi(E+\rho\omega
L,\rho L)$ $\displaystyle\Psi_{\eta}(\rho,(E+\rho\omega L,\rho
L),(X,W,\sigma))\,dEdL.$

∎

###  5.5 Renormalised unknowns and their equations

This section follows closely [17]. Nonetheless, we detail the computations and
the arguments in order to be self contained.
In order to apply the fixed point theorem in a neighbourhood of a fixed Kerr
solution, we introduce new quantities which are normalised with respect to the
Kerr metric. This choice of variables was first considered in [17] and allows
to subtract off the leading order singular behaviour near the axis and the
horizon.
We begin by defining an adapted Ernst potential, $Y:\mathscr{B}\to\mathbb{R}$.
Let us review the construction in the vacuum case just for comparison.
In vacuum, the twist one-form associated to
$\displaystyle\Phi=\frac{\partial}{\partial\phi}$ vanishes:

$d\theta=0.$

Therefore, since $\mathscr{B}$ is simply connected, there exists a function
$Y_{vacuum}:\mathscr{B}\to\mathbb{R}$, which satisfies

$\theta=dY_{vacuum}.$ (5.33)

This leads to a harmonic map system in $(X,Y_{vacuum})$ which decouples from
the remaining metric data. Moreover, the requirements of asymptotic flatness
and regular extensions to the axis and to the horizon lead to boundary
conditions for $X$ and $Y_{vacuum}$. This allows one to determine uniquely
$(X,Y_{vacuum})$ (See [55], [12]). Now, given $(X,Y_{vacuum})$ and the
boundary conditions, the rest of the metric components are uniquely determined
by quadratures, see Section 2.2.2 of the introduction and [28, Chapter 10] for
details.
In the presence of matter, the twist one-form is no longer closed. We recall
from (5.25) that $\theta$ satisfies

$d\theta=2\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(\rho,z)d\rho\wedge dz.$
(5.34)

However, we can still define an "Ernst potential" $Y$, which will define,
together with $X$ a harmonic map system. The idea is to split the twist one-
form $\theta$ into an Ernst potential piece $dY$ and an other one-form $B$,
defined on $\overline{\mathscr{B}}$ which verifies

$dB=2\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(\rho,z)d\rho\wedge
dz\quad\text{on}\quad\mathscr{B}.$ (5.35)

  1. 1.

We begin by extending the two-form $\omega\in\Omega^{2}(\mathscr{B})$ defined
by

$\omega:=\omega_{\rho z}d\rho\wedge
dz:=2\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(\rho,z)d\rho\wedge dz$

to a two-form $\overline{\omega}$ defined on $\overline{\mathscr{B}}$ so that

$\overline{\omega}=\omega\quad\quad\text{on}\quad\mathscr{B}.$

First, assume that $\omega_{\rho z}$ can be extended to a function
$\overline{\omega}_{\rho z}$ 212121We identify $\overline{\omega}_{\rho z}$
with its coordinate representations in every chart
$\overline{\mathscr{B}_{N}}$, $\overline{\mathscr{B}_{S}}$,
$\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$. defined on
$\overline{\mathscr{B}}$. Now, let $(\xi_{N},\xi_{S},1-\xi_{N}-\xi_{S})$ be
the partition of unity subordinate to
$(\overline{\mathscr{B}_{N}},\overline{\mathscr{B}_{S}},\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}})$
and let $p\in\overline{\mathscr{B}}$

     * •

If $p\in(\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}})$, then

$\omega_{\rho z}(p)=\overline{\omega}_{\rho z}^{A}(\rho,z)$

     * •

If $p\in\overline{\mathscr{B}_{N}}$, then

$\omega_{\rho z}(p)=\overline{\omega}_{\rho z}^{N}(s,\chi)$

     * •

If $p\in\overline{\mathscr{B}_{S}}$, then

$\omega_{\rho z}(p)=\overline{\omega}_{\rho z}^{S}(s^{\prime},\chi^{\prime})$

where $\overline{\omega}_{\rho z}^{A}$, $\overline{\omega}_{\rho z}^{N}$ and
$\overline{\omega}_{\rho z}^{S}$ are defined on
$\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$,
$\overline{\mathscr{B}_{N}}$ and $\overline{\mathscr{B}_{S}}$ respectively. We
have

