#  A complete characterisation of the orbital shapes of the non-circular Kerr
geodesic solutions with circular orbit constants of motion

Andrew Mummery  Steven Balbus  Oxford University, Dept. of Physics, Beecroft
Building, Keble Road, Oxford OX1 3 RH, United Kingdom

###### Abstract

We present analytical solutions describing a family of both inwardly and
outwardly spiralling orbits in the Kerr spacetime. The solutions are exact,
and remarkable for their simplicity. These orbits all have the angular
momentum and energy of a circular orbit at some radius $r_{c}$, but are not
restricted to remaining on that circular orbit, a property not possible in
Newtonian gravity. We demonstrate that there are five distinct orbital
solutions which terminate at the black hole singularity, and three solutions
which either escape to infinity or remain bound. The different orbital
solutions are characterised entirely by the black hole spin $a$ and the
location of $r_{c}$. Photon orbits spiralling into or out of their (unstable)
circular orbit radii are also analysed. These have properties similar to the
hyperbolic class of massive particle orbits discussed herein.

##  I  Introduction

Despite the complexity of the Kerr spacetime, classes of exactly solvable
timelike geodesics (orbital solutions) are known. Examples include the
familiar circular orbits, radial plunges rev1 , “zoom-whirl” orbits, and
homoclinic orbits, which separate bound and plunging orbital domains LP1 . A
useful compendium of such orbits may be found in LP2 . The study of
relativistic test-particle orbits characterised by the energy and angular
momentum of a circular orbit, but which are not in fact circular, is by no
means new. As early as 1959, Darwin solved for some of these solutions in
Schwarzschild geometry Darwin . The homoclinic orbits LP1  are another example
of such “pseudo-circular” orbits, as are the inspirals from the innermost
stable circular orbit of Kerr black holes derived recently MB . Still other
studies of geodesic motion in black hole spacetimes are summarised in
Chandrasekhar Chand . While many such particular solutions are known, and
completely general solutions of, e.g., the bound orbits of the Kerr spacetime
are known in terms of elliptic integrals FH , a complete characterisation of
the entire family of the much simpler “pseudo-circular” orbital solutions is
not present in the literature; rather, incomplete sets of solutions are found
in different papers.

More than mathematical completeness is involved here. Exact geodesic solutions
can be a powerful and extremely useful theoretical tool, and as such often
surface in astrophysical studies. These include, for example, the study of
black hole accretion flows NT ; PT ; Rey , and the starting point for
analysing the gravitational radiation from extreme mass ratio inspirals D ;
DH1 ; DH2 ; DH3 ; CH ; LH ; GHK . The latter is likely to be an important
gravitational wave source for the LISA mission G . Additions to the known
family of simple orbital solutions are therefore of general interest, as they
offer insight into the internal dynamics of more complex situations, which are
otherwise accessible only through numerical means.

The purpose of the current study is to provide a systematic and detailed
analysis of those orbits for which the effective potential of a Kerr black
hole presents a double root and factors cleanly, a result of the underlying
symmetries of the circular constants of motion. This condition lends itself to
integrability in terms of elementary functions. As noted above, not all of the
solutions presented in this paper are new, but the majority of them are. In
particular, we present a formal analysis of those orbits about extremal Kerr
black holes $a=\pm M$ which do not appear to have been studied previously in
the literature. In particular, we demonstrate that eight distinct classes of
these “pseudo-circular” orbital solutions exist, five of which plunge to the
singularity, and three of which move to larger radii. There, they may either
remain bound or escape to infinity. We highlight a number of interesting
physical properties of these solutions.

The layout of this paper is as follows. In §2 we introduce the general
properties (stability, boundless, etc.) of circular test particle orbits in
the Kerr metric. In §3 we demonstrate that associated with each of these
circular orbits is a “pseudo-circular” orbit, which shares the same constants
of motion (angular momentum and energy) as these circular orbits, but is not
moving on a circular orbit. In §3 we derive the general properties of the
radial velocity of these quasi-circular orbits. In §4 we derive the solutions
of the entire family of these orbits evolving in the Schwarzschild metric,
before generalising to the Kerr metric in §5, and extremal Kerr black holes in
§6. In §7, we discuss how photon orbits fit into this general set of
solutions, and show that they are analogous to a hyperbolic class of massive
particle orbits. We conclude in §8.

##  II Preliminaries

We shall use geometric units in which the speed of light $c$ and the Newtonian
gravitational constant $G$ are both set equal to unity. In coordinates
$x^{\mu}$, the invariant line element ${\rm d}\tau$ is given by

${\rm d}\tau^{2}=-g_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}$ (1)

where $g_{\mu\nu}$ is the usual covariant metric tensor with spacetime indices
$\mu,\nu$. We shall use the standard Boyer-Lindquist coordinates
$(t,r,\theta,\phi)$, where $t$ is time as measured at infinity and the other
symbols have their usual quasi-spherical interpretation. We shall work
exclusively in the Kerr midplane $\theta=\pi/2$. For black hole mass $M$, and
angular momentum $a$ (both having dimensions of length in our choice of
units), the non-vanishing $g_{\mu\nu}$ and $g^{\mu\nu}$ we require for our
calculation may be summarised:

$\displaystyle g_{00}$ $\displaystyle=-1+2M/r,\quad
g^{00}=-{1\over\Delta}\left(r^{2}+a^{2}+{2Ma^{2}\over r}\right),$
$\displaystyle g_{0\phi}$ $\displaystyle=g_{\phi 0}=-2Ma/r,\quad
g^{0\phi}=g^{\phi 0}=-{2Ma}/{r\Delta},$ $\displaystyle g_{\phi\phi}$
$\displaystyle=r^{2}+a^{2}+2Ma^{2}/r,\quad
g^{\phi\phi}={1\over\Delta}\left(1-{2M\over r}\right),$ $\displaystyle g_{rr}$
$\displaystyle=r^{2}/\Delta,\quad\Delta\equiv r^{2}-2Mr+a^{2}.$ (2)

The 4-velocity vectors are denoted by $U^{\mu}={\rm d}x^{\mu}/{\rm d}\tau$.
For circular orbits, only $U^{0}$ and $U^{\phi}$ are present, their
$r$-dependence may be deduced from the two coupled algebraic equations:

$-1=g_{\mu\nu}U^{\mu}U^{\nu},\quad 0=U^{\mu}U^{\nu}\partial_{r}g_{\mu\nu}$ (3)

where $\partial_{r}$ is the standard partial derivative with respect to $r$.
(The second equation is a consequence of the geodesic equation for ${\rm
d}U_{r}/{\rm d}\tau$; see e.g., HEL ). Solving these equations gives

$U^{\phi}={M^{1/2}\over r^{3/2}{\cal D}},\quad
U^{0}={1+aM^{1/2}/r^{3/2}\over{\cal D}},$ (4)

where ${\cal D}^{2}=1-3M/r+2aM^{1/2}/r^{3/2}$.

