$\varepsilon^{2}>1$.

  * •

The sign of $a_{2}$ is positive by (4.32) and by $q<0$.

Depending on the sign of $a_{1}$, either we have zero roots if $a_{1}>0$ or
one root if $a_{1}<0$, by Descartes’s rule of sign.This ends the proof. ∎

Since we are interested in trapped geodesics, we will henceforth study the
case where $q$ is non-negative. The case of vanishing $q$ is of particular
interest. More precisely,

######  Lemma 12.

Let $(\gamma,I)$ be a timelike future-directed geodesic moving in the exterior
region with constants of motion $(\varepsilon,\ell_{z},q)$. Then, $q=0$ is a
necessary and sufficient condition for a motion initially in the equatorial
plane to remain in the equatorial plane for all time.

###### Proof.

If $\gamma$ is confined to the equatorial plane, then $\forall\tau\in I$

$\theta(\tau)=\frac{\pi}{2}\quad\text{and}\quad
v^{\theta}(\tau)=\dot{\theta}(\tau)=0.$

Therefore, by (4.1), we have $q=0$. Now, assume that $q=0$ and
$\theta(0)=\frac{\pi}{2}$. Then, there exists a unique
$\gamma:I\to\mathcal{O}$ solution to (4.29) with initial conditions
$\left(r(0),\frac{\pi}{2},v_{r}(0),0\right)$. Therefore, by (4.12), we obtain

$\forall\tau\in I\;,\;v_{\theta}(\tau)=0.$

∎

Now, we study the roots of $T(\cdot,\varepsilon,\ell_{z},q)$ in the region
$]-1,1[$ for
$(\varepsilon,\ell_{z},q)\in\mathbb{R}\times\mathbb{R}\times[0,\infty[$. We
state the following lemma

######  Lemma 13.

Assume that
$(\varepsilon,\ell_{z},q)\in\mathbb{R}\times\mathbb{R}\times[0,\infty[$ and
consider the equation in $Y$

$T(Y,\varepsilon,\ell_{z},q)=0.$ (4.33)

on $]-1,1[$. Then, $T$ has

  * •

two roots counted with their multiplicity in the region $]-1,1[$ if and only
if $d^{2}(\varepsilon^{2}-1)\leq\ell_{z}^{2}$. They are given by

$Y=\pm\sqrt{y_{+}},$

  * •

two simple roots in the region $]-1,1[$ if and only if
$d^{2}(\varepsilon^{2}-1)>\ell_{z}^{2}$ and $q>0$. They are given by

$Y=\pm\sqrt{y_{-}},$

  * •

one double root given by $0$ and two simple roots in the region $]-1,1[$ if
and only if $d^{2}(\varepsilon^{2}-1)>\ell_{z}^{2}$ and $q=0$. The simple
roots are given by

$Y=\pm\sqrt{y_{-}},$

where

$y=\left\\{\begin{aligned}
&y_{+}=\frac{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q+\sqrt{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q)^{2}-4qd^{2}(1-\varepsilon^{2})})}{2d^{2}(1-\varepsilon^{2})}\quad\text{if}\;\varepsilon^{2}<1\\\
&y_{-}=\frac{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q-\sqrt{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q)^{2}-4qd^{2}(1-\varepsilon^{2})})}{2d^{2}(1-\varepsilon^{2})}\quad\text{if}\;\varepsilon^{2}>1\\\
&\frac{q}{\ell_{z}^{2}+q}\quad\text{if}\;\varepsilon^{2}=1.\\\
\end{aligned}\right.$ (4.34)

Moreover, if $Y$ is a double root of $T$ in $]-1,1[$, then $\displaystyle Y=0$
and $q=0$.

###### Proof.

Let
$(\varepsilon,\ell_{z},q)\in]0,\infty[\times\mathbb{R}^{\ast}\times[0,\infty[$
and let $Y\in]-1,1[$ be a solution of (4.33). Then, $Y$ verifies

$q=F(Y,\varepsilon,\ell_{z})\quad\text{where}\quad
F(Y,\varepsilon,\ell_{z}):=Y^{2}\left(-d^{2}(\varepsilon^{2}-1)+\frac{\ell_{z}^{2}}{1-Y^{2}}\right).$

We have $\forall(\varepsilon,\ell_{z})\in\mathbb{R}\times\mathbb{R}^{\ast}$

  * •

$\lim_{|Y|\to 1}\,F(Y,\varepsilon,\ell_{z})=+\infty,$

  * •

$\displaystyle F(0,\varepsilon,\ell_{z})=0,$

  * •

$\displaystyle\frac{\partial F}{\partial Y}$
$\displaystyle=\frac{2Y}{(1-Y^{2})^{2}}\left(\ell_{z}^{2}-d^{2}(\varepsilon^{2}-1)(1-Y^{2})\right).$

Therefore, $Y=0$ is always a critical point for
$F(\cdot,\varepsilon,\ell_{z})$ and

    * –

if $d^{2}(\varepsilon^{2}-1)\leq\ell_{z}^{2}$, then $0$ is the unique critical
point.

    * –

Otherwise, there are two more critical points given by

$y_{c}=\pm\left(1-\sqrt{\frac{\ell_{z}^{2}}{d^{2}(\varepsilon^{2}-1)}}\right),$

and they verify

$-1<y_{c}^{-}<0<y_{c}^{+}<1.$

Therefore, if $q\geq 0$, then the equation $q=F(Y,\varepsilon,\ell_{z})$
admits

  * •

if $\displaystyle d^{2}(\varepsilon^{2}-1)\leq\ell_{z}^{2}$, two roots in the
region $]-1,1[$ symmetric about $0$. These roots coincide if and only if $q=0$
and they are given by $0$.

  * •

Otherwise,

    * –

four roots if and only if $q=0$: one double root given by $0$ and two simple
roots symmetric about $0$,

    * –

two simple roots otherwise.

In order to write $Y$ in terms of $(\varepsilon,\ell_{z},q)$, we make the
following change of variables $Y=\sqrt{y}$ and we consider
$\overline{T}(y):=T(\sqrt{y})$ on $[0,1[$. Since $\overline{T}$ is quadratic
and its discriminant is always positive, its roots are given by

$\left\\{\begin{aligned}
y&=y_{\pm}:=\frac{\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q\pm\sqrt{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q)^{2}-4qd^{2}(1-\varepsilon^{2})}}{2d^{2}(1-\varepsilon^{2})}\quad\varepsilon\neq
1.\\\ &=\frac{q}{\ell_{z}^{2}+q}\quad\;\varepsilon^{2}=1.\end{aligned}\right.$

  * •

If $d^{2}(\varepsilon^{2}-1)\leq\ell_{z}^{2}$, then $y_{+}>0$ and $y_{-}<0$.
Therefore,

$y=y_{+}.$

  * •

Otherwise,

$y=y_{-}$

∎

Finally, we state necessary and sufficient conditions for the occurence of
spherical orbits.

######  Lemma 14.

Let $\gamma:I\to\mathcal{O}$ be a timelike future-directed geodesic with
constants of motion $(\varepsilon,\ell_{z},q)$. Then $\gamma$ is spherical of
radius $r_{c}$ and confined to the equatorial plane if and only if

  * •

$\gamma$ starts at some point
$\displaystyle\left(t,\phi,r_{c},\frac{\pi}{2}\right)$.

  * •

$q=0$ and $r_{c}$ is a double root of $R(\cdot,\varepsilon,\ell_{z},q)$.

###### Proof.

If $\gamma$ spherical of radius $r_{c}$ then by Lemma 9, $r_{c}$ is a double
root of $R(\cdot,\varepsilon,\ell_{z},q)$. If $\gamma$ is confined in the
equatorial plane, then by Lemma 12, $q=0$.
Reciprocally, if $\gamma$ satisfies the conditions of Lemma 14, then by Lemma
12, $\gamma$ is confined to the equatorial plane. Now, if $r_{c}$ is a double
root of $R(\cdot,\varepsilon,\ell_{z},q)$ and starts at $r(0)=r_{c}$, then the
point $\displaystyle(r_{c},\frac{\pi}{2},0,0)$ is a critical point for the
reduced system (4.29). Therefore, $\gamma$ has a constant radius. ∎

######  Lemma 15.

Let $\gamma:I\to\mathcal{O}$ be a timelike future-directed geodesic with
constants of motion $(\varepsilon,\ell_{z},q)$. Then $\gamma$ is spherical of
radius $r_{c}$ if and only if

  * •

$\gamma$ starts at some point $(t,\phi,r_{s},\theta)\in\mathcal{O}$.

  * •

$r_{s}$ is a double root of $R(\cdot,\varepsilon,\ell_{z},q)$.

###### Proof.

The proof is similar to the previous lemma. ∎

####  4.1.2 Circular orbits confined in the equatorial plane
$\displaystyle\theta=\frac{\pi}{2}$

We present here a detailed study of circular geodesic motion in the equatorial
plane $\theta$. We note that the study of circular orbits is included in
classical books of general relativity. See for example [15, Chapter 6]. We
state the main result of this section

######  Proposition 5.

Let $\gamma:I\to\mathcal{O}$ be a timelike future directed geodesic with
constants of motion
$(\varepsilon,\ell_{z},q)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$.
Assume that $\gamma$ is a circular orbit of radius $r_{c}$ confined in the
equatorial plane. Then,

$(\varepsilon,\ell_{z},q)\in\mathcal{A}_{circ}^{+,\leq
1}\cup\mathcal{A}_{circ}^{+,\geq 1}\cup\mathcal{A}_{circ}^{-,\leq
1}\cup\mathcal{A}_{circ}^{-,\geq 1}=:\mathcal{A}_{circ}$

where

$\displaystyle\mathcal{A}_{circ}^{+,\leq 1}:=$ (4.35)
$\displaystyle\left\\{(\varepsilon,\ell_{z},0)\;,\;\varepsilon\in]\varepsilon^{+}_{min},1[\;,\;\ell_{z}=\ell_{lb}^{+}(\varepsilon)\right\\}\sqcup\left\\{(\varepsilon,\ell_{z},0)\;,\;\varepsilon\in]\varepsilon^{+}_{min},1[\;,\;\ell_{z}=\ell_{ub}^{+}(\varepsilon)\right\\}\sqcup\left\\{(\varepsilon^{+}_{min},\ell^{+}_{min},0)\right\\},$
$\displaystyle\mathcal{A}_{circ}^{-,\leq 1}:=$ (4.36)
$\displaystyle\left\\{(\varepsilon,\ell_{z},0)\;,\;\varepsilon\in]\varepsilon^{-}_{min},1[\;,\;\ell_{z}=\ell_{lb}^{-}(\varepsilon)\right\\}\sqcup\left\\{(\varepsilon,\ell_{z},0)\;,\;\varepsilon\in]\varepsilon^{-}_{min},1[\;,\;\ell_{z}=\ell_{ub}^{-}(\varepsilon)\right\\}\sqcup\left\\{(\varepsilon^{-}_{min},\ell^{-}_{min},0)\right\\},$
$\mathcal{A}_{circ}^{+,\geq
1}:=\left\\{(\varepsilon,\ell_{z},0)\;,\;\varepsilon\in[1,\infty[\;,\;\ell_{z}=\ell_{lb}^{+}(\varepsilon)\right\\},$
(4.37)

and

$\mathcal{A}_{circ}^{-,\geq
1}:=\left\\{(\varepsilon,\ell_{z},0)\;,\;\varepsilon\in[1,\infty[\;,\;\ell_{z}=\ell_{lb}^{-}(\varepsilon)\right\\}$
(4.38)

and where $\varepsilon^{\pm}_{min}$, $\ell_{min}^{\pm}$,
$\ell_{ub}^{\pm}(\varepsilon)$ and $\ell_{lb}^{\pm}(\varepsilon)$ are given by
(4.47), (4.48) and Definition 15 respectively.
Moreover, $r_{c}$ is given by

  1. 1.

if $(\varepsilon,\ell_{z},q)\in\mathcal{A}_{circ}^{+,\leq 1}$, then

$r_{c}\in\left\\{r_{ms}^{+},\tilde{r}_{max}^{+}(\varepsilon),r_{min}^{+}(\varepsilon)\right\\},$

  2. 2.

if $(\varepsilon,\ell_{z},q)\in\mathcal{A}_{circ}^{-,\leq 1}$, then

$r_{c}\in\left\\{r_{ms}^{-},\tilde{r}_{max}^{-}(\varepsilon),r_{min}^{-}(\varepsilon)\right\\},$

  3. 3.

if $(\varepsilon,\ell_{z},q)\in\mathcal{A}_{circ}^{+,\geq 1}$, then

$r_{c}=r_{max}^{+}(\varepsilon),$

  4. 4.

if $(\varepsilon,\ell_{z},q)\in\mathcal{A}_{circ}^{-,\geq 1}$, then

$r_{c}=r_{max}^{-}(\varepsilon),$

where $r^{\pm}_{ms}$ is defined by (14), $\tilde{r}^{\pm}_{max}$ and
$r^{\pm}_{min}$ are defined in Lemma 20.

The remaining of this section is devoted to the proof of Proposition 5. We
start with the following lemma

######  Lemma 16.

Let
$\gamma:I\to\mathcal{O}\,,\,\tau\mapsto(t(\tau),\phi(\tau),r(\tau),\theta(\tau))$
be a circular orbit of radius $r_{c}>r_{H}$ confined in the equatorial plane
and let
$(\varepsilon,\ell_{z},q)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ be its
associated integrals of motion. Then

$q=0$

and $(r_{c},\varepsilon,\ell_{z})$ satisfies the following system of equations

$\displaystyle
3r_{c}^{4}-\frac{r_{c}^{2}}{\varepsilon^{2}}(3r_{c}^{2}-4r_{c}+d^{2})+r_{c}^{2}d^{2}$
$\displaystyle=r_{c}^{2}\ell_{z}^{2},$ $\displaystyle
r_{c}^{4}-\frac{r_{c}^{3}}{\varepsilon^{2}}(r_{c}-1)-d^{2}r_{c}$
$\displaystyle=r_{c}(\ell^{2}-2d\ell_{z}).$

###### Proof.

$\gamma$ is confined to the equatorial plane. Since $\gamma$ is circular, by
Lemma 14, $r_{c}$ is a double root of $R(\cdot,\varepsilon,\ell_{z},q)$ and
$q=0$. Therefore, $(r_{c},\varepsilon,\ell_{z},q=0)$ satisfies

$\displaystyle R(r_{c},\varepsilon,\ell_{z},q=0)$ $\displaystyle=0,$ (4.39)
$\displaystyle\frac{\partial R}{\partial r}(r_{c},\varepsilon,\ell_{z},q=0)$
$\displaystyle=0,$

which is equivalent to131313The system of equations is obtained by a linear
combination of (4.39). We refer to the computations performed in Section 4.1.3
for the general case (non-vanishing q).

$\displaystyle
3r_{c}^{4}-\frac{r_{c}^{2}}{\varepsilon^{2}}(3r_{c}^{2}-4r_{c}+d^{2})+r_{c}^{2}d^{2}$
$\displaystyle=r_{c}^{2}\ell_{z}^{2},$ (4.40) $\displaystyle
r_{c}^{4}-\frac{r_{c}^{3}}{\varepsilon^{2}}(r_{c}-1)-d^{2}r_{c}$
$\displaystyle=r_{c}(\ell^{2}-2d\ell_{z}).$

∎

We use the equations (4.40) (See [15, Chapter 6 ], [7], [8]) to express
$\ell_{z}$ and $\varepsilon$ in terms of $r_{c}$:

$\displaystyle\varepsilon$
$\displaystyle=\frac{r_{c}^{\frac{3}{2}}-2r_{c}^{\frac{1}{2}}\pm
d}{r_{c}^{\frac{3}{4}}\sqrt{r_{c}^{\frac{3}{2}}-3r_{c}^{\frac{1}{2}}\pm
2d}}=:\Phi_{\pm}(r_{c}),$ (4.41) $\displaystyle\ell_{z}$
$\displaystyle=\pm\frac{r_{c}^{2}\mp
2dr_{c}^{\frac{1}{2}}+d^{2}}{r_{c}^{\frac{3}{4}}\sqrt{r_{c}^{\frac{3}{2}}-3r_{c}^{\frac{1}{2}}\pm
2d}}=:\Psi_{\pm}(r_{c}).$ (4.42)

Figure 2: Shape of the the energy and the angular momentum in terms of the
radius of circular motion for retrograde and direct orbit at $d=0.8$. The
horizontal lines represent the energy levels at $\varepsilon=0.97$,
$\varepsilon=1$ and $\varepsilon=1.2$.

Here, the upper sign refers to direct orbits $(d\ell_{z}>0)$ and the lower
sign refers to retrograde orbits $(d\ell_{z}<0)$. In order to determine the
admissible values of $(\varepsilon,\ell_{z},r_{c})$ for circular orbits, we
study the mappings $\Phi_{\pm}(\cdot,d)$ and $\Psi_{\pm}(\cdot,d)$ on the
region $]r_{H},\infty[$. First, we study their properties and we state the
following lemma

######  Lemma 17.

  1. 1.

The mappings $\Phi_{+}$ and $\Psi_{+}$ given respectively by (4.41) and (4.42)
(see also Figure 2) are well-defined on $]r_{ph}^{+},\infty[$ and they are
smooth,

  2. 2.

the mappings $\Phi_{-}(\cdot,d)$ and $\Psi_{-}(\cdot,d)$ given respectively by
(4.41) and (4.42) (see also Figure 2) are well-defined on
$]r_{ph}^{-},\infty[$ and they are smooth.

Here $r^{\pm}_{ph}$ are defined by

$r^{\pm}_{ph}(d):=2\left(1+\cos\left(\frac{2}{3}\cos^{-1}(\mp
d)\right)\right).$

Moreover, we have the following asymptotics:

$\displaystyle\lim_{r\to\infty}\Phi_{\pm}(r)=1\quad\text{and}\quad\lim_{r\to\infty}\Psi_{\pm}(r)=\pm\infty,$
$\displaystyle\lim_{r\to
r^{\pm}_{ph}}\Phi_{\pm}(r)=+\infty\quad\text{and}\quad\lim_{r\to
r^{\pm}_{ph}}\Psi_{\pm}(r)=\pm\infty,$

###### Proof.

The mappings $\Phi_{\pm}$ and $\Psi_{\pm}$ are well-defined if and only if

$r^{\frac{3}{2}}-3r^{\frac{1}{2}}\pm 2d>0.$

Now, we consider the equations

$r^{\frac{3}{2}}-3r^{\frac{1}{2}}\pm 2d=0$

on $]r_{H},\infty[$. After the change of variable $u=\sqrt{r}$, we have

$u^{3}-3u\pm 2d=0.$

We use Cardano’s formula to express the unique real root in $]r_{H},\infty[$
of the above equation:

$u^{2}=r^{\pm}_{ph}(d):=2\left(1+\cos\left(\frac{2}{3}\cos^{-1}(\mp
d)\right)\right).$

Therefore $\Phi_{+}$, $\Psi_{+}$ are well-defined if and only if

$r>r^{+}_{ph}(d)$

and $\Phi_{-}$, $\Psi_{-}$ are well-defined if and only if

$r>r^{-}_{ph}(d).$

As for the asymptotics, it is clear that

$\lim_{r\to r^{\pm}_{ph}}\Phi_{\pm}(r)=+\infty\quad\text{and}\quad\lim_{r\to
r^{\pm}_{ph}}\Psi_{\pm}(r)=\pm\infty.$

In a neighbourhood of $+\infty$, we have

$\Phi_{\pm}(r)\sim 1\quad\text{and}\quad\Psi_{\pm}(r)\sim\pm\sqrt{r}.$

∎

######  Remark 15.

$r^{\pm}_{ph}=r^{\pm}_{ph}(d)$ is a photon orbit. There are no circular direct
orbits of radius $\displaystyle r<r^{+}_{ph}$ and there are no circular
retrograde orbits of radius $\displaystyle r<r^{-}_{ph}$. When $d=0$,
$\displaystyle r^{+}_{ph}=r^{-}_{ph}=3$, which the location of the photon
sphere with $M=1$.

Now, we claim that

######  Lemma 18.

$\displaystyle\forall
d\in[0,1]\quad:\quad\exists!r^{\pm}_{ms}(d)\in]r^{\pm}_{ph},+\infty[$ such
that
$\Phi_{\pm}^{\prime}(r^{\pm}_{ms}(d))=\Psi_{\pm}^{\prime}(r^{\pm}_{ms}(d))=0$.
It is given by

$r_{ms}^{\pm}(d)=3+Z_{2}(d)\mp\sqrt{(3-Z_{1})(3+Z_{1}(d)+2Z_{2}(d))},$ (4.44)

where

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

###### Proof.

  * We will prove the result for $\Phi_{+}$ and $\Psi_{+}$. The other case is treated in the same manner. Along the proof, we omit $+$ from the expression of the latter functions in order to lighten the expressions.

  * •

First, we claim that $\Phi$ and $\Psi$ have the same critical points. In fact,
let $r\in]r_{ph},\infty[$ then $(r,\Phi(r),\Psi(r),0)$ verifies:

$R(r,\Phi(r),\Psi(r),0)=0\quad\text{and}\quad\partial_{r}R(r,\Phi(r),\Psi(r),0)=0.$

We differentiate the first equation with respect to $r$ to obtain:

$\Phi^{\prime}(r)\frac{\partial
R}{\partial\varepsilon}(r,\Phi(r),\Psi(r),0)+\Psi^{\prime}(r)\frac{\partial
R}{\partial\ell_{z}}(r,\Phi(r),\Psi(r),0)=0.$ (4.45)

Now, we show that $\displaystyle\frac{\partial
R}{\partial\varepsilon}(r,\Phi(r),\Psi(r),0)$ and $\displaystyle\frac{\partial
R}{\partial\ell_{z}}(r,\Phi(r),\Psi(r),0)$ do not vanish on $]r_{H},\infty[$:

    * –

We compute

$\displaystyle\frac{\partial R}{\partial\ell_{z}}(r,\Phi(r),\Psi(r),0)$
$\displaystyle=-2r(2d\Phi(r)+(r-2)\Psi(r))$
$\displaystyle=-2r\frac{2d(r^{\frac{3}{2}}-2r^{\frac{1}{2}}+d)+(r-2)(r^{2}-2dr^{\frac{1}{2}}+d^{2})}{r^{\frac{3}{4}}\sqrt{r^{\frac{3}{2}}-3r^{\frac{1}{2}}+2d}}$
$\displaystyle=-2r^{2}\frac{\Delta}{r^{\frac{3}{4}}\sqrt{r^{\frac{3}{2}}-3r^{\frac{1}{2}}+2d}}.$

The latter can not vanish since $r>r_{H}$.

