type of motion, thus verifying in a concrete way one of the main consequences
of complete integrability of the system.

In this chapter, we will study Kerr polar motion in full generality, but leave
the radial motion aside. Its study is actually much more involved, since the
associated potential is of degree four, whereas it is only of degree two for
the polar motion. Instead, we will analyze in details the radial geodesic
motion in both NHEK and near-NHEK spacetimes. This analysis turns out to be
relevant for at least three reasons: (i) it will provide us with a deep
comprehension of the peculiarities of the motion near highly spinning black
holes, and in particular of the behaviour of their spherical geodesics. (ii)
As we will see, any geodesic can be obtained by applying a well-chosen
conformal transformation to a spherical geodesic. This enables to use the
computational scheme extensively applied in [58] to generate the waveforms
emitted by objects moving along geodesics in the near-horizon region. Such an
analysis can allow to unravel “smoking gun” signatures for the existence of
highly spinning black holes in Nature. Finally, (iii) this radial
classification was also the work which paved the road to the one of generic
Kerr radial motion, which was completed by G. Compère, Y. Liu and J. Long
[64].

All along this chapter, we will adopt a different notation than the one used
in all the rest of this thesis for Kerr geodesic conserved quantities. The
main reason for this switch is to match with the conventions used in [20]. We
also take the opportunity to restore the dependence of the equations upon
$\mu$. This enables our classification scheme to hold for null geodesics as
well as for timelike ones. The temporary convention is given by the following
dictionary

$\displaystyle\begin{split}&\hat{t}\to t,\quad,\hat{r}\to
r\quad,\hat{\varphi}\to\phi,\quad E_{0}\to\frac{\hat{E}}{\mu},\quad
L_{0}\to\frac{\ell}{\mu},\\\ &L_{*}\to\frac{\ell_{*}}{\mu},\quad
L_{\circ}\to\frac{\ell_{\circ}}{\mu},\quad
Q_{0}\to\frac{Q}{\mu^{2}}.\end{split}$ (3.2)

####  3.1 Classification of polar motion

This section aims to describe in full detail the classification of the polar
geodesic motion around a Kerr black hole. The classification scheme used has
been first introduced in [59], and subsequently completed in [20]. The
taxonomy is depicted in Figs. 3.2 and 3.2. The phenomenology of the polar
behavior is principally governed by the sign of $Q$. The details of the
classification are summarized in Tables 3.1 and 3.2.

Let us recall that, due to mirror symmetry between the two hemispheres, the
natural variable to describe the polar motion is $z\triangleq\cos^{2}\theta$.
Let us define

$\epsilon_{0}(\hat{E},\mu)\triangleq a^{2}(\hat{E}^{2}-\mu^{2}),$ (3.3)

and

$z_{\pm}\triangleq\Delta_{\theta}\pm\text{sign}\,{\epsilon_{0}}\sqrt{\Delta_{\theta}^{2}+\frac{Q}{\epsilon_{0}}},\qquad\Delta_{\theta}\triangleq\frac{1}{2}\quantity(1-\frac{Q+\ell^{2}}{\epsilon_{0}}),\qquad
z_{0}\triangleq\frac{Q}{Q+\ell^{2}}.$ (3.4)

The classification is determined by the roots of the polar potential (1.25c).
Assuming $a\neq 0$, we can rewrite it as

$\Theta(z)=-\ell^{2}z+(Q+\epsilon_{0}z)(1-z)=\left\\{\begin{array}[]{ll}\epsilon_{0}(z_{+}-z)(z-z_{-}),&\epsilon_{0}\neq
0;\\\ (Q+\ell^{2})(z_{0}-z),&\epsilon_{0}=0.\end{array}\right.$ (3.7)

Our definition of the roots $z_{\pm}$ implies the ordering $z_{-}<z_{+}$ (and
respectively $z_{+}<z_{-}$) for $\epsilon_{0}>0$ (respectively
$\epsilon_{0}<0$). This is a convenient convention because in both cases the
maximal angle will be related to $z_{+}$. The positivity of the polar
potential implies that the poles $z=1$ ($\theta=0,\pi$) can only be reached if
$\ell=0$. Note that when a geodesic crosses a pole, its $\varphi$ coordinates
discontinuously jump by $\pi$. The invariance of the polar geodesic equation
under $(\hat{E},\ell)\to(-\hat{E},-\ell)$ allows us to reduce the analysis to
prograde $\ell\geq 0$ orbits. We distinguish the orbits with angular momentum
$\ell\neq 0$ and without, $\ell=0$:

###### I. Nonvanishing angular momentum $\ell\neq 0$.

We must consider the following cases:

  1. 1.

$\boldsymbol{-(\ell-a\hat{E})^{2}\leq Q<0}$ can only occur if
$\epsilon_{0}>0$, otherwise leading to $\Theta<0$. For $\epsilon_{0}>0$, the
motion is vortical; i.e., it takes place only in one of the two hemispheres
without crossing the equatorial plane and is bounded by

$0<z_{-}\leq z\leq z_{+}<1.$ (3.8)

This vortical motion can only occur provided
$\ell^{2}\leq\quantity(\sqrt{\epsilon_{0}}-\sqrt{-Q})^{2}$.

  2. 2.

$\boldsymbol{Q>0}$ leads to motion crossing the equator and symmetric with
respect to it, bounded by

$\displaystyle 0$ $\displaystyle\leq z\leq z_{+}<1\qquad(\epsilon_{0}\neq 0),$
(3.9a) $\displaystyle 0$ $\displaystyle\leq z\leq
z_{0}<1\qquad(\epsilon_{0}=0).$ (3.9b)

We will refer to such a motion as pendular;

  3. 3.

$\boldsymbol{Q=0}$ allows us to write

$\Theta(z)=\epsilon_{0}\,z\,(1-\frac{\ell^{2}}{\epsilon_{0}}-z).$ (3.10)

If $\epsilon_{0}\leq 0$, the positivity of the polar potential enforces the
motion to be equatorial. For $\epsilon_{0}\geq 0$, equatorial motion exists at
$z=0$. For $\epsilon_{0}\geq 0$ and $\ell^{2}\leq\epsilon_{0}$, another motion
exists bounded by

$0<z\leq 1-\frac{\ell^{2}}{\epsilon_{0}}<1,$ (3.11)

which is a marginal case separating the pendular and vortical regimes; the
motion then admits only one turning point and asymptotes to the equator both
at future and at past times. Since we could not find a terminology for such a
motion in the literature, we propose to call it _equator-attractive_ 111This
neologism accurately reflects the fact that the motion is polar and that the
equator is an attractor. The terminology “homoclinic” is already used in the
literature to refer to radial motion.. In the special case where $z=0$ at the
initial time, the motion remains $z=0$ at all times: it is _equatorial_.

###### II. Vanishing angular momentum $\ell=0$.

The polar potential reduces to

$\Theta(z)=\left\\{\begin{array}[]{ll}\epsilon_{0}\quantity(\frac{Q}{\epsilon_{0}}+z)\quantity(1-z),&\epsilon_{0}\neq
0\\\ Q\quantity(1-z),&\epsilon_{0}=0.\end{array}\right.$ (3.14)

We distinguish the following cases:

  1. 1.

$\boldsymbol{\epsilon_{0}=0}$ leads to motion over the whole polar range
$0\leq z\leq 1$ for $Q>0$; we called it polar motion. The only turning point
is located at $z=1$. For $Q=0$, the potential vanishes identically and the
polar angle remains constant; we call it azimuthal motion; for $Q<0$ the
potential is positive only if the motion takes place along the black hole axis
$z=1$; we call it axial motion.

  2. 2.

$\boldsymbol{\epsilon_{0}>0}$ leads to a polar motion $0\leq z\leq 1$ for
$Q>0$. For $Q=0$ and $z=0$, the motion is equatorial. For $Q=0$ and $z\neq 0$,
$z=0$ is an asymptotic attractor of the motion which only takes place in one
of the hemispheres. It is therefore a special case of equator-attractive
motion where the turning point is at the pole $z=1$. For $Q<0$, the motion is
either vortical ($0<-\frac{Q}{\epsilon_{0}}\leq z\leq 1$) for
$-\epsilon_{0}<Q<0$ or axial with $z=1$ for $Q\leq-\epsilon_{0}<0$.

  3. 3.

$\boldsymbol{\epsilon_{0}<0}$ leads to a polar motion $0\leq z\leq 1$ for
$Q\geq-\epsilon_{0}>0$ and to a pendular one ($0\leq
z\leq-\frac{Q}{\epsilon_{0}}<1$) for $0<Q<-\epsilon_{0}$. For $Q=0$, the
motion is either equatorial or axial for the potential to be positive. For
$Q<0$, the motion also has to take place along the axis.

Let us finally notice that, for any value of $\epsilon_{0}$ and
$Q\geq-(aE_{0})^{2}$, an axial motion is always possible.

Energy | Carter constant | Polar range | Denomination
---|---|---|---
$\epsilon_{0}<0$ ($|\hat{E}|<\mu$) | $-(\ell-a\hat{E})^{2}\leq Q<0$ | $\emptyset$ |  $\emptyset$
| $Q=0$ | $z=0$ |  Equatorial$(\hat{E})$
| $Q>0$ | $0\leq z\leq z_{+}<1$ |  Pendular$(\hat{E},Q)$
$\epsilon_{0}=0$ ($|\hat{E}|=\mu$) | $-(\ell-a\hat{E})^{2}\leq Q<0$ | $\emptyset$ |  $\emptyset$
| $Q=0$ | $z=0$ |  Equatorial∘
| $Q>0$ | $0\leq z\leq z_{0}<1$ |  Pendular${}_{\circ}(Q)$
$\epsilon_{0}>0$ ($|\hat{E}|>\mu$) | $-(\ell-a\hat{E})^{2}\leq Q<0$ | $0<z_{-}\leq z\leq z_{+}<1$ |  Vortical$(\hat{E},Q)$
|  $Q=0$ and $\epsilon_{0}\geq\ell^{2}$ |  |  $0<z\leq 1-\frac{\ell^{2}}{\epsilon_{0}}<1$
---
($\text{sign}\,{\cos\theta}$ fixed)
|  Equator-
---
attractive$(\hat{E})$
|  | $z=0$ |  Equatorial$(\hat{E})$
| $Q>0$ | $0\leq z\leq z_{+}<1$ |  Pendular$(\hat{E},Q)$
Table 3.1: Polar taxonomy of Kerr geodesics with $\ell\geq 0$. The orbits with
$\ell<0$ are obtained from $\ell>0$ by flipping the signs of both $\hat{E}$
and $\ell$.

a

Energy | Carter constant | Polar range | Denomination
---|---|---|---
$\epsilon_{0}<0$ ($|\hat{E}|<\mu$) | $-(a\hat{E})^{2}\leq Q<0$ | $z=1$ |  Axial${}^{0}(\hat{E},Q)$
| $Q=0$ | $z=0,1$ |  |  Equatorial${}^{0}(\hat{E})$
---
Axial${}^{0}(\hat{E})$
| $0<Q<-\epsilon_{0}$ | $0\leq z\leq-\frac{Q}{\epsilon_{0}}<1$ |  Pendular${}^{0}(\hat{E},Q)$
| $0<-\epsilon_{0}\leq Q$ | $0\leq z\leq 1$ |  Polar${}^{0}(\hat{E},Q)$
$\epsilon_{0}=0$ ($|\hat{E}|=\mu$) | $-(a\hat{E})^{2}\leq Q<0$ | $z=1$ |  Axial${}^{0}_{\circ}(Q)$
| $Q=0$ | $z=\text{constant}$ |  Azimuthal${}^{0}_{\circ}$
| $Q>0$ | $0\leq z\leq 1$ |  Polar${}^{0}_{\circ}(Q)$
$\epsilon_{0}>0$ ($|\hat{E}|>\mu$) | $-(a\hat{E})^{2}\leq Q\leq-\epsilon_{0}<0$ | $z=1$ |  Axial${}^{0}(\hat{E},Q)$
| $-\epsilon_{0}<Q<0$ | $0<-\frac{Q}{\epsilon_{0}}\leq z\leq 1$ |  Vortical${}^{0}(\hat{E},Q)$
| $Q=0$ |  $0<z\leq 1$ ($\text{sign}\,{\cos\theta}$ fixed) |  | Equator-
---
attractive${}^{0}(\hat{E})$
|  | $z=0$ |  Equatorial${}^{0}(\hat{E})$
| $Q>0$ | $0\leq z\leq 1$ |  Polar${}^{0}(\hat{E},Q)$
$\epsilon_{0}\in\mathbb{R}$ | $Q\geq-(a\hat{E})^{2}$ | $z=1$ |  Axial${}^{0}(E_{0},Q)$
Table 3.2: Polar taxonomy of Kerr geodesics with $\ell=0$.

Figure 3.1: Polar taxonomy of $\ell\neq 0$ Kerr geodesics. Equator-
attractive$(\hat{E})$ orbits become Equatorial$(\hat{E})$ orbits when the
initial angle is at the equator.

Figure 3.2: Polar taxonomy of $\ell=0$ Kerr geodesics. In addition to the
possible motions depicted in the figure, an axial motion is always possible
for any value of $\epsilon_{0}$ and $Q\geq-(aE_{0})^{2}$.

#####  3.1.1 Solution to the polar integrals

After having classified the different types of motion allowed, we will provide
manifestly real and positive explicit solutions in terms of elliptic integrals
for each type of polar motion with $\ell\neq 0$, in line with the recent
analysis [59]. All such integrals will turn out to agree with Ref. [59], but
our presentation will be slightly simpler.

The solution to the polar integrals (1.28a) and (1.28b) can be organized in
terms of the categories of polar motion with $\ell\neq 0$:

| Vortical | Equator-attractive | Pendular
---|---|---|---
$\boldsymbol{\epsilon_{0}<0}$ | $\emptyset$ | $\emptyset$ |  Pendular$(\hat{E},Q)$
$\boldsymbol{\epsilon_{0}=0}$ | $\emptyset$ | $\emptyset$ |  Pendular${}_{*}(Q)$
$\boldsymbol{\epsilon_{0}>0}$ |  Vortical$(\hat{E},Q)$ |  Equator-attractive$(\hat{E})$ |  Pendular$(\hat{E},Q)$

Each type of motion yields to a specific decomposition of the line integrals
$\scriptsize\rotatebox[origin={c}]{-90.0}{$\backslash$}\int$ in terms of basic
integrals. In order to simplify the notations, we drop the “$f$” indices
labeling the final event and define $h\equiv\text{sign}\,{\cos\theta}$,
$\theta_{a}\triangleq\arccos\sqrt{z_{a}}$ ($a=+,-,0$), as well as the initial
and final signs $\eta_{i}$, $\eta$:

$\displaystyle\eta_{i}\triangleq-
s_{\theta}^{i}\,\text{sign}\,{\cos\theta_{i}},\qquad\eta\triangleq-(-1)^{m}s_{\theta}^{i}\,\text{sign}\,{\cos\theta}.$
(3.52)

We are now ready to perform the explicit decomposition:

  1. 1.

Pendular motion. We have $0<z_{+}\leq 1$, and $\theta$ therefore belongs to
the interval $\theta_{+}\leq\theta\leq\pi-\theta_{+}$. The polar integral can
be written (see Ref. [59])

$\displaystyle\rotatebox[origin={c}]{-90.0}{$\backslash$}\int_{\cos\theta_{i}}^{\cos\theta}$
$\displaystyle=2m\left|\int_{0}^{\cos\theta_{+}}\right|-\eta\left|\int_{0}^{\cos\theta}\right|+\eta_{i}\left|\int_{0}^{\cos\theta_{i}}\right|,\qquad\epsilon_{0}\neq
0,$ (3.53a)
$\displaystyle\rotatebox[origin={c}]{-90.0}{$\backslash$}\int_{\cos\theta_{i}}^{\cos\theta}$
$\displaystyle=2m\left|\int_{0}^{\cos\theta_{0}}\right|-\eta\left|\int_{0}^{\cos\theta}\right|+\eta_{i}\left|\int_{0}^{\cos\theta_{i}}\right|,\qquad\epsilon_{0}=0.$
(3.53b)

It is useful to note that our definitions of the roots imply

$\epsilon_{0}\,z_{-}<0,\qquad\epsilon_{0}(z-z_{-})>0,\qquad\frac{z_{+}}{z_{-}}\leq
1.$ (3.54)

  2. 2.

Vortical motion. We have $\epsilon_{0}>0$ and $0<z_{-}\leq\cos^{2}\theta\leq
z_{+}<1$. The motion therefore never reaches the equator. The sign of
$\cos\theta$ is constant and determines whether the motion takes place in the
northern or the southern hemisphere. Without loss of genericity, let us focus
on the northern hemisphere:
$0\leq\theta_{+}\leq\theta\leq\theta_{-}<\frac{\pi}{2}$; we denote again as
$m$ the number of turning points at Mino time $\lambda$. The polar integral
can be written (see Ref. [59] and Appendix A of Ref. [60]):

$\displaystyle\rotatebox[origin={c}]{-90.0}{$\backslash$}\int_{\cos\theta_{i}}^{\cos\theta}=\left(m-\eta_{i}\frac{1-(-1)^{m}}{2}\right)\left|\int_{\cos\theta_{-}}^{\cos\theta_{+}}\right|-\eta\left|\int_{\cos\theta_{-}}^{\cos\theta}\right|+\eta_{i}\left|\int_{\cos\theta_{-}}^{\cos\theta_{i}}\right|.$
(3.55)

  3. 3.

Equator-attractive motion. This is a limit case of the vortical motion reached
in the limit $z_{-}\to 0$, $z_{+}\to 2\Delta_{\theta}$. As detailed in Ref.
[59], the turning point $z_{-}=0$ corresponds to a non-integrable singularity
of the polar integrals and the motion exhibits consequently at most one
turning point at $z_{+}=2\Delta_{\theta}$, leading to the line-integral
decomposition

$\rotatebox[origin={c}]{-90.0}{$\backslash$}\int_{\cos\theta_{i}}^{\cos\theta}=\eta\absolutevalue{\int_{\cos\theta_{+}}^{\cos\theta}}-\eta_{i}\absolutevalue{\int_{\cos\theta_{+}}^{\cos\theta_{i}}}.$
(3.56)

In all cases but the equator-attractive case, the polar motion is periodic.
Denoting by $\Lambda_{\theta}$ its period, one can easily give an explicit
formula for the number of turning points $m$ as a function of the Mino time:

$m(\lambda)=\left\\{\begin{array}[]{ll}\rule{0.0pt}{6.0pt}\left\lfloor\frac{2}{\Lambda_{\theta}}(\lambda-\lambda_{i}^{\theta})+\frac{1}{2}\right\rfloor,&Q>0\\\
\rule{0.0pt}{16.0pt}\left\lfloor\frac{2}{\Lambda_{\theta}}(\lambda-\lambda_{i}^{\theta})\right\rfloor+\left\lfloor\frac{2}{\Lambda_{\theta}}(\lambda_{i}^{\theta}-\lambda_{i})\right\rfloor+\frac{3-s^{i}_{\theta}}{2},&Q<0\end{array}\right.$
(3.59)

with
$\lambda_{i}^{\theta}\triangleq\lambda_{i}-s^{i}_{\theta}\int_{0}^{\cos\theta_{i}}\frac{\differential\cos\theta}{\sqrt{\Theta(\cos^{2}\theta)}}$
and where the floor function is defined as $\left\lfloor
x\right\rfloor\triangleq\max\quantity{n\in\mathbb{Z}|n\leq x}$. For the
equator-attractive case, one has simply
$m(\lambda)=\theta(\lambda-\lambda_{i}^{\theta})$ where $\theta$ is here the
Heaviside step function.