$\displaystyle\omega(p)$ $\displaystyle=\omega_{\rho z}(p)d\rho\wedge
dz=\overline{\omega}_{\rho z}(s,\chi)(s^{2}+\chi^{2})ds\wedge
d\chi\quad\forall p\in\mathscr{B}_{N}.$

Since the coordinate system $(s,\chi)$ is defined on
$\overline{\mathscr{B}_{N}}$, we set:

$\overline{\omega}(p):=\overline{\omega}^{N}_{\rho
z}(s,\chi)(s^{2}+\chi^{2})ds\wedge d\chi\quad\forall
p\in\overline{\mathscr{B}_{N}}.$

Similarly, we set

$\overline{\omega}(p):=\overline{\omega}^{S}_{\rho
z}(s^{\prime},\chi^{\prime})((s^{\prime})^{2}+(\chi^{\prime})^{2})ds^{\prime}\wedge
d\chi^{\prime},\quad\forall p\in\overline{\mathscr{B}_{S}}.$

Finally, we set

$\displaystyle\overline{\omega}(p)$ $\displaystyle:=\overline{\omega}_{\rho
z}(p)d\rho\wedge dz,\quad\forall
p\in\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}.$

This defines $\overline{\omega}$ on $\overline{\mathscr{B}}$.

  2. 2.

In order to construct $B$, or rather to find the equations for $B$, we first
make the following ansatz:

$B:=B^{(A)}+\xi_{N}B^{(N)}+\xi_{S}B^{(S)}$ (5.36)

where $B^{(N)}$, $B^{(S)}$ and $B^{(A)}$ are one-forms that are defined on
$\overline{\mathscr{B}_{N}}$, $\overline{\mathscr{B}_{S}}$ and
$\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$ respectively.
Therefore, solving the equations for $B^{(N)}$, $B^{(S)}$ and $B^{(A)}$ will
allow us to determine $B$. Hence, we determine the equations for $B^{(N)}$,
$B^{(S)}$ and $B^{(A)}$ based on (5.35):

     * •

On $\overline{\mathscr{B}_{N}}$, we construct $B^{(N)}$ such that

$dB^{(N)}=\xi_{N}\overline{\omega}.$

In the local coordinates $(s,\chi)$, we have

$B^{(N)}=B^{(N)}_{s}ds+B^{(S)}_{\chi}d\chi.$

This implies

$dB^{(N)}=\partial_{\chi}B^{(N)}_{s}d\chi\wedge
ds+\partial_{s}B_{\chi}^{(N)}ds\wedge d\chi.$

Therefore,

$\xi_{N}\overline{\omega}_{\rho z}(s,\chi)(s^{2}+\chi^{2})ds\wedge
d\chi=\partial_{\chi}B^{(N)}_{s}d\chi\wedge
ds+\partial_{s}B_{\chi}^{(N)}ds\wedge d\chi.$

We make the following gauge choice:

$B^{(N)}_{s}(s,\chi)=0\;,\;B_{\chi}^{(N)}(0,\chi)=B_{\chi}^{(N)}(s,0)=0.$

Moreover, we require $B^{(N)}_{s}$ to satisfy the equation.