Circular values of angular momentum and energy constants are calculated in
their usual manner ($U_{\mu}=g_{\mu\nu}U^{\nu}$) and are given by

$\displaystyle{J}\equiv U_{\phi}$
$\displaystyle=(Mr)^{1/2}{(1+a^{2}/r^{2}-2aM^{1/2}/r^{3/2})\over\sqrt{1-3M/r+2aM^{1/2}/r^{3/2}}},$
(5) $\displaystyle{\gamma}\equiv-U_{0}$
$\displaystyle={(1-2M/r+aM^{1/2}/r^{3/2})\over\sqrt{1-3M/r+2aM^{1/2}/r^{3/2}}}.$
(6)

##  III Radial velocity of pseudo-circular orbits

###  III.1 The radial velocity equation

The radial velocity of a massive particle is given by the solution of the
geodesic equation $g_{\mu\nu}U^{\mu}U^{\nu}=-1$, or explicitly

$({U^{r}})^{2}+{{J}\over r^{2}}\left({2Ma{\gamma}\over r}+\left(1-{2M\over
r}\right){J}\right)\\\ -{{\gamma}\over
r^{2}}\left(\left(r^{2}+a^{2}+{2Ma^{2}\over r}\right){\gamma}-{2Ma{J}\over
r}\right)\\\ +1+{a^{2}\over r^{2}}-{2M\over r}=0,$ (7)

where we have expressed all azimuthal $U^{\phi}$ and temporal $U^{0}$
4-velocity components in terms of their (conserved) 4-momenta counterparts $J$
and $\gamma$ MB . Note that equation 7 is of the form $({U^{r}})^{2}+V_{\rm
eff}(r)=0$, which defines an effective potential $V_{\rm eff}$, cubic in
$1/r$, which may generally be factored:

$V_{\rm eff}(r)=-V_{0}\left({r_{1}\over r}-1\right)\left({r_{2}\over
r}-1\right)\left({r_{3}\over r}-1\right),$ (8)

where $r_{1},r_{2}$ and $r_{3}$ are the general (possibly complex) roots of
${U^{r}}$. For a circular orbit at radius $r_{c}$, both $V_{\rm eff}(r_{c})=0$
and $\partial_{r}V_{\rm eff}(r_{c})=0$, meaning $r_{c}$ must be a double root
of $V_{\rm eff}$, i.e., $r_{1}=r_{2}=r_{c}$. The normalisation constant
$V_{0}$ may be found by going back to equation (7), and evaluating
$r^{3}({U^{r}})^{2}$ in the limit $r\rightarrow 0$. Taking this limit we find

$V_{0}={2M\over r_{1}r_{2}r_{3}}({J}-a{\gamma})^{2}.$ (9)

On the other hand, we may consider the $r\rightarrow\infty$ limit, which leads
to

$V_{0}=1-{\gamma}^{2}.$ (10)

Taking ${\gamma}$ and ${J}$ to be the energy and angular momentum of a
circular orbit at $r_{c}$, and writing $r_{1}=r_{2}=r_{c}$, we can constrain
the other root $r_{3}$ with these two expressions

$r_{3}={2M\over r_{c}^{2}}{({J}-a{\gamma})^{2}\over 1-{\gamma}^{2}},$ (11)

which gives the radial velocity equation

$U^{r}=-\sqrt{1-{\gamma}^{2}}\left({r_{c}\over r}-1\right)\sqrt{{r_{3}\over
r}-1}.$ (12)

For any given black hole spin and mass, $\gamma$, $J$ and $r_{3}$ are fully
determined by the choice of $r_{c}$. As $r_{c}$ descends through smaller
radii, the behaviour of pseudo-circular orbits qualitatively changes as
$r_{c}$ passes through a series of characteristic radii, which we discuss
below.

###  III.2 Characteristic radii

The largest characteristic radius is the innermost stable circular orbit, or
ISCO, denoted $r_{I}$. Circular orbits with $r_{c}>r_{I}$ are stable and
approach their Newtonian behaviour at large radii; orbits with $r_{c}<r_{I}$
are dynamically unstable. The ISCO itself corresponds to a radius of marginal
stability.

The second key radius is the innermost bound circular orbit, $r_{\rm ib}$, or
IBCO. Circular orbits with $r_{\rm ib}<r_{c}<r_{I}$ remain bound though they
are unstable. Physically, this instability results in a class of pseudo-
circular orbits that describe a particle infinitesimally perturbed outwards
from $r_{c}$, then traveling out to a finite radius ($r_{3}$) before returning
to $r_{c}$ once again in the limit $t\to\infty$.

The final key circular orbit radius is the photon radius $r_{p}$, lying
interior to $r_{\rm ib}$. The photon radius corresponds to the unique radius
at which photons are able to undergo (unstable) circular motion. Massive
particle circular orbits with $r_{p}<r_{c}<r_{\rm ib}$ are possible, but are
both unstable and ultimately unbound. The region between $r_{p}$ and the outer
event horizon $r_{+}$ does not host any circular orbit solutions, for either
massive particles or for photons, as this would require superluminal
velocities.