    * –

Now we compute

$\displaystyle\frac{\partial R}{\partial\varepsilon}(r,\Phi(r),\Psi(r),0)$
$\displaystyle=2r(-2d\Psi(r)+\Phi(r)(r^{3}+d^{2}(r+2)))$
$\displaystyle=2r\frac{d^{3}+dr(r-2)+d^{2}r^{\frac{3}{2}}+(r-2)r^{\frac{5}{2}}}{r^{\frac{3}{4}}\sqrt{r^{\frac{3}{2}}-3r^{\frac{1}{2}}+2d}}$
$\displaystyle=2r^{\frac{5}{4}}\frac{\Delta(d+r^{\frac{3}{2}})}{\sqrt{r^{\frac{3}{2}}-3r^{\frac{1}{2}}+2d}}.$

The latter can not vanish since $r>r_{H}$.

Therefore, by (4.45), $\Phi$ and $\Psi$ have the same critical points.

  * •

If $r_{c}$ is a critical point for $\Phi$, then $r_{c}$ satisfies:

$\frac{\partial^{2}R}{\partial r^{2}}(r_{c},\Phi(r_{c}),\Psi(r_{c}),0)=0.$

Indeed, we have $\forall r>r_{ph}\,,\,$

$\frac{\partial}{\partial r}R(r,\Phi(r),\Psi(r),0)=0,$

We differentiate this expression with respect to $r$ in order to obtain

$\frac{\partial^{2}R}{\partial
r^{2}}(r,\Phi(r),\Psi(r),0)+\Phi^{\prime}(r)\frac{\partial^{2}R}{\partial\varepsilon\partial
r}(r,\Phi(r),\Psi(r),0)+\Psi^{\prime}(r)\frac{\partial^{2}R}{\partial\ell_{z}\partial
r}(r,\Phi(r),\Psi(r),0)=0.$

If $r=r_{c}$, then it follows

$\frac{\partial^{2}R}{\partial r^{2}}(r_{c},\Phi(r_{c}),\Psi(r_{c}),0)=0.$

  * •

Now we compute

$\displaystyle\frac{\partial^{2}R}{\partial
r^{2}}(r_{c},\Phi(r_{c}),\Psi(r_{c}),0)$
$\displaystyle=2(d^{2}(\Phi^{2}(r)-1)-\Psi(r)^{2}+6r+6r^{2}(\Phi^{2}(r)-1))$
$\displaystyle=-2\sqrt{r}\frac{-3d^{2}+8d\sqrt{r}+r(r-6)}{2d+\sqrt{r}(r-3)}.$

  * •

It remains to solve the equation

$-3d^{2}+8d\sqrt{r}+r(r-6)=0$

on $]r_{ph},\infty[$. Again, we introduce the change of variables $u=\sqrt{r}$
and we obtain the quartic polynomial:

$u^{4}-6u^{2}+8du-3d^{2}=0.$

The latter is a depressed quartic equation that can be solved explicitly using
Ferrari’s method. The condition that $r>r_{ph}$ yields a unique root given by
$r_{ms}$.

∎

In the following, it is useful to introduce the quantity

$\rho^{\pm}_{ms}(d):=\sqrt{\Delta(r^{\pm}_{ms}(d))}.$ (4.46)

$\displaystyle r_{ms}^{\pm}(d)$ is a minimiser for
$\displaystyle\Phi_{\pm},\displaystyle\Psi_{+}$ and $-\Psi_{-}$ and their
minima are respectively given by

$\varepsilon_{min}^{\pm}(d):=\Phi_{\pm}(r^{\pm}_{ms}(d))$ (4.47)

and

$\ell_{min}^{+}(d):=\Psi_{+}(r^{+}_{ms})\quad\text{and}\quad\ell_{min}^{-}(d):=-\Psi_{-}(r^{-}_{ms})$
(4.48)

Moreover, we have

$\varepsilon_{min}^{+}\leq\varepsilon_{min}^{-}$ (4.49)

and

$\varepsilon_{min}^{+}=\varepsilon_{min}^{-}$ (4.50)

if and only if $d=0$. In this case,
$\displaystyle\varepsilon_{min}^{\pm}(0)=\sqrt{\frac{8}{9}}$. From now on, we
do not write the dependence of the above quantities on $d$. From the previous
lemma, we obtain the following monotonicity properties for $\Phi_{\pm}$ and
$\Psi_{\pm}$:

######  Lemma 19.

  1. 1.

$\Phi_{\pm}$ is monotonically decreasing on $]r^{\pm}_{ph},r^{\pm}_{ms}[$ and
monotonically increasing on $]r^{\pm}_{ms}(d),\infty[$,

  2. 2.

$\Psi_{+}$ is monotonically decreasing on $]r^{+}_{ph},r^{+}_{ms}[$ and
monotonically increasing on $]r^{+}_{ms},\infty[$,

  3. 3.

$\Psi_{-}$ is monotonically increasing on $]r^{-}_{ph},r^{-}_{ms}[$ and
monotonically decreasing on $]r^{-}_{ms},\infty[$.

Now, we go back to the equations (4.41) and (4.42): let
$\varepsilon\in]0,\infty[$ and consider the equations

$\varepsilon=\Phi_{+}(r,d)\quad\text{on}\quad]r^{+}_{ph},\infty[$ (4.51)

and

$\varepsilon=\Phi_{-}(r,d)\quad\text{on}\quad]r^{-}_{ph},\infty[$ (4.52)

By the monotonicity properties of $\Phi_{\pm}$, we obtain the following lemma

######  Lemma 20.

Let $\varepsilon\in\mathbb{R}$ and141414The case of $d=0$ corresponds to the
Schwarzschild case and was already tackled. See Proposition LABEL:Propo1.
$d\in]0,1]$, we have the following cases:

  1. 1.

If $\displaystyle\varepsilon<\varepsilon_{min}^{+}$, then the equations (4.51)
and (4.52) do not have solutions.

  2. 2.

If $\displaystyle\varepsilon=\varepsilon_{min}^{+}$ , then (4.51) admits a
unique solution given by $r=r_{ms}^{+}$ and (4.52) does not have solutions.

  3. 3.

If $\displaystyle\varepsilon_{min}^{+}<\varepsilon<\varepsilon_{min}^{-}$,
then (4.51) admits two solutions $\displaystyle r^{+}_{max}(\varepsilon)$ and
$\displaystyle r^{+}_{min}(\varepsilon)$ which satisfy

$\displaystyle r^{+}_{max}(\varepsilon)<r^{+}_{ms}<r^{+}_{min}(\varepsilon).$

and the second one does not have solutions.

  4. 4.

If $\displaystyle\varepsilon=\varepsilon_{min}^{-}$, then (4.51) admit two
solutions and (4.52) admits one solution given by $r=r_{ms}^{-}$.

  5. 5.

If $\varepsilon_{min}^{-}<\displaystyle\varepsilon<1$, then both equations
(4.51) and (4.52) admit two distinct solutions $\displaystyle
r^{\pm}_{max}(\varepsilon)$ and $\displaystyle r^{\pm}_{min}(\varepsilon)$
which satisfy

$\displaystyle
r^{\pm}_{max}(\varepsilon)<r^{\pm}_{ms}<r^{\pm}_{min}(\varepsilon).$

  6. 6.

If $\displaystyle\varepsilon\geq 1$, then both equations (4.51) and (4.52)
admit a unique solution $\displaystyle r^{\pm}_{max}(\varepsilon,d)$
satisfying

$\displaystyle r^{\pm}_{max}(\varepsilon)<r^{\pm}_{ms}.$

We note that there exists exactly two circular orbits with energy
$\varepsilon=1$ which we denote by $\displaystyle r_{mb}^{\pm}(d)$. They
correspond to the radii of the marginally bound circular direct and retrograde
orbits.
In order to compute $r_{mb}^{\pm}$, we solve the equation

$\Phi_{\pm}(r_{mb}^{\pm})=1.$

They are given by:

$r^{\pm}_{mb}(d):=2\mp d+2\sqrt{1\mp d}.$ (4.53)

Therefore, by Lemma 19,

  * •

If $\varepsilon>1$, then there exist exactly one direct circular orbit
$\displaystyle r<r_{mb}^{+}(d)$ and one retrograde circular orbit
$\displaystyle r>r_{mb}^{-}(d)$.

  * •

Otherwise, there exists at most four circular orbits and we refer to Lemma 20
for details.

It is useful to rewrite the previous lemma in terms of the implicit functions:

######  Lemma 21.

  1. 1.

There exists a unique smooth function $\displaystyle
r^{+}_{max}:[\varepsilon_{min}^{+},\infty[\to]r^{+}_{ph}(d),r^{+}_{ms}[$ such
that $r^{+}_{max}(\varepsilon)$ solves (4.51),

  2. 2.

there exists a unique smooth function $\displaystyle
r^{-}_{max}:[\varepsilon_{min}^{-},\infty[\to]r^{-}_{ph}(d),r^{+}_{ms}[$ such
that $r^{-}_{max}(\varepsilon)$ solves (4.52),

  3. 3.

there exists a unique smooth function $\displaystyle
r^{+}_{min}:[\varepsilon_{min}^{+},1[\to]r^{+}_{ms},\infty[$ such that
$r^{+}_{min}(\varepsilon)$ solves (4.51),

  4. 4.

there exists a unique smooth function $\displaystyle
r^{-}_{min}:[\varepsilon_{min}^{-},1[\to]r^{-}_{ms},\infty[$ such that
$r^{-}_{min}(\varepsilon)$ solves (4.52).

Now, we state the monotonicity properties of the functions defined in the
previous lemma

######  Lemma 22.

  1. 1.

$\displaystyle r^{+}_{max}$ is monotonically decreasing on
$]\varepsilon_{min}^{+},\infty[$,

  2. 2.

$\displaystyle r^{-}_{max}$ is monotonically decreasing on
$]\varepsilon_{min}^{-},\infty[$,

  3. 3.

$\displaystyle r^{+}_{min}$ is monotonically increasing on
$]\varepsilon_{min}^{+},1[$,

  4. 4.

$\displaystyle r^{-}_{min}$ is monotonically increasing on
$]\varepsilon_{min}^{-},1[$.

###### Proof.

The proof is straightforward using Lemma 19. First, we have

$\varepsilon=\Phi_{\pm}(r^{\pm}_{max}(\varepsilon))=\Phi_{\pm}(r^{\pm}_{min}(\varepsilon)).$

We differentiate with respect to $\varepsilon$:

$1=\frac{\partial
r^{\pm}_{max}(\varepsilon)}{\partial\varepsilon}\frac{\Phi_{\pm}(r^{\pm}_{max}(\varepsilon))}{\partial
r}=\frac{\partial
r^{\pm}_{min}(\varepsilon)}{\partial\varepsilon}\frac{\partial\Phi_{\pm}(r^{\pm}_{min}(\varepsilon))}{\partial
r}.$

By monotonicity properties of $\Phi_{\pm}$, we obtain the desired result. ∎

It remains to find conditions on $\ell_{z}$ for the circular motion. We recall
that a circular orbit of radius $r_{c}$, and constants of motion
$(\varepsilon,\ell_{z},q=0)$ must satisfy (4.41) and (4.42). Therefore,

  1. 1.

If $d\ell_{z}>0$, then
$\displaystyle\varepsilon\in[\varepsilon^{+}_{min}(d),\infty[$ and $r$ is
given by Lemma 21. Moreover $\ell_{z}$ must satisfy

$\ell_{z}=\Psi_{+}(r).$ (4.54)

If

    1. (a)

$\displaystyle\varepsilon\in[1,\infty[$, then $r=r^{+}_{max}(\varepsilon)$ and

$\ell_{z}=\Psi_{+}(r^{+}_{max}(\varepsilon)).$ (4.55)

    2. (b)

Otherwise, $\ell_{z}$ verifies

$\ell_{z}=\Psi_{+}(\tilde{r}^{+}_{max}(\varepsilon))\quad\text{or}\quad\ell_{z}=\Psi_{+}(r^{+}_{min}(\varepsilon))$
(4.56)

where $\tilde{r}^{+}_{max}$ denotes the restriction of $r^{+}_{max}$ on
$[\varepsilon^{+}_{min}(d),1[$.

  2. 2.

If $d\ell_{z}<0$, then $\varepsilon\in]\varepsilon^{-}_{min},\infty[$ and $r$
is given by Lemma 21. Moreover $\ell_{z}$ must satisfy

$\ell_{z}:=\Psi_{-}(r).$ (4.57)

If

    1. (a)

$\displaystyle\varepsilon\in[1,\infty[$, then $r=r^{-}_{max}(\varepsilon)$ and

$\ell_{z}=\Psi_{-}(r^{-}_{max}(\varepsilon)).$ (4.58)

    2. (b)

Otherwise, $\ell_{z}$ verifies

$\ell_{z}=\Psi_{-}(\tilde{r}^{-}_{max}(\varepsilon))\quad\text{or}\quad\ell_{z}=\Psi_{-}(r^{-}_{min}(\varepsilon))$
(4.59)

where $\tilde{r}^{-}_{max}$ denotes the restriction of $r^{-}_{max}$ on
$[\varepsilon^{-}_{min},1[$

Moreover, note that when $\varepsilon=\varepsilon^{\pm}_{min}$, one has
$r=r^{\pm}_{ms}$ and $\ell_{z}=\ell^{\pm}_{min}$. Now, we introduce the
following functions

######  Definition 15.

We define the functions
$\ell_{lb}^{\pm}:[\varepsilon^{\pm}_{min}(d),\infty[\times]0,1]\to\mathbb{R}$
and $\ell_{ub}^{\pm}:[\varepsilon^{\pm}_{min}(d),1[\times]0,1]\to\mathbb{R}$
by

$\ell_{lb}^{\pm}(\varepsilon,d):=\Psi_{\pm}(r_{max}^{\pm}(\varepsilon,d))\quad,\quad\ell_{ub}^{\pm}(\varepsilon,d):=\Psi_{\pm}(r_{min}^{\pm}(\varepsilon,d)).$
(4.60)

The proof of Proposition 5 now follows from lemmas 16, 18, 21 and Definition
15.

######  Lemma 23.

  1. 1.

$\ell_{ub}^{+}$ is monotonically increasing from $\ell^{+}_{min}$ to $\infty$
when $\varepsilon$ goes from $\varepsilon_{min}^{+}$ to $1$.

  2. 2.

$\ell_{lb}^{+}$ is monotonically increasing from $\ell^{+}_{min}$ to $\infty$
when $\varepsilon$ goes from $\varepsilon_{min}^{+}$ to $\infty$.

  3. 3.

$\ell_{ub}^{-}$ is monotonically decreasing from $\ell^{-}_{min}$ to $-\infty$
when $\varepsilon$ goes from $\varepsilon_{min}^{-}$ to $1$.

  4. 4.

$\ell_{lb}^{-}$ is monotonically decreasing from $\ell^{-}_{min}$ to $-\infty$
when $\varepsilon$ goes from $\varepsilon_{min}^{-}$ to $\infty$.

###### Proof.

The proof is a straightforward application of Lemma 22 and the monotonicity
properties of $\Psi_{\pm}$. ∎

From now on, the dependence of the above quantities in $d$ will not be
written.
It will be useful in the remaining of our work (see Section 4.2.1) to write
$\varepsilon$ in terms of $\ell_{z}$. Following Lemma (23), we introduce the
functions

$\varepsilon_{s}^{\pm}(\ell_{z},d):={\ell_{lb}^{\pm}}(\cdot,d)^{-1}(\ell_{z})\quad\text{on}\quad[\ell_{min}^{\pm},\infty[$
(4.61)

and

$\varepsilon_{m}^{\pm}(\ell_{z},d):={\ell_{ub}^{\pm}}(\cdot,d)^{-1}(\ell_{z})\quad\text{on}\quad[\ell_{min}^{\pm},\infty[$
(4.62)

and we state the following lemma

######  Lemma 24.

  * •

$\displaystyle\varepsilon^{+}_{s}$ increases monotonically from
$\varepsilon^{+}_{min}$ to $\infty$ when $\ell_{z}$ grows from
$\ell_{min}^{+}$ to $\infty$.

  * •

$\displaystyle\varepsilon^{-}_{s}$ increases monotonically from
$\varepsilon^{-}_{min}$ to $\infty$ when $\ell_{z}$ grows from
$\ell_{min}^{+}$ to $\infty$.

  * •

$\displaystyle\varepsilon^{+}_{m}$ increases monotonically from
$\varepsilon^{+}_{min}$ to $1$ when $\ell_{z}$ grows from $\ell_{min}^{+}$ to
$\infty$.

  * •

$\displaystyle\varepsilon^{-}_{m}$ increases monotonically from
$\varepsilon^{-}_{min}$ to $1$ when $\ell_{z}$ grows from $\ell_{min}^{+}$ to
$\infty$.

######  Remark 16.

We recover the values found in [30] when $d=0$. In particular:

$\varepsilon_{min}^{\pm}(0)=\sqrt{\frac{8}{9}}\quad\text{and}\quad\ell_{min}^{\pm}(0)=\sqrt{12}.$
(4.63)

The necessary conditions of Proposition 5 are also sufficient in the following
sense

######  Proposition 6.

Let $\gamma:I\to\mathcal{O}$ be a timelike future directed geodesic with
constants of motion
$(\varepsilon,\ell_{z},q)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$.
Assume that $(\varepsilon,\ell_{z},q)\in\mathcal{A}_{circ}$. If
$\displaystyle(r,\theta)(0)=\left(r_{c},\frac{\pi}{2}\right)$ where $r_{c}$ is
given by one of the above cases, then $\gamma$ is circular of radius $r_{c}$
and confined in the equatorial plane.

####  4.1.3 Orbits with constant radial motion

In this section, we are interested in spherical orbits, given by Definition
13. We recall that spherical orbits are circular if they are confined in the
equatorial plane and their classification was obtained in the previous
section. In the general case, by Lemma 15, spherical orbits of radius
$r_{s}>r_{H}$ occur if and only if

  * •

$\gamma$ starts at some point $(t,\phi,r_{s},\theta)$,

  * •

$r_{s}$ is a double root of $R(\cdot,\varepsilon,\ell_{z},q)$,

We note that the study of spherical orbits was is included in classical books
of general relativity. See for example [15, Chapter 6] and [39, Chapter 4].
Now, we state the main result of this section

######  Proposition 7.

Let $\gamma:\tau\ni I\to\mathcal{O}$ be a timelike future directed geodesic
and let $\displaystyle(\varepsilon,\ell_{z},q)$ be its associated integrals of
motion. Assume that that $\gamma$ is spherical of radius
$r_{s}\in]r_{H},\infty[$. Then,

$(\varepsilon,\ell_{z},q)\in\mathcal{A}_{spherical}$ (4.64)

defined by (4.108).

The remaining of this section is devoted to the proof of Proposition 7. We
start with the following lemma

######  Lemma 25.

Let
$\gamma:I\to\mathcal{O}\,,\,\tau\mapsto(t(\tau),\phi(\tau),r(\tau),\theta(\tau))$
be a spherical orbit of radius $r_{s}>r_{H}$. Let
$(\varepsilon,\ell_{z},q)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ be its
associated integrals of motion. Then, the quadruplet
$(r_{s},\varepsilon,\ell_{z},q)$ satisfies

$\displaystyle\ell$ $\displaystyle=\ell_{c}^{\pm}(r_{s},\varepsilon^{2}),$
(4.65) $\displaystyle d^{2}(r_{s}-1)\eta$
$\displaystyle=d^{2}(r_{s}-1)\eta_{c}^{\pm}(r_{s},\varepsilon^{2}),$

where

$\ell:=\frac{\ell_{z}}{\varepsilon}\quad\text{and}\quad\eta:=\frac{q}{\varepsilon^{2}},$
(4.66)
$\ell^{\pm}_{c}(r_{s},\varepsilon^{2}):=\frac{1}{d(r_{s}-1)}\left((r_{s}^{2}-d^{2})\pm
r_{s}(r_{s}^{2}-2r_{s}+d^{2})\sqrt{1-\frac{1}{\varepsilon^{2}}(1-\frac{1}{r_{s}})}\right)$
(4.67)

and

$\displaystyle\eta^{\pm}_{c}(r_{s},\varepsilon^{2})$
$\displaystyle:=\frac{r_{s}^{3}}{r_{s}-1}\left(4d^{2}-r_{s}(r_{s}-3)^{2}\right)+\frac{r_{s}^{2}}{\varepsilon^{2}}\left(r_{s}(r_{s}-2)^{2}-d^{2}\right)$
(4.68)
$\displaystyle-\frac{2r_{s}^{3}}{r_{s}-1}(r_{s}^{2}-2r_{s}+d^{2})\left(1\pm\sqrt{1-\frac{1}{\varepsilon^{2}}\left(1-\frac{1}{r_{s}}\right)}\right).$

###### Proof.

By Lemma 15, the quadruplet $(r_{s},\varepsilon,\ell_{z},q)$ verifies

$\displaystyle R(r,\varepsilon,\ell_{z},q,d)$ $\displaystyle=0$ (4.69)
$\displaystyle\frac{\partial R}{\partial r}(r,\varepsilon,\ell_{z},q,d)$
$\displaystyle=0.$ (4.70)

Moreover, by Lemma 11, we require

$q\geq 0.$ (4.71)

Solving the equations (4.69) and (4.70) simultaneously eliminates two of the
four unknowns $r_{s},\varepsilon,\ell_{z}$ and $q$. We introduce the following
change of variables

$\ell=\frac{\ell_{z}}{\varepsilon}\quad\text{and}\quad\eta=\frac{q}{\varepsilon^{2}}$
(4.72)

After developing the above equations, we obtain

$\displaystyle
r_{s}^{4}-\frac{r_{s}^{2}}{\varepsilon^{2}}(r_{s}^{2}-2r_{s}+d^{2})-\eta(r_{s}^{2}-2r_{s}+d^{2})+(d^{2}-\ell^{2})r_{s}^{2}+2(d^{2}+\ell^{2}-2d\ell)r_{s}$
$\displaystyle=0,$ $\displaystyle
4r_{s}^{3}-\frac{2r_{s}}{\varepsilon^{2}}(r_{s}^{2}-2r_{s}+d^{2})-\frac{2r_{s}^{2}}{\varepsilon^{2}}(r_{s}-1)-2\eta(r_{s}-1)+2r_{s}(d^{2}-\ell^{2})+2(d^{2}+\ell^{2}-2d\ell)$
$\displaystyle=0.$

The linear combinations
$\displaystyle\frac{1}{\varepsilon^{2}}\left(r_{s}\frac{\partial R}{\partial
r}(r_{s})-R(r_{s})\right)$ and
$\displaystyle\frac{1}{\varepsilon^{2}}\left(\frac{r_{s}}{2}\frac{\partial
R}{\partial r}(r_{s})-R(r_{s})\right)$ yield

$\displaystyle
3r_{s}^{4}-\frac{r_{s}^{2}}{\varepsilon^{2}}(3r_{s}^{2}-4r_{s}+d^{2})+r_{s}^{2}d^{2}-\eta(r_{s}^{2}-d^{2})$
$\displaystyle=r_{s}^{2}\ell^{2},$ (4.73) $\displaystyle
r_{s}^{4}-\frac{r_{s}^{3}}{\varepsilon^{2}}(r_{s}-1)-d^{2}r_{s}+\eta(d^{2}-r_{s})$
$\displaystyle=r_{s}(\ell^{2}-2d\ell).$ (4.74)

We solve for $\ell$ first by eliminating $\eta$. Hence, we multiply the first
equation by $(d^{2}-r_{s})$ and the second one by $(r_{s}^{2}-d^{2})$ and we
obtain a second order polynomial in $\ell$. Straightforward computations yield
the following solutions for $\ell$:

$\ell=\frac{1}{d(r_{s}-1)}\left((r_{s}^{2}-d^{2})\pm
r_{s}(r_{s}^{2}-2r_{s}+d^{2})\sqrt{1-\frac{1}{\varepsilon^{2}}(1-\frac{1}{r_{s}})}\right)=:\ell^{\pm}_{c}(r_{s},\varepsilon^{2}).$
(4.75)

Now, we replace $\ell$ with its expression in (4.73) in order to obtain

$\displaystyle d^{2}(r_{s}-1)\eta$
$\displaystyle=\frac{r_{s}^{3}}{r_{s}-1}\left(4d^{2}-r_{s}(r_{s}-3)^{2}\right)+\frac{r_{s}^{2}}{\varepsilon^{2}}\left(r_{s}(r_{s}-2)^{2}-d^{2}\right)-\frac{2r_{s}^{3}}{r_{s}-1}(r_{s}^{2}-2r_{s}+d^{2})$
(4.76)
$\displaystyle\left(1\pm\sqrt{1-\frac{1}{\varepsilon^{2}}\left(1-\frac{1}{r_{s}}\right)}\right)$
$\displaystyle=:d^{2}(r_{s}-1)\eta^{\pm}_{c}(r_{s},\varepsilon^{2}).$

∎

We have shown so far that (4.69) and (4.70) imply (4.75) and (4.76). Tedious
but straightforward computations imply that if
$(\ell,\eta)=(\ell^{\pm}_{c}(r_{s},\varepsilon^{2}),\eta^{\pm}_{c}(r_{s},\varepsilon^{2}))$,
then the quadruplets
$(r_{s},\varepsilon^{2},\ell^{\pm}_{c}(r_{s},\varepsilon^{2}),$
$\eta^{\pm}_{c}(r_{s},\varepsilon^{2}))$ (4.69) and (4.70) solve the equations
(4.69) and (4.70).
In the following, we will determine the set of admissible parameters
$(r_{s},\varepsilon^{2})$ so that solutions of (4.69)-(4.70) are given by a
two parameter family indexed by $(r_{s},\varepsilon^{2})$.