The integrals introduced above are solved explicitly in Appendix B.1. For each
case, the corresponding solutions are detailed below and schematically
depicted in Fig. 3.3.

|
---|---
(a) Pendular$(\hat{E},Q)$ |  (b) Pendular${}_{\circ}(Q)$
|
(c) Vortical$(\hat{E},Q)$ |  (d) Equator-attractive$(\hat{E})$
Figure 3.3: Angular taxonomy of $\ell\neq 0$ Kerr geodesics. The angular
behavior is depicted in spherical coordinates on the unit sphere: the polar
angle is $\theta(\lambda)$, and the azimuthal angle is the purely angular part
of the Kerr azimuthal angle
$(\ell-a\hat{E})(\lambda-\lambda_{i})+\ell\Phi_{\theta}(\lambda)$.

###### Pendular$(\hat{E},Q)$ motion.

The motion exhibits a positive Carter constant $Q$ and can occur for any
$\epsilon_{0}\neq 0$; our definition of the roots $z_{\pm}$ allows us to treat
simultaneously the two cases $\epsilon_{0}<0$ and $\epsilon_{0}>0$, which is a
simplification with respect to the analysis carried out in Ref. [59]. The
period of the polar motion (comprising two turning points) in Mino time is
given by

$\Lambda_{\theta}=4\int_{0}^{\cos\theta_{+}}\frac{\text{d}\cos\theta}{\sqrt{\Theta(\cos^{2}\theta)}}\triangleq
4I^{(0)}(\sqrt{z_{+}})=\frac{4}{\sqrt{-\epsilon_{0}z_{-}}}K\quantity(\frac{z_{+}}{z_{-}}).$
(3.64)

Using the basic integrals of Appendix A, one can write (1.28a) as

$\displaystyle\lambda-\lambda_{i}$
$\displaystyle=\frac{1}{\sqrt{-\epsilon_{0}z_{-}}}\left[2mK\quantity(\frac{z_{+}}{z_{-}})+s^{i}_{\theta}(-1)^{m}F\quantity(\Psi^{+}(\cos\theta),\frac{z_{+}}{z_{-}})\right.$
$\displaystyle~{}\left.-s_{\theta}^{i}F\quantity(\Psi^{+}(\cos\theta_{i}),\frac{z_{+}}{z_{-}})\right]$
(3.65)

where we define
$\Psi^{+}(x)\triangleq\arcsin\quantity(\frac{x}{\sqrt{z_{+}}})$. Using
(9.126n), one can invert (3.65) as

$\cos\theta=s^{i}_{\theta}(-1)^{m}\sqrt{z_{+}}\text{sn}\quantity(\sqrt{-\epsilon_{0}z_{-}}\quantity(\lambda-\lambda_{i}^{\theta})-2mK\quantity(\frac{z_{+}}{z_{-}}),\frac{z_{+}}{z_{-}})$
(3.66)

where we introduce

$\displaystyle\lambda_{i}^{\theta}$
$\displaystyle\triangleq\lambda_{i}-\frac{s^{i}_{\theta}}{\sqrt{-\epsilon_{0}z_{-}}}F\quantity(\Psi^{+}(\cos\theta_{i}),\frac{z_{+}}{z_{-}}).$
(3.67)

This expression matches with Eq. (38) of Ref. [54]. Using the periodicity
property (9.126da) of the elliptic sine, we can further simplify it to

$\cos\theta(\lambda)=s^{i}_{\theta}\sqrt{z_{+}}\text{sn}\quantity(\sqrt{-\epsilon_{0}z_{-}}(\lambda-\lambda_{i}^{\theta}),\frac{z_{+}}{z_{-}}).$
(3.68)

It consistently obeys $\cos\theta(\lambda_{i})=\cos\theta_{i}$ and
$\text{sign}\,{\cos\theta^{\prime}(\lambda_{i})}=s_{\theta}^{i}$. This formula
agrees with (53) of Ref. [59] but it is written in a simpler form. We also
obtain

$\displaystyle T_{\theta}$
$\displaystyle=\frac{-2z_{+}}{\sqrt{-\epsilon_{0}z_{-}}}\left[2mE^{\prime}\quantity(\frac{z_{+}}{z_{-}})+(\pm_{\theta})E^{\prime}\quantity(\Psi^{+}(\cos\theta),\frac{z_{+}}{z_{-}})\right.$
$\displaystyle\left.~{}-s_{\theta}^{i}E^{\prime}\quantity(\Psi^{+}(\cos\theta_{i}),\frac{z_{+}}{z_{-}})\right],$
(3.69) $\displaystyle\Phi_{\theta}$
$\displaystyle=\frac{1}{\sqrt{-\epsilon_{0}z_{-}}}\left[2m\Pi\quantity(z_{+},\frac{z_{+}}{z_{-}})+(\pm_{\theta})\Pi\quantity(z_{+},\Psi^{+}(\cos\theta),\frac{z_{+}}{z_{-}})\right.$
$\displaystyle~{}\left.-s_{\theta}^{i}\Pi\quantity(z_{+},\Psi^{+}(\cos\theta_{i}),\frac{z_{+}}{z_{-}})\right]-(\lambda-\lambda_{i}).$
(3.70)

where $\lambda-\lambda_{i}$ is given by (3.65). All quantities involved are
manifestly real. These final expressions agree with Ref. [59].

###### Pendular${}_{\circ}(Q)$ motion.

We now consider the critical case $|\hat{E}|=\mu$. The period of the polar
motion is

$\Lambda_{\theta}=4I^{(0)}(\sqrt{z_{0}})=2\pi\sqrt{\frac{z_{0}}{Q}}.$ (3.71)

In this critical case, (1.28a) leads to

$\lambda-\lambda_{i}=\sqrt{\frac{z_{0}}{Q}}\quantity[m\pi+s^{i}_{\theta}(-1)^{m}\arcsin{\frac{\cos\theta}{\sqrt{z_{0}}}}-s^{i}_{\theta}\arcsin{\frac{\cos\theta_{i}}{\sqrt{z_{0}}}}],$
(3.72)

which can be simply inverted as

$\cos\theta=s_{\theta}^{i}\sqrt{z_{0}}\,\sin\quantity(\sqrt{\frac{Q}{z_{0}}}(\lambda-\lambda_{i}^{\theta})),\qquad\lambda_{i}^{\theta}\triangleq\lambda_{i}-\sqrt{\frac{z_{0}}{Q}}\arcsin{\frac{\cos\theta_{i}}{\sqrt{z_{0}}}}.$
(3.73)

The other polar integrals are

$\displaystyle T_{\theta}$
$\displaystyle=\frac{1}{2}\quantity{z_{0}(\lambda-\lambda_{i})-\sqrt{\frac{z_{0}}{Q}}\quantity[(\pm_{\theta})\cos\theta\sqrt{z_{0}-\cos^{2}\theta}-s^{i}_{\theta}\cos\theta_{i}\sqrt{z_{0}-\cos^{2}\theta_{i}}]},$
$\displaystyle\Phi_{\theta}$
$\displaystyle=\sqrt{\frac{z_{0}}{Q(1-z_{0})}}\bigg{[}m\pi+(\pm_{\theta})\arcsin\quantity(\sqrt{\frac{1-z_{0}}{z_{0}}}\cot\theta)$
$\displaystyle\quad-s^{i}_{\theta}\arcsin\quantity(\sqrt{\frac{1-z_{0}}{z_{0}}}\cot\theta_{i})\bigg{]}-(\lambda-\lambda_{i}).$
(3.74)

###### Vortical$(\hat{E},Q)$ motion.

The period in Mino time is given by

$\Lambda_{\theta}=2\left|\int_{\cos\theta_{-}}^{\cos\theta_{+}}\frac{\text{d}\cos\theta}{\sqrt{\Theta(\cos^{2}\theta)}}\right|=\frac{2}{\sqrt{\epsilon_{0}z_{+}}}K\quantity(1-\frac{z_{-}}{z_{+}}).$
(3.75)

Using the basic integrals of Appendix B.1, one has

$\displaystyle\lambda-\lambda_{i}$
$\displaystyle=\frac{1}{\sqrt{\epsilon_{0}z_{+}}}\left[\quantity(m-hs^{i}_{\theta}\frac{1-(-1)^{m}}{2})K(\tilde{m})-s^{i}_{\theta}(-1)^{m}F\quantity(\Psi^{-}(\cos\theta),\tilde{m})\right.$
$\displaystyle~{}\left.+s^{i}_{\theta}F\quantity(\Psi^{-}(\cos\theta_{i}),\tilde{m})\right]$
(3.76)

where

$\tilde{m}\triangleq
1-\frac{z_{-}}{z_{+}},\qquad\Psi^{-}(x)=\arcsin{\sqrt{\frac{z_{+}-x^{2}}{z_{+}-z_{-}}}}.$
(3.77)

Using the inversion formula (9.126t) and the periodicity property (9.126db),
we obtain

$\cos\theta=h\sqrt{z_{+}}\text{dn}\quantity(\sqrt{\epsilon_{0}z_{+}}(\lambda-\lambda_{\theta}^{i}),\tilde{m})$
(3.78)

with

$\lambda_{i}^{\theta}\triangleq\lambda_{i}+\frac{s^{i}_{\theta}h}{\sqrt{\epsilon_{0}z_{+}}}F\quantity(\Psi^{-}(\cos\theta_{i}),\tilde{m}).$
(3.79)

Again, one has $\cos\theta(\lambda_{i})=\cos\theta_{i}$ and
$\text{sign}\,{\cos\theta^{\prime}(\lambda_{i})}=s_{\theta}^{i}$. The two
other polar integrals are

$\displaystyle T_{\theta}$
$\displaystyle=\sqrt{\frac{z_{+}}{\epsilon_{0}}}\left[\quantity(m-hs^{i}_{\theta}\frac{1-(-1)^{m}}{2})E(\tilde{m})-(\pm_{\theta})E\quantity(\Psi^{-}(\cos\theta),\tilde{m})\right.$
$\displaystyle~{}\left.+s^{i}_{\theta}E\quantity(\Psi^{-}(\cos\theta_{i}),\tilde{m})\right],$
(3.80) $\displaystyle\Phi_{\theta}$
$\displaystyle=\frac{1}{(1-z_{+})\sqrt{\epsilon_{0}z_{+}}}\left[\quantity(m-hs^{i}_{\theta}\frac{1-(-1)^{m}}{2})\Pi\quantity(\frac{z_{-}-z_{+}}{1-z_{+}},\tilde{m})\right.$
$\displaystyle~{}\left.-(\pm_{\theta})\Pi\quantity(\frac{z_{-}-z_{+}}{1-z_{+}},\Psi^{-}(\cos\theta),\tilde{m})+s^{i}_{\theta}\Pi\quantity(\frac{z_{-}-z_{+}}{1-z_{+}},\Psi^{-}(\cos\theta_{i}),\tilde{m})\right]$
$\displaystyle~{}-(\lambda-\lambda_{i})$ (3.81)

in agreement with the results of Ref. [59].

###### Equator-attractive$(\hat{E})$ motion.

This is the only polar motion which is not periodic. One has

$\lambda-\lambda_{i}=\frac{h}{\sqrt{\epsilon_{0}z_{+}}}\quantity[-(\pm_{\theta})\,\text{arctanh}\sqrt{1-\frac{\cos^{2}\theta}{z_{+}}}+s^{i}_{\theta}\,\text{arctanh}\sqrt{1-\frac{\cos^{2}\theta_{i}}{z_{+}}}]$
(3.82)

leading to

$\displaystyle\cos\theta$
$\displaystyle=h\sqrt{z_{+}}\,\text{sech}\quantity(\sqrt{\epsilon_{0}z_{+}}(\lambda-\lambda_{i}^{\theta})),$
(3.83a) $\displaystyle\lambda_{i}^{\theta}$
$\displaystyle\triangleq\lambda_{i}+\frac{s^{i}_{\theta}h}{\sqrt{\epsilon_{0}z_{+}}}\text{arctanh}\sqrt{1-\frac{\cos^{2}\theta_{i}}{z_{+}}}.$
(3.83b)

The polar integrals are

$\displaystyle T_{\theta}$
$\displaystyle=\frac{h}{\sqrt{\epsilon_{0}}}\quantity[-(\pm_{\theta})\sqrt{z_{+}-\cos^{2}\theta}+s^{i}_{\theta}\sqrt{z_{+}-\cos^{2}\theta_{i}}],$
(3.84a) $\displaystyle\Phi_{\theta}$
$\displaystyle=\frac{h}{\sqrt{\epsilon_{0}(1-z_{+})}}\bigg{[}-(\pm_{\theta})\arctan\sqrt{\frac{z_{+}-\cos^{2}\theta}{1-z_{+}}}$
$\displaystyle\quad+s^{i}_{\theta}\arctan\sqrt{\frac{z_{+}-\cos^{2}\theta_{i}}{1-z_{+}}}\bigg{]}.$
(3.84b)

This agrees with the results of Ref. [59].

####  3.2 Classification of near-horizon motion for high spin Kerr black holes

In this section, we derive a complete classification of timelike and null
geodesic trajectories lying in the near-horizon region of a quasi extremal
Kerr black hole. We will provide explicit manifestly real analytic expressions
for all geodesic trajectories. We will present the classification in terms of
the geodesic energy, angular momentum, and Carter constant $Q$. We will also
illustrate each radial motion in NHEK with a Penrose diagram.

Partial classifications were performed in Refs. [58] and [59]. In Ref. [58],
equatorial timelike prograde incoming (i.e. that originate from the Kerr
exterior geometry) geodesics were classified. Such geodesics reach the spatial
boundary of the near-horizon region at infinite past proper time and therefore
physically reach the asymptotically flat Kerr region once the near-horizon is
glued back to the exterior Kerr region. It turns out that bounded geodesics in
the near-horizon Kerr region also arise in the study of gravitational waves
since they correspond to the end point of the transition motion [96]. Timelike
outgoing geodesics originating from the white hole horizon and reaching the
near-horizon boundary are also relevant for particle emission within the near-
horizon region [80]. In addition, null outgoing geodesics are relevant for
black hole imaging around high-spin black holes [37].

The generic non-equatorial geodesics were obtained in Ref. [59]. In
particular, real forms were obtained for each angular integral involved in
geodesic motion. However, zero-measure sets of parameters were discarded.
These zero-measure sets include in particular the separatrix between bounded
and unbounded radial motion which plays a key role in EMRIs.

In the following, we do not make any assumption on the geodesic parameters. We
will treat both timelike and null geodesics, prograde or retrograde, and with
any boundary conditions. Without loss of genericity, we will consider future-
directed orbits. Past-directed geodesics can be obtained from future-directed
geodesics using the $\mathbb{Z}_{2}$ map:

$\displaystyle T\rightarrow-T,\quad\Phi\rightarrow-\Phi,\quad
E\rightarrow-E,\quad\ell\rightarrow-\ell,$ (3.85)

which will play an important role in Sec. 3.4. We will denote it as the
$\uparrow\\!\downarrow$-flip.

#####  3.2.1 NHEK

Future orientation of the geodesic is equivalent to $\differential
T/\differential\lambda>0$ or

$\displaystyle E+L_{0}R>0.$ (3.86)

Future-oriented geodesics with $L_{0}=0$ have $E>0$. For $L_{0}\neq 0$, we
define the critical radius as in [59]:

$\displaystyle R_{c}=-\frac{E}{L_{0}}.$ (3.87)

Future-orientation of the orbit requires

$\displaystyle R<R_{c}\quad\text{for}\quad L_{0}<0,\quad\text{and}\quad
R>R_{c}\quad\text{for}\quad L_{0}>0.$ (3.88)

###### Polar behavior

The results derived in Sec. 3.1 in the context of generic Kerr still hold in
the near-horizon high-spin limit which is obtained by the scaling limit
$\lambda\rightarrow 0$ taken in the near-horizon coordinates (1.42). We
anticipate that the results also hold in the distinct near-NHEK limit
$\lambda\rightarrow 0$ taken in the near-horizon coordinates (1.63). Due to
the high-spin limit, the following substitution can be made:

$\displaystyle a\mapsto
M,\qquad\hat{E}\mapsto\frac{\ell}{2M},\qquad\epsilon_{0}\mapsto\mathcal{C}_{\circ}\triangleq\frac{\ell^{2}-\ell_{\circ}^{2}}{4},$
(3.89) $\displaystyle\Theta(z)\mapsto
v_{\theta}(z),\qquad\hat{\Phi}_{\theta}-\frac{1}{4}\hat{T}_{\theta}\mapsto\Phi_{\theta}.$
(3.90)

Notice that the dependence on $\hat{E}$ of $\epsilon_{0}$ has been changed
into a dependence in $\ell$, the Kerr energy being the same at zeroth order on
$\lambda$ for all trajectories. Therefore, the quadratic term of the polar
potential vanishes at the critical value $\ell_{\circ}$ of the angular
momentum $\ell$.

One of the most striking features of the near-horizon polar motion is that $Q$
is non-negative as a consequence of the reality of polar motion, as noticed in
Ref. [59]:

######  Proposition 3.1.

$\forall z\in[0,1]:v_{\theta}(z)\geq 0\Rightarrow Q\geq 0.$ (3.91)

_Proof._ This property is a consequence of the dependence on $Q$ of
$\mathcal{C}$ defined in (1.54). Indeed, using the fact that
$z=\cos^{2}\theta\in[0,1]$ one can write

$\displaystyle
Q=\mathcal{C}+\frac{3}{4}\ell^{2}-M^{2}\mu^{2}\geq\mathcal{C}+(1-\Lambda^{-2})\ell^{2}-M^{2}\mu^{2}\geq
v_{\theta}(z)\geq 0.$ (3.92)

A direct consequence is that the near-horizon polar motion cannot be vortical
and is consequently either equatorial, pendular, polar or axial. We note that
the condition $\epsilon_{0}\geq\ell^{2}$ is never obeyed in the near-horizon
case after using the definition (3.3), $a=M$ and $\hat{E}=\frac{\ell}{2M}$.
The equator-attractive class is therefore discarded. The resulting polar
classes are listed in Table 3.3 and the phase space is represented in Fig.3.4.

###### Radial behavior

The radial behavior for generic inclined orbits can be solved using the
equatorial results [58] thanks to the following observation:

######  Proposition 3.2.

The radial integrals $T_{R}^{(i)}(R)$ $(i=0,1,2)$ only depend upon the NHEK
energy $E$ and angular momentum $\ell$ while all the dependence upon the mass
$\mu$ and Carter constant $Q$ is through
$\ell_{*}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q}$.

This simple observation has far-reaching consequences. For any timelike
geodesic with $Q\neq 0$, one could directly reuse the classification
established in Ref. [58], modulo the substitution
$\frac{2}{\sqrt{3}}M\mu\to\ell_{*}$ in every expression encountered. Moreover,
null geodesics with $\mu=0$ have $Q\geq 0$ from Proposition 3.1. We can
therefore reuse the classification established in Ref. [58] to classify null
geodesics modulo the substitution $M\mu\to Q$ in every expression encountered.
Overall, all radial integrals can be described in closed form for all cases by
keeping the dependence upon $\ell_{*}$ or, equivalently, upon the Casimir
invariant $\mathcal{C}$.