When normalised to $M$, each of these special radii depends only upon the Kerr
black hole’s angular momentum parameter $a$ (e.g., HEL ) via $a_{\star}=a/M$.
The ISCO radius is given explicitly by Bard

${r_{I}\over
M}=3+Z_{2}-{a_{\star}\over|a_{\star}|}\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})},$ (13)

where

$Z_{1}=1+\left(1-a_{\star}^{2}\right)^{1/3}\left[\left(1+a_{\star}\right)^{1/3}+\left(1-a_{\star}\right)^{1/3}\right],$
(14)

and

$Z_{2}=\sqrt{3a_{\star}^{2}+Z_{1}^{2}}.$ (15)

The IBCO radius is Chand

${r_{\rm ib}\over M}=\left(1+\sqrt{1-a_{\star}}\right)^{2}.$ (16)

The photon radius is given by HEL

${r_{p}\over M}=4\cos^{2}\left[{1\over 3}\cos^{-1}\left(-{a\over
M}\right)\right],$ (17)

and finally, the inner ($r_{-}$) and outer ($r_{+}$) event horizons of a Kerr
black hole are located at HEL

${r_{\pm}\over M}=1\pm\sqrt{1-a_{\star}^{2}}.$ (18)

The values of these five important radii are plotted as a function of black
hole spin in Fig. 1. In the formal $a_{\star}\to 1$ limit, each of these radii
converge on $r=M$. However, this is in actual fact a Boyer-Lindquist
coordinate effect. They key radii remain separated by a finite proper radial
distance. A free-falling observer would, just as for all other values of the
black hole spin parameter, first pass an ISCO, then the IBCO, then the photon
orbit, and finally the event horizon in this $a=M$ limit.

Figure 1: The key radii of the Kerr spacetime, plotted as a function of black
hole spin.

###  III.3 The relationship between $r_{3}$ and $r_{c}$.

Figure 1 shows that the ordering $r_{I}>r_{\rm ib}>r_{p}$ is always obeyed.
The nature of a circular orbit will then depend upon the magnitude of $r_{c}$
relative to these key quantities, and is best understood by using $r_{c}$ and
$r_{3}$ (a direct function of $r_{c}$) in tandem.

Consider first the large $r_{c}$ limit of the $r_{3}$ equation. We then have
${J}\rightarrow\sqrt{Mr_{c}}$ and $1-{\gamma}^{2}\rightarrow M/r_{c}$
following equations 5 and 6. Equation 11 then demonstrates that $r_{3}$
approaches $2M$, the Schwarzschild event horizon. The same limit holds for
Kerr black holes as well, and so lies outside $r_{+}$ for all values of
$a_{\star}$.

Figs. 2 and 3 are plots in the $(r_{c}-r_{\rm ib})/M$, $r/M$ plane, for a
Schwarzschild and $a=0.9$ Kerr black hole respectively. (We restrict the plots
to the $r_{c}>r_{\rm ib}$ domain.) Both $r_{c}$ and $r_{3}$ appear as labelled
curves, which by definition must intersect at $r_{c}=r_{I}$. It can be seen
that for $r_{c}>r_{I}$, the root $r_{3}$ is always located between $r_{+}$ and
$r_{I}$. In particular, as $r_{c}$ approaches $r_{I}$ from above, $r_{3}$
approaches $r_{I}$ from below. At the intersection, the right side of equation
12 for the radial velocity takes the form of a square root of a perfect cubic.
This is of some astrophysical interest MB . As $r_{c}$ approaches $r_{\rm
ib}$, the $r_{3}$ root increases monotonically, diverging as $r_{c}\rightarrow
r_{\rm ib}$.

The shaded zones of these plots denote regions accessible to a particle in
pseudo circular orbit for a given $r_{c}$ (or the associated $r_{3}$). The red
region indicates the domain of plunging orbits; the blue, homoclinic orbits.
(Homoclinic orbits will be discussed in more detail below.) In the white zone,
only $r=r_{c}$ is allowed, in conformance with Newtonian sensibilities! For
$r_{c}=r_{\rm ib}$, the particle may access all radii whilst retaining its
constant angular momentum and energy, either through an escape to infinity
with infinitesimal final velocity, or a plunge into the singularity. Finally,
we find $r_{3}<0$ when $r_{c}$ is located between the IBCO and photon radius,
$r_{p}<r_{c}<r_{\rm ib}$. (Not shown.) In this regime, all radii are similarly
accessible, the difference being that particles escaping to infinity do so
with finite velocity.

Figure 2: The properties of the third root $r_{3}$ of the Schwarzschild
($a=0$) velocity potential, plotted against the difference of the circular
radius $r_{c}$ from the IBCO $r_{\rm ib}=4M$. Different regions of radial
space available to different classes of pseudo-circular orbits are indicated
by the shaded regions. See text for further details.  Figure 3: As in Fig. 2,
except for Kerr spin parameter $a=0.9M$. Note that $r_{3}\geq r_{+}$ for all
values of both $r_{c}$ and $a$.

##  IV Schwarzschild solutions

The Schwarzschild metric is particularly simple as, unlike the finite $a$ Kerr
metric, the $\phi$ coordinate is well-behaved at the single event horizon. The
general solution required for characterising the various Schwarzschild orbits
starts with the simple expression for the $\phi$ component of the test
particles 4-velocity

${{\rm d}\phi\over{\rm d}\tau}=U^{\phi}={{J}\over r^{2}}.$ (19)

The shape of a particle’s orbit is then given by the solution of the following
ordinary differential equation

${{\rm d}\phi\over{\rm
d}r}={U^{\phi}\over{U^{r}}}=-{{J}\over\sqrt{1-{\gamma}^{2}}}{1\over
r^{2}\left({r_{c}/r}-1\right)\sqrt{{r_{3}/r}-1}},$ (20)

or explicitly

${\sqrt{1-{\gamma}^{2}}\over{J}}\phi=-\int{{\rm d}r\over
r^{2}}{1\over\left({r_{c}/r}-1\right)\sqrt{{r_{3}/r}-1}}.$ (21)

For three of the five radial regimes we may solve the integrals by defining
$\tilde{u}=r_{3}/r$, leaving

${\sqrt{1-{\gamma}^{2}}\over{J}}\phi={1\over
r_{3}}\int{1\over\left({r_{c}\tilde{u}/r_{3}}-1\right)\sqrt{{\tilde{u}}-1}}\,{\rm
d}\tilde{u}.$ (22)

Finally, with $y=\sqrt{\tilde{u}-1}$ we have

${\sqrt{1-{\gamma}^{2}}\over{J}}\phi={2\over
r_{c}}\int{1\over\left(1-r_{3}/r_{c}+y^{2}\right)}\,{\rm d}y,$ (23)

note $r=r_{3}/(1+y^{2})$. The exact solution of this final integral depends on
the value of the ratio $r_{3}/r_{c}$. The integral