######  Lemma 26.

  1. 1.

$\eta^{\pm}_{c}$ and $\ell^{\pm}_{c}$ are defined on the domain

$\displaystyle D_{s}$
$\displaystyle:=\left\\{(r,\varepsilon)\in]r_{H},\infty[\times[1,\infty[\right\\}\sqcup\left\\{(r,\varepsilon)\in]r_{H},\infty[\times]0,1[\;:\;r>(1-\varepsilon^{2})^{-1}\right\\}$
$\displaystyle:=D_{s}^{\geq 1}\sqcup D_{s}^{\leq 1}.$

  2. 2.

We have the following asymptotics for $\eta^{\pm}_{c}(\cdot,\varepsilon^{2})$:

     * •

$\forall\varepsilon^{2}>0$

$\lim_{r\to r_{H}}\eta(r,\varepsilon^{2})<0$ (4.77)

     * •

and

$\displaystyle\lim_{r\to(1-\varepsilon^{2})^{-1}}\eta(r,\varepsilon^{2})<0$
$\displaystyle\quad\text{if}\quad\varepsilon^{2}<1,$ (4.78)
$\displaystyle\lim_{r\to\infty}\eta(r,\varepsilon^{2})<0$
$\displaystyle\quad\text{if}\quad\varepsilon^{2}\geq 1,$

  3. 3.

$\eta_{c}^{+}$ is negative on $D_{s}$.

###### Proof.

  1. 1.

The first point is straightforward.

  2. 2.

For the second point, we compute

$\displaystyle\eta_{c}(r_{H},\varepsilon^{2})$
$\displaystyle=\frac{r_{H}^{3}}{r_{H}-1}\left(4d^{2}-r_{H}(r_{H}-3)^{2}\right)+\frac{r_{H}^{2}}{\varepsilon^{2}}\left(r_{H}(r_{H}-2)^{2}-d^{2}\right)$
$\displaystyle=\frac{r_{H}^{2}}{r_{H}-1}\left(r_{H}\left(4d^{2}-r_{H}(r_{H}-3)^{2}\right)+\frac{(r_{H}-1)}{\varepsilon^{2}}\left(r_{H}(r_{H}-2)^{2}-d^{2}\right)\right)$
$\displaystyle=\frac{r_{H}^{2}}{r_{H}-1}\left(I+II\right),$

where

$\displaystyle I$ $\displaystyle:=r_{H}\left(4d^{2}-r_{H}(r_{H}-3)^{2}\right)$
$\displaystyle=(1+\sqrt{1-d^{2}})(4d^{2}-(1+\sqrt{1-d^{2}})(\sqrt{1-d^{2}}-2)^{2})$
$\displaystyle=(1+\sqrt{1-d^{2}})^{2}(d^{2}-1)<0$

and

$\displaystyle II$
$\displaystyle:=\frac{(r_{H}-1)}{\varepsilon^{2}}\left(r_{H}(r_{H}-2)^{2}-d^{2}\right)$
$\displaystyle=\frac{\sqrt{1-d^{2}}}{\varepsilon^{2}}\left((1+\sqrt{1-d^{2}})(\sqrt{1-d^{2}}-1)^{2}-d^{2})\right)$
$\displaystyle=-\frac{d^{2}}{\varepsilon^{2}}(1-d^{2}).$

Therefore,

$\eta_{c}(r_{H},\varepsilon^{2})<0.$

Now, we compute $\forall\varepsilon^{2}<1\,,\,\forall d\in]0,1[$

$\eta_{c}((1-\varepsilon^{2})^{-1},\varepsilon^{2})=\left(\left((1-\varepsilon^{2})^{-1}-1\right)^{-1}\right)\left(a(\varepsilon^{2},d)+b(\varepsilon^{2},d)+c(\varepsilon^{2},d)\right)$

where

$\displaystyle a(\varepsilon^{2},d)$
$\displaystyle:=\frac{1}{(1-\varepsilon^{2})^{3}}\left(4d^{2}-\frac{1}{1-\varepsilon^{2}}\left(\frac{1}{1-\varepsilon^{2}}-3\right)^{2}\right)$
$\displaystyle=\frac{1}{(1-\varepsilon^{2})^{6}}(4d^{2}(1-\varepsilon^{2})^{3}-(2-3\varepsilon^{2})^{2}),$
$\displaystyle b(\varepsilon^{2},d)$
$\displaystyle:=\frac{1}{\varepsilon^{2}}\frac{1}{(1-\varepsilon^{2})^{2}}\left(\frac{1}{1-\varepsilon^{2}}-1\right)\left(\frac{1}{1-\varepsilon^{2}}\left(\frac{1}{1-\varepsilon^{2}}-2\right)^{2}-d^{2}\right)$
$\displaystyle=\frac{1}{(1-\varepsilon^{2})^{6}}\left((2\varepsilon^{2}-1)^{2}-d^{2}(1-\varepsilon^{2})^{3}\right),$
$\displaystyle c(\varepsilon^{2},d)$
$\displaystyle:=-\frac{2}{(1-\varepsilon^{2})^{3}}\left(\frac{1}{(1-\varepsilon^{2})^{2}}-\frac{2}{1-\varepsilon^{2}}+d^{2}\right)$
$\displaystyle=-\frac{2}{(1-\varepsilon^{2})^{5}}\left((2\varepsilon^{2}-1)+d^{2}(1-\varepsilon^{2})^{2}\right).$

Straightforward computations imply

$\eta_{c}((1-\varepsilon^{2})^{-1},\varepsilon^{2})=\frac{1}{(1-\varepsilon^{2})^{4}}\left(d^{2}(1-\varepsilon^{2})-1\right)<0.$

  3. 3.

Let $(r,\varepsilon)\in D_{s}$. Then, $\forall r\in]r_{H},\infty[$,
$\varepsilon$ must verify

$\varepsilon^{2}\geq\left(1-\frac{1}{r}\right).$

Now let $r\in]r_{H},\infty[$ and consider the function
$\tilde{\eta}_{c}^{+}(r,\cdot)$ defined on
$\left]\sqrt{\left(1-\frac{1}{r}\right)},\infty\right[$ by

$\tilde{\eta}_{c}^{+}(r,\varepsilon):=\frac{r-1}{r^{2}}\eta_{c}^{+}(r,\varepsilon).$

It is easy to see that the terms $r(r-2)^{2}-d^{2}$ and $r(r^{2}-2r+d^{2})$
are positive since $r>1+\sqrt{1-d^{2}}$ and $0<d^{2}<1$. Therefore
$\tilde{\eta}_{c}^{+}(r,\cdot)$ is monotonically decreasing on
$\left]\sqrt{\left(1-\frac{1}{r}\right)},\infty\right[$. Hence

$\forall\varepsilon\geq\sqrt{\left(1-\frac{1}{r}\right)}\;:\;\eta_{c}^{+}(r,\varepsilon)<\eta_{c}^{+}(r,\sqrt{\left(1-\frac{1}{r}\right)}).$

Straightforward computations imply

$\eta_{c}^{+}(r,\sqrt{\left(1-\frac{1}{r}\right)})=-r(r-d^{2})<0.$

Therefore, $\tilde{\eta}_{c}^{+}(r,\cdot)$ is negative and so is
$\eta_{c}^{+}$.

∎

Since $q\geq 0$, $\ell$ and $\eta$ must equal $\ell^{-}_{c}$ and
$\eta^{-}_{c}$. From now on, we omit the sign from the latter quantities so
that they are simply denoted by $\ell_{c}$ and $\eta_{c}$.

######  Remark 17.

Note that if $(r,\varepsilon)\in D_{s}^{\leq 1}$ then we have a lower bound on
$\varepsilon$. Indeed, $\forall r>r_{H}$, $\varepsilon^{2}$ satisfies

$\varepsilon^{2}\geq\left(1-\frac{1}{r}\right)>\left(1-\frac{1}{r_{H}}\right)=\frac{\sqrt{1-d^{2}}}{1+\sqrt{1-d^{2}}}.$

By the second point of the previous lemma and the positivity of $q$, not all
values of $r$ are allowed. In order to determine the allowed region for $r$,
we first look at the equation

$\eta_{c}(r,\varepsilon^{2})=0.$ (4.79)

The latter is equivalent to a vanishing Carter constant. This case corresponds
to circular orbits confined in the equatorial plane. Hence, by Proposition 5,
$(\varepsilon,\ell_{z},q)\in\mathcal{A}_{circ}$ and $r_{s}$ is given by one of
the cases in the latter proposition. More precisely, one of the following
cases is possible:

  1. 1.

If $\varepsilon<\varepsilon^{+}_{min}$, then the equation (4.79) does not
admit solutions.

  2. 2.

If $\varepsilon=\varepsilon^{+}_{min}$, then the equation (4.79) admits a
unique solution given by $r_{ms}^{+}$.

  3. 3.

If $\varepsilon^{+}_{min}<\varepsilon<\varepsilon^{-}_{min}$, then the
equation (4.79) admits two solutions $\tilde{r}_{max}^{+}(\varepsilon)$ and
$r_{min}^{+}(\varepsilon)$.

  4. 4.

If $\varepsilon=\varepsilon^{-}_{min}$, then the equation (4.79) admits three
solutions $\tilde{r}_{max}^{+}(\varepsilon)$, $r_{min}^{+}(\varepsilon)$ and
$r_{ms}^{-}$.

  5. 5.

If $\varepsilon^{-}_{min}<\varepsilon<1$, then the equation (4.79) admits four
solutions $\tilde{r}_{max}^{+}(\varepsilon)$, $r_{min}^{+}(\varepsilon)$,
$\tilde{r}_{max}^{-}(\varepsilon)$ and $r_{min}^{-}(\varepsilon)$.

  6. 6.

If $\varepsilon\geq 1$, then the equation (4.79) admits two solutions
$r_{max}^{+}(\varepsilon)$ and $r_{max}^{-}(\varepsilon)$.

Moreover, by the asymptotics of $\eta_{c}$ given by the second point of Lemma
26, the allowed regions for $r$ so that $\eta_{c}$ is positive are given by
the following lemma

######  Lemma 27.

Let $(r,\varepsilon)\in D_{s}$. Then $\eta_{c}$ is positive ($\geq 0$) if and
only if

$(r,\varepsilon)\in\overline{D}_{s}:=\overline{D}_{s}^{\leq
1}\sqcup\overline{D}_{s}^{\geq 1}$ (4.80)

where

$\overline{D}_{s}^{\geq 1}:=\left\\{(r,\varepsilon)\in D_{s}^{\geq
1}\;:\;r\in[r_{max}^{+}(\varepsilon),r_{max}^{-}(\varepsilon)]\right\\}.$
(4.81)

and

$\displaystyle\overline{D}_{s}^{\leq 1}$
$\displaystyle:=\left\\{(r_{ms}^{+},\varepsilon^{+}_{min})\right\\}\sqcup\left\\{(r,\varepsilon)\in
D_{s}^{\leq
1}(d)\;:\;\varepsilon\in]\varepsilon^{+}_{min},\varepsilon^{-}_{min}]\;\text{and}\;r\in[\tilde{r}_{max}^{+}(\varepsilon),\tilde{r}_{min}^{+}(\varepsilon)]\right\\}$
(4.82) $\displaystyle\sqcup\left\\{(r,\varepsilon)\in D_{s}^{\geq
1}(d)\;:\;\varepsilon\in]\varepsilon^{-}_{min},1[\;\text{and}\;r\in[\tilde{r}_{max}^{+}(\varepsilon),\tilde{r}_{max}^{-}(\varepsilon)]\sqcup[r_{min}^{-}(\varepsilon),r_{min}^{+}(\varepsilon)]\right\\}$

###### Proof.

The proof is straightforward using the above six cases and the asymptotics of
$\eta_{c}(\cdot,\varepsilon^{2})$. ∎

######  Lemma 28 (Critical points of $\eta_{c}(\cdot,\varepsilon^{2})$ when
$\varepsilon^{2}<1$).

Let $\varepsilon^{+}_{min}<\varepsilon<1$.Then, the critical points of
$\eta_{c}(\cdot,\varepsilon^{2})$ are:

  1. 1.

If $\displaystyle\varepsilon<\sqrt{\frac{8}{9}}$, the critical points of
$\eta_{c}(\cdot,\varepsilon^{2})$ are the critical points of
$\ell_{c}(\cdot,\varepsilon^{2})$.

  2. 2.

Otherwise,

     * •

the critical points of $\ell_{c}(\cdot,\varepsilon^{2})$,

     * •

the points $\overset{\circ}{r}_{\pm}(\varepsilon)$:

$\overset{\circ}{r}_{\pm}(\varepsilon):=\frac{-4+3\varepsilon^{2}\pm\varepsilon\sqrt{-8+9\varepsilon^{2}}}{2(\varepsilon^{2}-1)}.$

###### Proof.

Let $(r,\varepsilon)\in\overline{D}_{s}$ such that $r$ is a critical point for
$\eta_{c}(\cdot,\varepsilon^{2})$. We have

$R(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=0\quad\text{and}\quad\frac{\partial
R}{\partial
r}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=0.$

We differentiate the first equation with respect to $r$ to obtain:

$\varepsilon\partial_{r}\ell_{c}(r,\varepsilon^{2})\frac{\partial
R}{\partial\ell_{z}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))+\varepsilon^{2}\partial_{r}\eta_{c}(r,\varepsilon^{2})\frac{\partial
R}{\partial
q}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=0.$

Therefore,

$\partial_{r}\ell_{c}(r,\varepsilon^{2})=0\quad\text{or}\quad\frac{\partial
R}{\partial\ell_{z}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2})).$

We use the equations (4.69)-(4.70) in order to obtain

$\frac{\partial
R}{\partial\ell_{z}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=-2(2d\varepsilon+\varepsilon\ell_{c}(r,\varepsilon^{2})(r-2))r.$

Hence $r$ is a solution of the equation

$\ell_{c}(r,\varepsilon^{2})(r-2)=-2d.$

Now we plug the expression of $\ell_{c}$ in the latter equation to obtain

$\varepsilon((r^{2}-d^{2})(r-2)+2d^{2}(r-1))-\sqrt{r}\Delta\sqrt{r(\varepsilon^{2}-1)+1}(r-2)=0,$

which is equivalent to

$r\varepsilon\Delta-\sqrt{r}(r-2)\Delta\sqrt{r(\varepsilon^{2}-1)+1}=0.$

Therefore, $r$ satisfies the equation

$\sqrt{r}\varepsilon=(r-2)\sqrt{r(\varepsilon^{2}-1)+1}.$

We compute

$\displaystyle((r-2)\sqrt{r(\varepsilon^{2}-1)+1})^{2}-(\sqrt{r}\varepsilon)^{2}$
$\displaystyle=4-8r+4r\varepsilon^{2}+5r^{2}-4\varepsilon^{2}r^{2}-r^{3}+\varepsilon^{2}r^{3}-r\varepsilon^{2}$
$\displaystyle=(r-1)((\varepsilon^{2}-1)r^{2}+(4-3\varepsilon^{2})r-4).$

Since $r>r_{H}>1$, $r$ is solution of the second degree polynomial

$((\varepsilon^{2}-1)r^{2}+(4-3\varepsilon^{2})r-4).$

Straightforward computations imply that $r$ is given by
$\overset{\circ}{r}_{\pm}(\varepsilon)$. ∎

Now, we study the function $\ell_{c}$ on the domain $\overline{D}_{s}$. We
recall that $\ell_{c}$ is defined by

$\ell_{c}(r,\varepsilon^{2})=\frac{1}{d(r-1)}\left((r^{2}-d^{2})-r(r^{2}-2r+d^{2})\sqrt{1-\frac{1}{\varepsilon^{2}}\left(1-\frac{1}{r}\right)}\right).$
(4.83)

In the following, it is useful to introduce the following function
$\xi:\overline{D}_{s}\to\mathbb{R}$ defined by

$\xi(r,\varepsilon^{2}):=\eta_{c}(r,\varepsilon^{2})+(\ell_{c}(r,\varepsilon^{2})-d)^{2}.$
(4.84)

We state the following lemma on the critical points of
$\ell_{c}(\cdot,\varepsilon^{2})$ and $\xi(\cdot,\varepsilon^{2})$ when
$\varepsilon^{+}_{min}<\varepsilon^{2}<1$.

######  Lemma 29.

Let $\varepsilon^{+}_{min}<\varepsilon<1$. Then, the critical points of
$\xi(\cdot,\varepsilon^{2})$ are the critical points of
$\ell_{c}(\cdot,\varepsilon^{2})$. Moreover, there exists a unique
$r_{m}(\varepsilon)\in]\tilde{r}_{max}^{+}(\varepsilon),r_{min}^{+}(\varepsilon)[$
such that

$\partial_{r}\ell_{c}(r_{m}(\varepsilon),\varepsilon^{2})=\partial_{r}\xi(r_{m}(\varepsilon),\varepsilon^{2})=0.$
(4.85)

Moreover, $\ell_{c}(r_{m}(\varepsilon),\varepsilon^{2})$ is a global minimum
for $\ell_{c}(\cdot,\varepsilon^{2})$ and
$\xi(r_{m}(\varepsilon),\varepsilon^{2})$ is a global maximum for
$\xi(\cdot,\varepsilon^{2})$.

###### Proof.

Let $\displaystyle\varepsilon^{+}_{min}<\varepsilon<1$.

  1. 1.

Let $r$ be a critical point for $\ell_{c}(\cdot,\varepsilon^{2})$. Then,
$\partial_{r}\ell_{c}(r,\varepsilon^{2})=0$. Moreover, by Lemma 28, $r$ is
also a critical point for $\eta_{c}(\cdot,\varepsilon^{2})$. Therefore,

$\partial_{r}\xi(r,\varepsilon^{2})=\partial_{r}\eta_{c}(r,\varepsilon^{2})+2(\ell_{c}(r,\varepsilon^{2})-d)\partial_{r}\ell_{c}(r,\varepsilon^{2})=0.$

Now, let $r$ be a critical point for $\xi(\cdot,\varepsilon^{2})$. Then,
$(r,\varepsilon^{2})$ satisfies

$R(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon^{2}(\xi(r,\varepsilon^{2})-(d-\ell_{c}(r,\varepsilon^{2}))^{2}))=0.$
(4.86)

We differentiate the latter with respect to $r$ and we use the fact that $r$
is double root for $R$ to obtain

$\displaystyle\partial_{r}\ell_{c}(r,\varepsilon^{2})\frac{\partial
R}{\partial\ell_{z}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon^{2}(\xi(r,\varepsilon^{2})-(d-\ell_{c}(r,\varepsilon^{2}))^{2}))+$
$\displaystyle(\partial_{r}\xi(r,\varepsilon^{2})-2(\ell_{c}(r,\varepsilon^{2})-d)\partial_{r}\ell_{c}(r,\varepsilon^{2}))\frac{\partial
R}{\partial
q}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon^{2}(\xi(r,\varepsilon^{2})-(d-\ell_{c}(r,\varepsilon^{2}))^{2}))=0.$

Therefore,

$\partial_{r}\ell_{c}(r,\varepsilon^{2})\frac{\partial
R}{\partial\ell_{z}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon^{2}(\xi(r,\varepsilon^{2})-(d-\ell_{c}(r,\varepsilon^{2}))^{2}))+2(\ell_{c}(r,\varepsilon^{2})-d)\partial_{r}\ell_{c}(r,\varepsilon^{2})\Delta(r)=0.$

If $\partial_{r}\ell_{c}(r,\varepsilon^{2})\neq 0$, then

$\frac{\partial
R}{\partial\ell_{z}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon^{2}(\xi(r,\varepsilon^{2})-(d-\ell_{c}(r,\varepsilon^{2}))^{2}))+2(\ell_{c}(r,\varepsilon^{2})-d)\Delta(r)=0.$

We plug the expression of the $\ell_{z}$-derivative of $R$ in the above
equation in order to obtain

$\ell_{c}(r,\varepsilon^{2})r(r-2)+2rd=\Delta(r)\ell_{c}(r,\varepsilon^{2})-d\Delta(r).$

Therefore, $\ell_{c}(r,\varepsilon^{2})$ verifies

$\ell_{c}(r,\varepsilon^{2})=\frac{r^{2}}{d}+d.$

Furthermore, $(r,\varepsilon)$ must satisfy (4.86). We plug the expression of
$\ell_{c}(r,\varepsilon^{2})$ in the latter equation to obtain

$-\Delta(r)(r^{2}+\xi(r,\varepsilon^{2}))=0.$

Therefore,

$\xi(r,\varepsilon^{2})=-r^{2}<0,$

which is not possible since $\xi$ is positive. Therefore,
$\partial_{c}\ell_{c}(r,\varepsilon^{2})=0$.