Since the equatorial taxonomy of Ref. [58] did not consider bounded orbits and
only considered $\ell>0$, we will expand the taxonomy to the generic case. The
generic classification can be achieved by studying the roots of $v_{R}$ and
the range of $R$ where $v_{R}\geq 0$. We only consider orbits outside the
horizon, $R>0$. There are three broad categories depending on the angular
momentum: the supercritical case $|\ell|>\ell_{*}$ or equivalently
$\mathcal{C}<0$, the critical case $|\ell|=\ell_{*}$ or equivalently
$\mathcal{C}=0$ and the subcritical case $0\leq|\ell|<\ell_{*}$ or
$\mathcal{C}>0$.

The relative position of the critical radius (3.87) with respect to the roots
of $v_{R}$ may restrict the allowed classes of future-oriented orbits. As a
result of (3.88), subcritical $\ell^{2}<\ell^{2}_{*}$ orbits have either
$R_{+}<R_{c}$ for $\ell<0$ or $R_{c}<0$ for $\ell>0$, and all orbits are
future oriented. Critical orbits $\ell^{2}=\ell^{2}_{*}$ have either $R_{c}<0$
for $\ell=\ell_{*}$ or $R_{c}>R_{0}$ for $\ell=-\ell_{*}$. This restricts the
classes of orbits. Supercritical orbits $\ell^{2}>\ell^{2}_{*}$ with
$E,\ell>0$ are future directed. Supercritical orbits with $E>0$, $\ell<0$
admit $R_{-}<R_{c}<R_{+}$, and only bounded orbits with $R\leq R_{-}$ are
admissible. Finally, supercritical orbits with $E<0$ and $\ell>0$ obey $R\geq
R_{+}>R_{c}$ and are therefore deflecting.

After a simple analysis, we reach the following taxonomy, displayed in Table
3.4 and in Fig. 3.5. In comparison with Ref. [58], the classes
Outward$(E,\ell)$, Outward${}_{*}(E)$, Bounded${}_{>}(E,$ $\ell)$,
Bounded${}^{-}_{*}(E)$, and Bounded${}_{<}(E,\ell)$ are new, while all other
classes with $\ell>0$ appeared in Ref. [58]. The class
$\text{Osculating}(E,\ell)$ is now better called
$\text{Def}\text{lecting}(E,\ell)$. The classes with $\ell=\pm\ell_{*}$ will
be denoted with a subscript ∗. The Spherical∗ orbit with $\ell=\ell_{*}$ is
also the prograde ISSO. For $\ell\geq 0$, the conformal diagrams corresponding
to those orbits are depicted in Fig. 3.6 and their explicit forms are given in
Appendix B.2. Past-oriented geodesics (not depicted) are obtained from a
central symmetry around the origin $E=\ell=0$ as a result of the
$\uparrow\\!\downarrow$-flip (3.85).

Angular momentum | Carter constant | Polar range | Denomination
---|---|---|---
$\ell=0$ ($\mathcal{C}_{\circ}=-\ell_{\circ}^{2}/4$) | $Q=0$ | $z=0,~{}1$ |  |  Equatorial0
---
Axial0
| $Q>0$ | $z=1$ |  Axial${}^{0}(Q)$
$0<\ell<\ell_{\circ}$ ($-\frac{\ell_{\circ}^{2}}{4}<\mathcal{C}_{\circ}<0$) | $Q=0$ | $z=0$ |  Equatorial$(\ell)$
| $Q>0$ | $0\leq z\leq z_{+}$ |  Pendular$(Q,\ell)$
$\ell=\ell_{\circ}$ ($\mathcal{C}_{\circ}=0$) | $Q=0$ | $z=0$ |  Equatorial∘
| $Q>0$ | $0\leq z\leq z_{0}$ |  Pendular${}_{\circ}(Q)$
$\ell>\ell_{\circ}$ ($\mathcal{C}_{\circ}>0$) | $Q>0$ | $0\leq z\leq z_{+}$ |  Pendular$(Q,\ell)$
Table 3.3: Polar taxonomy of near-horizon geodesics with $\ell\geq 0$. The
orbits with $\ell<0$ are obtained from $\ell>0$ by flipping the sign of $\ell$
with the rest unchanged. Figure 3.4: Polar taxonomy of near-horizon geodesics.
For clarity, the scale is not respected on the horizontal axis. The dashed
blue curves $\ell^{2}=\ell^{2}_{*}$ represent the position of the spherical
orbits in parameter space. This figure contrasts with Figure 3.2.

Angular momentum (and Casimir) | Energy | Radial range | Denomination
---|---|---|---
Supercritical: $\ell>\ell_{*}$ ($-\ell^{2}<\mathcal{C}<0$) | $E>0$ | $0\leq R\leq\infty$ |  |  Plunging$(E,\ell)$
---
Outward$(E,\ell)$
| $E=0$ | $0<R\leq\infty$ |  Marginal$(\ell)$
| $E<0$ | $R_{+}\leq R\leq\infty$ |  Deflecting$(E,\ell)$
Critical: $\ell=\ell_{*}$ ($\mathcal{C}=0$) | $E>0$ | $0\leq R\leq\infty$ |  |  Plunging${}_{*}(E)$
---
Outward${}_{*}(E)$
| $E=0$ | $0<R\leq\infty$ |  Spherical∗ (ISSO)
Subcritical: $0\leq\ell^{2}<\ell_{*}^{2}$ ($0<\mathcal{C}\leq\frac{3\ell_{*}^{2}}{4}$) | $E>0$ | $0\leq R\leq R_{+}$ |  Bounded${}_{<}(E,\ell)$
Critical: $\ell=-\ell_{*}$ ($\mathcal{C}=0$) | $E>0$ | $0\leq R\leq R_{0}$ |  Bounded${}^{-}_{*}(E)$
Supercritical: $\ell<-\ell_{*}$ ($-\ell^{2}<\mathcal{C}<0$) | $E>0$ | $0\leq R\leq R_{-}$ |  Bounded${}_{>}(E,\ell)$
Table 3.4: Radial taxonomy of future-oriented geodesics in NHEK.  Figure 3.5: Radial taxonomy of future oriented geodesics in NHEK. For equatorial geodesics, $\ell_{*}=\frac{2}{\sqrt{3}}M\mu$, while for orbits with inclination, $\ell_{*}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q}$. |  |  |
---|---|---|---
(a) Spherical${}_{*}(ISSO)$ |  (b) Plunging${}_{*}(E)$ |  (c) Bounded${}^{-}_{*}(E)$ |  |  (d) Outward${}_{*}(E)$,
---
Outward$(E,\ell)$
|  |  |
|  |  |
(e) Bounded${}_{<}(E,\ell)$ |  |  (f) Bounded${}_{>}(E,\ell)$,
---
Deflecting$(E,\ell)$
(g) Marginal$(\ell)$ |  (h) Plunging$(E,\ell)$
Figure 3.6: Taxonomy of NHEK geodesics depicted in the global NHEK conformal
diagram. The upper (or respectively, lower) blue line represent the future
(respectively, past) event horizon $R=0$ and the dashed/dotted lines are the
roots of the radial potential. We used $M=1$, $E=\pm 1$ and $\ell=\pm
2\ell_{*}$ (and $\ell=\pm\frac{1}{2}\ell_{*}$, respectively) for supercritical
(and subcritical, respectively) trajectories.

a

|  Angular momentum
---
(and Casimir)
Energy | Radial range | Denomination
Supercritical: $\ell>\ell_{*}$ | $e>-\kappa\sqrt{-\mathcal{C}}$ | $\kappa\leq R\leq\infty$ |  |  Plunging$(e,\ell)$
---
Outward$(e,\ell)$
($-\ell^{2}<\mathcal{C}<0$) | $e=-\kappa\sqrt{-\mathcal{C}}<0$ | $R=\frac{\kappa\ell}{\sqrt{-\mathcal{C}}}$ |  Spherical$(\ell)$
| $e<-\kappa\sqrt{-\mathcal{C}}<0$ | $R_{+}\leq R\leq\infty$ |  Deflecting$(e,\ell)$
Critical: $\ell=\ell_{*}$ ($\mathcal{C}=0$) | $e>0$ | $\kappa\leq R\leq\infty$ |  |  Plunging${}_{*}(e)$
---
Outward${}_{*}(e)$
| $e=0$ | $\kappa\leq R\leq\infty$ |  |  Plunging∗
---
Outward∗
| $-\kappa\ell<e<0$ | $\kappa\leq R\leq R_{0}$ |  Bounded${}_{*}(e)$
|  Subcritical: $0\leq\ell^{2}<\ell_{*}^{2}$
---
($0<\mathcal{C}\leq\frac{3\ell_{*}^{2}}{4}$)
$e>-\kappa\ell$ | $\kappa\leq R\leq R_{+}$ |  Bounded${}_{<}(e,\ell)$
Critical: $\ell=-\ell_{*}$ ($\mathcal{C}=0$) | $e>-\kappa\ell>0$ | $\kappa\leq R\leq R_{0}$ |  Bounded${}_{*}^{-}(e)$
|  Supercritical: $\ell<-\ell_{*}$
---
($-\ell^{2}<\mathcal{C}<0$)
$e>-\kappa\ell>0$ | $\kappa\leq R\leq R_{-}$ |  Bounded${}_{>}(e,\ell)$
Table 3.5: Taxonomy of future-directed geodesics in near-NHEK.

#####  3.2.2 Near-NHEK

The only difference between NHEK and near-NHEK geodesic solutions lies in the
terms involving the radial coordinate. The proposition stating the equivalence
relation between the equatorial and inclined radial parts of the geodesic
motion takes the same form as in NHEK:

######  Proposition 3.3.

For a given normalization $\kappa$, the radial integrals
$t^{(i)}_{R;\kappa}(R)$ $(i=0,1,2)$ only depend upon the near-NHEK energy $e$
and angular momentum $\ell$ while all the dependence upon the mass $\mu$ and
Carter constant $Q$ is through
$\ell_{*}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q}$.

As in NHEK, the radial taxonomy of Ref. [58] is easily extended to bounded,
outward and/or retrograde orbits by studying the roots and the sign of
$v_{R;\kappa}(R)$. This leads to the classification displayed in Table 3.5 and
Figure 3.7.

The future-orientation condition (3.88) implies $e>-\kappa\ell$ for each orbit
that reaches the horizon at $R=\kappa$. In the case $\ell>\ell_{*}$ and $e<0$,
the condition $e\leq-\kappa\sqrt{-\mathcal{C}}$ implies $e+\kappa\ell\geq 0$
and therefore the parabola does not intersect the line. Past-oriented
geodesics (not depicted here) are obtained from a central symmetry around the
origin $e=\ell=0$ as a result of the $\uparrow\\!\downarrow$-flip (3.85). The
explicit expressions of all near-NHEK geodesics are listed in Appendix B.3.
Also notice that the energy range of the Deflecting$(e,\ell)$ class has been
corrected in this thesis with respect to the original derivation [20],
accordingly to the observation of [64].

Figure 3.7: Radial taxonomy of geodesics in near-NHEK. For equatorial
geodesics, $\ell_{*}=\frac{2}{\sqrt{3}}M\mu$, while for orbits with
inclination, $\ell_{*}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q}$.

#####  3.2.3 High-spin features of geodesic motion

Let us now discuss a few generic and universal features of near-horizon
geodesic motion holding in the high-spin case.

###### Radial motion

A first straightforward conclusion one can derive from the analysis of the
near-horizon radial geodesic motion is that

######  Proposition 3.4.

All radially unbounded NHEK or near-NHEK geodesics are prograde and either
critical or supercritical; _i.e._ , they satisfy $\ell\geq\ell_{*}$.

This feature of the near-horizon radial motion is directly visible in Figs.
3.5 and 3.7 and leads to remarkable consequences concerning the polar behavior
of such trajectories that we will derive in the following section.

The separatrix between bound and unbound motion is clearly visible in Figs.
3.5 and 3.7. It consists of the geodesic classes Plunging${}_{*}(E)$ and
Outward${}_{*}(E)$ for NHEK and the geodesic classes Plunging${}_{*}(e)$,
Outward${}_{*}(e)$, and Bounded${}_{*}(e)$ for near-NHEK that each lie at the
critical angular momentum line $\ell=\ell_{*}$.

###### Polar motion

The polar motion of both NHEK and near-NHEK trajectories is bounded in an
interval around the equator,
$\theta_{\text{\text{min}}}\leq\theta\leq\pi-\theta_{\text{\text{min}}}$,
where $\cos\theta_{\text{\text{min}}}=\sqrt{z_{+}}$ or
$\cos\theta_{\text{\text{min}}}=\sqrt{z_{0}}$. The maximal polar angle is
determined for $\ell^{2}\neq\ell^{2}_{\circ}=4M^{2}\mu^{2}$ as

$z_{+}(\ell,Q)=\frac{3\ell^{2}+4(Q+M^{2}\mu^{2})-\sqrt{9\ell^{4}+16(M^{2}\mu^{2}-Q)^{2}+8\ell^{2}(3M^{2}\mu^{2}+5Q)}}{2(4M^{2}\mu^{2}-\ell^{2})}$

and for $\ell=\pm\ell_{\circ}$ as

$\displaystyle z_{0}(Q)=\lim_{\ell\rightarrow\pm
2M\mu}z_{+}=\frac{Q}{Q+4M^{2}\mu^{2}}.$ (3.147)

Remember that $Q\geq 0$ by consistency of polar motion. The asymptotic values
are

$\displaystyle\lim_{\scriptsize{\begin{array}[]{l}Q\to 0\\\ \ell\text{
fixed}\end{array}}}z_{+}(\ell,Q)$
$\displaystyle=0,\qquad\qquad\qquad\qquad\qquad\;\;\;\;\;\;\;\;\lim_{\scriptsize{\begin{array}[]{l}Q\to\infty\\\
\ell\text{ fixed}\end{array}}}z_{+}(\ell,Q)=1,$ (3.152)
$\displaystyle\lim_{\scriptsize{\begin{array}[]{l}\ell\to 0\\\ Q\text{
fixed}\end{array}}}z_{+}(\ell,Q)$
$\displaystyle=\left\\{\begin{array}[]{cl}\frac{Q}{M^{2}\mu^{2}}&\text{ if
}Q<M^{2}\mu^{2}\\\ 1&\text{ if }Q\geq
M^{2}\mu^{2}\end{array}\right.,\qquad\lim_{\scriptsize{\begin{array}[]{l}\ell\to\infty\\\
Q\text{ fixed}\end{array}}}z_{+}(\ell,Q)=0.$ (3.159)

For fixed $\ell$, $z_{+}$ is a monotonic function of $Q$, and reciprocally
$z_{+}$ is monotonic in $\ell$ at fixed $Q$. The pendular oscillation around
the equatorial plane will explore a larger range of $\theta$ when
$\theta_{\text{\text{min}}}$ is smallest or $z_{+}$ closer to 1, which occurs
either for small $\ell$ and $Q\geq M^{2}\mu^{2}$ or large $Q$.

Now, one can check that for critical or supercritical angular momentum
$\ell^{2}\geq\ell^{2}_{*}(Q)$, one has $z_{+}<2\sqrt{3}-3$ for
$\ell\neq\ell_{\circ}(Q)$ and $z_{0}<2\sqrt{3}-3$ for
$\ell^{2}=\ell^{2}_{\circ}$. The special angle

$\displaystyle\theta_{\text{VLS}}\triangleq\arccos{\sqrt{2\sqrt{3}-3}}\approx
47^{\circ}$ (3.160)

is in fact the velocity-of-light surface in the NHEK geometry (1.43) (or near-
NHEK geometry) defined as the polar angle such that $\partial_{T}$ is null. It
obeys $\Lambda(\theta_{\text{VLS}})=1$. The polar region closer to either the
north or south poles admits a timelike Killing vector, namely $\partial_{T}$.
On the contrary, the polar region around the equator
$\theta\in]\theta_{\text{VLS}},\pi-\theta_{\text{VLS}}[$ does not admit a
timelike Killing vector. The velocity-of-light surface separates these two
polar regions. We have therefore proven the following property:

######  Proposition 3.5.

All critical or supercritical orbits $\ell^{2}\geq\ell^{2}_{*}$ in (near-)NHEK
geometry lie in the polar region
$\theta\in]\theta_{\text{VLS}},\pi-\theta_{\text{VLS}}[$ where there is no
timelike Killing vector. This applies in particular to all spherical orbits.

The subcritical orbits $\ell^{2}<\ell^{2}_{*}$ can explore all polar regions
of the (near)-NHEK geometry. As a consequence of Propositions 3.4 and 3.5, we
have

######  Proposition 3.6.

All radially unbounded geodesics in (near-)NHEK geometry lie in the polar
region $\theta\in]\theta_{\text{VLS}},\pi-\theta_{\text{VLS}}[$ bounded by the
velocity-of-light surface.

In particular, for null geodesics, this feature provides the “NHEKline” in the
imaging of light sources around a nearly extreme Kerr black hole [30, 80]. In
[79, 57], Proposition 3.6 was proven for null geodesics. Here, we show that it
is a generic property of all timelike geodesics as well.

####  3.3 Spherical geodesics

The spherical (near-)NHEK geodesics take a distinguished role among all
geodesics. First, a subclass of spherical geodesics in NHEK and near-NHEK
constitute the innermost stable spherical orbits (ISSOs) and the innermost
spherical bound orbits (ISBOs) in the high-spin limit, respectively. Our first
motivation is to fully characterize the ISSO, in order to generalize the
analysis of the inspiral/merger transition performed around the equatorial
plane in the high-spin limit [96, 97] to inclined orbits.

Second, as noticed in Ref. [58], the equatorial NHEK (resp. near-NHEK) orbits
are the simplest representatives for each equivalence class of prograde
incoming critical (respectively, supercritical) equatorial orbits under
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times\mathbb{Z}_{2}$ symmetry.
We will show in Sec. 3.4 that the spherical (near-)NHEK orbits are the
simplest representatives for each equivalence class of arbitrary timelike
(near-)NHEK geodesics under
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times(\mathbb{Z}_{2})^{3}$
symmetry without any restriction. These two reasons justify the comprehensive
study of the spherical geodesics.

#####  3.3.1 Innermost stable spherical orbits

The ISSOs are defined as the last stable spherical orbits of Kerr. They are
defined from the solutions to

$\displaystyle R(r)=R^{\prime}(r)=R^{\prime\prime}(r)=0$ (3.161)

where $R$ is defined in (1.25b). They admit a constant radius $r$ and a fixed
$\hat{E}$ and $\ell$, which can be obtained as solutions of polynomial
equations which we will not give explicitly. There are two branches at
positive $\hat{E}$ corresponding to prograde ($\ell\geq 0$) and retrograde
orbits ($\ell<0$). For the Schwarzschild black hole, the parameters on the two
branches of the ISSO are

$\displaystyle
r_{\text{ISSO}}=6M,\qquad\frac{\hat{E}_{\text{ISSO}}}{\mu}=\frac{2\sqrt{2}}{3},\qquad\frac{\ell_{\text{ISSO}}}{\mu
M}=\pm\sqrt{12-\frac{Q}{M^{2}\mu^{2}}},$ (3.162)

which implies the bound $Q\leq 12M^{2}\mu^{2}$.