$\partial_{s}B^{(N)}_{\chi}=\xi_{N}\overline{\omega}_{\rho
z}(s,\chi)(s^{2}+\chi^{2}).$

Consequently, we define the one-form $B^{(N)}$ on $\overline{\mathscr{B}_{N}}$
to be the solution the equations

$\displaystyle B_{\chi}^{(N)}(0,\chi)$ $\displaystyle=0,$ (5.37)
$\displaystyle\partial_{s}B^{(N)}_{\chi}$
$\displaystyle=2\xi_{N}\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(\rho,z)(s^{2}+\chi^{2}),$
$\displaystyle B^{(N)}_{s}(s,\chi)$ $\displaystyle=0.$

     * •

Similarly, we define the one-form $B^{(S)}$ on $\overline{\mathscr{B}_{S}}$ to
be the solution the equations

$\displaystyle B_{\chi^{\prime}}^{(S)}(0,\chi^{\prime})$ $\displaystyle=0,$
(5.38) $\displaystyle\partial_{s^{\prime}}B^{(S)}_{\chi^{\prime}}$
$\displaystyle=2\xi_{S}\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(s^{\prime},\chi^{\prime})((s^{\prime})^{2}+(\chi^{\prime})^{2}),$
$\displaystyle B^{(S)}_{s^{\prime}}(s^{\prime},\chi^{\prime})$
$\displaystyle=0.$

     * •

Finally, we use (5.36) in order to find the equations satisfied by $B^{(A)}$.
We have

$\displaystyle dB^{(A)}$
$\displaystyle=dB-d(\xi_{N}B^{(N)})-d(\xi_{S}B^{(S)})$
$\displaystyle=dB-\xi_{N}d(B^{(N)})-d\xi_{N}\wedge
B^{(N)}-\xi_{S}d(B^{(S)})-d\xi_{S}\wedge B^{(S)}$
$\displaystyle=(1-\xi_{N}^{2}-\xi_{S}^{2})dB-d\xi_{N}\wedge
B^{(N)}-d\xi_{S}\wedge B^{(S)}.$

On $\overline{\mathscr{B}_{A}}\cup\overline{\mathscr{B}_{H}}$, we have

$dB=2\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(\rho,z)d\rho\wedge dz,$
$B^{(N)}=B^{(N)}_{\rho}d\rho+B^{(S)}_{z}dz\quad\text{,}\quad
B^{(N)}=B^{(S)}_{\rho}d\rho+B^{(S)}_{z}dz\quad{,}\quad
B^{(A)}=B^{(A)}_{\rho}d\rho+B^{(A)}_{z}dz$

We compute

$dB^{(A)}=\partial_{z}B_{\rho}^{(A)}dz\wedge
d\rho+\partial_{\rho}B_{z}^{(A)}d\rho\wedge dz,$ $d\xi_{N}\wedge
B^{(N)}=(\partial_{\rho}\xi_{N})B_{z}^{(N)}-(\partial_{z}\xi_{N})B_{\rho}^{(N)}.$

We make the following gauge choice

$B_{\rho}^{(A)}=0\quad\text{,}\quad B_{z}^{(A)}(0,z)=0$

and we require $B_{z}^{(A)}$ to satisfy the equation

$\displaystyle\partial_{\rho}B_{z}^{(A)}(\rho,z)$
$\displaystyle=2(1-\xi_{N}^{2}-\xi_{S}^{2})\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(\rho,z)$
$\displaystyle-(\partial_{\rho}\xi_{N})B_{z}^{(N)}+(\partial_{z}\xi_{N})B_{\rho}^{(N)}-(\partial_{\rho}\xi_{S})B_{z}^{(S)}+(\partial_{z}\xi_{S})B_{\rho}^{(S)}.$

Therefore, we define the one-form $B^{(A)}$ to be the solution of the equation

$\displaystyle B_{z}^{(A)}(0,z)$ $\displaystyle=0,$ (5.39)
$\displaystyle\partial_{\rho}B_{z}^{(A)}(\rho,z)$
$\displaystyle=2(1-\xi_{N}^{2}-\xi_{S}^{2})\sigma^{-1}e^{2\lambda}F_{2}(W,X,\sigma)(\rho,z)$
$\displaystyle-(\partial_{\rho}\xi_{N})B_{z}^{(N)}+(\partial_{z}\xi_{N})B_{\rho}^{(N)}-(\partial_{\rho}\xi_{S})B_{z}^{(S)}+(\partial_{z}\xi_{S})B_{\rho}^{(S)},$
$\displaystyle B_{\rho}^{(A)}(\rho,z)$ $\displaystyle=0.$

  3. 3.