  2. 2.

For the uniqueness, we refer to Appendix II of [26] for a detailed proof.

∎

######  Lemma 30.

Let $\varepsilon^{2}>1$. Then $\ell_{c}(\cdot,\varepsilon^{2})$ does not have
critical points and $\eta_{c}(\cdot,\varepsilon^{2})$ admits a unique extremum
(maximum) at $r^{\geq
1}_{max}(\varepsilon)\in]r_{max}^{+}(\varepsilon),r_{max}^{-}(\varepsilon)[$.

###### Proof.

  1. 1.

Let $(r,\varepsilon)\in\overline{D}_{s}^{\geq 1}$ such that
$r=r^{c}(\varepsilon^{2})$ is a critical point for
$\ell_{c}(\cdot,\varepsilon^{2})$. We have

$R(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=0\quad\text{and}\quad\frac{\partial
R}{\partial
r}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=0.$

We differentiate the first equation with respect to $r$ to obtain:

$\varepsilon\partial_{r}\ell_{c}(r,\varepsilon^{2})\frac{\partial
R}{\partial\ell_{z}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))+\varepsilon^{2}\partial_{r}\eta_{c}(r,\varepsilon^{2})\frac{\partial
R}{\partial
q}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=0.$

Now we differentiate the second equation with respect to $r$ to obtain:

$\displaystyle\frac{\partial^{2}R}{\partial
r^{2}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))+\varepsilon\partial_{r}\ell_{c}(r,\varepsilon^{2})\frac{\partial^{2}R}{\partial
r\partial\ell_{z}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))$
$\displaystyle+\varepsilon^{2}\partial_{r}\eta_{c}(r,\varepsilon^{2})\frac{\partial^{2}R}{\partial
r\partial
q}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=0.$

Therefore, $r$ is a critical point of $\eta_{c}(\cdot,\varepsilon^{2})$ since
the derivative of $R$ with respect to $q$ does not vanish. Hence, $r$
verifies:

$\frac{\partial^{2}R}{\partial
r^{2}}(r,\varepsilon,\varepsilon\ell_{c}(r,\varepsilon^{2}),\varepsilon\eta_{c}(r,\varepsilon^{2}))=0.$
(4.87)

$r\in[r^{+}_{max}(\varepsilon),r^{+}_{min}(\varepsilon)]$. Therefore,
$r^{c}(\varepsilon)$ is a triple root of the polynomial $R$ with parameters
$(\varepsilon,\varepsilon\ell_{c}(\varepsilon),\varepsilon^{2}\eta_{c}(r^{c}(\varepsilon),\varepsilon^{2}))$.
This cannot happen because of Lemma 10.

  2. 2.

Since $\ell_{c}(\cdot,\varepsilon^{2})$ does not have critical points,
$\eta_{c}(\cdot,\varepsilon^{2})$ admits two critical points, given by
$\overset{\circ}{r}_{\pm}(\varepsilon)$. A study of the function
$\overset{\circ}{r}_{-}$ on $[1,\infty[$ shows that this function is always
negative. Since we are looking for critical points in the region
$]r_{max}^{+}(\varepsilon),r_{max}^{-}(\varepsilon)[\subset]0,\infty[$,
$\ell_{c}(\cdot,\varepsilon^{2})$ admits a unique critical point given by
$\overset{\circ}{r}_{+}(\varepsilon)\in]0,\infty[$. It remains to show that it
lies in the region $]r_{max}^{+}(\varepsilon),r_{max}^{-}(\varepsilon)[$. This
is straightforward by the mean value theorem. Indeed,
$\eta_{c}(\cdot,\varepsilon^{2})$ vanishes at $r_{max}^{+}(\varepsilon)$ and
$r_{max}^{-}(\varepsilon)$.

∎

Finally, we introduce the following notations:

$\tilde{\ell}_{min}(\varepsilon):=\varepsilon\ell_{c}(r_{m}(\varepsilon),\varepsilon^{2})\quad\text{for}\quad{\varepsilon\in]\varepsilon^{+}_{min},1[}$
(4.88)

and

$q_{max}(\varepsilon):=\varepsilon^{2}\eta_{c}(r^{\geq
1}_{max}(\varepsilon),\varepsilon^{2})\quad\text{for}\quad\varepsilon^{2}>1.$
(4.89)

######  Lemma 31 (Properties of $\tilde{\ell}_{min}$ and
$r_{m}(\varepsilon)$).

Let $\varepsilon\in[\varepsilon_{min}^{+},1[$. Then,

  * •

$r_{m}$ is monotonically increasing from $r_{ms}^{+}$ to $+\infty$ when
$\varepsilon$ grows monotonically from $\varepsilon_{min}^{+}$ to $1$.

  * •

$\displaystyle\tilde{\ell}_{min}$ is monotonically decreasing from
$\ell_{min}^{+}$ to $-\infty$ when $\varepsilon$ grows monotonically from
$\varepsilon_{min}^{+}$ to $1$.

In particular, if $\varepsilon=\varepsilon^{-}_{min}$, then

$r_{m}(\varepsilon)=r_{ms}^{-}\quad\text{and}\quad\tilde{\ell}_{min}(\varepsilon)=\ell_{min}^{-}.$

Moreover, if $\varepsilon\in]\varepsilon_{min}^{-},1[$, then $\displaystyle
r_{m}(\varepsilon)\in]r_{max}^{-}(\varepsilon),r_{min}^{-}(\varepsilon)[$.

###### Proof.

This follows from the definitions and the implicit function theorem. ∎

Based on the above lemmas, we derive the monotonicity properties of
$\ell_{c}(\cdot,\varepsilon^{2})$.

######  Lemma 32.

  1. 1.

If $\displaystyle(r,\varepsilon)=(r_{ms}^{+},\varepsilon^{+}_{min})$, then

$\ell_{c}(r,\varepsilon^{2})=\frac{\ell_{min}^{+}}{\varepsilon^{+}_{min}}.$
(4.90)

  2. 2.

If $\varepsilon_{min}^{+}<\varepsilon\leq\varepsilon_{min}^{-}$, then the
restriction of $\ell_{c}(\cdot,\varepsilon^{2})$ on the domain
$[\tilde{r}^{+}_{max}(\varepsilon),r^{+}_{min}(\varepsilon)]$ has the
following properties: $\ell_{c}(\cdot,\varepsilon^{2})$ is monotonically
decreasing on $]\tilde{r}^{+}_{max}(\varepsilon),r_{m}(\varepsilon)[$ from
$\displaystyle\frac{\ell_{lb}^{+}(\varepsilon)}{\varepsilon}$ to
$\displaystyle\frac{\tilde{\ell}_{min}(\varepsilon)}{\varepsilon}<\frac{\ell_{min}^{+}}{\varepsilon}$
and monotonically increasing on
$]r_{m}(\varepsilon),r^{+}_{min}(\varepsilon)[$ from
$\displaystyle\frac{\tilde{\ell}_{min}(\varepsilon)}{\varepsilon}$ to
$\displaystyle\frac{\ell_{ub}^{+}(\varepsilon)}{\varepsilon}$. Therefore,
there exists a unique
$r_{lb}(\varepsilon)\in]r_{m}(\varepsilon),r^{+}_{min}(\varepsilon)[$ such
that
$\displaystyle\ell_{c}(r_{lb}(\varepsilon),\varepsilon^{2})=\frac{\ell_{lb}^{+}(\varepsilon)}{\varepsilon}$.

  3. 3.

If $\varepsilon_{min}^{-}<\varepsilon<1$, then the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on the domain
$[\tilde{r}^{+}_{max}(\varepsilon),\tilde{r}^{-}_{max}(\varepsilon)]\sqcup[r^{-}_{min}(\varepsilon),r^{-}_{min}(\varepsilon)]$
has the following properties:

     * •

$\ell_{c}(\cdot,\varepsilon^{2})$ is monotonically decreasing on
$]\tilde{r}^{+}_{max}(\varepsilon),\tilde{r}^{-}_{max}(\varepsilon)[$ from
$\displaystyle\frac{\ell_{lb}^{+}(\varepsilon)}{\varepsilon}$ to
$\displaystyle\frac{\ell_{lb}^{-}(\varepsilon)}{\varepsilon}$.

     * •

$\ell_{c}(\cdot,\varepsilon^{2})$ is monotonically increasing on
$]r^{-}_{min}(\varepsilon),r^{+}_{min}(\varepsilon)[$ from
$\displaystyle\frac{\ell_{ub}^{-}(\varepsilon)}{\varepsilon}$ to
$\displaystyle\frac{\ell_{ub}^{+}(\varepsilon)}{\varepsilon}$

  4. 4.

If $\displaystyle(r,\varepsilon)\in\overline{D}_{s}^{\geq 1}$, then
$\forall\varepsilon\geq 1$, the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on the domain
$[r^{+}_{max}(\varepsilon),r^{-}_{max}(\varepsilon)]$ satisfies

     * •

$\ell_{c}(\cdot,\varepsilon^{2})$ is monotonically decreasing on
$]r^{+}_{max}(\varepsilon),r^{-}_{max}(\varepsilon)[$ from
$\displaystyle\frac{\ell_{lb}^{+}(\varepsilon)}{\varepsilon}$ to
$\displaystyle\frac{\ell_{lb}^{-}(\varepsilon)}{\varepsilon}$.

###### Proof.

  1. 1.

If $\varepsilon^{+}_{min}\leq\varepsilon<1$, then we use Lemmas 29 and 31 to
obtain the result.

  2. 2.

If $\varepsilon^{2}\geq 1$, then we use Lemma 30 to obtain the result.

∎

Now, instead of writing the solutions of the system of equations (4.69)-(4.70)
as a two parameter family $(\ell_{c},\eta_{c})(r,\varepsilon^{2})$ indexed by
$(r,\varepsilon^{2})\in\overline{D}_{s}$, we will write the solutions of
(4.69)-(4.70) as a two-parameter family
$(r,\eta_{c})(\varepsilon^{2},\ell_{z})$ indexed by
$(\varepsilon^{2},\ell_{z})\in\overline{D}_{sph}$ where $\overline{D}_{sph}$
is the set of admissible values of $(\varepsilon^{2},\ell_{z})$ which will be
determined. More precisely, we will fix $\varepsilon$ so that $\ell_{z}$
becomes a function of $r$ only, which will be inverted. This will allow us to
write $r$ in terms of $\ell_{c}$. We state the following lemma

######  Lemma 33.

Let $d\in]0,1]$ and let $(\varepsilon,\ell_{z})\in]0,\infty[\times\mathbb{R}$.
The system of equations $\eqref{root:simple}-\eqref{root:double}$ admits
solutions $(r,q)\in]r_{H},\infty[\times[0,\infty[$ if and only if one of the
following cases occur.

  1. 1.

$\varepsilon=\varepsilon^{+}_{min}$ and $\ell_{z}=\ell_{min}^{+}$. In this
case,

$(r,q)=(r_{ms}^{+},0).$ (4.91)

  2. 2.

$\varepsilon^{+}_{min}<\varepsilon\leq\varepsilon^{-}_{min}$ and
$\ell_{z}\in[\tilde{\ell}_{min}(\varepsilon),\ell_{ub}^{+}(\varepsilon)]$.

     * •

If $\ell_{z}\in[\tilde{\ell}_{min}(\varepsilon),\ell_{lb}^{+}(\varepsilon)]$,
there exist two solutions $(r_{s}^{1},q^{1}_{s})(\varepsilon,\ell_{z})$ and
$(r_{s}^{2},q^{2}_{s})(\varepsilon,\ell_{z})$ such that

       * –

$r^{1}_{s}(\varepsilon,\ell_{z})$ lies in the region
$[r_{max}^{+}(\varepsilon),r_{m}(\varepsilon)]$ and is given by

$r^{1}_{s}(\varepsilon,\ell_{z})=\ell_{c}^{-1}\left(\frac{\ell_{z}}{\varepsilon},\varepsilon^{2}\right),$
(4.92)

where $\ell_{c}^{-1}$ is the inverse of the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on
$]r_{max}^{+}(\varepsilon),r_{m}(\varepsilon)[$.

       * –

$r^{2}_{s}(\varepsilon,\ell_{z})$ lies in the region
$[r_{m}(\varepsilon),r_{lb}(\varepsilon)]$ and is given by

$r^{2}_{s}(\varepsilon,\ell_{z})=\ell_{c}^{-1}\left(\frac{\ell_{z}}{\varepsilon},\varepsilon^{2}\right),$
(4.93)

where $\ell_{c}^{-1}$ is the inverse of the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on
$]r_{m}(\varepsilon),r_{lb}(\varepsilon)[$.

       * –

$q^{i}_{s}(\varepsilon,\ell_{z})$ are given by

$q^{i}_{s}(\varepsilon,\ell_{z})=\varepsilon^{2}\eta_{c}(r^{i}_{s}(\varepsilon,\ell_{z}),\varepsilon^{2}).$

     * •

If $\ell_{z}\in[\ell_{lb}^{+}(\varepsilon),\ell_{ub}^{+}(\varepsilon)]$, there
exists a unique $(\tilde{r}^{+},\tilde{q}^{+})(\varepsilon,\ell_{z})$ such
that

       * –

$\tilde{r}^{+}(\varepsilon,\ell_{z})$ lies in the region
$[r_{lb}(\varepsilon,d),r_{min}^{+}(\varepsilon)]$ and is given by

$\tilde{r}^{+}(\varepsilon,\ell_{z})=\ell_{c}^{-1}\left(\frac{\ell_{z}}{\varepsilon},\varepsilon^{2}\right),$
(4.94)

where $\ell_{c}^{-1}$ is the inverse of the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on
$]r_{lb}(\varepsilon),r_{min}^{+}(\varepsilon)[$.

       * –

$\tilde{q}^{+}(\varepsilon,\ell_{z})$ is given by

$\tilde{q}^{+}(\varepsilon,\ell_{z})=\varepsilon^{2}\eta_{c}(\tilde{r}^{+}(\varepsilon,\ell_{z}),\varepsilon^{2}),$

  3. 3.

$\varepsilon^{-}_{min}<\varepsilon<1$ and
$\ell_{z}\in[\ell_{ub}^{-}(\varepsilon),\ell_{ub}^{+}(\varepsilon)]$.

     * •

If $\ell_{z}=\ell_{z}^{+}\geq\ell_{lb}^{+}(\varepsilon)$ or
$\ell_{z}=\ell_{z}^{-}\leq\ell_{lb}^{-}(\varepsilon)$, then there exists a
unique $(\tilde{r}^{\pm},\tilde{q}^{\pm})(\varepsilon,\ell_{z})$ such that

       * –

$\tilde{r}^{\pm}(\varepsilon,\ell_{z})$ lie in the region
$[r^{-}_{min}(\varepsilon),r^{+}_{min}(\varepsilon)]$ and are given by

$\tilde{r}^{\pm}(\varepsilon,\ell_{z})=\ell_{c}^{-1}\left(\frac{\ell_{z}^{\pm}}{\varepsilon},\varepsilon^{2}\right),$
(4.95)

where $\ell_{c}^{-1}$ is the inverse of the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on
$]r^{-}_{min}(\varepsilon),r^{+}_{min}(\varepsilon)[$ and

       * –

$\tilde{q}^{\pm}(\varepsilon,\ell_{z})$ are given by

$\tilde{q}^{\pm}(\varepsilon,\ell_{z}):=\varepsilon^{2}\eta_{c}(\tilde{r}^{\pm}(\varepsilon,\ell_{z},d),\varepsilon^{2}).$
(4.96)

     * •

If $\ell_{z}\in[\ell_{lb}^{-}(\varepsilon),\ell_{lb}^{+}(\varepsilon)]$, there
exist two solutions $(\tilde{r}_{1},\tilde{q}_{1})(\varepsilon,\ell_{z})$ and
$(\tilde{r}_{2},\tilde{q}_{2})(\varepsilon,\ell_{z})$ such that

       * –

$\tilde{r}_{1}(\varepsilon,\ell_{z})$ lies in the region
$[r^{+}_{max}(\varepsilon),r^{-}_{max}(\varepsilon)]$ and is given by

$\tilde{r}_{1}(\varepsilon,\ell_{z})=\ell_{c}^{-1}\left(\frac{\ell_{z}}{\varepsilon},\varepsilon^{2}\right),$
(4.97)

where $\ell_{c}^{-1}$ is the inverse of the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on
$]r^{+}_{max}(\varepsilon),r^{-}_{max}(\varepsilon)[$.

       * –

$\tilde{r}_{2}(\varepsilon,\ell_{z})$ lies in the region
$[r^{-}_{min}(\varepsilon),r^{+}_{min}(\varepsilon)]$ and is given by

$\tilde{r}_{2}(\varepsilon,\ell_{z})=\ell_{c}^{-1}\left(\frac{\ell_{z}}{\varepsilon},\varepsilon^{2}\right),$
(4.98)

where $\ell_{c}^{-1}$ is the inverse of the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on
$]r^{-}_{min}(\varepsilon),r^{+}_{min}(\varepsilon)[$.

       * –

$\tilde{q}_{i}(\varepsilon,\ell_{z})$ are given by

$\tilde{q}_{i}(\varepsilon,\ell_{z}):=\varepsilon^{2}\eta_{c}(\tilde{r}_{i}(\varepsilon,\ell_{z},d),\varepsilon^{2}).$
(4.99)

  4. 4.

$\varepsilon\geq 1$ and
$\ell_{z}\in\left]\ell^{-}_{lb}(\varepsilon,d),\ell^{+}_{lb}(\varepsilon)\right[$.
In this case, there a unique
$(\overline{r},\overline{q})(\varepsilon,\ell_{z})$ such that

     * •

$\overline{r}(\varepsilon,\ell_{z})$ lies in the region
$]r^{+}_{max}(\varepsilon),r^{-}_{max}(\varepsilon)[$ and is given by

$\overline{r}(\varepsilon,\ell_{z})=\ell_{c}^{-1}\left(\frac{\ell_{z}}{\varepsilon},\varepsilon^{2}\right),$
(4.100)

where $\ell_{c}^{-1}$ is the inverse of the restriction of
$\ell_{c}(\cdot,\varepsilon^{2})$ on
$]r^{+}_{max}(\varepsilon),r^{-}_{max}(\varepsilon)[$.

     * •

$\overline{q}(\varepsilon,\ell_{z})$ is given by

$\overline{q}(\varepsilon,\ell_{z})=\varepsilon^{2}\eta_{c}(\overline{r}(\varepsilon,\ell_{z}),\varepsilon^{2})$
(4.101)

and satisfies

$0\leq q\leq q_{max}(\varepsilon^{2}).$

###### Proof.

The proof is straightforward based on the monotonicity properties of
$\ell_{c}(\cdot,\varepsilon^{2})$ on the allowed regions for $r$ given in
Lemma 32. ∎

The previous lemma allows us to introduce the following subsets of
$\mathbb{R}\times\mathbb{R}\times[0,\infty[$:

$\mathcal{A}^{\geq
1}_{spherical}:=\left\\{(\varepsilon,\ell_{z},q)\in]0,\infty[\times\mathbb{R}\times[0,\infty[\;:\;\varepsilon\geq
1\;,\;\ell_{z}\in[\ell_{lb}^{-}(\varepsilon),\ell_{lb}^{+}(\varepsilon)\;:\;q=\overline{q}(\varepsilon,\ell_{z})]\right\\},$
(4.102) $\mathcal{A}^{\leq
1}_{spherical}:=\left\\{(\varepsilon_{min}^{+},\ell_{min}^{+},0)\right\\}\sqcup\mathcal{A}^{\leq
1}_{+}\sqcup\mathcal{A}^{\leq 1}_{-}\sqcup\mathcal{A}^{\leq
1}_{1}\sqcup\mathcal{A}^{\leq 1}_{2},$ (4.103)

where

$\displaystyle\mathcal{A}^{\leq 1}_{1}$
$\displaystyle:=\left\\{\varepsilon\in]\varepsilon^{+}_{min},\varepsilon^{-}_{min}]\;,\,\ell_{z}\in[\tilde{\ell}_{min}(\varepsilon),\ell_{lb}^{+}(\varepsilon)]\;,\;q=q_{s}^{1}(\varepsilon,\ell_{z})\right\\}$
(4.104)
$\displaystyle\sqcup\left\\{\varepsilon\in]\varepsilon^{+}_{min},\varepsilon^{-}_{min}]\;,\,\ell_{z}\in[\tilde{\ell}_{min}(\varepsilon),\ell_{lb}^{+}(\varepsilon)]\;,\;q=q_{s}^{2}(\varepsilon,\ell_{z})\right\\},$

$\displaystyle\mathcal{A}^{\leq 1}_{2}$
$\displaystyle:=\left\\{\varepsilon^{-}_{min}<\varepsilon<1\;,\,\ell_{z}\in[\ell_{lb}^{-}(\varepsilon),\ell_{lb}^{+}(\varepsilon)]\;,\;q=\tilde{q}^{1}(\varepsilon,\ell_{z})\right\\}$
(4.105)
$\displaystyle\sqcup\left\\{\varepsilon^{-}_{min}<\varepsilon<1\;,\,\ell_{z}\in[\ell_{lb}^{-}(\varepsilon),\ell_{lb}^{+}(\varepsilon)]\;,\;q=\tilde{q}^{2}(\varepsilon,\ell_{z})\right\\},$

$\mathcal{A}^{\leq
1}_{+}:=\left\\{\varepsilon\in]\varepsilon^{+}_{min},1[\;,\,\ell_{z}\in[\ell_{lb}^{+}(\varepsilon),\ell_{ub}^{+}(\varepsilon)]\;,\;q=\tilde{q}^{+}(\varepsilon,\ell_{z})\right\\}$
(4.106)

and

$\mathcal{A}^{\leq
1}_{-}:=\left\\{\varepsilon\in]\varepsilon^{-}_{min},1[\;,\,\ell_{z}\in[\ell_{ub}^{-}(\varepsilon),\ell_{lb}^{-}(\varepsilon)]\;,\;q=\tilde{q}^{-}(\varepsilon,\ell_{z})\right\\}.$
(4.107)

Finally, we introduce the set

$\mathcal{A}_{spherical}:=\mathcal{A}^{\geq
1}_{spherical}\sqcup\mathcal{A}^{\leq 1}_{spherical}.$ (4.108)

We conclude that if $\gamma$ is a timelike future directed spherical orbit
with constants of motion $(\varepsilon,\ell_{z},q)$, then

$(\varepsilon,\ell_{z},q)\in\mathcal{A}_{spherical}.$

This end the proof of Proposition 7. Reciprocally, we have

######  Proposition 8.

Let $\gamma:\tau\ni I\to\mathcal{O}$ be a timelike future directed geodesic
with constants of motion $\displaystyle(\varepsilon,\ell_{z},q)$. If
$(\varepsilon,\ell_{z},q)\in\mathcal{A}_{spherical}$ and $\gamma$ starts at
$(t,\phi,r_{s},\theta)$ where $r_{s}$ is determined by one of the cases of
Lemma 32. Then, $\gamma$ is spherical.