$I(y)=\int{1\over\left(1-r_{3}/r_{c}+y^{2}\right)}\,{\rm d}y,$ (24)

has solution

$\displaystyle
I(y)=\begin{cases}\tan^{-1}\left(y/\sqrt{1-r_{3}/r_{c}}\right)/\sqrt{1-r_{3}/r_{c}},\quad
r_{3}<r_{c},\\\ \\\ -1/y,\quad r_{3}=r_{c},\\\ \\\
-\tanh^{-1}\left(y/\sqrt{r_{3}/r_{c}-1}\right)/\sqrt{r_{3}/r_{c}-1},\quad
r_{3}>r_{c}.\end{cases}$ (25)

For the final radial regime $r_{p}<r<r_{\rm ib}$, we have ${\gamma}>1$ and
$r_{3}<0$ and the integral (eq. 21) should be re-written before the
substitution $\tilde{u}=-r_{3}/r=|r_{3}|/r$ is made. Writing all terms as a
manifestly positive,

${\sqrt{{\gamma}^{2}-1}\over{J}}\phi=-\int{{\rm d}r\over
r^{2}}{1\over\left({r_{c}/r}-1\right)\sqrt{{|r_{3}|/r}+1}}.$ (26)

We now follow a similar approach to the previous case with
$\tilde{u}=-r_{3}/r=|r_{3}|/r$, $y=\sqrt{1+\tilde{u}}$, and
$r=|r_{3}|/(y^{2}-1)$. We then find

${\sqrt{{\gamma}^{2}-1}\over{J}}\phi=\int{1\over\left({1+|r_{3}|/r_{c}}-y^{2}\right)}{\rm
d}y.$ (27)

Note

$\int{1\over\left({1+|r_{3}|/r_{c}}-y^{2}\right)}{\rm
d}y={\tanh^{-1}\left(y/\sqrt{|r_{3}|/r_{c}+1}\right)\over\sqrt{|r_{3}|/r_{c}+1}}.$
(28)

Note that $\tanh^{-1}(X)$ and $\tanh^{-1}(1/X)$ have exactly the same
derivative, so that the proper argument of this function must be chosen on the
basis of whether $X$ or its reciprocal is less than unity. This flexibility
embodies the difference between the plunging and out-spiralling orbits. We now
discuss each of the eight different solutions of these integrals in turn.

###  IV.1 $r_{c}>r_{I}$ plunging orbits

For $r_{c}>r_{I}$, we have $r_{3}<r_{c}$. We may then solve explicitly for
$r(\phi)$ from the first line of equation 25,

$r(\phi)={r_{3}\over 1+(1-r_{3}/r_{c})\tan^{2}(\phi/\phi_{\star})},$ (29)

where

$\phi_{\star}={2{J}\over\sqrt{r_{c}(r_{c}-r_{3})(1-{\gamma}^{2})}},$ (30)

with the convention that the orbit starts at $\phi=0$. Unlike other
Schwarzschild solutions, the $r_{c}>r_{I}$ orbits involve only a finite
rotation angle $\Delta\phi$ before reach the singularity $r=0$ (Fig. 4):

$\Delta\phi={\pi{J}\over\sqrt{r_{c}(r_{c}-r_{3})(1-{\gamma}^{2})}}.$ (31)

Note the divergence as $r_{3}\rightarrow r_{c}$ (i.e., $r_{c}\to r_{I}$).

Figure 4: A Schwarzschild plunging orbit, with $r_{c}=7M$.

As $r_{c}\rightarrow\infty$, we have

$\Delta\phi\rightarrow\pi\left(1-{2M\over r_{c}}\right)^{-1/2}>\pi,$ (32)

i.e. all orbits complete at least one half-rotation of the singularity,
independent of $r_{c}$. Interestingly, in this limit the plunging particle has
a formally divergent specific angular momentum
$J\rightarrow\sqrt{Mr_{c}}\rightarrow\infty$. Having an arbitrarily large
amount of angular momentum and starting outside the event horizon is not
sufficient for preventing the particle from eventually being swallowed by the
singularity!

###  IV.2 $r_{c}=r_{I}$ spiral orbits

The ISCO radius $r_{c}=r_{I}$ is the special location at which
$r_{3}=r_{c}=6M$, and the effective potential is a perfect cube. This
remarkably simple solution was discussed in detail in MB , and seems to have
been first written explicitly in Chand . It has the following form:

$r(\phi)={6M\over 1+12/\phi^{2}},$ (33)

with the convention that $\phi$ increases from $-\infty$ at $r=6M$ to $0$ at
the singularity $r=0$.

###  IV.3 $r_{\rm ib}<r_{c}<r_{I}$ homoclinic and plunging orbits

For $r_{\rm ib}<r_{c}<r_{I}$, two possible pseudo-circular orbits exist. These
correspond to orbits which either spiral outwards from $r_{c}$ towards
$r=r_{3}$, or inwards from $r=r_{c}$ towards $r=0$.

####  IV.3.1 $r>r_{c}$: homoclinic orbits

The homoclinic orbits where first discussed in full detail by LP1 . However,
the conceptual history of homoclinic orbits goes back much further in time.
Darwin Darwin  first wrote down an explicit Schwarzschild homoclinic orbit in
1959. (Interestingly, Darwin writes that he was motivated by his experience
with the relativistic Bohr model for hydrogen.) While these solutions are
therefore not new, we shall include a brief discussion for completeness.

For $r_{\rm ib}<r_{c}<r_{I}$, the third root of the radial velocity $r_{3}$ is
greater than $r_{c}$. Thus, orbits exist where the particle spirals outwards
from $r=r_{c}$ towards $r=r_{3}$. The following solution for $r(\phi)$
describes the test particles motion:

$r(\phi)={r_{3}\over 1+(r_{3}/r_{c}-1)\tanh^{2}(\phi/\phi_{\star})},$ (34)

where

$\phi_{\star}={2{J}\over\sqrt{r_{c}(r_{3}-r_{c})(1-{\gamma}^{2})}}.$ (35)

A homoclinic orbit begins with $\phi=-\infty$, at $r=r_{c}$. As the azimuthal
coordinate increase the radial coordinate grows until $\phi=0,r=r_{3}$, before
the orbit repeats itself, with the radius returning to $r=r_{c}$ as
$\phi\rightarrow+\infty$.

Figure 5: A series of many Schwarzschild homoclinic orbits, with circular
orbit radii ranging from $r_{c}=4.15M$ (purple) to $r_{c}=4.45M$ (red).