With this construction, $B$ is defined by
$\displaystyle\left(B_{\chi}^{(N)},B_{\chi^{\prime}}^{(S)},B_{z}^{(A)}\right)$
and verifies (5.35)

Therefore, we can define the Ernst potential $Y$:

######  Definition 19.

We define, up to a constant, the Ernst potential $Y:\mathscr{B}\to\mathbb{R}$
to be the function which satisfies:

$dY=\theta-B.$ (5.40)

Now, we introduce the renormalised unknowns to be the following set of
functions and a one-form, which are all assumed to be continuous on
$\overline{\mathscr{B}}$ and satisfy the following definitions on
$\mathscr{B}$:

$\displaystyle\overset{\circ}{\sigma}$
$\displaystyle:=\frac{\sigma-\sigma_{K}}{\sigma_{K}}\quad\sigma_{K}=\rho,$
(5.41) $\displaystyle\overset{\circ}{X}$
$\displaystyle:=X_{K}^{-1}(X-X_{K}),\quad\overset{\circ}{Y}:=X_{K}^{-1}(Y-Y_{K}),$
(5.42) $\displaystyle\overset{\circ}{\Theta}$
$\displaystyle:=X^{-1}W-X_{K}^{-1}W_{K},$ (5.43)
$\displaystyle\overset{\circ}{\lambda}$ $\displaystyle:=\lambda-\lambda_{K}.$
(5.44)

Henceforth, the quantities
$\displaystyle\left(\overset{\circ}{\sigma},\left(B_{\chi}^{(N)},B_{\chi^{\prime}}^{(S)},B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta},\overset{\circ}{\lambda}\right)$
will be called the "renormalised unknowns".
Now, given
$\displaystyle\left(\overset{\circ}{\sigma},\displaystyle\left(B_{\chi}^{(N)},B_{\chi^{\prime}}^{(S)},B_{z}^{(A)}\right),\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta},\overset{\circ}{\lambda}\right)$,
we can recover the original unknowns in the following way:

$\sigma=\sigma_{K}(1+\overset{\circ}{\sigma}),$ (5.45)
$X=X_{K}(1+\overset{\circ}{X}),\quad Y=X_{K}(1+\overset{\circ}{Y}),$ (5.46)
$X^{-1}W=\overset{\circ}{\Theta}+X_{K}^{-1}W_{K},$ (5.47)
$\lambda=\overset{\circ}{\lambda}+\lambda_{K}.$ (5.48)

In terms of the renormalised unknowns, the matter terms become:

$\displaystyle
F_{1}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma})(\rho,z):=-\frac{2\pi
e^{2\left(\overset{\circ}{\lambda}+\lambda_{K}\right)}}{1+\overset{\circ}{\sigma}}$
$\displaystyle\int_{D(\rho,z)}(X_{K}(1+\overset{\circ}{X})+2(\rho
L)^{2})\Phi(E+\rho\left(-\overset{\circ}{\Theta}+\omega_{K}\right)L,\rho L)$
$\displaystyle\Psi_{\eta}(\rho,(E+\rho(-\overset{\circ}{\Theta}+\omega_{K})L,\rho
L),(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma}))\,dEdL,$
$\displaystyle
F_{2}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma})(\rho,z):=\frac{2\pi}{1+\overset{\circ}{\sigma}}X_{K}(1+\overset{\circ}{X})$
$\displaystyle\int_{D(\rho,z)}\rho
LE\Phi(E+\rho\left(-\overset{\circ}{\Theta}+\omega_{K}\right)L,\rho L)$
$\displaystyle\Psi_{\eta}(\rho,(E+\rho\left(-\overset{\circ}{\Theta}+\omega_{K}\right)L,\rho
L),(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma}))\,dEdL,$
$\displaystyle
F_{3}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma})(\rho,z):=\frac{\rho^{4}}{X^{2}_{K}}\frac{2\pi(1+\overset{\circ}{\sigma})}{\left(1+\overset{\circ}{X}\right)^{2}}$
$\displaystyle\int_{D(\rho,z)}\left(\tilde{L}^{2}-L^{2}\right)\Phi(E+\rho\left(-\overset{\circ}{\Theta}+\omega_{K}\right)L,\rho
L)$
$\displaystyle\Psi_{\eta}(\rho,(E+\rho\left(-\overset{\circ}{\Theta}+\omega_{K}\right)L,\rho
L),(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma}))\,dEdL,$
$\displaystyle
F_{4}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma},\overset{\circ}{\lambda})(\rho,z)$
$\displaystyle:=-\frac{4\pi
e^{2\left(\overset{\circ}{\lambda}+\lambda_{K}\right)}}{1+\overset{\circ}{\sigma}}$
$\displaystyle\int_{D(\rho,z)}\left(\frac{X_{K}^{2}\left(1+\overset{\circ}{X}\right)^{2}}{\rho^{4}\left(1+\overset{\circ}{\sigma}\right)^{2}}E^{2}+\left(1-\frac{X_{K}}{\rho^{2}}\left(1+\overset{\circ}{X}\right)\right)\left(1+\frac{\rho^{2}}{X_{K}}\frac{1}{1+\overset{\circ}{X}}L^{2}\right)\right)$
$\displaystyle\Phi(E+\rho\left(-\overset{\circ}{\Theta}+\omega_{K}\right)L,\rho
L)\Psi_{\eta}(\rho,(E+\rho\left(-\overset{\circ}{\Theta}+\omega_{K}\right),\rho
L),(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma}))\,dEdL,$

We apply Proposition 18 in order to obtain the equations for the renormalised
unknows.

######  Proposition 19.

The renormalised unknowns
$\displaystyle\left(\overset{\circ}{\sigma},B,\left(\overset{\circ}{X},\overset{\circ}{Y}\right),\overset{\circ}{\Theta},\overset{\circ}{\lambda}\right)$verify
the following equations

  * •

$\overset{\circ}{\sigma}$ satisfies

$\Delta_{\mathbb{R}^{4}}{\overset{\circ}{\sigma}}=\rho^{-1}\sigma^{-1}{X}e^{2\lambda}F_{3}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\sigma})(\rho,z),$
(5.49)

where $\Delta_{\mathbb{R}^{4}}$ is the Laplacian corresponding to the flat
metric on $\mathbb{R}^{4}$ given by
$g_{\mathbb{R}^{4}}=d\rho^{2}+dz^{2}+\rho^{2}d\mathbb{S}^{2}$.

  * •

$B$ satisfies

$\displaystyle\partial_{\chi}B^{(N)}_{s}=2\xi_{N}(s\chi)^{-1}(1+\overset{\circ}{\sigma})^{-1}e^{2\lambda+2\overset{\circ}{\lambda}}F_{2}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\sigma})(s,\chi)(s^{2}+\chi^{2}),$
(5.50)
$\displaystyle\partial_{\chi^{\prime}}B^{(S)}_{s^{\prime}}=2\xi_{S}(s^{\prime}(\chi)^{\prime})^{-1}(1+\overset{\circ}{\sigma})^{-1}e^{2\lambda+2\overset{\circ}{\lambda}}F_{2}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\sigma})(s^{\prime},\chi^{\prime})((s^{\prime})^{2}+(\chi^{\prime})^{2}),$
$\displaystyle\partial_{\rho}B^{(A)}_{z}(\rho,z)+(\partial_{\rho}\xi_{N})B_{z}^{(N)}-(\partial_{z}\xi_{N})B_{\rho}^{(N)}+(\partial_{\rho}\xi_{S})B_{z}^{(S)}-(\partial_{z}\xi_{S})B_{\rho}^{(S)}=$
$\displaystyle
2(1-\xi_{N}^{2}-\xi_{S}^{2})\rho^{-1}(1+\overset{\circ}{\sigma})^{-1}e^{2\lambda+2\overset{\circ}{\lambda}}F_{2}(\overset{\circ}{X},\overset{\circ}{\Theta},\overset{\circ}{\sigma})(\rho,z),$