####  4.1.4 Roots of the fourth order polynomial $R$

In this section, we will determine the number of solutions in $r$ of the
equation

$R(r,\varepsilon,\ell_{z},q)=0$ (4.109)

in the region $]r_{H},\infty[$ based on the possible values of
$(\varepsilon,\ell_{z},q)$. This will allow us to compute the ZVCs associated
to timelike future-directed geodesics. First, we recall that $R$ is defined by

$R(r,\varepsilon,\ell_{z},q)=(\varepsilon(r^{2}+d^{2})-d\ell_{z})^{2}-(r^{2}-2r+d^{2})(r^{2}+(d\varepsilon-\ell_{z})^{2}+q).$

Let $(\varepsilon,\ell_{z},q)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$.
We will study the existence of roots of the polynomial
$R(\cdot,\varepsilon,\ell_{z},q)$ in the region $(r_{H},\infty)$. First of
all, we note that:

  1. 1.

By Lemma 11, if $q<0$, then $\varepsilon>1$ and the equation $\displaystyle
R(r,\varepsilon,\ell_{z},q)=0$ does not have roots in the region
$(r_{H},\infty)$.

  2. 2.

$\forall(r,\varepsilon,\ell_{z})\in(r_{H},\infty)\times\mathbb{R}\times\mathbb{R},$
the functions $R(r,\varepsilon,\ell_{z},\cdot)$ and $\frac{\partial
R}{\partial r}(r,\varepsilon,\ell_{z},\cdot,d)$ are monotonically decreasing
on $\mathbb{R}$.

  3. 3.

Let
$(\varepsilon,\ell_{z},q)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}_{+}$
and let $r(\varepsilon,\ell_{z},q)\in]r_{H},\infty[$ be a root of
$R(\cdot,\varepsilon,\ell_{z},q)$. Then, if $r$ is a simple root, then it
defines a smooth function of $(\varepsilon,\ell_{z},q)$. Indeed, we have

$\frac{\partial R}{\partial r}(\varepsilon,\ell_{z},q)\neq 0.$

By the implicit function theorem, $r$ is locally a smooth function of
$(\varepsilon,\ell_{z},q)$.

  4. 4.

Finally, we note that in general, the roots of
$R(\cdot,\varepsilon,\ell_{z},q)$ can be parametrised by functions of
$(\varepsilon,\ell_{z},q)$ which are merely continuous when double or triple
roots occur.

Based on the asymptotics of $R(\cdot,\varepsilon,\ell_{z},q)$ (4.31), we will
separate the cases $\varepsilon<1$ and $\varepsilon>1$. First of all, we
classify the roots of $R$ when $q=0$. We state

######  Proposition 9 (Roots of $R$ when $q=0$).

Let $(\varepsilon^{2},\ell_{z})\in[0,\infty[\times\mathbb{R}$, then the roots
of $R(\cdot,\varepsilon,\ell_{z},0)$ are summarised in Tables 2 and 1.

$\varepsilon=\varepsilon_{min}^{+}$ | $\varepsilon_{min}^{+}<\varepsilon<\varepsilon_{min}^{-}$ | $\varepsilon=\varepsilon_{min}^{-}$
---|---|---
$\ell_{z}\in\mathbb{R}$ |  $\ell_{z}=\ell_{min}^{+}$ |  $\ell_{z}\in\mathbb{R}\backslash\ell_{min}^{+}$ |  $\ell_{lb}^{+}(\varepsilon)\leq\ell_{z}\leq\ell_{ub}^{+}(\varepsilon)$ |  $\ell_{z}<\ell_{lb}^{+}(\varepsilon)$ and $\ell_{z}\neq\ell_{min}^{-}$ |  $\ell_{z}=\ell_{min}^{-}$ |  $\ell_{lb}^{+}(\varepsilon)\leq\ell_{z}\leq\ell_{ub}^{+}(\varepsilon)$
One root |  One tripe root $r^{+}_{ms}$ |  One root |  Three roots |  One root |  Three roots |  One triple root $r^{-}_{ms}$
Table 1: Possible roots of $R(\cdot,\varepsilon,\ell_{z},0)$ $\varepsilon<\varepsilon^{+}_{min}$ | $\varepsilon^{2}\geq 1$ | $\varepsilon_{min}^{-}<\varepsilon<1$
---|---|---
$\ell_{z}\in\mathbb{R}$ |  $\ell_{lb}^{-}(\varepsilon)<\ell_{z}<\ell_{lb}^{+}(\varepsilon)$ |  $\quad\ell_{z}\geq\ell_{lb}^{+}(\varepsilon)$ or $\ell_{z}\leq\ell_{lb}^{-}(\varepsilon)$ |  $\ell_{lb}^{-}(\varepsilon)<\ell_{z}<\ell_{lb}^{+}(\varepsilon)$ |  $\ell_{lb}^{+}(\varepsilon)\leq\ell_{z}\leq\ell_{ub}^{+}(\varepsilon)$ or $\ell_{ub}^{-}(\varepsilon)\leq\ell_{z}\leq\ell_{lb}^{-}(\varepsilon)$
One root |  No roots | Two roots |  One root |  Three roots
Table 2: Possible roots of $R(\cdot,\varepsilon,\ell_{z},0)$

###### Proof.

  1. 1.

We will write details for the case $\varepsilon^{2}>1$. The other cases follow
using the same method.

  2. 2.

Assume that $\varepsilon^{2}>1$. We recall that if
$\ell_{z}=\ell_{lb}^{\pm}(\varepsilon)$, then
$R(\cdot,\varepsilon,\ell_{z},0)$ admits a unique double root in the region
$]r_{h}(d),\infty[$ given by $r=r_{max}^{\pm}(\varepsilon)$.

  3. 3.

Now, in view of Lemma 10, it suffices to prove that if
$\ell_{z}\in]-\infty,\ell_{lb}^{-}(\varepsilon)]\cup[\ell_{lb}^{+}(\varepsilon),+\infty[$,
then there exists $r\in]r_{H},\infty[$ such that $\displaystyle
R(r,\varepsilon,\ell_{z},0)\leq 0$. Otherwise, $\displaystyle\forall
r\in]r_{H},\infty[\;,\;R(r,\varepsilon,\ell_{z},0)>0$.

To this end, we claim that $\displaystyle\forall\varepsilon^{2}>1$, the
function $R(r^{+}_{max}(\varepsilon),\varepsilon,\cdot,0)$ is monotonically
decreasing on $[\ell_{lb}^{+}(\varepsilon),+\infty[$ and the function
$R(r^{+}_{max}(\varepsilon),\varepsilon,\cdot,0)$ is monotonically increasing
on $]-\infty,\ell_{lb}^{-}(\varepsilon)]$,

     * •

Indeed, we note the following relation between
$\varepsilon,\ell_{lb}^{\pm}(\varepsilon)$ and $r_{max}^{\pm}(\varepsilon)$:

$\ell_{lb}^{\pm}(\varepsilon)-2\frac{\ell_{lb}^{\pm}(\varepsilon)-d\varepsilon}{r_{max}^{\pm}(\varepsilon)}=\mp\frac{\Delta\left(\frac{1}{r_{max}^{\pm}(\varepsilon)}\right)}{\sqrt{\frac{Q_{\mp}\left(r_{max}^{\pm}(\varepsilon)\right)}{r_{max}^{\pm}(\varepsilon)}}}\quad\text{where}\quad
Q_{\mp}(u):=1-3u\mp 2du^{\frac{3}{2}}.$

     * •

Moreover, we recall that $\forall\varepsilon>1,\forall
d\in[0,1]\;,\;r^{-}_{max}(\varepsilon)=r^{-}_{max}(\varepsilon,d)>2$ and there
exists $0<d_{0}<1$ such that $\forall\varepsilon>1,\forall
d\in[d_{0},1]\;,\;r^{+}_{max}(\varepsilon)=r^{+}_{max}(\varepsilon,d)<2$

     * •

Now, we compute

$\displaystyle\forall\ell_{z}\in\mathbb{R}\;,\;\frac{\partial
R}{\partial\ell_{z}}(r^{\pm}_{max}(\varepsilon),\varepsilon,\ell_{z})$
$\displaystyle=-2r^{\pm}_{max}(\varepsilon)(2d\varepsilon+(r^{\pm}_{max}(\varepsilon)-2)\ell_{z})$
$\displaystyle=-2(r^{\pm}_{max})^{2}(\varepsilon)\left(\ell_{z}-2\frac{\ell_{z}-d\varepsilon}{r_{max}^{\pm}(\varepsilon)}\right)$

Therefore,

       1. (a)

If $\ell_{z}<\ell_{lb}^{-}(\varepsilon)<0$, then

$\frac{\partial
R}{\partial\ell_{z}}(r^{-}_{max}(\varepsilon),\varepsilon,\ell_{z})>2(r^{-}_{max})^{2}\frac{\Delta\left(\frac{1}{r_{max}^{-}(\varepsilon)}\right)}{\sqrt{\frac{Q_{+}\left(r_{max}^{-}(\varepsilon)\right)}{r_{max}^{-}(\varepsilon)}}}>0.$

       2. (b)

If $\ell_{z}>\ell_{lb}^{+}(\varepsilon)>0$, then

          * –

if $r_{max}^{+}(\varepsilon)\geq 2$, then $\displaystyle\frac{\partial
R}{\partial\ell_{z}}(r^{+}_{max}(\varepsilon),\varepsilon,\ell_{z})<0$.

          * –

Otherwise, we have

$\frac{\partial
R}{\partial\ell_{z}}(r^{\pm}_{max}(\varepsilon),\varepsilon,\ell_{z})<-2(r^{+}_{max})^{2}\frac{\Delta\left(\frac{1}{r_{max}^{+}(\varepsilon)}\right)}{\sqrt{\frac{Q_{-}\left(r_{max}^{+}(\varepsilon)\right)}{r_{max}^{-}(\varepsilon)}}}<0.$

Hence,

     * •

For $\ell_{z}>\ell_{lb}^{+}(\varepsilon)$, we have $\displaystyle
R(r^{+}_{max}(\varepsilon),\varepsilon,\ell_{z},0)<R(r^{+}_{max}(\varepsilon),\varepsilon,\cdot,\ell_{lb}^{+}(\varepsilon))=0$.

     * •

For $\ell_{z}<\ell_{lb}^{-}(\varepsilon)$, we have $\displaystyle
R(r^{-}_{max}(\varepsilon),\varepsilon,\ell_{z},0)<R(r^{-}_{max}(\varepsilon),\varepsilon,\cdot,\ell_{lb}^{-}(\varepsilon))=0$.

This ends the proof.

∎

Now, we discuss the general case.

##### 4.1.4.1 Case $\varepsilon^{2}\geq 1$

######  Proposition 10.

Let $\varepsilon^{2}>1$. The possible number of roots of
$R(\cdot,\varepsilon,\ell_{z},q)$ are summarised in Table 3.

$\varepsilon^{2}\geq 1$
---
$\ell_{lb}^{-}(\varepsilon)<\ell_{z}<\ell_{lb}^{+}(\varepsilon)$ |  $\quad\ell_{z}\geq\ell_{lb}^{+}(\varepsilon)$ or $\ell_{z}\leq\ell_{lb}^{-}(\varepsilon)$
$q\geq\overline{q}(\varepsilon,\ell_{z})$ | $q<\overline{q}(\varepsilon,\ell_{z})$ | $q\geq 0$ | $q<0$
Two roots | No roots | Two roots | No roots
Table 3: Possible roots or $R$ in the unbounded case

###### Proof.

Let $(\varepsilon,\ell_{z},q)\in[1,\infty[\times\mathbb{R}\times\mathbb{R}$
and consider the equation in $r$

$R(r,\varepsilon,\ell_{z},q)=0.$ (4.110)

By Lemma 10, $R(\cdot,\varepsilon,\ell_{z},q)$ admits either zero roots or two
roots in the region $]r_{H},\infty[$. Moreover, by Lemma 11, if $q<0$, then
$R(\cdot,\varepsilon,\ell_{z})$ has no roots. Now, assume that $q\geq 0$.
Since $R(r,\varepsilon,\ell_{z},\cdot)$ is monotonically decreasing on
$\mathbb{R}$, we have: if $r_{0}(\varepsilon,\ell_{z},0)$ is a root of
$R(\cdot,\varepsilon,\ell_{z},0)$, then

$\forall
q>0\;:\;R(r_{0}(\varepsilon,\ell_{z},0),\varepsilon,\ell_{z},q)<R(r_{0}(\varepsilon,\ell_{z},0),\varepsilon,\ell_{z},0)=0.$

Hence, in view of the asymptotics, $R(\cdot,\varepsilon,\ell_{z},0)$ admits
two roots.
Now, by lemma 9 $R(\cdot,\varepsilon,\ell_{z},0)$ admits two roots if and only
if $\ell_{z}\geq\ell_{lb}^{+}(\varepsilon)$ or
$\ell_{z}\leq\ell_{lb}^{-}(\varepsilon)$. Hence, if
$\ell_{z}>\ell_{lb}^{+}(\varepsilon)$ or
$\ell_{z}<\ell_{lb}^{-}(\varepsilon)$, $R(\cdot,\varepsilon,\ell_{z},q)$
admits two simple roots for all $q\geq 0$. Moreover, if $q=0$ and
$\ell_{z}=\ell_{lb}^{\pm}(\varepsilon)$, then the roots coincide.
Now, assume that
$\ell_{z}\in]\ell_{lb}^{-}(\varepsilon),\ell_{lb}^{+}(\varepsilon)[$. Then, by
Proposition 7, $R(\cdot,\varepsilon,\ell_{z})$ admits a double root if and
only if $q=\overline{q}(\varepsilon,\ell_{z},d)$ and it is given by
$\overline{r}(\varepsilon,\ell_{z})$. By the monotonicity properties of
$R(r,\varepsilon,\ell_{z},\cdot)$, we have

$\forall
q>\overline{q}(\varepsilon,\ell_{z})\;,\;R(\overline{r}(\varepsilon,\ell_{z}),\varepsilon,\ell_{z},q)<R(\overline{r}(\varepsilon,\ell_{z}),\varepsilon,\ell_{z},\overline{q}(\varepsilon,\ell_{z}))=0$

Now assume that $q<\overline{q}(\varepsilon,\ell_{z})$. Then,

$\forall
r>r_{H}\;:\;R(r,\varepsilon,\ell_{z},q)>R(r,\varepsilon,\ell_{z},\overline{q}(\varepsilon,\ell_{z}))\geq
0.$

Therefore, $R$ has no roots. This ends the proof. ∎

##### 4.1.4.2 Case $\varepsilon^{2}<1$

######  Proposition 11.

Assume that
$(\varepsilon,\ell_{z},q)\in]-1,1[\times\mathbb{R}\times[0,\infty[$. The
possible number of roots of $R(\cdot,\varepsilon,\ell_{z},q)$ are summarised
in Table 4 and Table 5.

$0<\varepsilon<\varepsilon^{+}_{min}$ | $\varepsilon^{+}_{min}\leq\varepsilon\leq\varepsilon^{-}_{min}$
---|---
$\ell_{z}\in\mathbb{R}$ | $\ell_{z}<\tilde{\ell}_{min}(\varepsilon)$ | $\tilde{\ell}_{min}(\varepsilon)\leq\ell_{z}\leq\ell_{lb}^{+}(\varepsilon)$ | $\ell_{lb}^{+}(\varepsilon)\leq\ell_{z}\leq\ell_{ub}^{+}(\varepsilon)$
$q\geq 0$ |  $q\geq 0$ |  $0\leq q<q_{s}^{1}(\varepsilon,\ell_{z})$ |  $q_{s}^{1}(\varepsilon,\ell_{z})\leq q\leq q_{s}^{2}(\varepsilon,\ell_{z})$ |  $q>q_{s}^{2}(\varepsilon,\ell_{z})$ |  $0\leq q\leq\tilde{q}^{+}(\varepsilon,\ell_{z})$ |  $q>\tilde{q}^{+}(\varepsilon,\ell_{z})$
One root |  One root |  One root |  Three roots |  One root |  Three roots |  One root
$r_{abs}^{K}(\varepsilon,\ell_{z},q)$ |  $r_{abs}^{K}(\varepsilon,\ell_{z},q)$ |  $r_{abs}^{1}(\varepsilon,\ell_{z},q)$ |  $r_{abs}^{i}(\varepsilon,\ell_{z},q)\;,i\in\left\\{1,2,3\right\\}$ |  $r_{abs}^{3}(\varepsilon,\ell_{z},q)$ |  $r_{i}^{K}(\varepsilon,\ell_{z},q)\;,\;i\in\left\\{0,1,2\right\\}$ |  $r_{abs}^{0}(\varepsilon,\ell_{z},q)$
Table 4: Possible roots or $R$ in the bounded case
$\varepsilon^{-}_{min}<\varepsilon<1$
---
$\ell_{lb}^{-}(\varepsilon)\leq\ell_{z}\leq\ell_{lb}^{+}(\varepsilon)$ |  $\ell_{lb}^{+}(\varepsilon)\leq\ell_{z}\leq\ell_{ub}^{+}(\varepsilon)$ or $\ell_{ub}^{-}(\varepsilon)\leq\ell_{z}\leq\ell_{lb}^{-}(\varepsilon)$
$0\leq q<\tilde{q}^{1}(\varepsilon,\ell_{z})$ |  $\tilde{q}^{1}(\varepsilon,\ell_{z})$ $\leq q\leq\tilde{q}^{2}(\varepsilon,\ell_{z})$ |  $q>\tilde{q}^{2}(\varepsilon,\ell_{z})$ |  $0\leq q\leq\tilde{q}^{\pm}(\varepsilon,\ell_{z})$ |  $q>\tilde{q}^{\pm}(\varepsilon,\ell_{z})$
$r_{0}^{K}(\varepsilon,\ell_{z},q)$ |  $r_{i}^{K}(\varepsilon,\ell_{z},q)\;,i\in\left\\{0,1,2\right\\}$ |  $r_{0}^{K}(\varepsilon,\ell_{z},q)$ |  $r_{i}^{K}(\varepsilon,\ell_{z},q)\;,i\in\left\\{0,1,2\right\\}$ |  $r_{0}^{K}(\varepsilon,\ell_{z},q)$
Table 5: Possible roots or $R$ in the bounded case

###### Proof.

Let $(\varepsilon,\ell_{z},q)\in]0,1[\times\mathbb{R}\times[0,\infty[$. By
Lemma 10, $R$ admits either one root or three roots in the region
$]r_{H},\infty[$. Note also that $\partial_{r}R$ has either no roots or two
roots in the region$]r_{H},\infty[$.
By Lemma 9, $R(\cdot,\varepsilon,\ell_{z},0)$ admits three roots if and only
if $\varepsilon_{min}^{+}<\varepsilon<1$ and
$\ell_{z}\in[\ell_{lb}^{+}(\varepsilon),\ell_{ub}^{+}(\varepsilon)]$ or
$\varepsilon_{min}^{-}(d)<\varepsilon<1$ and
$\ell_{z}\in[\ell_{ub}^{-}(\varepsilon),\ell_{lb}^{-}(\varepsilon)]$. Hence,

  1. 1.

If $\varepsilon<\varepsilon_{min}^{+}$, then $\forall\ell_{z}\in\mathbb{R}$,
$R(\cdot,\varepsilon,\ell_{z},0)$ admits a unique root and
$\displaystyle\frac{\partial R}{\partial r}(\cdot,\varepsilon,\ell_{z},0)<0$.
Therefore,

$\forall q\geq 0\;:\;\frac{\partial R}{\partial
r}(\cdot,\varepsilon,\ell_{z},q)\leq\frac{\partial R}{\partial
r}(\cdot,\varepsilon,\ell_{z},0)<0.$

Hence, $R(\cdot,\varepsilon,\ell_{z},q)$ admits a unique root.

  2. 2.

If $\varepsilon_{min}^{+}(d)\leq\varepsilon<\varepsilon_{min}^{-}(d)$

    1. (a)

If $\ell_{z}\in[\ell_{lb}^{+}(\varepsilon),\ell_{ub}^{+}(\varepsilon)]$. Then,
$R(\cdot,\ell_{z},0)$ admits three roots and $\displaystyle\frac{\partial
R}{\partial r}(\cdot,\varepsilon,\ell_{z},0)$ admits two roots, denoted by
$\underline{r}_{i}(\varepsilon,\ell_{z})$.

       * •

If $q=\tilde{q}^{+}(\varepsilon,\ell_{z})$, then
$R(\cdot,\varepsilon,\ell_{z},q)$ admits a double root given by
$\tilde{r}^{+}(\varepsilon,\ell_{z})$.