####  IV.3.2 $r<r_{c}$: plunging orbits

The complementary test particle orbit which spirals inwards from $r=r_{c}$ to
the singularity is described by the following function:

$r(\phi)={r_{3}\over 1+(r_{3}/r_{c}-1)\coth^{2}(\phi/\phi_{\star})},$ (36)

where $\phi_{\star}$ is given once again by eq. once again by eq. 35. This
orbit is in many ways similar to the inspiralling trajectory starting from the
ISCO. The particle undergoes a formally infinite number of rotations about the
unstable circular orbit $r_{c}$ (the $\phi\to-\infty$ limit), before
transitioning to a plunge which in effect “begins” at an azimuthal angle
$\phi\sim-\phi_{\star}$ and terminates at $\phi=0$.

###  IV.4 $r_{c}=r_{\rm ib}$ parabolic orbits

The IBCO is defined by the circular orbit radius at which ${\gamma}(r_{\rm
ib})=1$, allowing an escape to infinity with vanishing velocity, and
$r_{3}\rightarrow\infty$. In this limit, the radial component of the test
particle 4-velocity is quite well-behaved, and simplifies to

$U^{r}_{\rm ib}=-{J\over r_{\rm ib}}\sqrt{2M\over r}\left({r_{\rm ib}\over
r}-1\right).$ (37)

For a Schwarzschild black hole, from equations 5 and 6 we find $J_{\rm
ib}=r_{\rm ib}=4M$, and $U^{\phi}=4M/r^{2}$. Thus

${{\rm d}\phi\over{\rm d}r}={U^{\phi}\over U^{r}}={2\sqrt{2M}\over
r^{3/2}}{1\over\left({4M/r}-1\right)}.$ (38)

The two solutions of this differential equation (corresponding to inspiralling
and outspiralling orbits) are

$r(\phi)=4M\tanh^{2}\left({\phi\over 2\sqrt{2}}\right),\quad(r<r_{\rm
ib}\,\,{\rm inspiral}),$ (39)

and

$r(\phi)=4M\coth^{2}\left({\phi\over 2\sqrt{2}}\right),\quad(r>r_{\rm
ib}\,\,{\rm outspiral}).$ (40)

These are displayed in Fig. 6. Note that by running the out-spiralling orbit
in a time reversed fashion, we see a particle in-falling from rest at
infinity, and ending on a circular orbit.

Figure 6: The Schwarzschild parabolic orbits, spiralling outwards (blue) or
inwards (red) from the IBCO ($r_{\rm ib}=4M$).

###  IV.5 $r_{p}<r_{c}<r_{\rm ib}$ hyperbolic orbits

For $r_{p}<r_{c}<r_{\rm ib}$ we have $\gamma>1,r_{3}<0$, and a test particle
can escape to infinity with non-zero radial velocity. The shape of the
particles orbit is the following

$r(\phi)={|r_{3}|\over(1+|r_{3}|/r_{c})\tanh^{2}(\phi/\phi_{\star})-1},\quad({\rm
outspiral})$ (41)

where

$\phi_{\star}={2{J}\over\sqrt{r_{c}(|r_{3}|+r_{c})({\gamma}^{2}-1)}}.$ (42)

A test particle undergoing this orbit escapes to infinity at a finite angle
(the orbit begins with $\phi\to-\infty$)

$\phi_{\infty}=-{2{J}\over\sqrt{r_{c}(|r_{3}|+r_{c})({\gamma}^{2}-1)}}\tanh^{-1}\left({1\over\sqrt{1+|r_{3}|/r_{c}}}\right).$
(43)

This escape angle is a measure of the finite radial velocity with which these
orbits approach infinity. These trajectories are plotted in Fig. 7. The
decrease in the escape angle as $r_{c}$ approaches the photon radius $3M$ is
clearly visible. Alternatively, the test particle may spiral inwards from
$r=r_{c}$ to the singularity at $r=0$ according to the following profile

$r(\phi)={|r_{3}|\over(1+|r_{3}|/r_{c})\coth^{2}(\phi/\phi_{\star})-1},\quad({\rm
inspiral}).$ (44)

with the convention where $\phi$ progresses from $-\infty$ to $0$ while $r$
progresses from $r_{c}$ to $0$.

Figure 7: A series of many Schwarzschild hyperbolic orbits, with circular
orbit radii ranging from $r_{c}=r_{\rm ib}-0.01M$ (purple) to
$r_{c}=r_{p}+0.01M$ (red).

##  V Kerr solutions

The shape of spiral in the Kerr metric for a finite mass particle differs from
the equivalent Schwarzschild orbits in two key ways: the first is that frame
dragging forces the particle to co-rotate with the black hole within the
ergosphere, which in the equatorial plane lies at the location $r_{E}=2M$. The
second is that the $\phi$ diverges, a purely bad-coordinate effect, at the two
event horizons of the black hole hole, at the radial locations
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$.

The general form of the Kerr azimuthal 4-velocity is the following

${{\rm d}\phi\over{\rm d}\tau}=U^{\phi}={2M{\gamma}a/r+{J}(1-2M/r)\over
r^{2}-2Mr+a^{2}},$ (45)

which implies that the orbital integral takes the form

$\sqrt{1-{\gamma}^{2}}\phi=\int{r^{3/2}(2M{\gamma}a/r+{J}(1-2M/r))\over(r-r_{+})(r-r_{-})(r-r_{c})\sqrt{r_{3}-r}}\,{\rm
d}r.$ (46)

The solution of this integral is clearly more complicated than its
Schwarzschild analogue. We define a “Kerr-type” integral to be of the form

$I_{\cal K}=\int{x^{n/2}\over(x-A)(x-B)(x-C)\sqrt{D-x}}\,{\rm d}x.$ (47)

Kerr-type integrals can be transformed into Schwarzschild-type integrals by
repeated use of the identity

${1\over(x-\alpha)(x-\beta)}={1\over\alpha-\beta}\left[{1\over
x-\alpha}-{1\over x-\beta}\right],$ (48)

where a Schwarzschild-type integral has the following form

$I_{\cal S}=\int{x^{n/2}\over(x-\Gamma)\sqrt{\Delta-x}}\,{\rm d}x.$ (49)

The solutions of these Schwarzschild-type integrals can then be found by
similar steps to those performed in section IV.