For arbitary spin, the innermost stable circular orbit (ISCO) is defined as
the prograde ISSO equatorial orbit, i.e. restricted to $Q=0$
($\theta=\frac{\pi}{2}$). The parameters are [30]

$\displaystyle\frac{\hat{E}_{\text{ISCO}}}{M\mu}=\frac{1-2/\tilde{r}_{\text{ISCO}}-\tilde{a}/\tilde{r}_{\text{ISCO}}^{3/2}}{\sqrt{1-3/\tilde{r}_{\text{ISCO}}-2\tilde{a}/\tilde{r}_{\text{ISCO}}^{3/2}}},\qquad\frac{\ell_{\text{ISCO}}}{\mu
M}=\frac{2}{\sqrt{3\tilde{r}_{\text{ISCO}}}}(3\sqrt{\tilde{r}_{\text{ISCO}}}+2\tilde{a}),$
(3.163)

where $\tilde{a}=a/M$ and

$\displaystyle\tilde{r}_{\text{ISCO}}$
$\displaystyle\triangleq\frac{r_{\text{ISCO}}}{M}=3+Z_{2}-\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})},$
(3.164a) $\displaystyle Z_{1}$ $\displaystyle\triangleq
1+(1-\tilde{a}^{2})^{1/3}[(1+\tilde{a})^{1/3}+(1-\tilde{a})^{1/3}],\qquad
Z_{2}\triangleq\sqrt{3\tilde{a}^{2}+(Z_{1})^{2}}.$ (3.164b)

###### Minimal polar angle

In the generic case $\ell\neq 0$, the polar motion is pendular – i.e.,
oscillating around the equator in the interval
$[\theta_{\text{\text{min}}},\pi-\theta_{\text{\text{min}}}]$. The minimal
angle as a function of the spin $a$ and ISCO radius $r_{\text{ISSO}}$ can
simply be found by solving numerically the three equations (3.161) that define
the ISSO together with the condition that there is a polar turning point,
$\Theta(\cos\theta_{\text{\text{min}}})=0$ where $\Theta(\cos^{2}\theta)$ is
defined in (1.25c). The resulting minimal angle is displayed in Fig. 3.8 for a
large range of spins including nearly extremal. This completes a similar plot
drawn in Ref. [98] for spins far from extremality.

Figure 3.8: $\cos\theta_{\text{\text{min}}}$ as a function of ISSO radius for
several black hole spins $a$. Figure 3.9: Euclidean embedding of the ISSO
using the Boyer-Lindquist radius $r$, azimuthal angle $\varphi$ and polar
angle $\theta$ for $a=0.9999$. A cone is drawn at the critical polar angle
beyond which the ISSO lies in the NHEK region. In the extremal limit, this
polar angle is $\theta\approx 65^{\circ}$.

We note that for high-spins, the radius asymptotes to $r=M$ and the minimal
angle reaches a critical value around $0.42$ radians or $65^{\circ}$. When the
motion reaches regions sufficiently far from the equatorial plane, the ISSO
radius increases steeply and leaves the near-horizon region $r\simeq M$.
Another graphical representation of this behavior is shown in Fig. 3.9. We
will explain these features in the next section.

#####  3.3.2 The NHEK spherical orbit and the high-spin ISSOs

In the high-spin limit $a\rightarrow M$, the prograde ISSOs are characterized
by the following Boyer-Lindquist energies and angular momentum:

$\displaystyle\hat{E}_{\text{ISSO}}=\frac{1}{\sqrt{3}M}\sqrt{M^{2}\mu^{2}+Q},\qquad\ell_{\text{ISSO}}=+2M\hat{E}_{\text{ISSO}}$
(3.165)

and the following Boyer-Lindquist radius:

$\displaystyle
r_{\text{ISSO}}=M+M\left(\frac{Q+M^{2}\mu^{2}}{-Q+\frac{M^{2}\mu^{2}}{2}}\right)^{1/3}\lambda^{2/3}+\mathcal{O}(\lambda^{4/3}).$
(3.166)

Given the scaling in $\lambda$, for the range

$\displaystyle 0\leq Q\leq\frac{M^{2}\mu^{2}}{2},$ (3.167)

the ISSOs belong to the NHEK geometry and admit the NHEK radius

$\displaystyle
R=R_{\text{ISSO}}\triangleq\left(\frac{Q+M^{2}\mu^{2}}{-Q+\frac{M^{2}\mu^{2}}{2}}\right)^{1/3}.$
(3.168)

In particular, the ISCO has the minimal radius $R_{\text{ISCO}}=2^{1/3}$. In
terms of NHEK quantities, the orbits admit a critical angular momentum and a
vanishing NHEK energy,

$\displaystyle\ell=\ell_{*}\triangleq\frac{2}{\sqrt{3}}\sqrt{Q+M^{2}\mu^{2}},\qquad
E=0.$ (3.169)

In the high-spin limit, the prograde ISSOs in the range (3.167) are therefore
exactly the $\text{Spherical}_{*}(Q)$ orbits in the classification of Sec.
1.3.2. The prograde ISSOs outside the range (3.167) and the retrograde ISSOs
do not belong to the near-horizon geometry and will not be described here.

In terms of polar behavior, $\text{Spherical}_{*}(Q)$ orbits are instances of
$\text{Pendular}(Q,\ell_{*})$ motion (except for $Q=0$, where they are just
equatorial orbits). In the range (3.167), they admit an $\epsilon_{0}$ as
defined in (3.3) given by $\epsilon_{0}=\frac{Q-2M^{2}\mu^{2}}{3}<0$, and the
angular momentum lies below the value $\ell_{\circ}$:

$\displaystyle\ell_{*}\leq\sqrt{2}M\mu<\ell_{\circ}.$ (3.170)

The main property of $\text{Pendular}(Q,\ell_{*})$ motion is that the polar
angle $\theta$ is bounded in an interval around the equator (see (3.68) and
(3.4)) :

$\theta\in[\theta_{\text{min}},\pi-\theta_{\text{min}}]$ (3.171)

where

$\cos\theta_{\text{min}}=\sqrt{z_{+}}=\sqrt{\frac{Q}{\frac{3}{4}\ell^{2}_{*}+\sqrt{\frac{9}{16}\ell_{*}^{4}-\frac{\ell_{*}^{2}Q}{2}+Q^{2}}}}.$
(3.172)

At fixed $M\mu$, $\theta_{\text{min}}(Q)$ is a monotonic function
interpolating between the equator $\theta=90^{\circ}$ at $Q=0$ and
$\theta_{\text{VLS}}\triangleq\arccos{\sqrt{2\sqrt{3}-3}}\approx 47^{\circ}$
for $Q\rightarrow\infty$. The special angle $\theta_{\text{VLS}}$ is the
velocity-of-light surface in the NHEK geometry (1.43) as described in Sec.
3.2.3. The ISSO therefore always lies in the region of NHEK spacetime around
the equator, where there is no timelike Killing vector. This is depicted in
Fig. 3.10.

|  |  |  … |
---|---|---|---|---

$Q$

Figure 3.10: For increasing $Q\geq 0$, $\text{Spherical}_{*}(Q)$ orbits can
explore a equator-centered band whose width becomes larger, finally reaching
for $Q\rightarrow\infty$ the angular range
$\theta\in[\theta_{\text{VLS}},\pi-\theta_{\text{VLS}}]$ bounded by the
velocity-of-light surface. The prograde IBSOs lie in the near-NHEK region for
$Q\leq 2M^{2}\mu^{2}$, which further bounds the angular range.

However, since the ISSO admits the range (3.167) due to its relationship to
the asymptotically flat Boyer-Lindquist radius (3.166), the limiting angle is
reached first for $Q=\frac{M^{2}\mu^{2}}{2}$ at
$\arccos{\sqrt{3-2\sqrt{2}}}\approx 65^{\circ}$. This explains the behavior
depicted in Fig. 3.8. This result was discovered simultaneously in [62, 20].
The limiting angle of the ISSO is given by
$\arcsin{\sqrt{2(\sqrt{2}-1)}}=\arccos{\sqrt{3-2\sqrt{2}}}\approx 65^{\circ}$.

#####  3.3.3 The near-NHEK spherical orbits and the high-spin IBSOs

The innermost bound spherical orbits (IBSOs) are determined by the equations

$\displaystyle R(r)=R^{\prime}(r)=0,\qquad\hat{E}=\mu.$ (3.174)

In the high-spin limit $\lambda\rightarrow 0$, the angular momentum and Boyer-
Lindquist radius of the prograde IBSOs are given by

$\displaystyle\ell$
$\displaystyle=\ell_{\circ}\left(1+\frac{\lambda}{\sqrt{2}}\sqrt{1-\frac{Q}{2M^{2}\mu^{2}}}+\mathcal{O}(\lambda^{2})\right),$
(3.175a) $\displaystyle r$
$\displaystyle=M\left(1+\frac{\sqrt{2}\lambda}{\sqrt{1-\frac{Q}{2M^{2}\mu^{2}}}}+\mathcal{O}(\lambda^{2})\right)$
(3.175b)

where $\ell_{\circ}\equiv 2M\mu$. In particular, for $Q=0$ we recover the
scaling of the innermost bound circular orbit (IBCO) [30]. Given the scaling
$\sim\lambda$, the prograde IBCOs therefore lie in the near-NHEK region for
all $Q<2M^{2}\mu^{2}$. Using (1.63)–(1.72), the angular momentum, near-NHEK
energy and near-NHEK radius are given in the high-spin limit by

$\displaystyle\ell$ $\displaystyle=\ell_{\circ},$ (3.176a)
$\displaystyle\frac{e}{\kappa}$ $\displaystyle=-\sqrt{2M^{2}\mu^{2}-Q},$
(3.176b) $\displaystyle\frac{r}{\kappa}$
$\displaystyle=\frac{\sqrt{2}\lambda}{\sqrt{1-\frac{Q}{2M^{2}\mu^{2}}}}.$
(3.176c)

The prograde IBCOs in the range $0\leq Q<2M^{2}\mu^{2}$ are described by
instances of Spherical$(\ell)$ orbits. In terms of polar motion, $Q=0$ are
equatorial and $Q>0$ are pendular of class Pendular${}_{\circ}(Q)$; see Table
3.3. The polar range is determined as
$\theta_{\text{\text{min}}}\leq\theta\leq\pi-\theta_{\text{\text{min}}}$ where

$\displaystyle\theta_{\text{\text{min}}}=\arccos\sqrt{\frac{Q}{Q+\ell_{\circ}^{2}}}.$
(3.177)

The maximal polar angle reachable within the near-NHEK region by IBSOs is
obtained for the limiting value $Q=2M^{2}\mu^{2}$ at

$\displaystyle\theta_{\text{\text{min}}}=\arccos\sqrt{1/3}=\arcsin{\sqrt{2/3}}\approx
55^{\circ}.$ (3.178)

This critical angle was also previously obtained in Refs. [78, 62]. Finally,
note that spherical photon orbits in the high-spin limit were also discussed
in Refs. [99, 100].

####  3.4 Conformal mappings between radial classes

The near-horizon region of near-extremal Kerr black holes admits four Killing
vectors forming the group $\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)$,
hereafter denoted as the conformal group $G$. The geodesic equations are
invariant under $G$ and the geodesics therefore transform under the action of
$G$. Moreover, a group generated by four $\mathbb{Z}_{2}$ symmetries exists
that preserve the geodesic equations. The subgroup preserving the domain $R>0$
for NHEK (or $r>0$ for near-NHEK) is generated by the
$\uparrow\\!\downarrow$-flip (3.85), which flips the geodesic orientation, and
two additional $\mathbb{Z}_{2}$ transformations that preserve the geodesic
orientation: namely, the parity flip

$\displaystyle\theta\rightarrow\pi-\theta,\qquad\Phi\rightarrow\Phi+\pi,\qquad
s_{\theta}^{i}\rightarrow-s_{\theta}^{i},$ (3.179)

and the $\rightleftarrows$-flip

$\displaystyle
T\rightarrow-T,\qquad\Phi\rightarrow-\Phi,\qquad\lambda\rightarrow-\lambda,\qquad
s_{R}^{i}\rightarrow-s_{R}^{i},\qquad s_{\theta}^{i}\rightarrow-
s_{\theta}^{i}.$ (3.180)

The last discrete transformation that we use as a basis is the
$\rightleftarrows$ -flip

$\displaystyle
R\rightarrow-R,\qquad\Phi\rightarrow-\Phi,\qquad\ell\rightarrow-\ell,\qquad
s^{i}_{R}\rightarrow-s^{i}_{R}.$ (3.181)

The parity transformation defined in (3.179) leaves each motion invariant and
will not be considered further. The $\rightleftarrows$-flip changes the
boundary conditions of the geodesics, which may affect their denomination. It
maps bounded orbits to bounded orbits, and deflecting orbits to deflecting
orbits, but plunging orbits to outward orbits, as illustrated in Fig. 3.11.
For bounded orbits, the part before the turning point is mapped to the part
after the turning point, and vice-versa. The  $\rightleftarrows$ -flip can be
used as follows: one first continues a geodesic defined in $R>0$ beyond the
horizon $R=0$ and the resulting geodesic with $R<0$ is then mapped to a
geodesic in the $R>0$ region using the  $\rightleftarrows$ -flip. Together
with the action of (3.180), it allows us to map plunging orbits with $\ell>0$
to bounded orbits with $\ell<0$. This process is illustrated in Fig. 3.12.

|
---|---
(a) | (b)
Figure 3.11: Penrose diagram of NHEK spacetime depicting the action of the $\rightleftarrows$-flip on (a) plunging and (b) bounded geodesics. Under this transformation, a trajectory belonging to the patch I is mapped to an orbit of the patch I’. While plunging geodesics become outward ones, bounded motion remains bounded. The energy and angular momentum of the trajectory are unchanged. |  |
---|---|---
(a) | (b) | (c)
Figure 3.12: Penrose diagram representation of the construction of a critical
NHEK bounded geodesic from a plunging one. (a) Continuation of the trajectory
beyond the horizon in patch I’ until the radial potential root (depicted with
dashes); (b)

$\rightleftarrows$

-flip which brings the part of the bounded geodesic before the turning point in the NHEK Poincaré patch I; (c) $\rightleftarrows$-flip, which maps the part of the bounded geodesic before the turning point to the part after the turning point.

The equivalence classes of equatorial critical and supercritical prograde
timelike geodesics under the action of
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times\uparrow\\!\downarrow$
symmetry were derived in Ref. [58] following earlier work [73, 74, 75, 77]. In
this section, we will perform the decomposition of arbitrary geodesics into
equivalence classes under the action of
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times\uparrow\\!\downarrow\times\rightleftarrows\times$
$\rightleftarrows$ .

The Casimir $\mathcal{C}$ of $\textsf{SL}(2,\mathbb{R})$ cannot vary upon
acting with $G\triangleq\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)$
transformations. Moreover, the action of the group $G$ acts trivially on the
polar coordinate $\theta$. These two properties imply that both $Q$ and $\ell$
are invariant under the action of $G$. In particular, critical, supercritical
or subcritical geodesics form distinct classes under $G$. On the contrary, the
(near-)NHEK energy $E$ (or $e$) can vary under conformal transformations.
Conformal transformations can map NHEK to near-NHEK orbits, and vice-versa. As
a result of Propositions 3.2 and 3.3, null geodesics can be treated on the
same footing as timelike geodesics.

A conformal transformation belonging to
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)$ maps (near)-NHEK spacetime
parametrized by $(T,R,\theta,\Phi)$ to (near-)NHEK spacetime parametrized by
$(\bar{T},\bar{R},\theta,\bar{\Phi})$222We denote here without distinction
NHEK and near-NHEK coordinates with capital letters. where

$\displaystyle\overline{T}$ $\displaystyle=\overline{T}(T,R),$
$\displaystyle\overline{R}$ $\displaystyle=\overline{R}(T,R),$ (3.186)
$\displaystyle\overline{\Phi}$ $\displaystyle=\Phi+\delta\bar{\Phi}(T,R).$

The geodesic equations in (near)-NHEK imply $T=T(R)$. Therefore, the action of
conformal symmetries reduces to an action on the radial motion, leaving the
polar motion unchanged. More precisely, in the decomposition of
$\Phi(\lambda)$ (1.60c)–(1.74c) in terms of a radial part and a polar part,
the polar part will remain untouched by conformal transformations.

It was shown in Ref. [58] that each equivalence class of equatorial prograde
critical (respectively, supercritical) geodesics with incoming boundary
conditions under $G\times\uparrow\\!\downarrow$ admits a distinguished simple
representative, namely the NHEK (respectively, near-NHEK) circular orbits.
After analysis, we obtain that each geodesic equivalence class under
$G\times\uparrow\\!\downarrow\times\rightleftarrows\times$ $\rightleftarrows$
admits a spherical orbit as the simplest representative as illustrated in Fig.
3.13. Past directed geodesics must be considered as intermediate steps in
order to relate each future directed geodesic to spherical geodesics.
Supercritical orbits ($\ell^{2}>\ell^{2}_{*}$) admit the near-NHEK
Spherical$(\ell)$ orbit as a representative and critical orbits
($\ell=\pm\ell_{*}$) admit the NHEK Spherical∗ orbit as a representative. No
subcritical spherical geodesic exists. However, we introduce an analytically
continued complex subcritical geodesic by continuing the radius $R_{0}\mapsto
iR_{0}$ and show that it generates the subcritical class.

The explicit formulas for the three categories of equivalence classes of
orbits under $G\times\uparrow\\!\downarrow\times\rightleftarrows\times$
$\rightleftarrows$ are given in the following sections. We will denote the
final coordinates and orbital parameters reached by the conformal mappings
with bars.

#####  3.4.1 Critical $\mathcal{C}=0$

###### Spherical∗ $\Leftrightarrow$ Plunging${}_{*}(E)$ (NHEK/NHEK).

The conformal mapping is given by

$\displaystyle\bar{T}$ $\displaystyle=-\frac{R^{2}T}{R^{2}T^{2}-1},$
$\displaystyle\bar{R}$ $\displaystyle=\frac{R^{2}T^{2}-1}{R},$ (3.187)
$\displaystyle\bar{\Phi}$ $\displaystyle=\Phi+\log\frac{RT+1}{RT-1}-i\pi.$

It maps a (future-directed) NHEK spherical trajectory of radius $R_{0}$ to a
(future-directed) critical plunge of energy
$\bar{E}=\frac{2\ell_{*}}{R_{0}}>0$.