       * •

By monotonicity properties of $\partial_{r}R(r,\varepsilon,\ell_{z},\cdot)$,
we have

$\forall q<\tilde{q}^{+}(\varepsilon,\ell_{z})\;,\;\frac{\partial R}{\partial
r}(\tilde{r}^{+}(\varepsilon,\ell_{z}),\varepsilon,\ell_{z},q)>\frac{\partial
R}{\partial
r}(\tilde{r}^{+}(\varepsilon,\ell_{z}),\varepsilon,\ell_{z},\tilde{q}^{+}(\varepsilon,\ell_{z}))=0.$

Thus, $\displaystyle\frac{\partial R}{\partial
r}(\cdot,\varepsilon,\ell_{z},q)$ admits two roots. This yields to three roots
for $R(\cdot,\ell_{z},q)$.

       * •

We have,

$\forall q>\tilde{q}^{+}(\varepsilon,\ell_{z})\;,\;\forall
r>r_{H}\;,\;\frac{\partial R}{\partial
r}(\tilde{r}^{+}(,\varepsilon,\ell_{z},q)<\frac{\partial R}{\partial
r}(r,\varepsilon,\ell_{z},\tilde{q}^{+}(\varepsilon,\ell_{z}))\leq 0.$

The latter inequality is due to the fact that
$\tilde{r}^{+}(\varepsilon,\ell_{z})$ is a global maximum for
$\displaystyle\frac{\partial R}{\partial
r}(\cdot,\varepsilon,\ell_{z},\tilde{q}^{+}(\varepsilon,\ell_{z}))$. Hence,
$\displaystyle\frac{\partial R}{\partial r}(\cdot,\varepsilon,\ell_{z},q)$
does not change sign. In this case, $R(\cdot,\varepsilon,\ell_{z},q)$ admits
only one root.

    2. (b)

The remaining cases $\ell_{z}\in]0,\tilde{\ell}^{+}_{min}(\varepsilon)[$ and
$\ell_{z}\in[\tilde{\ell}^{+}_{min}(\varepsilon),\ell^{+}_{lb}(\varepsilon)[$
follow using similar arguments.

  3. 3.

We use similar arguments for the remaining case
$\varepsilon^{-}_{min}\leq\varepsilon<1$

∎

####  4.1.5 Roots of the fourth order polynomial $T$

In this section, we recall the solutions of

$T(Y,\varepsilon,\ell_{z},q)=0$ (4.111)

in the region $]-1,1[$ at a given
$(\varepsilon,\ell_{z},q)\in]0,\infty[\times\mathbb{R}\times\mathbb{R}$.

  1. 1.

If $q\geq 0$, then by Lemma 13, $T$ admits two roots given by (4.34) which
coincide if and only if $q=0$.

  2. 2.

Otherwise, $\varepsilon^{2}>1$. Then $T$ admits two distinct roots
$\mu_{\pm}(\varepsilon,\ell_{z},q)$ in the region $]-1,1[$ which satisfy

$-1<\mu_{-}<0<\mu_{+}<1$

and are given by

$\cos\mu_{\pm}(\varepsilon,\ell_{z},q)=\pm\frac{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q+\sqrt{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q)^{2}-4qd^{2}(1-\varepsilon^{2})})}{2d^{2}(1-\varepsilon^{2})}$

We end this section by defining the following angles:

  * •

$\theta_{1}(\varepsilon,\ell_{z})\in\left(0,\frac{\pi}{2}\right]$ defined on
the domain
$\displaystyle\left\\{(\varepsilon,\ell_{z})\;:\;\varepsilon_{min}^{+}<\varepsilon<\varepsilon_{min}^{-}\>,\;\ell_{z}\in]\ell_{min}^{+},\ell_{lb}^{+}(\varepsilon)[\right\\}$
by the following expression:

$\cos\theta_{1}(\varepsilon,\ell_{z}):=\frac{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q^{1}_{s}(\varepsilon,\ell_{z})-\sqrt{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+q^{1}_{s}(\varepsilon,\ell_{z}))^{2}-4q^{1}_{s}(\varepsilon,\ell_{z})d^{2}(1-\varepsilon^{2})})}{2d^{2}(1-\varepsilon^{2})},$
(4.112)

  * •

$\tilde{\theta}_{1}(\varepsilon,\ell_{z})\in\left(0,\frac{\pi}{2}\right]$
defined on the domain
$\displaystyle\left\\{(\varepsilon,\ell_{z})\;:\;\varepsilon_{min}^{-}<\varepsilon<1\>,\;\ell_{z}\in]\ell_{lb}^{-}(\varepsilon),\ell_{lb}^{+}(\varepsilon)[\right\\}$
by the following expression:

$\cos\tilde{\theta}_{1}(\varepsilon,\ell_{z}):=\frac{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+\tilde{q}^{1}(\varepsilon,\ell_{z})-\sqrt{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+\tilde{q}^{1}(\varepsilon,\ell_{z}))^{2}-4\tilde{q}^{1}(\varepsilon,\ell_{z})d^{2}(1-\varepsilon^{2})})}{2d^{2}(1-\varepsilon^{2})},$
(4.113)

  * •

$\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\in\left(0,\frac{\pi}{2}\right)$
defined on the domain
$\displaystyle\left\\{(\varepsilon,\ell_{z})\;:\;\varepsilon_{min}^{+}<\varepsilon\leq\varepsilon_{min}^{-}\>,\;\ell_{z}\in]\ell_{lb}^{+}(\varepsilon),\ell_{ub}^{+}(\varepsilon)[\right\\}$
by the following expression:

$\cos\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z}):=\frac{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+\tilde{q}^{+}(\varepsilon,\ell_{z})-\sqrt{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+\tilde{q}^{+}(\varepsilon,\ell_{z}))^{2}-4\tilde{q}^{+}(\varepsilon,\ell_{z})d^{2}(1-\varepsilon^{2})})}{2d^{2}(1-\varepsilon^{2})},$
(4.114)

  * •

$\overline{\theta}_{max}^{\geq
1}(\varepsilon,\ell_{z})\in\left(0,\frac{\pi}{2}\right)$ defined on the domain
$\displaystyle\left\\{(\varepsilon,\ell_{z})\;:\;\varepsilon>1\>,\;\ell_{z}\in]\ell_{lb}^{-}(\varepsilon),\ell_{lb}^{+}(\varepsilon)[\right\\}$
by the following expression:

$\cos\overline{\theta}_{max}^{\geq
1}(\varepsilon,\ell_{z}):=\frac{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+\overline{q}(\varepsilon,\ell_{z})+\sqrt{(\ell_{z}^{2}+d^{2}(1-\varepsilon^{2})+\overline{q}(\varepsilon,\ell_{z}))^{2}-4\overline{q}(\varepsilon,\ell_{z})d^{2}(1-\varepsilon^{2})})}{2d^{2}(1-\varepsilon^{2})},$
(4.115)

######  Remark 18.

A timelike geodesic with negative $q$ must either in the region
$]r_{H},\infty[\times]\arccos{\mu_{-}(\varepsilon,\ell_{z})},\pi[$ or in the
region $]r_{H},\infty[\times]0,\arccos{\mu_{+}(\varepsilon,\ell_{z})},[$

####  4.1.6 Stationary solutions of the geodesic equation

In this section, we compute stationary solutions corresponding to the free
particle (future directed) Hamiltonian moving in the exterior region of a Kerr
spacetime. This will allow us to analyze the geodesic motion in Weyl
coordinates. See Section 4.2.
We state the main result of this section

######  Lemma 34.

Let $\gamma:I\to\mathcal{O}$ be a solution of (3.4) with constants of motion
$(\varepsilon,\ell_{z},q)$ such that $\forall\tau\in
I\;,\;(\gamma,\dot{\gamma})(\tau)\in\Gamma$.

  * •

The system (3.4) admits direct stationary solutions
$(\gamma_{s},\dot{\gamma}_{s})=(x_{s},v_{s})$ if and only if
$\varepsilon\geq\varepsilon^{+}_{min}$, $d\ell_{z}>0$ and $q=0$. In this case,

    1. 1.

$(r_{s},\theta_{s})$ is given by

$\theta_{s}=\frac{\pi}{2}$

and

       * –

if $\varepsilon^{2}\geq 1$, $r_{s}=r^{+}_{max}(\varepsilon)$,

       * –

if $\varepsilon^{2}<1$,
$r_{s}\in\left\\{r^{+}_{min}(\varepsilon),r^{+}_{max}(\varepsilon)\right\\}$,

where $r^{+}_{max}(\varepsilon)$ and $r^{+}_{min}(\varepsilon)$ are given by
Lemma 21.

    2. 2.

Moreover, $\ell_{z}$ is given by

       * –

if $\varepsilon^{2}\geq 1$, $\ell_{z}=\ell^{+}_{lb}(\varepsilon)$

       * –

if $\varepsilon^{2}<1$, $\ell_{z}=\ell^{+}_{lb}(\varepsilon)$ if
$r_{s}=r^{+}_{max}(\varepsilon)$ and $\ell_{z}=\ell^{+}_{ub}(\varepsilon)$ if
$r_{s}=r^{+}_{min}(\varepsilon),$

where $\ell_{lb}^{+}(\varepsilon)$ and $\ell_{ub}^{+}(\varepsilon)$ are
defined in (4.60). Furthermore, we have a lower bound on $\ell_{z}$:

$\ell_{z}\geq\ell^{+}_{min}.$

  * •

The system (3.4) admits retrograde stationary solutions
$(\gamma_{s},\dot{\gamma}_{s})=(x_{s},v_{s})$ if and only if
$\varepsilon\geq\varepsilon^{-}_{min}$, $d\ell_{z}<0$ and $q=0$. In this case,

    1. 1.

$(r_{s},\theta_{s})$ is given by

$\theta_{s}=\frac{\pi}{2}$

and

       * –

if $\varepsilon^{2}\geq 1$, $r_{s}=r^{-}_{max}(\varepsilon)$,

       * –

if $\varepsilon^{2}<1$,
$r_{s}\in\left\\{r^{-}_{min}(\varepsilon),r^{-}_{max}(\varepsilon)\right\\}$.

    2. 2.

Moreover, $\ell_{z}$ is given by

       * –

if $\varepsilon^{2}\geq 1$, $\ell_{z}=\ell^{-}_{lb}(\varepsilon)$

       * –

if $\varepsilon^{2}<1$, $\ell_{z}=\ell^{-}_{lb}(\varepsilon)$ if
$r_{s}=r^{-}_{max}(\varepsilon)$ and $\ell_{z}=\ell^{-}_{ub}(\varepsilon)$ if
$r_{s}=r^{-}_{min}(\varepsilon)$.

Furthermore, we have an upper bound on $\ell_{z}$:

$\ell_{z}\leq\ell^{-}_{min}.$

###### Proof.

The proof follows from the previous results:

  1. 1.

By Lemma 7, stationary solutions $(x_{s},v_{s})$ verify
$v_{s}^{r}=v^{\theta}_{s}=0$, $r_{s}$ is a double root of
$R(\cdot,\varepsilon,\ell_{z},q)$ and $\cos\theta_{s}$ is a double root of
$T(\cdot,\varepsilon,\ell_{z},q)$.

  2. 2.

Now, by Lemma 13, $q=0$ and $\theta_{s}=\frac{\pi}{2}$.

  3. 3.

Therfore, stationary solutions of (3.4) are the circular orbits confined in
the equatorial plane.

  4. 4.

We conclude using Proposition 5 and Proposition 6.

∎

####  4.1.7 Classification of timelike future directed geodesics

In this section, we classify the timelike future directed geodesics according
to the roots of $R(\cdot,\varepsilon,\ell_{z},q)$. First of all, we determine
$Z^{K}(\varepsilon,\ell_{z})$, which is a curve in the $(r,\theta)-$plane with
possibly different connected components, associated to a timelike future
directed geodesic with constants of motion $(\varepsilon,\ell_{z})$. Moreover,
the condition (4.30) yields restrictions on the initial position
$(r(0),\theta(0))$. The classification is based on the possible values of
$(\varepsilon,\ell_{z},q)$ and $(r(0),\theta(0))$.
Let $\gamma:I\ni 0\to\mathcal{O}$ be a timelike future-directed geodesic with
constants of motion $(\varepsilon,\ell_{z})\in\mathbb{R}\times\mathbb{R}$ and
$\tilde{\gamma}(\tau):=(r(\tau),\theta(\tau))$ be its projection in the
$(r,\theta)$-plane. We recall that by separability of the geodesics equation,
the radial motion decouples from the motion in $\theta$ direction. Moreover,
$\tilde{\gamma}$ satisfies the system of equations

$\displaystyle\Sigma^{4}\dot{r}^{2}$
$\displaystyle=R(r,\varepsilon,\ell_{z},q)$
$\displaystyle\Sigma^{4}\sin^{2}\theta\dot{\theta}^{2}$
$\displaystyle=T(\cos\theta,\varepsilon,\ell_{z},q)$

where $q\in\mathbb{R}$ is the Carter constant, the fourth integral of motion
associated to $\gamma$. In order to determine $Z^{K}(\varepsilon,\ell_{z})$,
we will use $q$ as a parameter for the curve $Z^{K}(\varepsilon,\ell_{z})$. We
state the first result of this section concerning the classification of
$Z^{K}(\varepsilon,\ell_{z})$ associated to $\gamma$.

######  Proposition 12.

(Shape of Zero-velocity curves) Let $d\in]0,1[$ and let
$\gamma:I\to\mathcal{O}$ be a timelike future directed geodesic with constants
of motion $(\varepsilon,\ell_{z})\in\mathbb{R}\times\mathbb{R}$.

  1. 1.

If $0<\varepsilon\leq\varepsilon^{+}_{min}$,

    1. (a)

If $\varepsilon=\varepsilon^{+}_{min}$ and $\ell_{z}=\ell_{min}^{+}$,
$Z^{K}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{R}$ with a singular
point at $\displaystyle\left(r^{+}_{ms},\frac{\pi}{2}\right)$. (See Figure 3)

    2. (b)

Otherwise, $Z^{K}(\varepsilon,\ell_{z})$ is a smooth curve diffeomorphic to
$\mathbb{R}$ with the following properties: it is symmetric with respect to
the equatorial plane and intersects it at a unique point
$r_{0}^{K}(\varepsilon,\ell_{z},0)$, the unique root of the polynomial
$\displaystyle R(\cdot,\varepsilon,\ell_{z},0)$.

  2. 2.

If $\varepsilon^{+}_{min}<\varepsilon\leq\varepsilon^{-}_{min}$,

    1. (a)

If $\varepsilon=\varepsilon^{-}_{min}$ and $\ell_{z}=\ell_{min}^{-}$,
$Z^{K}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{R}$ with a singular
point at $\displaystyle\left(r^{-}_{ms},\frac{\pi}{2}\right)$.

    2. (b)

Otherwise,

      1. i.

If $\ell_{z}<\ell_{lb}^{+}(\varepsilon)$, then $Z^{K}(\varepsilon,\ell_{z})$
is a smooth curve diffeomorphic to $\mathbb{R}$ with the following properties:
it is symmetric with respect to the equatorial plane and intersects it at a
unique point $r_{2}^{K}(\varepsilon,\ell_{z},0)$, the unique root of the
polynomial $\displaystyle R(\cdot,\varepsilon,\ell_{z},0)$. (See Figure 4)

      2. ii.

If $\ell_{z}=\ell_{lb}^{+}(\varepsilon)$, then $Z^{K}(\varepsilon,\ell_{z})$
is a self-intersecting smooth curve with the following properties: it is
symmetric with respect to the equatorial plane and intersects it at two points
$r_{0}^{K}(\varepsilon,\ell_{z},0)<r_{2}^{K}(\varepsilon,\ell_{z},0)$, the
roots of the polynomial $\displaystyle R(\cdot,\varepsilon,\ell_{z},0)$.
Moreover, $Z^{K}(\varepsilon,\ell_{z})$ admits one singular point
$\displaystyle\left(r^{K}_{0}(\varepsilon,\ell_{z},0),\frac{\pi}{2}\right)$
where it self-intersects. (See Figure 6)

      3. iii.

If $\ell_{lb}^{+}(\varepsilon)<\ell_{z}<\ell_{ub}^{+}(\varepsilon)$, then
$Z^{K}(\varepsilon,\ell_{z})$ is a smooth curve with two connected components:
$Z^{K,trapped}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{S}^{1}$ and
$Z^{K,abs}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{R}$. The latter
curves are symmetric with respect to the equatorial plane.
$Z^{K,trapped}(\varepsilon,\ell_{z})$ intersects it at
$r_{1}^{K}(\varepsilon,\ell_{z},0)$ and $r_{2}^{K}(\varepsilon,\ell_{z},0)$
and $Z^{K,abs}(\varepsilon,\ell_{z})$ intersects it at
$r^{K}_{0}(\varepsilon,\ell_{z},0)$.

      4. iv.

$\ell_{z}=\ell_{ub}^{+}(\varepsilon)$, then $Z^{K}(\varepsilon,\ell_{z})$ is a
smooth curve with two connected components: the point
$\displaystyle\left(r_{2}^{K}(\varepsilon,\ell_{z},0),\frac{\pi}{2}\right)$
and $Z^{K,abs}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{R}$. The
latter curve is symmetric with respect to the equatorial plane and intersects
it at $r_{0}^{K}(\varepsilon,\ell_{z},0)$. (See Figure 8)

  3. 3.

If $\varepsilon^{-}_{min}<\varepsilon<1$,

    1. (a)

If $\ell_{lb}^{-}(\varepsilon)<\ell_{z}<\ell_{lb}^{+}(\varepsilon)$, then
$Z^{K}(\varepsilon,\ell_{z})$ is a smooth curve diffeomorphic to $\mathbb{R}$
with the following properties: it is symmetric with respect to the equatorial
plane and intersects it at a unique point $r_{2}^{K}(\varepsilon,\ell_{z},0)$.

    2. (b)

If $\ell_{z}=\ell_{lb}^{+}(\varepsilon)$ or
$\ell_{z}=\ell_{lb}^{-}(\varepsilon)$, then $Z^{K}(\varepsilon,\ell_{z})$ is a
smooth self-intersection curve with the following properties: it is symmetric
with respect to the equatorial plane and intersects it at two points
$r_{0}^{K}(\varepsilon,\ell_{z},0)<r_{2}^{K}(\varepsilon,\ell_{z},0)$.
Moreover, $Z^{K}(\varepsilon,\ell_{z})$ admits one singular point
$\displaystyle\left(r^{K}_{0}(\varepsilon,\ell_{z},0),\frac{\pi}{2}\right)$
where it self-intersects.

    3. (c)

If $\ell_{lb}^{+}(\varepsilon)<\ell_{z}<\ell_{ub}^{+}(\varepsilon)$ or
$\ell_{ub}^{-}(\varepsilon)<\ell_{z}<\ell_{lb}^{-}(\varepsilon)$, then
$Z^{K}(\varepsilon,\ell_{z})$ is a smooth curve with two connected components:
$Z^{K,trapped}(\varepsilon,\ell_{z})$ is diffeomorphic to $S^{1}$ and
$Z^{K,abs}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{R}$. The latter
curves are symmetric with respect to the equatorial plane.
$Z^{K,trapped}(\varepsilon,\ell_{z})$ intersects it at
$r_{1}^{K}(\varepsilon,\ell_{z},0)$ and $r_{2}^{K}(\varepsilon,\ell_{z},0)$
and $Z^{K,abs}(\varepsilon,\ell_{z})$ intersects it at
$r^{K}_{0}(\varepsilon,\ell_{z},0)$.

    4. (d)

If $\ell_{z}=\ell_{ub}^{+}(\varepsilon)$ or
$\ell_{z}=\ell_{ub}^{-}(\varepsilon)$, then $Z^{K}(\varepsilon,\ell_{z})$ is a
smooth curve with two connected components: the point
$\displaystyle\left(r_{2}^{K}(\varepsilon,\ell_{z},0),\frac{\pi}{2}\right)$
and $Z^{K,abs}(\varepsilon,\ell_{z})$ is diffeomorphic to $\mathbb{R}$. The
latter curve is symmetric with respect to the equatorial plane and intersects
it at $r_{0}^{K}(\varepsilon,\ell_{z},0)$.

  4. 4.

If $\varepsilon>1$

    1. (a)

If $\ell_{lb}^{-}(\varepsilon)<\ell_{z}<\ell_{lb}^{+}(\varepsilon)$, then
$Z^{K}(\varepsilon,\ell_{z})$ is a smooth curve with two connected components
$Z^{K,z>0}(\varepsilon,\ell_{z})$ and $Z^{K,z<0}(\varepsilon,\ell_{z})$ which
are diffeomorphic to $\mathbb{R}$ and which do not intersect the equatorial
plane. $Z^{K}(\varepsilon,\ell_{z})$ is symmetric with respect to the
equatorial plane. (See Figure 5)

    2. (b)

If $\ell_{z}=\ell_{lb}^{+}(\varepsilon)$ or
$\ell_{z}=\ell_{lb}^{-}(\varepsilon)$, then $Z^{K}(\varepsilon,\ell_{z})$ has
the following properties: It consists of the union of two connected curves
diffeomorphic to $\mathbb{R}$ which intersect at the point
$\displaystyle\left(r_{0}^{K}(\varepsilon,\ell_{z},0),\frac{\pi}{2}\right)$.
They are symmetric with respect to the equatorial plane and intersect at the
point $r_{0}^{K}(\varepsilon,\ell_{z},0)$ . (See Figure 7)

    3. (c)

If $\ell_{z}>\ell_{lb}^{+}(\varepsilon)$ or
$\ell_{z}<\ell_{lb}^{-}(\varepsilon)$, then $Z^{K}(\varepsilon,\ell_{z})$ is a
smooth curve with two connected components: $Z^{K,scat}(\varepsilon,\ell_{z})$
is diffeomorphic to $\mathbb{R}$ and $Z^{K,abs}(\varepsilon,\ell_{z})$ is
diffeomorphic to $\mathbb{R}$. The latter curves are symmetric with respect to
the equatorial plane. $Z^{K,scat}(\varepsilon,\ell_{z})$ intersects it at
$r_{1}^{K}(\varepsilon,\ell_{z},0)$ and $Z^{K,abs}(\varepsilon,\ell_{z})$
intersects it at $r^{K}_{0}(\varepsilon,\ell_{z},0)$. (See Figure 9)

###### Proof.

Let $\gamma:I\to\mathcal{O}$ be a timelike future directed geodesic with
$(\varepsilon,\ell_{z})\in\mathcal{A}^{admissible}$. Let $(r,\theta)\in
Z^{K}(\varepsilon,\ell_{z})$. By Lemma 8, we have

$R(r,\varepsilon,\ell_{z},q(\theta,\varepsilon,\ell_{z}))=0,$ (4.116)

where

$q(\theta,\varepsilon,\ell_{z})=\cos^{2}\theta\left(d^{2}(1-\varepsilon^{2})+\frac{\ell_{z}^{2}}{\sin^{2}\theta}\right).$
(4.117)

$q(\cdot,\varepsilon,\ell_{z})$ is monotonically decreasing on
$\displaystyle\left(0,\frac{\pi}{2}\right)$ from $\infty$ to $0$ if
$\ell_{z}\neq 0$ and from $d^{2}(1-\varepsilon^{2})$ to $0$ if $\ell_{z}=0$
and monotonically increasing on $\displaystyle\left(\frac{\pi}{2},\pi\right)$
from $0$ to $\infty$.