The following Kerr-integral solutions will be useful

$\displaystyle I_{1}$
$\displaystyle=\int{x^{3/2}\over(x-A)(x-B)(x-C)\sqrt{D-x}}\,{\rm d}x$
$\displaystyle={2A^{3/2}\over(A-B)(C-A)\sqrt{D-A}}\tanh^{-1}\left(\sqrt{D/x-1\over
D/A-1}\right)$
$\displaystyle-{2B^{3/2}\over(A-B)(C-B)\sqrt{D-B}}\tanh^{-1}\left(\sqrt{D/x-1\over
D/B-1}\right)$
$\displaystyle-{2C^{3/2}\over(C-A)(C-B)\sqrt{D-C}}\tanh^{-1}\left(\sqrt{D/x-1\over
D/C-1}\right),$ (50)

and

$\displaystyle I_{2}$
$\displaystyle=\int{x^{1/2}\over(x-A)(x-B)(x-C)\sqrt{D-x}}\,{\rm d}x$
$\displaystyle=-{2A^{1/2}\over(A-B)(A-C)\sqrt{D-A}}\tanh^{-1}\left(\sqrt{D/x-1\over
D/A-1}\right)$
$\displaystyle+{2B^{1/2}\over(A-B)(B-C)\sqrt{D-B}}\tanh^{-1}\left(\sqrt{D/x-1\over
D/B-1}\right)$
$\displaystyle-{2C^{1/2}\over(C-A)(C-B)\sqrt{D-C}}\tanh^{-1}\left(\sqrt{D/x-1\over
D/C-1}\right).$ (51)

It is important here to note that these solutions assume $0<A<B<C<D$, and
$A,B,C<x<D$. For the case $C>D$ the identity $-i\tanh^{-1}(iz)=\tan^{-1}(z)$
allows the final term to be appropriately re-written. For $D=C$ some care must
be taken, which is discussed further below. The arguments of the $\tanh^{-1}$
functions should be inverted for $x<A,B,C$.

The general form of the Kerr orbital solution is therefore given by a sum of
three functions,

$\phi(r)=f_{+}(r)+f_{-}(r)+f_{0}(r),$ (52)

where the functions $f_{\pm}$ are defined as the functions which encapsulate
the divergent $\phi$ behaviour at $r_{\pm}$, while $f_{0}$ will be
qualitatively similar in character to the Schwarzschild solutions discussed
above. For $r_{c}>r_{\rm ib}$, $f_{\pm}(r)$ have the following form

$f_{\pm}(r)=\pm\left[{2{J}r_{\pm}^{3/2}+4Mr_{\pm}^{1/2}(a{\gamma}-{J})\over(r_{+}-r_{-})(r_{c}-r_{\pm})\sqrt{r_{3}-r_{\pm}}}\right]\\\
\times{1\over\sqrt{1-{\gamma}^{2}}}\tanh^{-1}\left(\sqrt{r_{3}/r-1\over
r_{3}/r_{\pm}-1}\right).$ (53)

Note that for $a=0$, where $r_{+}=2M,r_{-}=0$, we find $f_{\pm}=0$. For
$r_{c}\leq r_{\rm ib}$ the form of $f_{\pm}(r)$ changes slightly, as will be
discussed below. Note that for radii within the event horizons $r<r_{\pm}$ the
arguments of the $\tanh^{-1}$ functions in $f_{\pm}$ should be inverted.

###  V.1 $r_{c}>r_{I}$ plunging orbits

For $r_{c}>r_{I}$, the third root of the radial velocity polynomial $r_{3}$ is
small $r_{+}<r_{3}<r_{I}<r_{c}$. In this limit the only solution for
$f_{0}(r)$ is the following

$f_{0}(r)=\left[{2{J}r_{c}^{3/2}+4Mr_{c}^{1/2}(a{\gamma}-{J})\over(r_{c}-r_{+})(r_{c}-r_{-})\sqrt{r_{c}-r_{3}}}\right]\\\
\times{1\over\sqrt{1-{\gamma}^{2}}}\tan^{-1}\left(\sqrt{r_{3}/r-1\over
1-r_{3}/r_{c}}\right),$ (54)

which is entirely analogous to the Schwarzschild plunge discussed above. Note
that frame dragging and the poorly behaved $\phi$ coordinate on the Kerr event
horizons mean that there is no well-defined “rotated angle” which this orbit
traverses over its infall. However, there remain, in the formal limit
$r_{c}\rightarrow\infty$, orbital solutions of the Kerr equations which start
outside of the event horizon at $r=2M$, before subsequently reaching $r=0$,
despite the particle having an arbitrarily large specific angular momentum.

###  V.2 $r_{c}=r_{I}$ spiral orbits

For $r_{c}=r_{I}$, the third root of the radial velocity polynomial $r_{3}$ is
equal to the ISCO $r_{3}=r_{c}=r_{I}$. In this limit the solution $f_{0}(r)$
is the following (first derived in MB )

$f_{0}(r)=\sqrt{6r_{I}\over M}\ \left({2M(J-a\gamma)-r_{I}J\over
r^{2}_{I}-2Mr_{I}+a^{2}}\right)\sqrt{r\over r_{I}-r}.$ (55)

###  V.3 $r_{\rm ib}<r_{c}<r_{I}$ homclinic orbits

For $r_{\rm ib}<r_{c}<r_{I}$, the third root of the radial velocity polynomial
$r_{3}$ is greater than the ISCO $r_{3}>r_{I}>r_{c}$. In this limit there are
two solutions for $f_{0}(r)$. The first, the homoclinic solution, is the
following (as first derived in LP1 )

$f_{0}(r)=-\left[{2{J}r_{c}^{3/2}+4Mr_{c}^{1/2}(a{\gamma}-{J})\over(r_{c}-r_{+})(r_{c}-r_{-})\sqrt{r_{3}-r_{c}}}\right]\\\
\times{1\over\sqrt{1-{\gamma}^{2}}}\tanh^{-1}\left(\sqrt{r_{3}/r-1\over
r_{3}/r_{c}-1}\right).$ (56)

The second, a plunging orbit, has the following form:

$f_{0}(r)=-\left[{2{J}r_{c}^{3/2}+4Mr_{c}^{1/2}(a{\gamma}-{J})\over(r_{c}-r_{+})(r_{c}-r_{-})\sqrt{r_{3}-r_{c}}}\right]\\\
\times{1\over\sqrt{1-{\gamma}^{2}}}\tanh^{-1}\left(\sqrt{r_{3}/r_{c}-1\over
r_{3}/r-1}\right).$ (57)