###### Spherical∗ $\Leftrightarrow$ Plunging∗ (NHEK/near-NHEK).

One performs the NHEK/near-NHEK diffeomorphism
$(T,R,\theta,\Phi)\to(\bar{t},\bar{R},\theta,\bar{\phi})$, whose explicit form
is

$\displaystyle T$
$\displaystyle=-\exp\quantity(-\kappa\bar{t})\frac{\bar{R}}{\sqrt{\bar{R}^{2}-\kappa^{2}}},$
$\displaystyle R$
$\displaystyle=\frac{1}{\kappa}\exp\quantity(\kappa\bar{t})\sqrt{\bar{R}^{2}-\kappa^{2}},$
(3.188) $\displaystyle\Phi$
$\displaystyle=\phi-\frac{1}{2}\log\frac{\bar{R}-\kappa}{\bar{R}+\kappa}.$

Its inverse is

$\displaystyle\bar{t}$
$\displaystyle=\frac{1}{\kappa}\log\frac{R}{\sqrt{R^{2}T^{2}-1}},$
$\displaystyle\bar{R}$ $\displaystyle=-\kappa RT,$ (3.189)
$\displaystyle\bar{\phi}$
$\displaystyle=\Phi+\frac{1}{2}\log\frac{RT+1}{RT-1}$

for $R>0$ and $RT<-1$. The orbital parameters are related as

$\displaystyle R_{0}=\frac{1}{\kappa}\exp\quantity(\kappa
t_{0}),\qquad\Phi_{0}=\phi_{0}-\frac{3}{4}.$ (3.190)

###### Plunging∗ $\Leftrightarrow$ Outward∗ (near-NHEK/near-NHEK).

The orbits are related by the $\rightleftarrows$-flip (3.180).

###### Plunging∗ $\Leftrightarrow$ Plunging${}_{*}(e)$ (near-NHEK/near-NHEK).

The two (future-directed) orbits are related via the diffeomorphism

$\displaystyle\bar{t}$
$\displaystyle=\frac{1}{2\kappa}\log\frac{\sqrt{R^{2}-\kappa^{2}}\cosh{\kappa
t}-R}{\sqrt{R^{2}-\kappa^{2}}\cosh{\kappa t}+R}-\frac{i\pi}{\kappa},$
$\displaystyle\bar{R}$ $\displaystyle=\sqrt{R^{2}-\kappa^{2}}\sinh{\kappa t},$
(3.191) $\displaystyle\bar{\phi}$
$\displaystyle=\phi+\frac{1}{2}\log\frac{R\sinh\kappa t+\kappa\cosh\kappa
t}{R\sinh\kappa t-\kappa\cosh\kappa t}.$

The energy of the new trajectory is a function of the initial time $t_{0}$ of
the former one:

$\bar{e}=\kappa^{2}\ell_{*}\exp\quantity(-\kappa t_{0})>0.$ (3.192)

###### Plunging${}_{*}(e)$ $\Leftrightarrow$ Outward${}_{*}(e)$ (near-
NHEK/near-NHEK).

The orbits are related by the $\rightleftarrows$-flip.

###### Plunging${}_{*}(E)$ $\Leftrightarrow$ Bounded${}_{*}^{-}(E)$
(NHEK/NHEK).

The critical bounded orbit is obtained from the plunging orbit by a
continuation of the trajectory beyond the horizon ($R<0$) combined with
$\mathbb{Z}_{2}$ flips. One must proceed in three steps:

  1. 1.

Continue the plunge defined from the physical domain $0\leq R\leq\infty$ to
its whole domain of definition $R_{0}\leq R\leq\infty$ (i.e., up to the root
of the radial potential $R_{0}=-\frac{e}{2\ell_{*}}$) and consider now only
the part of the trajectory located beyond the horizon $R_{0}\leq R\leq 0$.

  2. 2.

Apply the  $\rightleftarrows$ -flip to the latter part of the solution. This
transformation restores the positivity of the radial coordinate. It preserves
the time orientation of the geodesic but flips the sign of its angular
momentum $\ell_{*}\to-\ell_{*}$. The new domain of definition of the
trajectory is consequently $0\leq R\leq\frac{E}{2\ell_{*}}$.

  3. 3.

The procedure outlined above only leads to the part of the geodesic with
$R^{\prime}(\lambda)>0$, which is located before the turning point. As
outlined in Appendix B.2, the part of a bounded trajectory located after the
turning point can be obtained from the one located before it by a
$\rightleftarrows$-flip.

This whole procedure is represented in Fig. 3.12.

###### Plunging${}_{*}(e)$ $\Leftrightarrow$ Bounded${}_{*}^{-}(e)$ (near-
NHEK/near-NHEK).

The mapping is similar to the one outlined above using the  $\rightleftarrows$
-flip. One subtlety is that one should start with the Plunging${}_{*}(e)$
orbit with $e>\kappa\ell_{*}$ in order to obtain the future-directed
Bounded${}_{*}^{-}(e)$ orbit.

###### Plunging${}_{*}(e)$ $\Leftrightarrow$ Bounded${}_{*}(-e)$ (near-
NHEK/near-NHEK).

We apply the  $\rightleftarrows$ -flip as outlined in the previous paragraph,
but now choosing $0<e<\kappa\ell_{*}$. This leads to a retrograde past-
directed bounded orbit. The future-directed prograde geodesic is then reached
using the $\uparrow\\!\downarrow$-flip.

#####  3.4.2 Supercritical $\mathcal{C}<0$

###### Spherical$(\ell)$ $\Leftrightarrow$ Marginal$(\ell)$ (near-NHEK/NHEK).

One applies the NHEK/near-NHEK diffeomorphism

$\displaystyle T$
$\displaystyle=-\exp\quantity(-\kappa\bar{t})\frac{\bar{R}}{\sqrt{\bar{R}^{2}-\kappa^{2}}},$
$\displaystyle R$
$\displaystyle=\frac{1}{\kappa}\exp\quantity(\kappa\bar{t})\sqrt{\bar{R}^{2}-\kappa^{2}},$
(3.193) $\displaystyle\Phi$
$\displaystyle=\phi-\frac{1}{2}\log\frac{\bar{R}-\kappa}{\bar{R}+\kappa}$

which maps the orbit Spherical$(\ell)$ on the past-directed Marginal$(-\ell)$
orbit. The future-directed Marginal$(\ell)$ orbit is recovered by composing
this transformation with a $\uparrow\\!\downarrow$-flip.

###### Marginal$(\ell)$ $\Leftrightarrow$ Plunging$(E,\ell)$ or
Deflecting$(E,\ell)$ (NHEK/NHEK).

One performs the transformation ($\zeta\neq 0$)

$\displaystyle\bar{T}$
$\displaystyle=\frac{1}{\bar{R}}\frac{2R^{2}T\cos\zeta-(1+R^{2}(1-T^{2}))\sin\zeta}{2R},$
$\displaystyle\bar{R}$
$\displaystyle=\frac{R^{2}(1+T^{2})-1+(1+R^{2}(1-T^{2}))\cos\zeta+2R^{2}T\sin\zeta}{2R},$
(3.194) $\displaystyle\bar{\Phi}$
$\displaystyle=\Phi+\log\frac{\cos\frac{\zeta}{2}R+\sin\frac{\zeta}{2}(RT+1)}{\cos\frac{\zeta}{2}R+\sin\frac{\zeta}{2}(RT-1)}.$

As outlined in Ref. [58], this mapping can be viewed as the action on Poincaré
NHEK coordinates of a shift of the global NHEK time $\tau\to\tau-\zeta$. The
energy of the final orbit is

$\displaystyle\bar{E}=\sqrt{-\mathcal{C}}\quantity(\sin\zeta+T_{0}(\cos\zeta-1)).$
(3.195)

We directly see that any energy $E\neq 0$ can be reached by conveniently
choosing the values of $T_{0}$ and $\zeta$.

###### Plunging$(E,\ell)$ $\Leftrightarrow$ Outward$(E,\ell)$ (NHEK/NHEK).

The orbits are related by the $\rightleftarrows$-flip.

###### Plunging$(E,\ell)$ $\Leftrightarrow$ Bounded${}_{>}(E,-\ell)$
(NHEK/NHEK).

The mapping consists in extending the radial range of the plunging orbit
beyond the horizon, $R<0$, then using the  $\rightleftarrows$ -flip, which
leads to the Bounded${}_{>}(E,-\ell)$ orbit.

###### Spherical$(\ell)$ $\Leftrightarrow$ Plunging$(e,\ell)$ or
Deflecting$(e,\ell)$ (near-NHEK/ near-NHEK).

One uses the diffeomorphism ($\chi\neq\pm 1$)

$\displaystyle t$
$\displaystyle=\frac{1}{\kappa}\log\frac{\sqrt{\bar{R}^{2}-\kappa^{2}}\cosh\kappa\bar{t}-\bar{R}}{\sqrt{R^{2}-\kappa^{2}}},$
$\displaystyle R$
$\displaystyle=\sqrt{\bar{R}^{2}-\kappa^{2}}\quantity(\sinh\kappa\bar{t}+\chi\cosh\kappa\bar{t})-\chi\bar{R},$
(3.196) $\displaystyle\phi$
$\displaystyle=\bar{\phi}-\frac{1}{2}\log\quantity[\frac{\sqrt{\bar{R}^{2}-\kappa^{2}}-\bar{R}\cosh\kappa\bar{t}+\kappa\sinh\kappa\bar{t}}{\sqrt{\bar{R}^{2}-\kappa^{2}}-\bar{R}\cosh\kappa\bar{t}-\kappa\sinh\kappa\bar{t}}\frac{R+\kappa}{R-\kappa}].$

This mapping can be seen as a NHEK global time shift written in near-NHEK
coordinates; see Refs. [58, 77]. The explicit inversion formula can be found
in Ref. [77]. The energy of the new trajectory reads as

$\displaystyle\bar{e}=\kappa\sqrt{-\mathcal{C}}\,\chi.$ (3.197)

For $-\frac{\ell}{\sqrt{-\mathcal{C}}}<\chi<-1$, the orbit reached is future-
directed and deflecting. The trajectory becomes plunging for $\chi>-1$. Note
that for $\absolutevalue{\chi}>1$,
$\bar{t}_{0}=-\frac{1}{2\kappa}\log\frac{1+\chi}{1-\chi}$ is complex and one
has to perform an additional shift on $\bar{t}$ to make it real.

###### Plunging$(e,\ell)$ $\Leftrightarrow$ Outward$(e,\ell)$ (near-NHEK/near-
NHEK).

The orbits are related by the $\rightleftarrows$-flip.

###### Plunging$(e,\ell)$ $\Leftrightarrow$ Bounded${}_{>}(e,-\ell)$ (near-
NHEK/near-NHEK).

The mapping consists in extending the radial range of the plunging orbit with
$e>\kappa\ell$ beyond the horizon, $r<0$, then using the  $\rightleftarrows$
-flip, which leads to the Bounded${}_{>}(e,-\ell)$ orbit.

#####  3.4.3 Subcritical $\mathcal{C}>0$

There is no near-NHEK spherical geodesic for $\mathcal{C}>0$. We can
nevertheless introduce the formal class of complex spherical trajectories

$\displaystyle\begin{split}t(\lambda)&=-i\frac{\ell}{R_{0}}\lambda,\\\
R(\lambda)&=iR_{0},\qquad
R_{0}\triangleq\frac{\kappa\ell}{\sqrt{\mathcal{C}}},\\\
\phi(\lambda)&=\phi_{0}-\frac{3}{4}\ell\lambda+\ell\Phi_{\theta}(\lambda)\end{split}$
(3.198)

which is a formal (but nonphysical) solution of the near-NHEK geodesic
equations, of complex near-NHEK “energy” $e=-i\kappa\sqrt{\mathcal{C}}$. We
will denote this class of solutions as Spherical${}_{\mathbb{C}}(\ell)$ and
show that it can be used to generate all subcritical bounded trajectories by
acting on it with properly chosen conformal transformations. The parametrized
form of the orbit reads as

$\displaystyle\begin{split}R&=iR_{0},\\\
\phi(t)&=\phi_{0}-\frac{3}{4}iR_{0}t+\ell\Phi_{\theta}(\lambda(t)).\end{split}$
(3.199)

###### Spherical${}_{\mathbb{C}}(\ell)$ $\Leftrightarrow$
Bounded${}_{<}(E,\ell)$.

One has to proceed in two steps, mimicking the procedure used to obtain the
NHEK Plunging$(E,\ell)$ class:

  * $\blacktriangleright$

We apply the near-NHEK/NHEK diffeomorphism (3.193) to a
Spherical${}_{\mathbb{C}}(\ell)$ orbit, leading to another complex NHEK
geodesic of null energy parametrized by

$\displaystyle\begin{split}T(R)&=-\frac{i\ell}{\sqrt{C}R},\\\
\Phi(R)&=\Phi_{0}-\frac{3i\ell}{8\sqrt{C}}\log\frac{\mathcal{C}R^{2}}{\mathcal{C}+\ell^{2}}\end{split}$
(3.200)

with the initial azimuthal angle
$\Phi_{0}\triangleq\phi_{0}-\frac{3\pi\ell}{8\sqrt{C}}-\frac{1}{2}\log\quantity(1-\frac{2\sqrt{C}}{\sqrt{C}+i\ell})$.
We denote this class as Marginal${}_{\mathbb{C}}(\ell)$.

  * $\blacktriangleright$

Second, we apply to the trajectory found above the global time shift (3.194),
but upgraded with an imaginary parameter $\zeta\to i\zeta$. This leads to the
Bounded${}_{<}(E,\ell)$ class with orbital parameters

$\displaystyle\bar{E}$ $\displaystyle=\sqrt{\mathcal{C}}\sinh\zeta,$ (3.201a)
$\displaystyle\bar{\Phi}_{0}$
$\displaystyle=\phi_{0}-\frac{3\pi\ell}{8\sqrt{\mathcal{C}}}-\log\quantity(\sqrt{\mathcal{C}}-i\ell)+\frac{3i\ell}{8\sqrt{\mathcal{C}}}\log\quantity[\mathcal{C}(\mathcal{C}+\ell^{2})\quantity(1+\sqrt{\frac{\mathcal{C}+E^{2}}{\mathcal{C}}})^{2}]$
$\displaystyle~{}-\frac{3\ell}{8\sqrt{C}}\log\quantity[E^{2}(\mathcal{C}+\ell^{2})]+\arctan\frac{\sqrt{\mathcal{C}}}{\ell}.$
(3.201b)

Note that choosing $\zeta>0$ is sufficient to reach the full range of energies
allowed for such a geodesic ($E>0$). Any geodesic of orbital parameters
($T_{0},\tilde{\Phi}_{0}$) can finally be obtained by performing the
transformation $T\to T+T_{0}$, $\Phi\to\Phi-\bar{\Phi}_{0}+\tilde{\Phi}_{0}$,
which also removes the unphysical imaginary part of the azimuthal coordinate.

###### Spherical${}_{\mathbb{C}}(\ell)$ $\Leftrightarrow$
Bounded${}_{<}(e,\ell)$.

We apply to the Spherical${}_{\mathbb{C}}(\ell)$ class the near-NHEK global
time shift (3.196) upgraded with an imaginary parameter $\chi\to i\chi$
($\chi\neq\pm 1$), leading to a Bounded${}_{<}(e,\ell)$ orbit of parameters

$\displaystyle\begin{split}\bar{e}&=\kappa\sqrt{\mathcal{C}}\,\chi,\\\
\bar{t}_{0}&=t_{0}+\frac{i}{\kappa}\arctan\frac{\kappa\sqrt{\mathcal{C}}}{e},\\\
\bar{\phi}_{0}&=\bar{\phi}_{0}(\phi_{0},e,\ell,\mathcal{C},\kappa).\end{split}$
(3.202)

The explicit value of $\bar{\phi}_{0}$ is easily calculable, but too long to
be reproduced here. To reach a manifestly real orbit of orbital parameters
$(\tilde{t}_{0},\tilde{\phi}_{0})$, one has to perform the final shift

$t\to
t-\bar{t}_{0}+\tilde{t}_{0},\qquad\phi\to\phi-\bar{\phi}_{0}+\tilde{\phi}_{0}.$
(3.203)

Figure 3.13: Schematic overview of the action of the group
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times\uparrow\\!\downarrow\times\rightleftarrows\times\\!$

$\rightleftarrows$

on near-horizon geodesics.

##  Part II Test bodies in Curved Spacetime:
Theoretical Foundations

Figure II.1: The two main pictures for the description of extended test bodies
used in this text: the gravitational skeletonization (left) and the Lagrangian
formulation (right).

Let us consider the motion of a object described by some smooth stress-energy
tensor $T_{\mu\nu}$ in a fixed background metric $g_{\mu\nu}$, thus neglecting
self-force effects. Provided that the body has a finite spatial extension, its
stress-energy tensor is supported on compact slices for any 3+1 decomposition
of the spacetime. Such an object will be referred to as an extended test body.
We have in mind the motion of a “small” astrophysical object (stellar mass
black hole or neutron star) around a hypermassive black hole. In this
situation, the former can be viewed as a perturbation of the spacetime
geometry created by the later.

While the geodesic equations describe the motion of a structureless monopole
test body in a fixed background spacetime, an important generalization is to
allow the test body (while still having negligible mass and thus negligible
influence on the gravitational field) to have a finite size and a nontrivial
structure. All these effects – departing from a bare geodesic motion – are
known as finite size effects.

###### Worldline description of extended test bodies

In the case where the extended test body is compact, that is, if its typical
size $\ell$ is small compared to the radius of curvature $r$ of the background
($\ell\ll r$), there exists various equivalent approximation schemes for
describing its motion in a somehow simpler way than considering its full
stress-energy tensor. Both approaches end into characterizing the body by a
centroid worldline $\gamma=\quantity{z^{\mu}(\lambda)}$ along which a tower of
gravitational multipole moments $I^{\mu\nu\alpha_{1}\ldots\alpha_{n}}$
replacing the original stress-energy tensor are defined, see Figure II.1.
These moments can be understood as spatial integrals of $T^{\mu\nu}$,

$I^{\mu\nu\alpha_{1}\ldots\alpha_{n}}\triangleq\int_{x^{0}=\text{cst}}\differential[3]{x}\sqrt{-g}T^{\mu\nu}\delta
x^{\alpha_{1}}\ldots\delta x^{\alpha_{n}},$

where $\delta x^{\mu}\triangleq x^{\mu}-z^{\mu}(\lambda)$.

The first of these schemes is known as the gravitational skeletonization: the
body is described by a distributional stress-energy tensor, which is non-
vanishing only on a certain worldline, and contains the aforementioned tower
of multipole moments. This tensor must be conserved within the background,
$\nabla_{\mu}T^{\mu\nu}=0$, and one can show that it implies that the monopole
$p_{\mu}$ and dipole $S_{\mu\nu}$ must evolve according to the Mathisson-
Papapetrou-Dixon (MPD) equations [101, 102, 103]

$\displaystyle\frac{\text{D}p^{\mu}}{\differential\tau}$
$\displaystyle=-\frac{1}{2}R^{\mu}_{\phantom{\mu}\nu\alpha\beta}v^{\nu}S^{\alpha\beta}+\ldots,\qquad\frac{\text{D}S^{\mu\nu}}{\differential\tau}=2p^{[\mu}v^{\nu]}+\ldots,$

where the dots represent corrections due to the quadrupole and higher
multipole moments. The monopole $p^{\mu}$ takes the interpretation of the
linear momentum of the body, whereas the dipole $S^{\mu\nu}$ can be seen as
its skew-symmetric spin tensor, describing the relativistic angular momentum
of the body. This last object will play a central role in our description,
which is the reason why we will sometimes refer to extended test bodies as
spinning test bodies. In terms of the original smooth stress-energy
distribution, one can show that the two first moments are related to the
original stress-energy tensor by

$\displaystyle p^{\mu}$
$\displaystyle\triangleq\int_{x^{0}=\text{constant}}\differential[3]{x}\sqrt{-g}\,T^{\mu
0},$ $\displaystyle S^{\mu\nu}$
$\displaystyle\triangleq\int_{x^{0}=\text{constant}}\differential[3]{x}\sqrt{-g}\,\quantity(\delta
x^{\mu}T^{\nu 0}-\delta x^{\nu}T^{\mu 0}).$

This approach has been investigated since the late thirties. The leading-order
EOMs were first derived in the seminal works of M. Mathisson [101, 104] and A.
Papapetrou [102]. They have been subsequently generalized to higher multipolar
orders by W.G. Dixon [105, 106, 107, 108]. Despite its elegance and rigour,
this approach appears to be quite long to perform and technically involved,
discarding it from being a well-suited viewpoint for exposing comprehensively
the problem in an introductory text like the present one.