By the definition of $R$ and (4.16), it is easy to express $q$ in terms of the
remaining variables:

$q=\overline{q}(r):=\frac{((r^{2}+d^{2})\varepsilon-d\ell_{z})^{2}}{\Delta}-(r^{2}+(\ell_{z}-d\varepsilon)^{2}).$
(4.118)

Moreover, and $\forall r\in]r_{H},\infty[$

$\frac{\partial q}{\partial
r}(r,\varepsilon,\ell_{z})=\Delta^{-1}(r)\frac{\partial R}{\partial
r}(r,\varepsilon,\ell_{z},\overline{q}(r,\varepsilon,\ell_{z})).$

In the following, we will use (4.117) and (4.118) in order to eliminate $q$
from the equations. As a consequence, either $\theta$ will be seen as a
function of $r$ or $r$ as a function of $\theta$. This will determine
$Z^{K}(\varepsilon,\ell_{z})$.

  1. 1.

If $\displaystyle 0<\varepsilon\leq\varepsilon_{min}^{+}$

    1. (a)

If $\varepsilon=\varepsilon_{min}^{+}$ and $\ell_{z}=\ell_{min}^{+}$, then

       * •

if $q=0$, there exists a unique triple root
$r_{abs}(\varepsilon,\ell_{z},0)=r^{+}_{ms}$ which solves (4.116). See

       * •

if $q>0$, there exists a unique simple root $r_{abs}(\varepsilon,\ell_{z},q)$
which solves (4.116).

The application $r_{abs}(\varepsilon,\ell_{z},\cdot)$ is well defined
continuous on $[0,\infty[$ and is monotonically decreasing on $]0,\infty[$
from $r^{+}_{ms}$ to $r_{H}$. Moreover, $r_{abs}(\varepsilon,\ell_{z},\cdot)$
is smooth on $]0,\infty[$ and we have

$\forall q>0\;,\;\frac{\partial r_{abs}}{\partial
q}(\varepsilon,\ell_{z},q)=-\frac{\frac{\partial R}{\partial
q}(r_{abs}(\varepsilon,\ell_{z},q),\varepsilon,\ell_{z},q)}{\frac{\partial
R}{\partial r}(r_{abs}(\varepsilon,\ell_{z},q),\varepsilon,\ell_{z},q)}.$

Now, by (4.117) and the above properties, the function $\theta\ni]0,\pi[\to
r_{abs}(\varepsilon,\ell_{z},q(\theta))$ is smooth on
$]0,\pi[\backslash\frac{\pi}{2}$ with

$\left.\frac{\partial}{\partial\theta}r_{abs}(\varepsilon,\ell_{z},q(\theta))\right|_{\frac{\pi}{2}^{-}}=-\infty\quad\text{and
}\quad\left.\frac{\partial}{\partial\theta}r_{abs}(\varepsilon,\ell_{z},q(\theta))\right|_{\frac{\pi}{2}^{+}}=+\infty.$

Therefore,

$Z^{K}(\varepsilon,\ell_{z})=Graph(r_{abs}(\varepsilon,\ell_{z},q(\cdot))),$

with a singular point at $(r^{+}_{ms},\frac{\pi}{2})$ and which is
diffeomorphic to $\mathbb{R}$.

    2. (b)

Otherwise, $\forall q\geq 0$, there exists a unique simple root
$r_{abs}(\varepsilon,\ell_{z},q)$ which solves (4.116). Moreover, the
application $r_{abs}(\varepsilon,\ell_{z},\cdot)$ is well defined, smooth and
monotonically decreasing on $[0,\infty[$ from
$r^{K}_{0}(\varepsilon,\ell_{z},0)$ to $r_{H}$.
Hence

$Z^{K}(\varepsilon,\ell_{z})=Graph(r_{abs}(\varepsilon,\ell_{z},q(\cdot))),$

which is diffeomorphic to $\mathbb{R}$.

  2. 2.

If $\displaystyle\varepsilon_{min}^{+}<\varepsilon\leq\varepsilon_{min}^{-}$,

    1. (a)

$\varepsilon=\varepsilon_{min}^{-}$ and $\ell_{z}=\ell_{min}^{-}$, then we use
similar arguments to those of Case 1.a.

    2. (b)

Otherwise,

      1. i.

If $\ell_{z}<\tilde{\ell}_{min}(\varepsilon)$, then $\forall q\geq 0$, there
exists a unique simple root $r_{abs}(\varepsilon,\ell_{z},q,d)$ which solves
(4.116). Moreover, the application $r_{abs}(\varepsilon,\ell_{z},\cdot,d)$ is
well defined on $[0,\infty[$ and is monotonically decreasing from
$r^{K}_{0}(\varepsilon,\ell_{z},0)$ to $r_{H}$ and we refer to Case 1.b for
conclusions.

      2. ii.

If $\tilde{\ell}_{min}(\varepsilon)<\ell_{z}<\ell_{lb}^{+}(\varepsilon)$, then
$\forall q\geq 0$, the solutions in $r$ of (4.116) are given by

$\left\\{\begin{aligned}
&r^{1}_{abs}(\varepsilon,\ell_{z},q)\quad\text{if}\quad 0\leq
q<q_{s}^{1}(\varepsilon,\ell_{z}),\\\
&r^{1}_{abs}(\varepsilon,\ell_{z},q),r^{2}_{abs}(\varepsilon,\ell_{z},q),r^{3}_{abs}(\varepsilon,\ell_{z},q)\quad\text{if}\quad
q_{s}^{1}(\varepsilon,\ell_{z})\leq q\leq q_{s}^{2}(\varepsilon,\ell_{z}),\\\
&r^{3}_{abs}(\varepsilon,\ell_{z},q)\quad\text{if}\quad
q>q_{s}^{2}(\varepsilon,\ell_{z})\end{aligned}\right.$

These solutions satisfy

         * •

$r^{2}_{abs}(\varepsilon,\ell_{z},q)=r^{3}_{abs}(\varepsilon,\ell_{z},q)=r_{s}^{1}(\varepsilon,\ell_{z})$
if and only if $q=q_{s}^{1}(\varepsilon,\ell_{z})$,

         * •

$r^{1}_{abs}(\varepsilon,\ell_{z},q)=r^{2}_{abs}(\varepsilon,\ell_{z},q)=r_{s}^{2}(\varepsilon,\ell_{z})$
if and only if $q=q_{s}^{2}(\varepsilon,\ell_{z})$,

         * •

$\displaystyle r^{3}_{abs}(\varepsilon,\ell_{z},q)\leq
r_{s}^{1}(\varepsilon,\ell_{z})\leq r^{2}_{abs}(\varepsilon,\ell_{z},q)\leq
r_{s}^{2}(\varepsilon,\ell_{z})\leq r^{1}_{abs}(\varepsilon,\ell_{z},q)$,

where $q_{s}^{i}(\varepsilon,\ell_{z})$ are defined in Lemma 33. Moreover,
seen as functions of $q$, $r^{i}_{abs}(\varepsilon,\ell_{z},\cdot)$ have the
following monotonicity properties

         * •

$r^{1}_{abs}(\varepsilon,\ell_{z},\cdot)$ is monotonically decreasing on
$]0,q_{s}^{2}(\varepsilon,\ell_{z})[$ from $r_{0}^{K}(\varepsilon,\ell_{z})$
to $r_{s}^{2}(\varepsilon,\ell_{z})$,

         * •

$r^{2}_{abs}(\varepsilon,\ell_{z},\cdot)$ is monotonically decreasing on
$]q_{s}^{1}(\varepsilon,\ell_{z}),q_{s}^{2}(\varepsilon,\ell_{z})[$ from
$r_{s}^{2}(\varepsilon,\ell_{z})$ to $r_{s}^{1}(\varepsilon,\ell_{z})$,

         * •

$r^{3}_{abs}(\varepsilon,\ell_{z},\cdot)$ is monotonically decreasing on
$]q_{s}^{1}(\varepsilon,\ell_{z}),\infty[$ from
$r_{s}^{1}(\varepsilon,\ell_{z})$ to $r_{H}$,

Now, we construct an atlas for $Z^{K}(\varepsilon,\ell_{z})$:

         * •

We recall the angles
$\theta_{i}(\varepsilon,\ell_{z})\in\left]0,\frac{\pi}{2}\right[$ defined by
(4.112).

         * •

By monotonicity properties of $q$ as a function of $\theta$, we can define the
following functions

           1. A.

$r^{1}_{abs}(\varepsilon,\ell_{z},q(\cdot)):\left]\theta_{2}(\varepsilon,\ell_{z}),\pi-\theta_{2}(\varepsilon,\ell_{z})\right[\to[r_{0}^{K}(\varepsilon,\ell_{z}),r^{2}_{s}(\varepsilon,\ell_{z})[$
defined by

$r^{1}_{abs}(\varepsilon,\ell_{z},q(\theta)).$

Here, $r^{1}_{abs}(\varepsilon,\ell_{z},q(\cdot))$ is symmetric with respect
to the equatorial plane and has a maximum at $\frac{\pi}{2}$ given by
$r_{0}^{K}(\varepsilon,\ell_{z})$.

           2. B.

$r^{2,a}_{abs}(\varepsilon,\ell_{z},q(\cdot)):\left]\theta_{1}(\varepsilon,\ell_{z}),\theta_{2}(\varepsilon,\ell_{z})\right[\to]r_{s}^{1}(\varepsilon,\ell_{z}),r^{2}_{s}(\varepsilon,\ell_{z})[$
defined by

$r^{2}_{abs}(\varepsilon,\ell_{z},q(\theta)).$

           3. C.

$r^{2,b}_{abs}(\varepsilon,\ell_{z},q(\cdot)):\left]\pi-\theta_{2}(\varepsilon,\ell_{z}),\pi-\theta_{1}(\varepsilon,\ell_{z})\right[\to]r_{s}^{1}(\varepsilon,\ell_{z}),r^{2}_{s}(\varepsilon,\ell_{z})[$
defined by

$r^{2}_{abs}(\varepsilon,\ell_{z},q(\theta)).$

           4. D.

$r^{3,a}_{abs}(\varepsilon,\ell_{z},q(\cdot)):\left]0,\theta_{2}(\varepsilon,\ell_{z})\right[\to]r_{H},r^{1}_{s}(\varepsilon,\ell_{z})[$
defined by

$r^{3}_{abs}(\varepsilon,\ell_{z},q(\theta)),$

           5. E.

$r^{3,b}_{abs}(\varepsilon,\ell_{z},q(\cdot)):\left]\pi-\theta_{2}(\varepsilon,\ell_{z}),\pi\right[\to]r_{H},r^{1}_{s}(\varepsilon,\ell_{z})[$
defined by

$r^{3}_{abs}(\varepsilon,\ell_{z},q(\theta)),$

         * •

It remains to cover the points
$\displaystyle\left(r^{i}_{s}(\varepsilon,\ell_{z}),\theta^{i}(\varepsilon,\ell_{z})\right)$
and
$\displaystyle\left(r^{i}_{s}(\varepsilon,\ell_{z}),\pi-\theta^{i}(\varepsilon,\ell_{z})\right)$.
To this end, we introduce the following functions

           1. A.

$\theta^{a}_{abs}(\varepsilon,\ell_{z},q(\cdot)):]r_{s}^{1}(\varepsilon,\ell_{z}),r^{2}_{s}(\varepsilon,\ell_{z})[\to\left]\theta_{1}(\varepsilon,\ell_{z}),\theta_{2}(\varepsilon,\ell_{z})\right[$
defined by

$\theta^{a}_{abs}(\varepsilon,\ell_{z},q(r)):=\left(\left.q\right|_{\left(0,\frac{\pi}{2}\right)}(\cdot,\varepsilon,\ell_{z},d)\right)^{-1}(\overline{q}(r))$

where $\overline{q}(r)$ is given by (4.118) and
$\displaystyle\left(\left.q\right|_{\left(0,\frac{\pi}{2}\right)}(\cdot,\varepsilon,\ell_{z},d)\right)^{-1}$
is the inverse of the restriction of $q(\cdot,\varepsilon,\ell_{z})$ on
$\displaystyle\left]0,\frac{\pi}{2}\right[$.

           2. B.

$\theta^{b}_{abs}(\varepsilon,\ell_{z},q(\cdot)):]r_{H},r^{2}_{s}(\varepsilon,\ell_{z})[\to\left]0,\frac{\pi}{2}\right[$
defined by

$\theta^{b}_{abs}(\varepsilon,\ell_{z},q(r)):=\left(\left.q\right|_{\left(0,\frac{\pi}{2}\right)}(\cdot,\varepsilon,\ell_{z})\right)^{-1}(\overline{q}(r))$

where $\overline{q}(r)$ is given by (4.118) and
$\displaystyle\left(\left.q\right|_{\left(0,\frac{\pi}{2}\right)}(\cdot,\varepsilon,\ell_{z},d)\right)^{-1}$
is the inverse of the restriction of $q(\cdot,\varepsilon,\ell_{z})$ on
$\displaystyle\left]0,\frac{\pi}{2}\right[$. In particular, when
$r=r_{s}^{1}(\varepsilon,\ell_{z})$ we have
$\displaystyle\theta^{b}_{abs}(\varepsilon,\ell_{z},q(r))=\theta_{1}(\varepsilon,\ell_{z})$.

           3. C.

We also introduce
$\displaystyle\pi-\theta^{a}_{abs}(\varepsilon,\ell_{z},q(\cdot))$ and
$\displaystyle\pi-\theta^{b}_{abs}(\varepsilon,\ell_{z},q(\cdot))$ whose
images lie in $\displaystyle\left]\frac{\pi}{2},\pi\right[$.

         * •

Therefore, $Z^{K}(\varepsilon,\ell_{z})$ is given by

$\displaystyle Z^{K}(\varepsilon,\ell_{z})$
$\displaystyle=Graph(r^{1}_{abs}(\varepsilon,\ell_{z},q(\cdot)))\cup
Graph(r^{2,a}_{abs}(\varepsilon,\ell_{z},q(\cdot)))\cup
Graph(r^{2,b}_{abs}(\varepsilon,\ell_{z},q(\cdot)))$ $\displaystyle\cup
Graph(r^{3,a}_{abs}(\varepsilon,\ell_{z},q(\cdot)))\cup
Graph(r^{3,b}_{abs}(\varepsilon,\ell_{z},q(\cdot)))\cup
Graph(\theta^{a}_{abs}(\varepsilon,\ell_{z},q(\cdot)))$ $\displaystyle\cup
Graph(\theta^{b}_{abs}(\varepsilon,\ell_{z},q(\cdot)))\cup
Graph(\pi-\theta^{a}_{abs}(\varepsilon,\ell_{z},q(\cdot)))$ $\displaystyle\cup
Graph(\pi-\theta^{b}_{abs}(\varepsilon,\ell_{z},q(\cdot))).$

      3. iii.

If $\ell_{z}=\tilde{\ell}^{+}_{min}(\varepsilon,d)$, the above analysis remain
valid and we obtain the same result. The only difference between the two cases
is that $r_{s}^{1}(\varepsilon,\ell_{z})=r_{s}^{2}(\varepsilon,\ell_{z})$ and
$q_{s}^{1}(\varepsilon,\ell_{z})=q_{s}^{2}(\varepsilon,\ell_{z})=q_{m}(\varepsilon,\ell_{z})$,
where,

$q_{m}(\varepsilon,\ell_{z})=\varepsilon^{2}\eta_{c}(r_{m}(\varepsilon),\varepsilon^{2})\quad\text{and
$r_{m}(\varepsilon)$ is defined in Lemma \ref{critical:l:c}.}$

Therefore, $r^{i}_{abs}$ coincide and we can describe
$Z^{K}(\varepsilon,\ell_{z})$ as the graph of one function
$r^{1}_{abs}(\varepsilon,\ell_{z},q(\cdot))$ defined on $(0,\pi)$ which
satisfies

$\frac{\partial
r^{1}_{abs}(\varepsilon,\ell_{z},q(\cdot))}{\partial\theta}(\theta_{m}(\varepsilon))=0\quad\text{and}\quad\frac{\partial^{2}r^{1}_{abs}(\varepsilon,\ell_{z},q(\cdot))}{\partial\theta^{2}}(\theta_{m}(\varepsilon))=0,$

where $\theta_{m}(\varepsilon)$ is the angle in
$\displaystyle\left(0,\frac{\pi}{2}\right)$ which satisfies

$q(\theta)=q_{m}(\varepsilon,\ell_{z}).$

      4. iv.

If $\ell_{lb}^{+}(\varepsilon)<\ell_{z}<\ell_{ub}^{+}(\varepsilon)$. We
proceed as in Case 2.b.ii: we have $\forall q\geq 0$, the solutions in $r$ of
(4.116) are given by

$\left\\{\begin{aligned}
&r^{0}_{abs}(\varepsilon,\ell_{z},q)\quad\text{if}q>\tilde{q}^{+}(\varepsilon,\ell_{z}),\\\
&r^{0}_{abs}(\varepsilon,\ell_{z},q),r^{1}_{tr}(\varepsilon,\ell_{z},q),r^{2}_{tr}(\varepsilon,\ell_{z},q)\quad\text{if}\quad
0\leq q\leq\tilde{q}^{+}(\varepsilon,\ell_{z}),\\\ \end{aligned}\right.$

These solutions satisfy

         * •

$r^{1}_{tr}(\varepsilon,\ell_{z},q)=r^{2}_{tr}(\varepsilon,\ell_{z},q)=\tilde{r}^{+}(\varepsilon,\ell_{z})$
if and only if $q=\tilde{q}^{+}(\varepsilon,\ell_{z})$,

         * •

$\displaystyle
r^{0}_{abs}(\varepsilon,\ell_{z},q)<r_{tr}^{1}(\varepsilon,\ell_{z})\leq\tilde{r}^{+}(\varepsilon,\ell_{z})\leq
r^{2}_{tr}(\varepsilon,\ell_{z},q)$,

where $\tilde{q}^{+}(\varepsilon,\ell_{z})$ and
$\tilde{r}^{+}(\varepsilon,\ell_{z})$ are defined in Lemma 33. Moreover, seen
as functions of $q$, $r^{0}_{abs}(\varepsilon,\ell_{z},\cdot)$ and
$r^{i}_{tr}(\varepsilon,\ell_{z},q)$ have the following monotonicity
properties

         * •

$r^{0}_{abs}(\varepsilon,\ell_{z},\cdot)$ is monotonically decreasing on
$]0,\infty[$ from $r_{0}^{K}(\varepsilon,\ell_{z})$ to $r_{H}$,

         * •

$r^{1}_{tr}(\varepsilon,\ell_{z},\cdot)$ is monotonically increasing on
$]0,\tilde{q}^{+}(\varepsilon,\ell_{z})[$ from
$r_{1}^{K}(\varepsilon,\ell_{z})$ to $\tilde{r}^{+}(\varepsilon,\ell_{z})$,

         * •

$r^{2}_{tr}(\varepsilon,\ell_{z},\cdot)$ is monotonically decreasing on
$]0,\tilde{q}^{+}(\varepsilon,\ell_{z})[$ from
$r_{2}^{K}(\varepsilon,\ell_{z})$ to $\tilde{r}^{+}(\varepsilon,\ell_{z})$,

Now, we construct an atlas for $Z^{K}(\varepsilon,\ell_{z})$. The properties
that $Z^{K}(\varepsilon,\ell_{z})$ has two connected components such that one
is diffeomorphic to $\mathbb{R}$ and the other is diffeomorphic to
$\mathbb{S}^{1}$ will follow from the construction. To this end,

         * •

We recall the angle $\displaystyle\theta\in\left]0,\frac{\pi}{2}\right[$
defined by (4.114).

         * •

We define the following mappings

           1. A.

$\displaystyle r^{0}_{abs}(\varepsilon,\ell_{z},q(\cdot)):$
$\displaystyle\left]0,\frac{\pi}{2}\right[\to]r_{H},r_{0}^{K}(\varepsilon,\ell_{z})]$
(4.119) $\displaystyle\theta\mapsto
r^{0}_{abs}(\varepsilon,\ell_{z},q(\theta)).$

           2. B.

$\displaystyle
r^{1}_{tr}(\varepsilon,\ell_{z},q(\cdot)):\left]\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z}),\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\right[$
$\displaystyle\to[r_{1}^{K}(\varepsilon,\ell_{z}),\tilde{r}^{+}(\varepsilon,\ell_{z})[$
(4.120) $\displaystyle\theta$ $\displaystyle\mapsto
r^{1}_{tr}(\varepsilon,\ell_{z},q(\theta)),$

           3. C.

$r^{2}_{tr}(\varepsilon,\ell_{z},q(\cdot)):\left]\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z}),\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\right[\to]\tilde{r}^{+}(\varepsilon,\ell_{z}),r_{2}^{K}(\varepsilon,\ell_{z})]$
defined by

$r^{2}_{tr}(\varepsilon,\ell_{z},q(\theta)).$ (4.121)

         * •

In order to cover the points
$\displaystyle\left(\tilde{r}^{+}(\varepsilon,\ell_{z}),\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\right)$
and
$\displaystyle\left(\tilde{r}^{+}(\varepsilon,\ell_{z}),\pi-\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})\right)$,
we introduce the mapping
$\displaystyle\theta_{tr}(\varepsilon,\ell_{z},q(\cdot)):]r_{1}^{K}(\varepsilon,\ell_{z}),r_{2}^{K}(\varepsilon,\ell_{z})[\to\left[\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z}),\frac{\pi}{2}\right[$
defined by

$\theta_{tr}(\varepsilon,\ell_{z},q(r)):=\left(\left.q\right|_{\left(0,\frac{\pi}{2}\right)}(\cdot,\varepsilon,\ell_{z},d)\right)^{-1}(q(r))$
(4.122)

and its symmetric with respect to the equatorial plane
$\pi-\theta_{tr}(\varepsilon,\ell_{z},q(\cdot))$. In particular, when
$r=\tilde{r}^{+}(\varepsilon,\ell_{z})$, we have
$\displaystyle\theta_{tr}(\varepsilon,\ell_{z},q(r))=\overline{\theta}_{max}^{<1}(\varepsilon,\ell_{z})$.