###  V.4 $r_{c}=r_{\rm ib}$ IBCO parabolic orbits

For $r_{c}=r_{\rm ib}$, we have simultaneously $1-{\gamma}^{2}=0$ and
$r_{3}\rightarrow\infty$. The combination
$\sqrt{(1-{\gamma}^{2})(r_{3}-r_{c})}$ converges to $\sqrt{2M}({J}-a)/r_{\rm
ib}$, and the three functions have the following form

$f_{\pm}(r)=\pm\left[{2{J}r_{\rm ib}r_{\pm}^{3/2}+4Mr_{\pm}^{1/2}r_{\rm
ib}(a-{J})\over\sqrt{2M}(r_{+}-r_{-})(r_{\rm ib}-r_{\pm})(J-a)}\right]\\\
\times\tanh^{-1}\left(\sqrt{r_{\pm}\over r}\right).$ (58)

and

$f_{0}(r)=-\left[{2{J}r_{\rm ib}^{5/2}+4Mr_{\rm
ib}^{3/2}(a-{J})\over\sqrt{2M}(r_{\rm ib}-r_{+})(r_{\rm
ib}-r_{-})(J-a)}\right]\\\ \times\tanh^{-1}\left(\sqrt{r_{\rm ib}\over
r}\right).$ (59)

for orbits which spiral outwards $r>r_{\rm ib}$. For inspiralling ($r<r_{\rm
ib}$) orbits, we instead have

$f_{0}(r)=-\left[{2{J}r_{\rm ib}^{5/2}+4Mr_{\rm
ib}^{3/2}(a-{J})\over\sqrt{2M}(r_{\rm ib}-r_{+})(r_{\rm
ib}-r_{-})(J-a)}\right]\\\ \times\tanh^{-1}\left(\sqrt{r\over r_{\rm
ib}}\right).$ (60)

For the inspiralling orbits, and for radii within the event horizons
$r<r_{\pm}$, the arguments of the $\tanh^{-1}$ functions in $f_{\pm}$ should
be inverted.

###  V.5 $r_{p}<r_{c}<r_{\rm ib}$ hyperbolic orbits

For $r_{p}<r_{c}<r_{\rm ib}$ we have $\gamma>1,r_{3}<0$, and a test particle
can escape to infinity with non-zero radial velocity. The shape of the
particle orbit is:

$f_{\pm}(r)=\pm\left[{2{J}r_{\pm}^{3/2}+4Mr_{\pm}^{1/2}(a{\gamma}-{J})\over(r_{+}-r_{-})(r_{c}-r_{\pm})\sqrt{|r_{3}|+r_{\pm}}}\right]\\\
\times{1\over\sqrt{{\gamma}^{2}-1}}\tanh^{-1}\left(\sqrt{|r_{3}|/r+1\over|r_{3}|/r_{\pm}+1}\right),$
(61)

and

$f_{0}(r)=-\left[{2{J}r_{c}^{3/2}+4Mr_{c}^{1/2}(a{\gamma}-{J})\over(r_{c}-r_{+})(r_{c}-r_{-})\sqrt{|r_{3}|+r_{c}}}\right]\\\
\times{1\over\sqrt{{\gamma}^{2}-1}}\tanh^{-1}\left(\sqrt{|r_{3}|/r+1\over|r_{3}|/r_{c}+1}\right),$
(62)

for orbits which spiral outwards $r>r_{\rm c}$. The inspiralling orbits
instead have

$f_{0}(r)=-\left[{2{J}r_{c}^{3/2}+4Mr_{c}^{1/2}(a{\gamma}-{J})\over(r_{c}-r_{+})(r_{c}-r_{-})\sqrt{|r_{3}|+r_{c}}}\right]\\\
\times{1\over\sqrt{{\gamma}^{2}-1}}\tanh^{-1}\left(\sqrt{|r_{3}|/r_{c}+1\over|r_{3}|/r+1}\right).$
(63)

##  VI Extremal Kerr Black holes, $a=\pm M$

For $a=\pm M$, the two event horizons of the Kerr black hole coincide
$r_{\pm}=M$, and so the partial fractions approach to solving the Kerr-type
integrals must be modified. As far as the authors are aware, none of the
pseudo-circular orbital solutions have been studied before in this extremal
limit (except the ISCO inspiral MB ).

###  VI.1 Extremal prograde spin $a=+M$

For $a=+M$, circular orbits in the Boyer-Lindquist coordinate system are
stable at all radii down to the event horizon $r_{+}=M$. (In these
coordinates, this is also the ISCO radius). We therefore have
$r_{\pm}<r_{3}<r_{c}$ for all $r_{c}$, and there is only one valid pseudo-
circular orbital solution, a plunging solution starting at $r_{3}$. The
solution no longer has two $f_{\pm}(r)$ solution, as the two event horizons
coincide, but is of the form

$\sqrt{1-{\gamma}^{2}}\phi=\int{r^{3/2}(2M^{2}{\gamma}/r+{J}(1-2M/r))\over(r-M)^{2}(r-r_{c})\sqrt{r_{3}-r}}\,{\rm
d}r.$ (64)

The solution is again most simply written as a sum of three functions:

$\phi(r)=f_{0}(r)+f_{H}(r)+\tilde{f}(r)$ (65)

where

$f_{0}(r)=\left[{2{J}r_{c}^{3/2}+4Mr_{c}^{1/2}(M{\gamma}-{J})\over(r_{c}-M)^{2}\sqrt{r_{c}-r_{3}}}\right]\\\
\times{1\over\sqrt{1-{\gamma}^{2}}}\tan^{-1}\left(\sqrt{r_{3}/r-1\over
1-r_{3}/r_{c}}\right),$ (66)

the analogue of the Schwarzschild plunge, and

$f_{H}(r)=-{M({J}-2M{\gamma})\over(r_{c}-M)(r_{3}-M)\sqrt{1-{\gamma}^{2}}}{\sqrt{r(r_{3}-r)}\over
r-M},$ (67)

which encapsulates the divergent $\phi$ behaviour of the event horizon, and
finally

$\tilde{f}(r)={M^{1/2}(1-{\gamma}^{2})^{-1/2}\over(r_{3}-M)^{3/2}(r_{c}-M)^{2}}\tanh^{-1}\left(\sqrt{r_{3}/r-1\over
r_{3}/M-1}\right)\\\ \times\bigg{[}{J}(r_{c}r_{3}-2r_{c}M-3r_{3}M+4M^{2})\\\
+2M{\gamma}(r_{c}r_{3}+Mr_{3}-2M^{2})\bigg{]},$ (68)

for $r_{3}>r>M$, and with inverted $\tanh^{-1}$ argument for $r<M$. In the
formal limit $r_{c}\rightarrow\infty$, $r_{3}\to 2M$, and the above solution
describes an orbital plunge from $2M$ to the singularity. An example
trajectory, with circular orbit parameter $r_{c}=10^{6}M$, is displayed in
Fig. 8.