We will instead follow the Lagrangian approach, whose generic formulation in
curved spacetime is due to I. Bailey and W. Israel in 1975 [109].
Nevertheless, this method was pioneered for Special Relativity in earlier
works, notably by A.J. Hanson and T. Regge (see e.g. [110, 111]). It consists
in formulating a generic action principle for the extended body modelled as a
worldline, representing the motion of some “center” of its stress-energy
distribution, and endowed with an orthonormal tetrad rigidly attached to it,
whose evolution describes the orientation of the body. This approach also
leads to the very same MPD equations. Having two equivalent descriptions of
the same problem is extremely fruitful, since each of them turns out to be
more appropriated for different purposes: as we will see, the skeletonization
will be powerful for providing us with physical insights about the
interpretation of multipole moments, while the Lagrangian approach will reveal
particularly useful when we will turn to the Hamiltonian description of
extended test bodies.

Others routes yielding the same equations of motion have also been followed. A
supersymmetric description of classical spinning particles has been provided
in 1993 by G.W. Gibbons, R.H. Rietdijk and J.W. van Holten [112], and recently
extended to include quadrupole effects [113]. Another recent (and somehow
elegant) formulation accounting for the description of finite size effects is
due to A. Harte [114, 115], using the concept of “generalized Killing vector
fields”. Its main interest it that it allows naturally to account for the
inclusion of gravitational back-reaction effects.

###### Spin supplementary conditions

There is a technical subtlety arising when studying the motion of test bodies
described by the MPD equations. In order to obtain a closed system of
equations, they have to be supplemented by an algebraic condition of the form
$\mathcal{V}_{\mu}S^{\mu\nu}=0$, for some timelike vector field
$\mathcal{V}^{\mu}$. Such conditions are known as spin supplementary
conditions. Physically, enforcing this kind of condition amounts to fix a
choice of centroid worldline, setting to zero the mass-dipole moment in the
proper frame whose timelike vector is aligned $\mathcal{V}^{\mu}$. The
discussion of what is really happening is quite subtle, the main reason being
that the notion of center of mass is observer-dependent in relativity.

###### Truncation of the multipole expansion

Another point of concern is to understand if the multipole expansion
introduced above can be consistently truncated as some desired order, that is,
if there exists a small parameter such that the magnitude of the successive
multipoles decreases when the order of the multipoles increases. As we will
see in Chapter 5, for compact objects, this small parameter will be the ratio
between the typical size of the object and the typical radius of the
background curvature, which is small by assumption.

As discussed in the introduction of the thesis, we will always truncate the
expansion at the quadrupole order. This is the first order in the multipole
expansion where the internal structure of the body begins to matter. At pole-
dipole order, the motion of finite size bodies is universal, in the sense that
it is independent of the nature of the object. Because we exclude the self-
force in our description, our expansion will thus be valid at zeroth order in
the mass ratio and at second order in the spin.

MPD equations promote the two first multipole $p^{\mu}$ and $S^{\mu\nu}$ to
the rank of dynamical variables, but leave the higher order multipoles acting
as sources. The latter shall consequently be prescribed depending on the
internal structure of the test body. In this thesis, we will only be concerned
with multipole moments induced by the proper rotation of the object, also
known as spin-induced multipoles. We therefore discard tidal and other type of
contributions to the multipole structure. As we will see, this is the relevant
description for modelling a binary black hole system evolving in vacuum, and
this is the choice of multipole structure that will allow the existence of the
largest number of conserved quantities along the motion.

Actually, for compact test bodies, they are several equivalent way of thinking
about the truncation of the multipole expansion, which are consistent one to
another, as we will check explicitly: (i) as a truncation of the number of
multipoles that we use to describe the stress-energy tensor, which is the
viewpoint of gravitational skeletonization that will be discussed in Chapter
5; (ii) as a truncation of the number of the derivatives of the background
Riemann tensor upon which the action of the Lagrangian formulation can depend
upon, as will be described in Chapter 4; and finally (iii) as an expansion in
integer powers of the magnitude of the spin dipole $\mathcal{S}$. This latter
viewpoint turns out to be consistent with the two former for spin-induced
multipoles, as will be reviewed in Chapter 7.

###### Plan of the text

This part of the thesis is organized as follows: Chapter 4 will describe the
Lagrangian formulation for extended test bodies in full generality, up to
quadrupole order. In Chapter 5, we will discuss a particularly simple form of
the gravitational skeletonization up to dipole order, which will enable to
gain more intuition about the physical meaning of the monopole and the dipole
moments. Our discussion will be however specialized to a specific choice of
coordinates. Chapter 6 will describe the problem of enforcing the
aforementioned spin supplementary conditions, as well as their physical
interpretation. Finally, Chapter 7 will be devoted to spin-induced multipoles,
and will focus on the explicit construction of the spin-induced quadrupole
moment.

###  Chapter 4 Lagrangian Formulation

This chapter discusses the Lagrangian formalism for spinning test bodies in
General Relativity. In this text, we will always restrict our derivations up
to quadrupole order. Nevertheless, higher orders can be reached, see e.g.
[116]. This chapter mainly follows the excellent exposition of Marsat [116].

The core idea of the Lagrangian approach is to construct the most generic
worldline Lagrangian action $S=\int L\,\differential\lambda$ describing the
motion of a spinning test body in curved spacetime. As we will see, the form
of the allowed Lagrangian $L$ can be highly constrained from very generic
symmetry arguments. Like in any classical mechanics problem, the equations of
motion can then be derived from the associated first order variational
principle $\delta S=0$ [66, 67].

####  4.1 Rotational degrees of freedom

The two main questions one should ask for building an action are

  1. 1.

What are the relevant degrees of freedom that should be introduced for
describing a spinning body in curved spacetime ?

  2. 2.

What are the symmetries under which the action should be invariant?

This section aims to tackle the first of them.

Let us denote $z^{\mu}(\lambda)$ the body’s worldline. Here, $\lambda$ is an
arbitrary “time” parameter describing the evolution of the motion. We also
define the four-velocity

$\displaystyle v^{\mu}\triangleq\derivative{z^{\mu}}{\lambda}.$ (4.1)

For any physical massive object, the four-velocity will be a timelike vector,
$v_{\mu}v^{\mu}<0$. In the canonical language of Lagrangian mechanics, the
four components $v^{\mu}$ will play the role of the velocities describing the
position of the test body and associated to the coordinates $z^{\mu}$.

All along this discussion, the specific form of $\lambda$ as well as the
normalization of the four-velocity will be left arbitrary at the level of the
action; they will only be constrained later at the level of the equations of
motion, setting $\lambda$ to be the proper time and consequently yielding the
standard normalization $v_{\mu}v^{\mu}=-1$.

We are left with the problem of choosing the degrees of freedom that will
account for the rotational orientation of the test body. Following the
proposition of Hanson and Regge for Special Relativity [111], the spin (that
is, the rotational) degrees of freedom of the body will be represented by an
orthonormal tetrad $e_{A}^{\phantom{A}\mu}(\lambda)$ rigidly attached to the
body’s worldline. Its orientation at any value of $\lambda$ will be measured
thanks to the introduction of another background orthonormal tetrad frame
$\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}(x)$. At any point
of the worldline, these two tetrads are related by a Lorentz transformation
$\Lambda^{A}_{\phantom{A}\underline{A}}(\lambda)$:

$\displaystyle\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}\quantity(z(\lambda))=\Lambda^{A}_{\phantom{A}\underline{A}}(\lambda)e_{A}^{\phantom{A}\mu}(\lambda).$
(4.2)

Of course, we have the standard relations for Lorentz matrices

$\displaystyle\Lambda_{A}^{\phantom{A}\underline{A}}(\lambda)\Lambda_{B\underline{A}}(\lambda)=\eta_{AB},\qquad\Lambda_{A\underline{A}}(\lambda)\Lambda^{A}_{\phantom{A}\underline{B}}(\lambda)=\eta_{\underline{A}\,\underline{B}},$
(4.3)

and for tetrad frames

$\displaystyle e_{A}^{\phantom{A}\mu}(\lambda)e_{B\mu}(\lambda)$
$\displaystyle=\eta_{AB},\quad
e_{A}^{\phantom{A}\mu}(\lambda)e^{A\nu}(\lambda)=g^{\mu\nu}(z(\lambda)),$
(4.4a)
$\displaystyle\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}(x)\underline{e}_{\underline{B}\mu}(x)$
$\displaystyle=\eta_{\underline{A}\,\underline{B}},\quad\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}(x)\underline{e}^{\underline{A}\nu}(x)=g^{\mu\nu}(x).$
(4.4b)

The evolution of the body’s tetrad will be described using the standard
antisymmetric rotation coefficients $\Omega^{\mu\nu}$ (see e.g. [43])

$\displaystyle\frac{\text{D}e_{A}^{\phantom{A}\mu}}{\differential\lambda}\triangleq-\Omega^{\mu\nu}e_{A\nu}\qquad\Leftrightarrow\qquad\Omega^{\mu\nu}\triangleq
e^{A\mu}\frac{\text{D}e_{A}^{\phantom{A}\nu}}{\differential\lambda}.$ (4.5)

Here and in the remaining of this text, we use the notation
$\text{D}/\differential\lambda\triangleq v^{\alpha}\nabla_{\alpha}$.

The Lorentz matrices
$\Lambda^{\underline{A}}_{\phantom{\underline{A}}A}(\lambda)$ encode all the
informations regarding the orientation of the body’s tetrad with respect to
the background. As any homogeneous Lorentz transformation, they contain 6
degrees of freedom: three of them represent spatial rotations, and the three
others relativistic boosts. Intuitively, one can see the three rotational
degrees of freedom (DOFs) as being the spin ones, whereas the three boosts
originate from the fact that one has not chosen yet the exact position of the
worldline $z^{\mu}(\lambda)$ inside the body’s worldtube. This ambiguity will
be extensively discussed and resolved in Chapter 6, by enforcing a so-called
spin supplementary condition (SSC).

####  4.2 Constraining the action

It is now time to write down an action for our theory. It seems natural to
require the following symmetry requirements to hold [117]:

  * $\blacktriangleright$

Spacetime diffeomorphisms: as any GR scalar expression, the action should be
invariant under any generic spacetime diffeomorphism $x^{\mu}\to
x^{\mu^{\prime}}\quantity(x^{\mu})$. The Lagrangian should consequently be a
tensorial scalar, in the sense that all the spacetime indices of the objects
it is built from should be properly contracted between themselves;

  * $\blacktriangleright$

Lorentz transformations: the action should be invariant under local Lorentz
transformations, which transform the body and the background tetrad as

$\displaystyle
e_{A}^{\phantom{A}\mu}\to\Lambda^{A}_{\phantom{A}B}e_{B}^{\phantom{B}\mu},\qquad\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}\to\bar{\Lambda}^{\underline{A}}_{\phantom{\underline{A}}\underline{B}}\underline{e}_{\underline{B}}^{\phantom{\underline{B}}\mu}.$
(4.6)

It amounts to require all the Lorentz indices of the tetrads to be properly
contracted;

  * $\blacktriangleright$

Reparametrization invariance: the time parameter $\lambda$ being arbitrary,
the action (4.7) should be invariant under any reparametrization
$\lambda\to\lambda^{\prime}(\lambda)$ of the trajectory.

In order to actually describe a spinning body, the Lagrangian should
kinematically depend on the worldline velocity $v^{\mu}$ and on the rotation
coefficients $\Omega^{\mu\nu}$, but not on the “positions” ($z^{\mu}$ and
$e_{A}^{\phantom{A}\mu}$) themselves, for the purpose of ensuring general
covariance. Moreover, we forbid any dependence in the background structure
$\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}$, so that our
description depends only on degrees of freedom intrinsic to the body. The
prescribed Lagrangian should account for finite size effects, that is,
dynamical effects originating from the coupling between the body’s spin and
the background’s curvature. The later is accounted for by the Riemann tensor
and its derivatives. Notice that the background metric $g_{\mu\nu}$ is assumed
to appear only for the purpose of contracting indices, thus allowing to
construct scalars from the other tensorial objects in a natural way. We
however forbid any dependence upon first derivatives of the metric (that is,
upon Christoffel symbols). Derivatives of the metric are only allowed to enter
in the action through the Riemann tensor and its derivatives.

Excluding any coupling with other external fields and given the discussion
above, the generic action for an extended test body is then assumed takes the
form:

$\displaystyle
S\quantity[z^{\mu},e_{A}^{\phantom{A}\mu}]=\int_{\gamma}L\quantity(v^{\mu},\Omega^{\mu\nu},g_{\mu\nu}(z),R_{\mu\nu\rho\sigma}(z),\nabla_{\lambda}R_{\mu\nu\rho\sigma}(z),\ldots)\differential\lambda.$
(4.7)

The subscript $\gamma$ just refers to the fact that the integration over
$\lambda$ is actually an integration over the worldline $\gamma$.

###### Homogeneity condition

The next step will be to constrain the generic form of the action Eq. (4.7).
Actually, a very simple argument allows to provide a simple explicit – but
still non-uniquely fixed – expression for the Lagrangian. As we have just
mentioned, the action Eq. (4.7) should be invariant under any
reparametrization of the trajectory; in particular, it should be invariant
under a scaling $\lambda\to\Delta\lambda$ ($\Delta\neq 0)$. This implies that
the Lagrangian must be homogeneously linear in $v^{\mu}$ and
$\Omega^{\mu\nu}$, which both scale as $\Delta^{-1}$ under this
transformation. Euler’s theorem on homogeneous functions111 Let
$f:\mathbb{R}^{n}\to\mathbb{R}$ be a positively homogeneous function of degree
$k\in\mathbb{Z}$, i.e. $\displaystyle\forall\Delta>0:f(\Delta
x_{1},\ldots\Delta x_{n})=\Delta^{k}f(x_{1},\ldots,x_{n})$ which is
continuously differentiable in some open subset
$\mathcal{U}\subset\mathbb{R}^{n}$. Then,  $\displaystyle
kf(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}x_{i}\partialderivative{f}{x_{i}}\quantity(x_{1},\ldots,x_{n}),\qquad\forall\quantity(x_{1},\ldots,x_{n})\in\mathcal{U}.$
then implies that

$\displaystyle
L(v^{\mu},\Omega^{\mu\nu},g_{\mu\nu},R_{\mu\nu\rho\sigma},\nabla_{\lambda}R_{\mu\nu\rho\sigma},\ldots)=\partialderivative{L}{v^{\mu}}v^{\mu}+\partialderivative{L}{\Omega^{\mu\nu}}\Omega^{\mu\nu}.$
(4.8)

Defining the conjugate momenta222The factor $2$ in the definition of
$S_{\mu\nu}$ is present for consistency with the conventions used in the
literature. (respectively referred to as the linear momentum and the spin
tensor)

$\displaystyle p_{\mu}\triangleq\partialderivative{L}{v^{\mu}},\qquad
S_{\mu\nu}\triangleq 2\partialderivative{L}{\Omega^{\mu\nu}},$ (4.9)

one can write

$\displaystyle L=p_{\mu}v^{\mu}+\frac{1}{2}S_{\mu\nu}\Omega^{\mu\nu}.$ (4.10)

Be careful: we have not provided a unique expression for the Lagrangian of our
theory. The momenta $p_{\mu}$ and $S_{\mu\nu}$ remain here arbitrary functions
of $v^{\mu}$, $\Omega^{\mu\nu}$, the Riemann tensor and its derivatives. They
will be fixed when a spin supplementary condition will be enforced, which will
provide us with an explicit relation between the linear momentum $p_{\mu}$ and
the four-velocity $v^{\mu}$. Nevertheless, the form of the Lagrangian (4.10)
is relevant to mention for later purposes (e.g. Hamiltonian description of the
present problem); in particular, it is the same regardless to the finite size
interactions allowed in the theory, i.e. regardless to the dependence of $L$
in the Riemann tensor and its derivatives that is allowed.

###### Quadrupole approximation

We will now constrain our theory by restricting the functional dependency of
the Lagrangian in the (derivatives of the) Riemann tensor. In the continuation
of this text, we will always restrict ourselves to the so-called quadrupole
approximation which consists into allowing $L$ to depends on the Riemann
tensor, but not on its derivatives:

$\displaystyle
L=L\quantity(v^{\mu},\Omega^{\mu\nu},g_{\mu\nu},R_{\mu\nu\rho\sigma}).$ (4.11)

More generically, allowing the Lagrangian to depend in the Riemann tensor up
to its $n^{\text{th}}$ derivative will lead to the appearance of
$2^{n+2}$-pole moments terms in the equations of motion. For an action of the
form (4.7), the multipole moments are only sourced by the spin of the body.
For spin-induced moments, one can show (see Chapter 7) that the $2^{n}$-pole
moment scales as $\mathcal{O}\quantity(\mathcal{S}^{n})$, where the spin
magnitude $\mathcal{S}$ is defined as

$\displaystyle\mathcal{S}^{2}\triangleq\frac{1}{2}S_{\mu\nu}S^{\mu\nu}.$
(4.12)

The aforementioned approximation makes sense, because (i) the spin magnitude
will turn out to be a constant of motion, regardless to the multipole order we
are working with and because (ii) $\mathcal{S}$ can be assumed to be small in
astrophysically realistic situations. It can consequently be used as an
expansion parameter for setting up a perturbative treatment of the generic
problem.

From a physical viewpoint, the quadrupole approximation amounts to consider
deformations induced by the proper rotation of the object in our description
up to quadrupole order, while neglecting higher order corrections.

Finally, let us notice that only the (mass) monopole moment $p_{\mu}$ and the
(spin) dipole moment $S_{\mu\nu}$ are dynamical variables, because they are
the only multipole moments present in the explicit form of the Lagrangian
(4.10). The higher moments are non-dynamical and entirely written in terms of
these two first moments. They will act as sources in the equations of motion.

####  4.3 Equations of motion

Before deriving the equations of motion, it is useful to explicit some results
concerning the first-order variations of the Lagrangian. First, in the
quadrupole approximation, a generic variation of $L$ takes the form

$\displaystyle\delta L=p_{\mu}\delta
v^{\mu}+\frac{1}{2}S_{\mu\nu}\delta\Omega^{\mu\nu}+\partialderivative{L}{g_{\mu\nu}}\delta
g_{\mu\nu}-\frac{1}{6}J^{\mu\nu\rho\sigma}\delta R_{\mu\nu\rho\sigma}.$ (4.13)

Here, we have defined – up to a proportionality factor – the quadrupole moment
$J^{\mu\nu\rho\sigma}$ as the conjugate moment to the Riemann tensor:

$\displaystyle
J^{\mu\nu\rho\sigma}\triangleq-6\partialderivative{L}{R_{\mu\nu\rho\sigma}}.$
(4.14)

Notice that the variation (4.13) is independent of the explicit form (4.10) of
the Lagrangian.