         * •

Now, we set

$Z^{K,abs}(\varepsilon,\ell_{z})=Graph(r^{0}_{abs}(\varepsilon,\ell_{z},\cdot))$

and

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

         * •

In order to show that $\displaystyle Z^{K,abs}(\varepsilon,\ell_{z})$ and
$\displaystyle Z^{K,trapped}(\varepsilon,\ell_{z})$ are disjoint, we use the
monotonicity properties of $r^{0}_{abs}(\varepsilon,\ell_{z},\cdot)$ and
$r^{1}_{tr}(\varepsilon,\ell_{z},q(\cdot))$ and the fact that

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

         * •

Moreover, it is easy to see that $Z^{K,trapped}(\varepsilon,\ell_{z})$ is
closed.

      5. v.

If $\ell_{z}=\ell_{lb}^{+}(\varepsilon)$, we use the same arguments as in the
previous case. The only difference is that when
$\ell_{z}=\ell_{lb}^{+}(\varepsilon)$ we have

$r_{0}^{K}(\varepsilon,\ell_{z})=r_{1}^{K}(\varepsilon,\ell_{z}).$

More precisely, when $q=0$, the equation (4.116) admits one double root given
by
$r_{0}^{K}(\varepsilon,\ell_{z})=r_{1}^{K}(\varepsilon,\ell_{z})=r_{max}^{+}(\varepsilon)$
and one simple root given by $r_{2}^{K}(\varepsilon,\ell_{z})$.
Therefore, $Z^{K,trapped}(\varepsilon,\ell_{z})$ and
$Z^{K,abs}(\varepsilon,\ell_{z})$ intersect at the point
$\left(r_{max}^{+}(\varepsilon),\frac{\pi}{2}\right)$.

  3. 3.

The remain cases follow in the same manner

    1. (a)

At a given $(\varepsilon,\ell_{z})$, we use Proposition 11 and Proposition 10
to determine the roots to the equation (4.116). These roots are functions of
$q$, which can also be seen as a function of $\theta$.

    2. (b)

We construct an atlas that covers the set of solutions, i.e.
$Z^{K}(\varepsilon,\ell_{z})$. Locally, it is either the graph of $r$
depending on $\theta$ or of $\theta$ depending on $r$ using similar arguments
as above.

∎

Following the above proposition, we introduce the following sets that will be
used along this work.

######  Definition 16.

$\displaystyle\mathcal{A}_{bound}^{+}:=$
$\displaystyle\left\\{(\varepsilon,\ell_{z})\in\left(\varepsilon^{+}_{min},1\right)\times({\ell}_{min}^{+},+\infty)\;:\;\quad{\ell}_{lb}^{+}(\varepsilon)<\ell_{z}<{\ell}_{ub}^{+}(\varepsilon)\;\right\\},$
(4.123)

$\displaystyle\mathcal{A}_{bound}^{-}:=$
$\displaystyle\left\\{(\varepsilon,\ell_{z})\in\left(\varepsilon^{-}_{min},1\right)\times(-\infty,{\ell}_{min}^{-})\;:\;\quad{\ell}_{ub}^{-}(\varepsilon)<\ell_{z}<{\ell}_{lb}^{-}(\varepsilon)\;\right\\},$
(4.124)

$\mathcal{A}_{bound}:=\mathcal{A}_{bound}^{+}\cup\mathcal{A}_{bound}^{-},$
(4.125)

$\displaystyle\mathcal{A}^{+}_{scattered}:=$
$\displaystyle\left\\{(\varepsilon,\ell_{z})\in\left[1,\infty\right)\times({\ell}_{min}^{+},+\infty)\;:\;\quad\ell_{z}>{\ell}_{lb}^{+}(\varepsilon)\right\\},$
(4.126)

$\displaystyle\mathcal{A}^{-}_{scattered}:=$
$\displaystyle\left\\{(\varepsilon,\ell_{z})\in\left[1,\infty\right)\times(-\infty,{\ell}_{min}^{-})\;:\;\quad\ell_{z}<{\ell}_{lb}^{-}(\varepsilon)\right\\},$
(4.127)

$\mathcal{A}_{scattered}:=\mathcal{A}_{scattered}^{+}\cup\mathcal{A}_{scattered}^{-},$
(4.128)

$\displaystyle\mathcal{A}^{+}_{circ}:=$
$\displaystyle\left\\{\left(\varepsilon^{+}_{min},\ell^{+}_{min}\right)\right\\}\cup\left\\{(\varepsilon,\ell_{z})\in\left]\varepsilon_{min}^{+},\infty\right[\times({\ell}_{min}^{+},+\infty)\;:\;\quad\ell_{z}={\ell}_{lb}^{+}(\varepsilon)\right\\}$
(4.129)
$\displaystyle\cup\left\\{(\varepsilon,\ell_{z})\in\left]\varepsilon_{min}^{+},\infty\right[\times({\ell}_{min}^{+},+\infty)\;:\;\quad\ell_{z}={\ell}_{ub}^{+}(\varepsilon)\quad,\quad\varepsilon<1\right\\},$

$\displaystyle\mathcal{A}^{-}_{circ}:=$
$\displaystyle\left\\{\left(\varepsilon^{-}_{min},\ell^{-}_{min}\right)\right\\}\cup\left\\{(\varepsilon,\ell_{z})\in\left]\varepsilon_{min}^{-},\infty\right[\times({\ell}_{min}^{-},+\infty)\;:\;\quad\ell_{z}={\ell}_{lb}^{-}(\varepsilon)\right\\}$
(4.130)
$\displaystyle\cup\left\\{(\varepsilon,\ell_{z})\in\left]\varepsilon_{min}^{-},\infty\right[\times(-\infty,{\ell}_{min}^{-})\;:\;\quad\ell_{z}={\ell}_{ub}^{-}(\varepsilon)\quad,\quad\varepsilon<1\right\\},$

$\mathcal{A}_{circ}:=\mathcal{A}_{circ}^{+}\cup\mathcal{A}_{circ}^{-},$
(4.131)

We also define the remaining subsets of $]0,\infty[\times\mathbb{R}$:

$\mathcal{A}^{<1}_{abs}:=\left\\{(\varepsilon,\ell_{z})\in\mathcal{A}^{admissible}\;:\;\varepsilon<1\quad\text{and}\quad(\varepsilon,\ell_{z})\notin\left(\mathcal{A}_{bound}\cup\mathcal{A}_{circ}\right)\right\\},$
(4.132)

$\mathcal{A}^{\geq
1}_{abs}:=\left\\{(\varepsilon,\ell_{z})\in\mathcal{A}^{admissible}\;:\;\varepsilon\geq
1\quad\text{and}\quad(\varepsilon,\ell_{z})\notin\left(\mathcal{A}_{bound}\cup\mathcal{A}_{circ}\right)\right\\},$
(4.133)

$\mathcal{A}_{abs}:=\mathcal{A}^{<1}_{abs}\cup\mathcal{A}^{\geq 1}_{abs}.$
(4.134)

It is easy to see that

$\mathcal{A}^{admissible}=\mathcal{A}_{abs}\sqcup\mathcal{A}_{circ}\sqcup\mathcal{A}_{scattered}\sqcup\mathcal{A}_{bound}.$

Only $(\varepsilon,\ell_{z})$ are required to characterise the zero velocity
curves. This allows one to determine the allowed regions for a timelike
future-directed geodesic with constants of motion $(\varepsilon,\ell_{z})$.
However, these constants of motion are not enough to determine the nature of
orbits. An orbit with same $(\varepsilon,\ell_{z})$ but starting at a two
different positions in the $(r,\theta)-$plane can have two different
behaviours, as we shall see in the remaining of this section. In this context,
we note that the above definitions of the different subsets can be misleading.
For example, assuming that $(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$ does
not necessarily imply that the orbit is trapped. One needs more assumptions to
obtain boundedness. Moreover, there exist trapped geodesics such that
$(\varepsilon,\ell_{z})\notin\mathcal{A}_{bound}$.
Our definition of $\mathcal{A}_{bound}$ is such that the associated allowed
region has a compact connected component.
A full classification of timelike future directed orbits is achieved by
finding a foliation of the mass shell $\Gamma$ indexed by
$(\varepsilon,\ell_{z},q)$ and by studying the initial position
$(t(0),\phi(0),r(0),\theta(0))$(See Lemma 5).

Figure 3: Shape of the zero velocity curve associated to a direct orbit with
$(\varepsilon,\ell_{z})=(\varepsilon_{min}^{+},\ell_{min}^{+})$ when $d=0.9$
Figure 4: Shape of the zero velocity curve associated to a direct orbit with
$(\varepsilon,\ell_{z})\in\mathcal{A}_{abs}^{<1}$ when $d=0.9$ Figure 5: Shape
of the zero velocity curve associated to a direct orbit with
$(\varepsilon,\ell_{z})\in\mathcal{A}_{abs}^{\geq 1}$ when $d=0.9$ Figure 6:
Shape of the zero velocity curve associated to a direct orbit with
$(\varepsilon,\ell_{z})\in\mathcal{A}_{circ}$ and $\varepsilon<1$ when $d=0.9$
Figure 7: Shape of the zero velocity curve associated to a direct orbit with
$(\varepsilon,\ell_{z})\in\mathcal{A}_{circ}$ and $\varepsilon\geq 1$ when
$d=0.9$ Figure 8: Shape of the zero velocity curve associated to a direct
orbit with $(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$ when $d=0.9$ Figure
9: Shape of the zero velocity curve associated to a direct orbit with
$(\varepsilon,\ell_{z})\in\mathcal{A}_{scattered}$ when $d=0.9$

Now, we determine the allowed region $A^{K}(\varepsilon,\ell_{z})$ associated
to $\gamma$. By Lemma 12, we obtain

######  Lemma 35.

Let $(\gamma,I\ni 0)$ be a timelike future-directed geodesic with constants of
motion $(\varepsilon,\ell_{z})$ and let $A^{K}(\varepsilon,\ell_{z})$ be the
allowed region associated to $\gamma$:

  * •

If $(\varepsilon,\ell_{z})\in\mathcal{A}_{bound}$,
$A^{K}(\varepsilon,\ell_{z})$ consists of two connected components. The
component which frontier in $]r_{H},\infty[\times]0,\pi[$ is
$Z^{K,trapped}(\varepsilon,\ell_{z})$ is compact and it will be denoted by
$A^{K,trapped}(\varepsilon,\ell_{z})$ and the component which frontier in
$]r_{H},\infty[\times]0,\pi[$ is $Z^{K,abs}(\varepsilon,\ell_{z})$ will be
denoted by $A^{K,abs}(\varepsilon,\ell_{z})$

  * •

If $(\varepsilon,\ell_{z})\in\mathcal{A}_{scattered}$,
$A^{K}(\varepsilon,\ell_{z})$ consists of two connected components. The
component which frontier in $]r_{H},\infty[\times]0,\pi[$ is
$Z^{K,scat}(\varepsilon,\ell_{z})$ is closed and unbounded and it will be
denoted by $A^{K,scat}(\varepsilon,\ell_{z})$ and the component which frontier
in $]r_{H},\infty[\times]0,\pi[$ is $Z^{K,abs}(\varepsilon,\ell_{z})$ will be
denoted by $A^{K,abs}(\varepsilon,\ell_{z})$.

  * •

If $(\varepsilon,\ell_{z})\in\mathcal{A}_{circ}$, then

    * –

if
$(\varepsilon,\ell_{z})\in\mathcal{A}_{circ}\cap\left\\{(\varepsilon,\ell_{z})\;:\;\varepsilon<1\right\\}$,
then $A^{K}(\varepsilon,\ell_{z})$ is the union of two connected components
which intersect at the point
$\displaystyle\left(r^{\pm}_{s}(\varepsilon,\ell_{z}),\frac{\pi}{2}\right)$.
The first component is bounded by $Z^{K,abs}(\varepsilon,\ell_{z})$ and the
second component is compact and bounded by
$Z^{K,trapped}(\varepsilon,\ell_{z})$.

    * –

if
$(\varepsilon,\ell_{z})\in\mathcal{A}_{circ}\cap\left\\{(\varepsilon,\ell_{z})\;:\;\varepsilon\geq
1\right\\}$, then $A^{K}(\varepsilon,\ell_{z})$ consists of two connected
components which intersect at the point
$\displaystyle\left(r^{\pm}_{s}(\varepsilon,\ell_{z}),\frac{\pi}{2}\right)$.
The first component is bounded by $Z^{K,abs}(\varepsilon,\ell_{z})$ and the
second component is bounded by $Z^{K,scat}(\varepsilon,\ell_{z})$.

  * •

If $(\varepsilon,\ell_{z})\in\mathcal{A}^{\leq 1}_{abs}$,
$A^{K}(\varepsilon,\ell_{z})$ consists of one connected component which
frontier is given by $Z^{K}(\varepsilon,\ell_{z})$.
$A^{K}(\varepsilon,\ell_{z})$ will be denoted by
$A^{K,abs}(\varepsilon,\ell_{z})$.

  * •

Otherwise, $A^{K}(\varepsilon,\ell_{z})$ is $]r_{H},\infty[\times]0,\pi[$. In
this case, we split $A^{K}(\varepsilon,\ell_{z})$ in three regions

    * –

$A^{K,z>0}(\varepsilon,\ell_{z})$, the region in $\mathscr{B}$ located between
the first component of the axis of symmetry and
$Z^{K,z>0}(\varepsilon,\ell_{z})$,

    * –

$A^{K,z>0}(\varepsilon,\ell_{z})$, the region in $\mathscr{B}$ located between
the second component of the axis of symmetry and
$Z^{K,z<0}(\varepsilon,\ell_{z})$

    * –

$\tilde{A}^{K,abs}(\varepsilon,\ell_{z})$, the remaining region which contains
the equatorial plane and which frontier in $\mathscr{B}$ is given by
$Z^{K,z>0}(\varepsilon,\ell_{z})\sqcup Z^{K,z<0}(\varepsilon,\ell_{z})$

Now, we announce the second main result of this section

######  Proposition 13.

(Classification of timelike future-directed geodesics) Let $\gamma:I\ni
0\to\mathcal{O}$ be a timelike future-directed geodesic with constants of
motion $(\varepsilon,\ell_{z},q)\in\mathcal{A}^{admissible}\times\mathbb{R}$
and let $\tilde{\gamma}$ be its projection in the $(r,\theta)-$plane.

  1. 1.

If $0<\varepsilon\leq\varepsilon^{+}_{min}$

    1. (a)

if $\varepsilon=\varepsilon^{+}_{min}$, $\ell_{z}=\ell^{+}_{min}$ and
$\tilde{\gamma}$ starts at
$\displaystyle\left(r_{ms}^{+},\frac{\pi}{2}\right)$, then $\tilde{\gamma}$ is
a circle confined in the equatorial plane.

    2. (b)

Otherwise, the orbit $\tilde{\gamma}$ starts at some point,
$(r_{0},\theta_{0})$ in the region $A^{K,abs}(\varepsilon,\ell_{z})\subset$
$]r_{H},r_{0}^{K}(\varepsilon,\ell_{z})]\times]0,\pi[$ and reaches the horizon
$r=r_{H}$ in a finite proper time while oscillating around the equatorial
plane between $\theta_{0}$ and $\pi-\theta_{0}$, where
$r_{0}^{K}(\varepsilon,\ell_{z})$ is the unique root of the equation (4.109).

  2. 2.

If $\varepsilon^{+}_{min}<\varepsilon\leq\varepsilon^{-}_{min}$,

    1. (a)

if $\varepsilon=\varepsilon^{-}_{min}$, $\ell_{z}=\ell^{-}_{min}$ and
$\tilde{\gamma}$ starts at
$\displaystyle\left(r_{ms}^{-},\frac{\pi}{2}\right)$, then $\tilde{\gamma}$ is
a circle confined in the equatorial plane.

    2. (b)

Otherwise,

      1. i.

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

         * •

If $\ell_{z}=\tilde{\ell}_{min}^{+}(\varepsilon)$ and $\tilde{\gamma}$ starts
at the point
$\displaystyle\left(r_{m}(\varepsilon),\theta_{m}(\varepsilon)\right)$, then
$\tilde{\gamma}$ is spherical of radius $r_{m}(\varepsilon)$.

         * •

If $\ell_{z}<\tilde{\ell}_{min}^{+}(\varepsilon)$ or
$\ell_{z}=\tilde{\ell}_{min}^{+}(\varepsilon)$ and starts at some point
different from
$\displaystyle\left(r_{m}(\varepsilon),\theta_{m}(\varepsilon)\right)$, then
$\tilde{\gamma}$ starts at some point in $A^{K,abs}(\varepsilon,\ell_{z})$ and
reaches the horizon in a finite proper time. See Case 1.b.

         * •

Otherwise, i.e
$\tilde{\ell}_{min}(\varepsilon)<\ell_{z}<\ell_{lb}^{+}(\varepsilon)$,

           * –

if $\tilde{\gamma}$ starts somewhere in the region
$\displaystyle\left\\{r^{1}_{s}(\varepsilon,\ell_{z})\right\\}\times]\theta_{1}(\varepsilon,\ell_{z}),\pi-\theta_{1}(\varepsilon,\ell_{z})[$
or in the region
$\displaystyle\left\\{r^{2}_{s}(\varepsilon,\ell_{z})\right\\}\times]\theta_{2}(\varepsilon,\ell_{z}),\pi-\theta_{2}(\varepsilon,\ell_{z})[$,
then $\tilde{\gamma}$ has a constant radius given by
$r^{i}_{s}(\varepsilon,\ell_{z})$ and oscillates between
$\theta_{i}(\varepsilon,\ell_{z})$ and $\pi-\theta_{i}(\varepsilon,\ell_{z})$
in the $\theta$ direction, where $r^{i}_{s}(\varepsilon,\ell_{z})$ are defined
in Lemma 33 and $\theta_{i}(\varepsilon,\ell_{z})$ are given by (4.112).

           * –

If $\tilde{\gamma}$ starts somewhere in the region $\displaystyle
A^{K,abs}(\varepsilon,\ell_{z})\cap(]r_{H},r^{1}_{s}(\varepsilon,\ell_{z})[\times]0,\pi[)$
or in the region $\displaystyle
A^{K,abs}(\varepsilon,\ell_{z})\cap(]r^{1}_{s}(\varepsilon,\ell_{z}),r^{K}_{0}(\varepsilon,\ell_{z})[\times]\theta_{1}(\varepsilon,\ell_{z}),\pi-\theta_{1}(\varepsilon,\ell_{z})[)$,
then it reaches the horizon in a finite proper time.

           * –

Otherwise, $\tilde{\gamma}$ is trapped.

      2. ii.

if $\ell_{z}=\ell_{lb}^{+}(\varepsilon)$,

         * •

If $\tilde{\gamma}$ starts at
$\displaystyle\left(r^{K}_{0}(\varepsilon,\ell_{z}\right),\frac{\pi}{2})$,
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 $\tilde{\gamma}$ is confined to the equatorial plane and starts with
positive radial velocity, then it approaches the circle of radius
$r^{K}_{0}(\varepsilon,\ell_{z})$ in an infinite proper time.

             * *

Otherwise, $\tilde{\gamma}$ reaches the horizon in a finite proper time,

           * –

If, $\tilde{\gamma}$ starts at some point in
$A^{K,trapped}(\varepsilon,\ell_{z})$, then it is either trapped or spherical
of radius $\tilde{r}^{+}(\varepsilon,\ell_{z})$.

      3. iii.

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

         * •

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

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

         * •

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.

  3. 3.

If $\varepsilon^{-}_{min}<\varepsilon<1$,

    1. (a)

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

       * •

if $\tilde{\gamma}$ starts somewhere in the region
$\displaystyle\left\\{\tilde{r}^{1}(\varepsilon,\ell_{z})\right\\}\times[\tilde{\theta}_{1}(\varepsilon,\ell_{z}),\pi-\tilde{\theta}_{1}(\varepsilon,\ell_{z})]$
or in the region
$\displaystyle\left\\{\tilde{r}^{2}(\varepsilon,\ell_{z})\right\\}\times[\tilde{\theta}_{2}(\varepsilon,\ell_{z}),\pi-\tilde{\theta}_{2}(\varepsilon,\ell_{z})]$,
then $\tilde{\gamma}$ has a constant radius given by
$\tilde{r}^{i}(\varepsilon,\ell_{z})$ and oscillates between
$\tilde{\theta}_{i}(\varepsilon,\ell_{z})$ and
$\pi-\tilde{\theta}_{i}(\varepsilon,\ell_{z})$ in the $\theta$ direction,
where $\tilde{r}^{i}(\varepsilon,\ell_{z})$ is defined in Lemma 33 and
$\tilde{\theta}_{i}(\varepsilon,\ell_{z})$ is defined by (4.113).

       * •

If $\tilde{\gamma}$ starts somewhere in the region $\displaystyle
A^{K,abs}(\varepsilon,\ell_{z})\cap(]r_{H},\tilde{r}^{1}(\varepsilon,\ell_{z})[\times]0,\pi[)$
or in the region $\displaystyle
A^{K,abs}(\varepsilon,\ell_{z})\cap(]\tilde{r}^{1}(\varepsilon,\ell_{z}),r_{0}^{K}(\varepsilon,\ell_{z})[\times]\tilde{\theta}_{1}(\varepsilon,\ell_{z}),\pi-\tilde{\theta}_{1}(\varepsilon,\ell_{z})[)$,
then it reaches the horizon in a finite proper time.

       * •

Otherwise, $\tilde{\gamma}$ is trapped.

    2. (b)

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

       * •

If $\tilde{\gamma}$ starts at
$\displaystyle\left(r^{K}_{0}(\varepsilon,\ell_{z}\right),\frac{\pi}{2})$,
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 $\tilde{\gamma}$ is confined to the equatorial plane and starts with
positive radial velocity, then it approaches the circle of radius
$r^{K}_{0}(\varepsilon,\ell_{z})$ in an infinite proper time.

           * *

Otherwise, $\tilde{\gamma}$ reaches the horizon in a finite proper time,

         * –

If, $\tilde{\gamma}$ starts at some point in
$A^{K,trapped}(\varepsilon,\ell_{z})$, then it is either trapped or spherical
of radius $r^{2}_{s}(\varepsilon,\ell_{z})$.

    3. (c)