  * •

$(\overset{\circ}{X},\overset{\circ}{Y})$ satisfies

$\displaystyle\Delta_{\mathbb{R}^{3}}\overset{\circ}{X}+\frac{2\partial
Y_{K}\cdot\partial\overset{\circ}{Y}}{X_{K}}-\frac{2|\partial
Y_{K}|^{2}}{X_{K}^{2}}\overset{\circ}{X}+2\frac{\partial X_{K}\cdot\partial
Y_{K}}{X_{K}^{2}}\overset{\circ}{Y}$
$\displaystyle=N_{X}:=N^{(1)}_{X}+N^{(2)}_{X},$ (5.51)
$\displaystyle\Delta_{\mathbb{R}^{3}}\overset{\circ}{Y}-\frac{2\partial
Y_{K}\cdot\partial\overset{\circ}{X}}{X_{K}}-\frac{(|\partial
X_{K}|^{2}+|\partial Y_{K}|^{2})}{X_{K}^{2}}\overset{\circ}{Y}$
$\displaystyle=N_{Y}:=N^{(1)}_{Y}+N^{(2)}_{Y},$

where

$N^{(1)}_{X}:=\frac{X_{K}^{2}(|\partial\overset{\circ}{X}|^{2}-|\partial\overset{\circ}{Y}|^{2})+(\overset{\circ}{X}\partial
Y_{K}-\overset{\circ}{Y}\partial
X_{K})\cdot(2X_{K}\partial\overset{\circ}{Y}-\overset{\circ}{X}\partial
Y_{K}+\overset{\circ}{Y}\partial X_{K})}{X_{K}^{2}(1+\overset{\circ}{X})},$
$N^{(1)}_{Y}:=\frac{\partial\overset{\circ}{X}\cdot\partial\overset{\circ}{Y}+2X_{K}(\overset{\circ}{Y}\partial
X_{K}-\overset{\circ}{X}\partial
Y_{K})\cdot\partial\overset{\circ}{X}}{X_{K}^{2}(1+\overset{\circ}{X})},$
$\displaystyle N^{(2)}_{X}$
$\displaystyle:=(\rho^{-1}-\sigma^{-1}\partial_{\rho}\sigma)\frac{\partial_{\rho}(X_{K}(1+\overset{\circ}{X}))}{X_{K}}-\sigma^{-1}\partial_{z}\sigma\frac{\partial_{z}(X_{K}(1+\overset{\circ}{X}))}{X_{K}}-\frac{2\partial_{\rho}(Y_{K}+X_{K}\overset{\circ}{Y})B_{\rho}}{X_{K}^{2}(1+\overset{\circ}{X})}$
$\displaystyle-\frac{2\partial_{z}(Y_{K}+X_{K}\overset{\circ}{Y})B_{z}}{X_{K}^{2}(1+\overset{\circ}{X})}-\frac{B_{\rho}^{2}+B_{z}^{2}}{X_{K}^{2}(1+\overset{\circ}{X})}+X^{-1}_{K}F_{1}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma})(\rho,z),$
$\displaystyle N^{(2)}_{Y}$
$\displaystyle:=(\rho^{-1}-\sigma^{-1}\partial_{\rho}\sigma)\partial_{\rho}(Y_{K}+X_{K}\overset{\circ}{Y})X_{K}^{-1}-\sigma^{-1}\partial_{\rho}\sigma
B_{\rho}X_{K}^{-1}-\sigma^{-1}\partial_{z}\sigma(\partial_{z}(Y_{K}+X_{K}\overset{\circ}{Y})+B_{z})X_{K}^{-1}$
$\displaystyle+\frac{B_{\rho}\partial_{\rho}(X_{K}(1+\overset{\circ}{X}))+B_{z}\partial_{z}(X_{K}(1+\overset{\circ}{X}))}{X_{K}^{2}(1+\overset{\circ}{X})},$

and where $\Delta_{\mathbb{R}^{3}}$ is the Laplacian corresponding to the flat
metric on $\mathbb{R}^{3}$ given by
$g_{\mathbb{R}^{3}}=d\rho^{2}+dz^{2}+\rho^{2}d\phi^{2}$.