In particular, the action should be invariant under an infinitesimal change of
coordinates $z^{\mu}\to z^{\mu}+\epsilon\xi^{\mu}$
($\absolutevalue{\epsilon}\ll 1$). Particularizing the variation (4.13) to
this case and working in a locally inertial frame yields

$\displaystyle\delta_{\xi}L$
$\displaystyle=\epsilon\quantity(p^{\nu}v^{\mu}+S^{\nu}_{\phantom{\nu}\lambda}\Omega^{\mu\lambda}-2\partialderivative{L}{g_{\mu\nu}}+\frac{2}{3}J^{\nu\alpha\beta\gamma}R^{\mu}_{\phantom{\mu}\alpha\beta\gamma})\partial_{\mu}\xi_{\nu}.$
(4.15)

This variation must vanish regardless to the value of $\xi^{\mu}$; therefore,
the following constraint must hold:

$\displaystyle
p_{\nu}v_{\mu}+S_{\nu\lambda}\Omega_{\mu}^{\phantom{\mu}\lambda}-2\partialderivative{L}{g_{\mu\nu}}+\frac{2}{3}J_{\nu}^{\phantom{\nu}\alpha\beta\gamma}R_{\mu\alpha\beta\gamma}=0.$
(4.16)

Taking the antisymmetric part of this expression allows to write

$\displaystyle
S^{\lambda[\mu}\Omega^{\nu]}_{\phantom{\nu]}\lambda}=p^{[\mu}v^{\nu]}+\frac{2}{3}R^{[\mu}_{\phantom{[\mu}\alpha\beta\gamma}J^{\nu]\alpha\beta\gamma},$
(4.17)

which is valid in any frame. This last relation will become extremely useful
in the following derivations.

###### Evolution equation for the spin tensor

The evolution equation for the spin tensor, also known as the precession
equation, is obtained by varying the action with respect to the body’s tetrad
$e_{A}^{\phantom{A}\mu}$. In the background frame, the variation of the
rotation coefficients takes the form

$\displaystyle\delta\Omega^{\underline{A}\,\underline{B}}=e^{\underline{A}}_{\phantom{\underline{A}}\mu}e^{\underline{B}}_{\phantom{\underline{B}}\nu}\frac{\text{D}\,\delta\theta^{\mu\nu}}{\differential\lambda}+\Omega^{\underline{A}}_{\phantom{\underline{A}}\underline{C}}\delta\theta^{\underline{C}\,\underline{B}}-\Omega^{\underline{B}}_{\phantom{\underline{B}}\underline{C}}\delta\theta^{\underline{C}\,\underline{A}}.$
(4.18)

For convenience, the variation of the tetrad has been entirely expressed in
terms of the object

$\displaystyle\delta\theta^{\underline{A}\,\underline{B}}\triangleq\Lambda^{A\underline{A}}\delta\Lambda_{A}^{\phantom{A}\underline{B}}.$
(4.19)

Plugging this result either in the explicit expression of the Lagrangian
(4.10) or in the generic variation (4.13), one obtains

$\displaystyle\delta_{\theta}L=\frac{1}{2}\quantity(-\frac{\text{D}\,S^{\mu\nu}}{\differential\lambda}+S^{\nu\rho}\Omega^{\mu}_{\phantom{\mu}\rho}-S^{\mu\rho}\Omega^{\nu}_{\phantom{\nu}\rho})\delta\theta_{\mu\nu}.$
(4.20)

Requiring this variation to be vanishing yields the evolution equation

$\displaystyle\frac{\text{D}\,S^{\mu\nu}}{\differential\lambda}=S^{\nu\rho}\Omega^{\mu}_{\phantom{\mu}\rho}-S^{\mu\rho}\Omega^{\nu}_{\phantom{\nu}\rho}.$
(4.21)

Before going further on, let us stress some points useful for the continuation
of this work.

  1. 1.

A direct computations shows that the identity

$\displaystyle S^{\rho}_{\phantom{\rho}\mu}\Omega^{\mu\nu}S_{\nu\rho}=0$
(4.22)

holds. It implies that the spin magnitude is conserved,

$\displaystyle\derivative{\mathcal{S}}{\lambda}=0.$ (4.23)

This conservation equation actually holds at any multipole order and is
independent of the spin supplementary condition (see e.g. [116] and references
therein).

  2. 2.

In the object’s frame, the evolution equation becomes

$\displaystyle\frac{\text{D}\,S^{AB}}{\differential\lambda}=0.$ (4.24)

The components of the spin tensor in the object’s frame $S^{AB}$ are thus
constant. This provides a posteriori a justification to the statement that the
tetrad $e_{A}^{\phantom{A}\mu}$ is “rigidly attached” to the compact object.

  3. 3.

Using Eq. (4.17), one can eliminate the dependence of the precession equation
in the rotation coefficients and make explicit its dependence in the Riemann
tensor. A straightforward computation yields

$\displaystyle\frac{\text{D}\,S^{\mu\nu}}{\differential\lambda}=2p^{[\mu}v^{\nu]}+\mathcal{L}^{\mu\nu},\qquad\mathcal{L}^{\mu\nu}\triangleq\frac{4}{3}R^{[\mu}_{\phantom{[\mu}\alpha\beta\gamma}J^{\nu]\alpha\beta\gamma}.$
(4.25)

This is the standard form of the precession equation that can be found in the
literature.

  4. 4.

Finally, contracting Eq. (4.25) with $v_{\nu}$, we remark that the momentum
and the four velocity are not aligned anymore when spin is present, by
contrast to the geodesic case:

$\displaystyle-v^{2}p^{\mu}=\mathfrak{m}v^{\mu}+p_{\perp}^{\mu},\qquad
p_{\perp}^{\mu}\triangleq\quantity(-\frac{\text{D}S^{\mu\nu}}{\differential\lambda}+\mathcal{L}^{\mu\nu})v_{\nu}.$
(4.26)

Here, $\mathfrak{m}\triangleq-v_{\alpha}p^{\alpha}$ denotes the body’s mass in
the frame attached to the worldline, also known as the kinetic mass. The norm
of the four-velocity can be set to $-1$ if the time evolution parameter is
chosen to be the body proper time. The orthogonal component of the momentum
$p^{\mu}_{\perp}$ can be expressed as a function of $x^{\mu}$, $v^{\mu}$ and
$S^{\mu\nu}$ solely when a spin supplementary condition has been enforced, see
Chapter 6.

###### Evolution equation for the linear momentum

The method for finding the evolution equation for the linear momentum is to
vary the action with respect to the worldline. The procedure is the very same
that the one which can be used for the derivation of the geodesic deviation
equation [83, 84]: let us consider an infinitesimal change of the worldline,
parametrized by a displacement vector $\xi^{\mu}(\lambda)$ which is Lie-
dragged along the worldline:

$\displaystyle\mathcal{L}_{v}\xi^{\mu}=0\quad\Leftrightarrow\quad\xi^{\lambda}\nabla_{\lambda}v^{\mu}=v^{\lambda}\nabla_{\lambda}\xi^{\mu}.$
(4.27)

In this case, the variation of the action takes the form

$\displaystyle\delta_{\xi}S=\int_{\gamma}\delta_{\xi}L\,\differential\lambda=\int_{\gamma}\xi^{\lambda}\partial_{\lambda}L\,\differential\lambda=\int_{\gamma}\xi^{\lambda}\nabla_{\lambda}L\,\differential\lambda.$
(4.28)

It is useful to notice that the following identities hold [116]:

$\displaystyle\xi^{\lambda}\nabla_{\lambda}v^{\mu}v^{\mu}$
$\displaystyle=\frac{\text{D}\,\xi^{\mu}}{\differential\lambda},$ (4.29a)
$\displaystyle\xi^{\lambda}\nabla_{\lambda}\Omega^{\mu\nu}$
$\displaystyle=-\frac{\text{D}}{\differential\lambda}\quantity(e^{A\mu}\delta_{\xi}e_{A}^{\phantom{A}\nu})+e^{A\mu}\frac{\text{D}\,\delta_{\xi}e_{A}^{\phantom{A}\nu}}{\differential\lambda}-e^{A\nu}\frac{\text{D}\,\delta_{\xi}e_{A}^{\phantom{A}\mu}}{\differential\lambda}-\xi^{\alpha}v^{\beta}R^{\mu\nu}_{\phantom{\mu\nu}\alpha\beta}.$
(4.29b)

Here, we have defined

$\displaystyle\delta_{\xi}e_{A}^{\phantom{A}\mu}\triangleq\xi^{\lambda}\nabla_{\lambda}e_{A}^{\phantom{A}\mu}$
(4.30)

The value of this quantity is actually is arbitrary, since we are left with
the freedom of choosing the way the tetrad is transported from the original
worldline $z^{\mu}(\lambda)$ to the perturbed one
$z^{\mu}(\lambda)+\xi^{\mu}(\lambda)$. Choosing the tetrad to be parallelly
transported between the two worldlines allows to set

$\displaystyle\delta_{\xi}e_{A}^{\phantom{A}\mu}=0.$ (4.31)

Gathering all the previous pieces, the evolution equation for the linear
momentum can then be derived – after integration by parts – from the
variational problem $\delta_{\xi}S=0$, yielding

$\displaystyle\frac{\text{D}\,p^{\mu}}{\differential\lambda}=-\frac{1}{2}R^{\mu}_{\phantom{\mu}\nu\alpha\beta}v^{\nu}S^{\alpha\beta}+\mathcal{F}^{\mu},\qquad\mathcal{F}^{\mu}\triangleq-\frac{1}{6}J^{\alpha\beta\gamma\delta}\nabla^{\mu}R_{\alpha\beta\gamma\delta}.$
(4.32)

This is the standard evolution equation of the linear momentum at quadrupole
order. Together with Eq. (4.25), these equations are the Mathisson-Papapetrou-
Dixon equations, restricted to quadrupole order. At higher orders, the
structure of the equations remains the same, the contribution of the higher
order multipoles being only contained in the force $\mathcal{F}^{\mu}$ and
torque $\mathcal{L}^{\mu\nu}$ terms [118].

####  4.4 Stress-energy tensor

An interesting computation to be performed at this stage of the discussion is
to write out the stress-energy tensor of the theory. It is advantageously
computed from the variation of the action with respect to the body’s tetrad
frame: first expressing the metric in terms of the tetrad in the action thanks
to Eq. (4.4a), the stress-energy tensor as be computed from

$\displaystyle T_{\mu\nu}\triangleq\frac{1}{\sqrt{-g}}e_{a(\mu}\frac{\delta
S}{\delta e_{a}^{\phantom{a}\nu)}}.$ (4.33)

The stress-energy tensor can be split as a sum over all the multipole orders
involved:

$\displaystyle
T_{\mu\nu}=T^{\text{pole}}_{\mu\nu}+T^{\text{dipole}}_{\mu\nu}+T^{\text{quad}}_{\mu\nu}.$
(4.34)

As usually when computing stress-energy tensors for point-like objects moving
along worldlines, the action should be written as an integral over the
spacetime by introducing a Dirac delta:

$\displaystyle
S=\int\differential[4]x\sqrt{-g}\int_{\gamma}\,L\quantity(v^{\mu},\Omega^{\mu\nu},e_{A\mu}e^{A}_{\phantom{A}\nu},R_{\mu\nu\rho\sigma})\delta_{4}(x,z)\differential\lambda.$
(4.35)

Here, the symbol $\delta_{4}(x,z)$ stands for the diffeomorphism invariant
Dirac distribution

$\displaystyle\delta_{4}(x,z)\triangleq\frac{\delta^{(4)}(x-z)}{\sqrt{-g}},$
(4.36)

where $\delta^{(4)}(x-z)$ is the standard four-dimensional Dirac distribution
[119]. After computation, the contributions of the right hand side are found
to be given by

$\displaystyle T^{\text{pole}}_{\mu\nu}$
$\displaystyle=\int_{\gamma}p_{(\mu}v_{\nu)}\delta_{4}(x,z)\differential\lambda,$
(4.37a) $\displaystyle T^{\text{dipole}}_{\mu\nu}$
$\displaystyle=-\nabla_{\lambda}\int_{\gamma}S^{\lambda}_{\phantom{\lambda}(\mu}v_{\nu)}\delta_{4}(x,z)\differential\lambda,$
(4.37b) $\displaystyle T^{\text{quad}}_{\mu\nu}$
$\displaystyle=\frac{1}{3}\int_{\gamma}R_{(\mu}^{\phantom{(\mu}\alpha\beta\gamma}J_{\nu)\alpha\beta\gamma}\delta_{4}(x,z)\differential\lambda$
$\displaystyle\quad-\frac{2}{3}\nabla_{\lambda}\nabla_{\rho}\int_{\gamma}J^{\lambda\phantom{(\mu\nu)}\rho}_{\phantom{\lambda}(\mu\nu)}\delta_{4}(x,z)\differential\lambda.$
(4.37c)

As expected, the stress energy tensor is not a function, but a distribution
which is only non-vanishing on the body’s worldline.

###  Chapter 5 Skeletonization of the stress-energy tensor

Until now, we have obtained the Mathisson-Papapetrou-Dixon equations governing
the motion of spinning test bodies in curved spacetime at quadrupole order,
which are given by Eqs. (4.25) and (4.32). This was performed through writing
down an action principle for our theory and deriving the associated equations
of motion from the associated variational principle.

In this chapter, we will see that a totally different method allows to recover
the very same equations of motion. It consists into replacing the smooth
stress-energy tensor of the physical extended body by a distributional one,
which is only supported on a single worldline encompassed in the body’s
worldtube. The equations of motion then follow from the stress-energy
conservation equation. Justifying rigorously this approximation from first
principles in GR is however very involved and technical. We refer the
interested reader to the references mentioned in the introduction of Part II
for more details (especially Dixon ones). In the present text, we will give
some insights about the coherence of this approximation by comparing the
(involved) GR situation to the (simpler) Newtonian one.

At quadrupole order, the computations associated to gravitational
skeletonization turn out to be very cumbersome. This chapter aiming to provide
a pedagogical introduction, we will restrict ourselves to the dipole order.
Explicit computations for the quadrupole may be found in [118, 120].

This chapter is organized as follows: Section 5.1 reviews the gravitational
skeletonization in Newtonian theory, which is then generalized to General
Relativity in Section 5.2. As we will see, they are several decompositions
that can be chosen for performing the skeletonization. In Section 5.3, we use
the Ellis decomposition to recover the MPD equations at dipole order. The
computations are carried out in a specific coordinates system, the adapted
coordinates, which enable to reduce dramatically the length and the
technicality of the derivation.

####  5.1 Invitation: gravitational skeleton in Newtonian gravity

This section is mainly based on [121, 118].

In order to acquire some feeling about the form of the Ansatz of the GR’s
gravitational skeleton of the stress-energy tensor, let us have a look at the
equivalent problem in Newtonian gravity. The gravitational potential
$U(t,\mathbf{x})$ created by an object of mass density $\rho(t,\mathbf{x})$
enclosed in a volume $\mathcal{V}\subset\mathbb{R}^{3}$ is solution of the
Poisson equation

$\displaystyle\Delta U(t,\mathbf{x})=-4\pi\rho(t,\mathbf{x}).$ (5.1)

This equation can be solved analytically, and its solution reads (up to a
trivial additive constant)

$\displaystyle
U(t,\mathbf{x})=\int_{\mathcal{V}}\differential[3]x^{\prime}\,\frac{\rho(t,\mathbf{x})}{\norm{\mathbf{x}-\mathbf{x}^{\prime}}}.$
(5.2)

This potential admit a convenient rewriting under the form of a multipole
decomposition above an arbitrary point $\mathbf{x}_{0}\in\mathcal{V}$. For any
$\mathbf{x}\notin\mathcal{V}$, one can write

$\displaystyle
U(t,\mathbf{x})=\sum_{l=0}^{+\infty}\frac{\quantity(-)^{l}}{l!}I^{i_{1}\ldots
i_{l}}(t,\mathbf{x}_{0})\partial_{i_{1}}\ldots\partial_{i_{l}}\norm{\mathbf{x}-\mathbf{x}_{0}}^{-1}$
(5.3)

where we have defined the multipole moments

$\displaystyle I^{i_{1}\ldots
i_{l}}(t,\mathbf{x}_{0})\triangleq\int_{\mathcal{V}}\differential[3]x\,\quantity(x-x_{0})^{i_{1}}\ldots\quantity(x-x_{0})^{i_{l}}\rho(t,\mathbf{x}).$
(5.4)

The proof of Eq. (5.3) is easily carried out by performing a Taylor expansion
of $\norm{\mathbf{x}-\mathbf{x}^{\prime}}$ with respect to
$\mathbf{x}^{\prime}$ above some point $\mathbf{x}_{0}\in\mathcal{V}$.

The gravitational skeletonization consists here in replacing the smooth mass
density distribution $\rho(t,\mathbf{x})$ (which is supported on a finite-size
region of space, $\text{supp}\,\rho\subseteq\mathcal{V}$) by a singular mass
density distribution – say $\rho_{\text{skel}}$ – which is supported on a
single point of space,
$\text{supp}\,\rho_{\text{skel}}=\quantity{\mathbf{x}_{0}}\in\mathcal{V}$. The
key result allowing such a skeletonization to be performed can be stated as
follows:

######  Proposition 5.1.

Let $\mathbf{x}_{0}$ be a point of $\mathcal{V}$. For any point
$\mathbf{x}\notin\mathcal{V}$ outside of the object, the distributional mass
density

$\displaystyle\rho_{\text{skel}}(t,\mathbf{x})\triangleq\sum_{l=0}^{+\infty}\frac{(-)^{l}}{l!}I^{i_{1}\ldots
i_{l}}(t,\mathbf{x}_{0})\partial_{i_{1}}\ldots\partial_{i_{l}}\delta^{(3)}(\mathbf{x}-\mathbf{x}_{0})$
(5.5)

generates the same potential $U(t,\mathbf{x})$ as the smooth mass density
$\rho(t,\mathbf{x})$.

###### Proof.

It is enough to recall ourselves that the identity

$\displaystyle\Delta\norm{\mathbf{x}-\mathbf{x}_{0}}^{-1}=-4\pi\delta^{(3)}\quantity(\mathbf{x}-\mathbf{x}_{0}).$
(5.6)

holds in the sense of distributions [119]. ∎

Of course, the explicit expression of $\rho_{\text{skel}}$ and all the related
equations have to be understood in the sense of distributions (see e.g. [119]
for a clear reminder of the meaning of this assertion).

In others words, when observed from outside, any localized gravitating object
can be replaced by a particle located at a single point of spacetime and
possessing an infinite tower of multipole moments. This replacement holds in
the sense that the gravitational potentials generated by these two systems are
identical as long as we remain outside of the object.