  * •

$\overset{\circ}{\Theta}$ satisfies

$\displaystyle\partial_{\rho}\overset{\circ}{\Theta}$
$\displaystyle=-\frac{\sigma}{X^{2}}(\partial_{z}Y+B_{z})+\frac{\rho}{X_{K}^{2}}\partial_{z}Y_{K},$
(5.52) $\displaystyle\partial_{z}\overset{\circ}{\Theta}$
$\displaystyle=\frac{\sigma}{X^{2}}(\partial_{\rho}Y+B_{\rho})-\frac{\rho}{X_{K}^{2}}\partial_{\rho}Y_{K}.$

  * •

$\overset{\circ}{\lambda}$ satisfies

$\displaystyle\partial_{\rho}\overset{\circ}{\lambda}$
$\displaystyle=\alpha_{\rho}-(\alpha_{K})_{\rho}-\frac{1}{2}\partial_{\rho}\log(1+\overset{\circ}{X}),$
(5.53) $\displaystyle\partial_{z}\overset{\circ}{\lambda}$
$\displaystyle=\alpha_{z}-(\alpha_{K})_{z}-\frac{1}{2}\partial_{z}\log(1+\overset{\circ}{X}),$

where

$\displaystyle(\alpha_{K})_{\rho}$ $\displaystyle=\frac{1}{4}\rho
X_{K}^{-2}((\partial_{\rho}X_{K})^{2}-(\partial_{z}X_{K})^{2}+(\partial_{\rho}Y_{K})^{2})-(\partial_{z}Y_{K})^{2},$
$\displaystyle(\alpha_{K})_{z}$ $\displaystyle=\frac{1}{4}\rho
X_{K}^{-2}((\partial_{\rho}X_{K})(\partial_{z}X_{K})+(\partial_{\rho}Y_{K})(\partial_{z}Y_{K})),$

where $\alpha_{\rho}$ and $\alpha_{z}$ satisfy

$\displaystyle((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})\alpha_{\rho}$
$\displaystyle=\frac{1}{4}(\partial_{\rho}\sigma)\sigma\frac{(\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}+(\partial_{\rho}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)+(\partial_{z}\sigma)((\partial^{2}_{\rho,z}\sigma))$
$\displaystyle+\frac{1}{2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z}))),$
$\displaystyle((\partial_{\rho}\sigma)^{2}+(\partial_{z}\sigma)^{2})\alpha_{z}$
$\displaystyle=-\frac{1}{4}(\partial_{z}\sigma)\sigma\frac{(\partial_{\rho}X)^{2}-(\partial_{z}X)^{2}+(\theta_{\rho})^{2}-(\theta_{z})^{2}}{X^{2}}-(\partial_{z}\sigma)(\partial^{2}_{\rho}\sigma-\partial^{2}_{z}\sigma)+(\partial_{\rho}\sigma)((\partial^{2}_{\rho,z}\sigma))$
$\displaystyle+\frac{1}{2}X^{-2}((\partial_{\rho}X)(\partial_{z}X)+(\theta_{\rho})(\theta_{z}))$

###### Proof.

  1. 1.

First, we derive the equation for $\overset{\circ}{\sigma}$: For this, we
compute the Laplacian of $\overset{\circ}{\sigma}$ with respect to the flat
metric $g=d\rho^{2}+dz^{2}+\rho^{2}d\mathbb{S}^{2}$:

$\displaystyle\Delta_{\mathbb{R}^{4}}\overset{\circ}{\sigma}$
$\displaystyle=\Delta_{\mathbb{R}^{4}}\left(\frac{\sigma}{\rho}\right)$
$\displaystyle=\frac{1}{\sqrt{\det g}}\partial_{i}\left(\sqrt{\det
g}\tensor{g}{{}^{ij}}\partial_{j}\left(\frac{\sigma}{\rho}\right)\right),\quad
i,j\in\left\\{1,\cdots 4\right\\}$
$\displaystyle=\frac{1}{\rho^{2}\sin\theta}\partial_{\rho}\left(\rho^{2}\sin\theta\tensor{g}{{}^{\rho\rho}}\partial_{\rho}\left(\frac{\sigma}{\rho}\right)\right)+\frac{1}{\rho^{2}\sin\theta}\partial_{z}\left(\rho^{2}\sin\theta\tensor{g}{{}^{zz}}\partial_{z}\left(\frac{\sigma}{\rho}\right)\right)$
$\displaystyle=\partial_{\rho\rho}\left(\frac{\sigma}{\rho}\right)+\frac{2}{\rho}\partial_{\rho}\left(\frac{\sigma}{\rho}\right)+\frac{1}{\rho}\partial_{zz}\sigma$
$\displaystyle=\frac{1}{\rho}(\partial_{\rho\rho}\sigma+\partial_{zz}\sigma)$

By (5.27), we have

$X^{-1}\exp{(-2\lambda)}\sigma(\partial^{2}_{\rho}\sigma+\partial^{2}_{z}\sigma)=F_{3}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma})(\rho,z),$

Therefore,

$\Delta_{\mathbb{R}^{4}}\overset{\circ}{\sigma}=\frac{X_{K}}{\rho^{2}}e^{2\lambda}\frac{1+\overset{\circ}{X}}{1+\overset{\circ}{\sigma}}F_{3}(\overset{\circ}{\Theta},\overset{\circ}{X},\overset{\circ}{\sigma})(\rho,z).$

  2. 2.

The equation for $B$ follows from the definition of
$\displaystyle\left(B_{\chi}^{(N)},B_{\chi^{\prime}}^{(S)},B_{z}^{(A)}\right)$
and the equations (5.37), (5.38), (5.39).

  3. 3.

We turn to the equations for $(\overset{\circ}{X},\overset{\circ}{Y})$:

     * •

First of all, we recall from the classical Carter Robinson theory (see for
example [55] )that $(X_{K},Y_{K})$ forms a harmonic map system and satisfies
the equations

$\left\\{\begin{aligned}
\rho^{-1}\partial_{\rho}(\rho\partial_{\rho}X_{K})+\rho^{-1}\partial_{z}(\rho\partial_{z}X_{K})&=\frac{(\partial_{\rho}X_{K})^{2}+(\partial_{z}X_{K})^{2}-(\partial_{\rho}Y_{K})^{2}-(\partial_{z}Y_{K})^{2}}{X_{K}},\\\
\rho^{-1}\partial_{\rho}(\rho\partial_{\rho}Y_{K})+\rho^{-1}\partial_{z}(\rho\partial_{z}Y_{K})&=\frac{2(\partial_{\rho}Y_{K})(\partial_{\rho}X_{K})+2(\partial_{z}Y_{K})(\partial_{z}X_{K})}{X_{K}}.\end{aligned}\right.$
(5.54)

     * •

Now, we compute

$\displaystyle\Delta_{\mathbb{R}^{3}}\overset{\circ}{X}$
$\displaystyle=\Delta_{\mathbb{R}^{3}}\left(\frac{X-X_{K}}{X_{K}}\right)$
$\displaystyle=\frac{1}{\rho}\partial_{\rho}\left(\rho\partial_{\rho}\left(\frac{X-X_{K}}{X_{K}}\right)\right)+\partial_{zz}\left(\frac{X-X_{K}}{X_{K}}\right)$