####  5.2 General Relativist Skeletons

We now consider an extended body within the framework of General Relativity.
This object is assumed to be described by a smooth stress-energy tensor
supported on some worldtube $\mathcal{T}$. In the same spirit, we replace its
smooth stress-energy tensor $T^{\mu\nu}(x)$ by a distributional stress-energy
tensor $T^{\mu\nu}_{\text{skel}}(x)$ supported on a single timelike worldline
$\gamma\subset\mathcal{T}$. By analogy with Eq. (5.5), we assume this stress-
energy tensor to take the form

$\displaystyle
T^{\mu\nu}_{\text{skel}}(x)=\sum_{l=0}^{+\infty}\frac{1}{l!}\int_{\gamma}\differential\lambda\,I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}(z)\mathcal{D}^{(l)}_{\alpha_{1}\ldots\alpha_{l}}\delta_{4}(x,z).$
(5.7)

Here, $\mathcal{D}^{(k)}_{\alpha_{1}\ldots\alpha_{l}}$ is some differential
operator which contains at most $l$ derivatives. $z^{\mu}(\tau)$ are
coordinates parametrizing the worldline $\gamma$ with respect to an affine
time parameter $\lambda$. We denote the tangent vector to the worldline
$v^{\mu}=\derivative{z^{\mu}}{\lambda}$. For $l=0$, we use the conventions
$\mathcal{D}^{(0)}=\text{Id}$ and
$I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}=I^{\mu\nu}$. At this level, the
multipoles $I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}$ are still arbitrary
functions.

###### Perturbative treatment

One can show that it makes sense to treat the expansion (5.7) perturbatively,
and consequently to truncate it at any desired order. By analogy with the non-
relativistic case, we do expect the multipole to scale as
$I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}\sim\mu\ell^{l}$, with $\mu$ and $\ell$
being respectively the typical mass and size of the extended body under
consideration. Moreover, we expect the multiple covariant derivative to scale
as $\nabla_{\alpha_{1}\ldots\alpha_{l}}\sim r^{-l}$, with $r$ the typical
curvature radius of the background metric. All in all, the $l^{\text{th}}$
term of the expansion (5.7) scales as $\mu\quantity(\frac{\ell}{r})^{l}$.
Then, if the object is assumed to be compact in the sense mentioned in the
introduction, we have $r\gg\ell$ and consequently $\frac{\ell}{r}\ll 1$. The
expansion (5.7) can thus be truncated in a perturbative sense. A truncation at
$l=0$ will correspond to the monopole approximation, $l=1$ to the pole-dipole
(or simply dipole) one, $l=2$ to the quadrupole, etc.

###### Dixon and Ellis representations

To go further, we shall choose an explicit form for the operator
$\mathcal{D}^{l}_{\alpha_{1}\ldots\alpha_{l}}$. Two equivalent choices have
been used in the literature [120]. The most common is the Dixon representation
[106, 107, 108]

$\displaystyle\mathcal{D}^{(k)}_{\alpha_{1}\ldots\alpha_{k}}$
$\displaystyle\triangleq\nabla_{\alpha_{1}}\ldots\nabla_{\alpha_{k}}.$ (5.8)

In this text, we will instead make another choice, the Ellis representation,
defined by [122]

$\displaystyle\mathcal{D}^{(l)}_{\alpha_{1}\ldots\alpha_{l}}$
$\displaystyle=\left\\{\ \begin{array}[]{ll}0&\text{ if }l<N\\\
\partial_{\alpha_{1}}\ldots\partial_{\alpha_{N}}&\text{ if
}l=l_{\text{max}}\end{array}\right.,$ (5.11)

with $l_{\text{max}}$ the order in $l$ at which the multipole expansion Eq.
(5.7) is chosen to be truncated.

Both of these representations have advantages and drawbacks, which are nicely
reviewed in [120]. For our purposes, it will be more convenient to work in the
Ellis representation, since the computations involved at dipole order turn out
to be less technical, and thus more suited for an introductory exposition.

###### Reduction of the stress-energy tensor

The above skeletonization amounts to replace the smooth object by a collection
of multipole moments supported on a single worldline contained in the object
worldtube. Actually proving the validity of the decomposition (5.7) would
require to show that there exists some expressions of the multipole moments
$I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}$ such that both the smooth and the
distributional stress-energy tensors generate the same spacetime curvature
though the Einstein field equations $G_{\mu\nu}=8\pi T_{\mu\nu}$. This task
being extremely involved, we will not attempt to tackle it in the present
text, but rather consider (5.7) for granted. The interested reader would
fruitfully refer to [108] for a more formal exposition of the subject.

In this text, we will instead discuss the so-called reduction of the stress-
energy tensor: as it is always the case in GR, our stress-energy tensor must
be conserved and consequently obeys [83, 84]

$\displaystyle\nabla_{\mu}T^{\mu\nu}=0.$ (5.12)

As we will show, this conservation equation highly constrains the form of the
stress-energy tensor. At quadrupole order, one can show that it implies that
there must exist a vector $p^{\mu}$, and antisymmetric tensor $S^{\mu\nu}$ and
a rank four tensor $J^{\mu\nu\rho\sigma}$ exhibiting the same symmetries than
the Riemann tensor such that [118, 120]

$\displaystyle T^{\mu\nu}(x)$
$\displaystyle=\int_{\gamma}\differential\lambda\,\quantity[v^{(\mu}p^{\nu)}+\frac{1}{3}R^{(\mu}_{\phantom{(\mu}\alpha\beta\gamma}J^{\nu)\alpha\beta\gamma}]\delta_{4}(x,z)+\nabla_{\alpha}\int_{\gamma}\differential\lambda\,v^{(\mu}S^{\nu)\alpha}\delta_{4}(x,z)$
$\displaystyle\quad-\frac{2}{3}\nabla_{\alpha}\nabla_{\beta}\int_{\gamma}\differential\lambda\,J^{\alpha(\mu\nu)\beta}\delta_{4}(x,z).$
(5.13)

Moreover, during the reduction process, we find additional constraints taking
the form of differential equations for the quantities $p^{\mu}$, $S^{\mu\nu}$
and $J^{\mu\nu\rho\sigma}$ that turn out to be precisely the MPD equations.
Notice that Eq. (5.2) agrees exactly with the form of the stress-energy tensor
which has been found through the Lagrangian formulation, given in Eq. (4.37).

####  5.3 Ellis skeleton in adapted coordinates: to dipole order

This section mainly follows the exposition of [120].

Performing the explicit reduction of the stress-energy tensor at quadrupole
order is computationally quite involved, and can be found in [118, 120]. As a
proof of principle, we will instead perform the reduction at pole-dipole level
and show that it leads to the MPD equations at the corresponding order. In the
following, we will work with Ellis representation of multipoles. Explicitly,
Ellis representation truncated at order $N$ takes the form

$\displaystyle T^{\mu\nu}_{\text{skel}}(x)$
$\displaystyle=\frac{1}{N!}\int_{\gamma}\differential\lambda\,I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}(z)\partial_{\alpha_{1}}\ldots\partial_{\alpha_{N}}\delta_{4}\quantity(x,z)$
(5.14)

with $z^{\mu}=z^{\mu}(\lambda)$ the coordinates of the worldline.

Since $T_{\text{skel}}^{\mu\nu}$ is a distributional stress-energy tensor, Eq.
(5.14) shall be formally understood in the sense of distributions: for any
symmetric test function $\phi_{\mu\nu}$, the quantity

$\displaystyle\int_{\mathcal{T}}\differential[4]x\sqrt{-g}\,T_{\text{skel}}^{\mu\nu}(x)\phi_{\mu\nu}(x)$
(5.15)

is a real number. Integrating distributional quantities against test functions
will be the main tool we will use to show that the conservation of the stress-
energy tensor leads to the MPD equations.

Notice that since $T^{\mu\nu}$ is symmetric and since the partial derivatives
commute, the moment $I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}$ must obey the
algebraic symmetries

$\displaystyle
I^{\mu\nu\alpha_{1}\ldots\alpha_{k}}=I^{(\mu\nu)\alpha_{1}\ldots\alpha_{k}}=I^{\mu\nu(\alpha_{1}\ldots\alpha_{k})}.$
(5.16)

Finally, let us comment on two drawbacks of Ellis representation: (i) from
their very definitions, the moments $I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}$ are
not tensors, since they are contracted on their last $N$ indices with partial
derivatives, and since the total stress-energy tensor must be itself a
tensorial object. Moreover, (ii) for a given order of truncation $N$, the full
multipolar structure of the test object is represented by a single moment
$I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}$. There is thus no a priori
decomposition of this moment between a hierarchy of moments (monopole,
dipole…). As we will see in the following, such a split can be obtained by
introducing a specific system of coordinates, the so-called adapted
coordinates.

###### Ellis skeleton in adapted coordinates

We now turn to coordinates adapted to the worldline, $x^{\mu}\to
X^{\mu}=(X^{0},\mathbf{X})$, with
$\mathbf{X}\triangleq\quantity(X^{1},X^{2},X^{3})$. They are required to
satisfy

$\displaystyle X^{0}\evaluated{}_{\gamma}$ $\displaystyle=\lambda,\quad
X^{i}\evaluated{}_{\gamma}=0\qquad\Rightarrow\qquad
v^{0}\evaluated{}_{\gamma}=\derivative{X^{0}}{\lambda}\evaluated{}_{\gamma}=1,\quad
v^{i}\evaluated{}_{\gamma}=\derivative{X^{i}}{\lambda}\evaluated{}_{\gamma}=0.$
(5.17)

An example of explicit construction of this type of coordinates are Fermi
normal coordinates [123]. In what follows, we will systematically assume that
such coordinates can be constructed over the worldtube $\mathcal{T}$ of the
body. Using these coordinates, we can write

$\displaystyle\begin{split}&\int_{\mathcal{T}}\differential[4]X\sqrt{-g}\,T_{\text{skel}}^{\mu\nu}\phi_{\mu\nu}\\\
&=\frac{(-1)^{N}}{N!}\int_{\gamma}\differential\lambda\,I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}\partial_{\alpha_{1}}\ldots\partial_{\alpha_{N}}\phi_{\mu\nu}\evaluated{}_{\gamma}\\\
&=\sum_{k=0}^{N}\frac{(-1)^{N}}{N!}\frac{N!}{k!(N-k)!}\int_{\gamma}\differential\lambda\,I^{\mu\nu
i_{1}\ldots i_{k}0\ldots
0}\partial_{i_{1}}\ldots\partial_{i_{k}}\partial^{N-k}_{0}\phi_{\mu\nu}\evaluated{}_{\gamma}\\\
&=\sum_{k=0}^{N}\frac{(-1)^{k}}{k!(N-k)!}\int_{\gamma}\differential\lambda\,\partial^{N-k}_{0}I^{\mu\nu
i_{1}\ldots i_{k}0\ldots
0}\partial_{i_{1}}\ldots\partial_{i_{k}}\phi_{\mu\nu}\evaluated{}_{\gamma}.\end{split}$
(5.18)

It is therefore useful to define a new collection of moments

$\displaystyle\gamma_{(N)}^{i_{1}\ldots
i_{k}}\triangleq\frac{1}{(N-k)!}\partial^{N-k}_{0}I^{\mu\nu i_{1}\ldots
i_{k}0\ldots 0},\qquad k\leq N.$ (5.19)

They still satisfy the algebraic symmetries

$\displaystyle\gamma_{(N)}^{\mu\nu i_{1}\ldots
i_{k}}=\gamma_{(N)}^{(\mu\nu)i_{1}\ldots
i_{k}}=\gamma_{(N)}^{\mu\nu(i_{1}\ldots i_{k})}.$ (5.20)

In terms of the new moments (and still in adapted coordinates), Ellis
decomposition is equivalently given by

$\displaystyle
T^{\mu\nu}_{\text{skel}}=\frac{1}{\sqrt{-g}}\mathcal{T}^{\mu\nu},\qquad\mathcal{T}^{\mu\nu}\triangleq\sum_{k=0}^{N}\frac{1}{k!}\gamma_{(N)}^{\mu\nu
i_{1}\ldots
i_{k}}(\lambda)\partial_{i_{1}}\ldots\partial_{i_{k}}\delta^{(3)}\quantity(\mathbf{X}).$
(5.21)

Notice that since $T^{\mu\nu}_{\text{skel}}$ is a tensor,
$\mathcal{T}^{\mu\nu}$ is a tensor density of weight $-1$. The proof of Eq.
(5.21) consists into integrating this expression against an arbitrary,
symmetric test function $\phi_{\mu\nu}$:

$\displaystyle\begin{split}\int_{\mathcal{T}}\differential[4]X\sqrt{-g}\,T_{\text{skel}}^{\mu\nu}\phi_{\mu\nu}&=\sum_{k=0}^{N}\frac{1}{k!}\int_{\gamma}\differential\lambda\int\differential[3]X\,\gamma_{(N)}^{\mu\nu
i_{1}\ldots
i_{k}}\partial_{i_{1}}\ldots\partial_{i_{k}}\delta^{(3)}(\mathbf{X})\phi_{\mu\nu}\\\
&=\sum_{k=0}^{N}\frac{(-1)^{k}}{k!}\int_{\gamma}\differential\lambda\,\gamma_{(N)}^{\mu\nu
i_{1}\ldots
i_{k}}\partial_{i_{1}}\ldots\partial_{i_{k}}\phi_{\mu\nu}.\end{split}$ (5.22)

We therefore recover the expression obtained in Eq. (5.18) using the
definition Eq. (5.19).

Before turning to the derivation of the MPD equations using the conservation
of the stress-energy tensor, let us make a couple of remarks. As we will check
explicitly at the dipole level, the multipoles $\gamma$ are fully determined
by the distribution of stress-energy, whereas the $I$s are not, thus leading
to the appearance of a gauge freedom in their definition. This can be seen
from Eq. (5.19), since the construction of the $I$s from the $\gamma$s require
to integrate with respect to time, thus leading to the appearance of arbitrary
integration constants.

Moreover, as we will see when discussing the relation between distributional
and smooth stress-energy tensors, the split of
$I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}$ in $k$ moments $\gamma^{\mu\nu
i_{1}\ldots i_{k}}$ actually corresponds to a physical split between a
monopole, a dipole, etc. Notice that the price to pay for obtaining simple
computations in Ellis representation is that we are forced to work in a
specific coordinate system, the adapted coordinates. Moreover, the multipole
decomposition tuned to this system is not explicitly covariant. Nevertheless,
this framework allows to recover the results obtained with more involved
approaches, e.g. Dixon representation. Finally, notice that a coordinate-free
approach to multipoles can be formulated, see [120] for more details.

###### Conservation equation for stress-energy tensor density

In adapted coordinates, since the decomposition Eq. (5.21) is valid, it is
more convenient to write the conservation of the stress-energy tensor Eq.
(5.12) in terms of the tensor density $\mathcal{T}^{\mu\nu}$ introduced in Eq.
(5.21). Since $T^{\mu\nu}_{\text{skel}}$ is a tensor, the covariant derivative
appearing in Eq. (5.12) can be expanded in terms of partial derivatives and
Christoffel symbols. We get

$\displaystyle\begin{split}\nabla_{\mu}T^{\mu\nu}_{\text{skel}}&=\partial_{\mu}T^{\mu\nu}_{\text{skel}}+2\Gamma^{(\mu}_{\nu\rho}T^{\nu)\rho}_{\text{skel}}\\\
&=\partial_{\mu}\quantity(\frac{1}{\sqrt{-g}})\mathcal{T}^{\mu\nu}+\frac{1}{\sqrt{-g}}\quantity(\partial_{\mu}\mathcal{T}^{\mu\nu}+2\Gamma^{(\mu}_{\nu\rho}\mathcal{T}^{\nu)\rho})\\\
&=\frac{1}{\sqrt{-g}}\quantity(\partial_{\mu}\mathcal{T}^{\mu\nu}+\Gamma^{\nu}_{\mu\rho}\mathcal{T}^{\mu\rho}).\end{split}$
(5.23)

The last equality uses the identity

$\displaystyle\partial_{\mu}\quantity(\sqrt{-g})=\sqrt{-g}\Gamma^{\alpha}_{\mu\alpha}.$
(5.24)

At the end of the day, the conservation equation for the stress-energy tensor
Eq. (5.12) is equivalent to the following equation for the stress-energy
tensor density

$\displaystyle\partial_{\mu}\mathcal{T}^{\mu\nu}+\Gamma^{\nu}_{\mu\rho}\mathcal{T}^{\mu\rho}=0.$
(5.25)

###### MPD equations in adapted coordinates

Before investigating the consequences of this conservation equation, it is
useful to write down the form that the MPD equations take in adapted
coordinates. Since we will derive only the pole-dipole equations in next
section, we set the force and torque terms to zero in all the equations below.

Recalling that $v^{\mu}=\delta^{\mu}_{0}$ on the worldline, the evolution
equation for the spin (4.25) takes the form

$\displaystyle\nabla_{0}S^{\mu\nu}=2p^{[\mu}v^{\nu]}.$ (5.26)

Separating the spatial coordinates from the temporal one, it is equivalent to
the two following equations

$\displaystyle\nabla_{0}S^{0i}$
$\displaystyle=-p^{i},\qquad\nabla_{0}S^{ij}=0.$ (5.27)

Using the definition of the Riemann tensor, the evolution equation for the
momentum (4.32) takes the form

$\displaystyle\begin{split}\nabla_{0}p^{\mu}&=-\frac{1}{2}R^{\mu}_{\phantom{\mu}0\alpha\beta}S^{\alpha\beta}\\\
&=-\quantity(\partial_{\alpha}\Gamma^{\mu}_{0\beta}+\Gamma^{\mu}_{\alpha\lambda}\Gamma^{\lambda}_{0\beta})S^{\alpha\beta}\\\
&=-\dot{\Gamma}^{0}_{0\mu}S^{0\mu}-\partial_{i}\Gamma^{0}_{0\mu}S^{i\mu}-\Gamma^{\mu}_{\alpha\lambda}\Gamma^{\lambda}_{0\beta}S^{\alpha\beta},\end{split}$
(5.28)

where we use the notation $\dot{}\triangleq\partial_{0}$.

###### Recovering MPD equations from stress-energy conservation

We will now truncate the multipole expansion to dipole order, and study the
constraints enforced by the conservation equation (5.25). At dipole order,
Ellis decomposition takes the form

$\displaystyle\mathcal{T}^{\mu\nu}=\gamma^{\mu\nu}\delta^{(3)}(\mathbf{X})+\gamma^{\mu\nu
i}\partial_{i}\delta^{(3)}(\mathbf{X})$ (5.29)

To lighten the notations, we drop the subscript “$(N)$” from the multipoles
$\gamma$ in the continuation of this section. For such a choice of stress-
energy density, the conservation equation (5.25) is a distributional identity.
It will be satisfied provided that

$\displaystyle\int\differential[3]X\,\quantity(\partial_{\mu}\mathcal{T}^{\mu\nu}+\Gamma^{\nu}_{\mu\rho}\mathcal{T}^{\mu\rho})\phi_{\nu}=0$
(5.30)

for any arbitrary test function $\phi_{\nu}$. Inserting the decomposition Eq.
(5.29) in this equation and integrating by parts yields

$\displaystyle\begin{split}&\quantity(\dot{\gamma}^{0\nu}+\Gamma^{\nu}_{\mu\rho}\gamma^{\mu\rho}-\partial_{i}\Gamma^{\nu}_{\mu\rho}\gamma^{\mu\rho
i})\phi_{\nu}\\\ &\quad-\quantity(\dot{\gamma}^{0\nu
i}+\gamma^{i\nu}+\Gamma^{\nu}_{\mu\rho}\gamma^{\mu\rho
i})\partial_{i}\phi_{\nu}+\gamma^{j\nu
i}\partial_{i}\partial_{j}\phi_{\nu}=0\end{split}$ (5.31)

Because this identity shall be valid for any choice of test function
$\phi_{\nu}$, it is equivalent to the set of three equations