Figure 8: An extremal prograde ($a=+M$) Kerr plunging orbit, with
$r_{c}=10^{6}M$. The colours denote the trajectory inside (orange) and outside
(blue) the event horizons $r_{\pm}=M$.

###  VI.2 Extremal retrograde spin $a=-M$

The extremal retrograde spin limit $a\to-M$ is a limit of great interest, as
it highlights some of the most interesting frame-dragging effects of the Kerr
metric quite simply.

For $a=-M$, the key orbital radii take the following values, $r_{I}=9M$,
$r_{\rm ib}=(3+\sqrt{8})M/2$, $r_{p}=4M$ and event horizons $r_{\pm}=M$. In
this limit, note that all key radii are distinct and a full set of orbital
solutions is available.

The orbital shape integral is of the form

$\sqrt{1-{\gamma}^{2}}\phi=\int{r^{3/2}({J}(1-2M/r)-2M^{2}{\gamma}/r)\over(r-M)^{2}(r-r_{c})\sqrt{r_{3}-r}}\,{\rm
d}r.$ (69)

Once again the general solution comprises of three functions:

$\phi(r)=f_{0}(r)+f_{H}(r)+\tilde{f}(r).$ (70)

The function

$f_{H}(r)=-{M({J}+2M{\gamma})\over(r_{c}-M)(r_{3}-M)\sqrt{1-{\gamma}^{2}}}{\sqrt{r(r_{3}-r)}\over
r-M},$ (71)

for all $r_{c}\neq r_{I},r_{c}>r_{\rm ib}$. It will, as will be discussed
shortly, be modified for $r_{c}<r_{\rm ib}$ and $r_{c}=r_{I}$. The second
function is equal to

$\tilde{f}(r)={M^{1/2}(1-{\gamma}^{2})^{-1/2}\over(r_{3}-M)^{3/2}(r_{c}-M)^{2}}\tanh^{-1}\left(\sqrt{r_{3}/r-1\over
r_{3}/M-1}\right)\\\ \times\bigg{[}{J}(r_{c}r_{3}-2r_{c}M-3r_{3}M+4M^{2})\\\
-2M{\gamma}(r_{c}r_{3}+Mr_{3}-2M^{2})\bigg{]},$ (72)

which again will be modified slightly for intra-IBCO circular orbits, and for
$r_{c}=r_{I}$. Note that for the special ISCO location, where
$r_{c}=r_{3}=r_{I}=9M$ (with ${J}=22\sqrt{3}M/9$, ${\gamma}=5\sqrt{3}/9$) the
function $\tilde{f}(r)=0$ as the final term in the square brackets vanishes.
We now discuss the eight distinct orbital solutions of the extremal retrograde
Kerr metric.

####  VI.2.1 $r_{c}>r_{I}$ plunging orbits

The plunging orbital solution for extremal $a=-M$ is given by:

$f_{0}(r)=\left[{2{J}r_{c}^{3/2}-4Mr_{c}^{1/2}(M{\gamma}+{J})\over(r_{c}-M)^{2}\sqrt{r_{c}-r_{3}}}\right]\\\
\times{1\over\sqrt{1-{\gamma}^{2}}}\tan^{-1}\left(\sqrt{r_{3}/r-1\over
1-r_{3}/r_{c}}\right).$ (73)

####  VI.2.2 $r_{c}=r_{I}$ spiral orbits

For $r_{c}=r_{I}$, the third root of the radial velocity polynomial $r_{3}$ is
equal to the ISCO $r_{3}=r_{c}=r_{I}$. In this limit the solution for
$\phi(r)$ is MB :

$\phi(r)={2\sqrt{2}\over 3M^{3/2}}{r^{3/2}\over(1-r/M)\sqrt{1-r/9M}},$ (74)

which is most easily found by returning to the governing integral (69) and
simplifying in this limit.

####  VI.2.3 $r_{\rm ib}<r_{c}<r_{I}$ homclinic orbits

The homoclinic orbital solution for extremal $a=-M$ is found to be:

$f_{0}(r)=\left[{2{J}r_{c}^{3/2}-4Mr_{c}^{1/2}(M{\gamma}+{J})\over(r_{c}-M)^{2}\sqrt{r_{3}-r_{c}}}\right]\\\
\times{1\over\sqrt{1-{\gamma}^{2}}}\tanh^{-1}\left(\sqrt{r_{3}/r-1\over
r_{3}/r_{c}-1}\right),$ (75)

as is the corresponding second orbit, a plunging orbit:

$f_{0}(r)=\left[{2{J}r_{c}^{3/2}-4Mr_{c}^{1/2}(M{\gamma}+{J})\over(r_{c}-M)^{2}\sqrt{r_{3}-r_{c}}}\right]\\\
\times{1\over\sqrt{1-{\gamma}^{2}}}\tanh^{-1}\left(\sqrt{r_{3}/r_{c}-1\over
r_{3}/r-1}\right),$ (76)

####  VI.2.4 $r_{c}=r_{\rm ib}$ IBCO parabolic orbits

For $r_{c}=r_{\rm ib}$, we have simultaneously $1-{\gamma}^{2}=0$ and
$r_{3}\rightarrow\infty$. The combination
$\sqrt{(1-{\gamma}^{2})(r_{3}-r_{c})}$ converges to $\sqrt{2M}({J}+M)/r_{\rm
ib}$, and the three functions have the following form (presented here for the
inspiralling case)

$\phi(r)=\alpha\tanh^{-1}\sqrt{M\over r}+\beta\tanh^{-1}\sqrt{r\over r_{\rm
ib}}+\gamma{r^{1/2}\over r-M}.$ (77)

The argument $\alpha\tanh^{-1}$ should of course be inverted within the event
horizon $r<M$. The three constants are

$\displaystyle\alpha$ $\displaystyle=-{r_{\rm ib}J(M-3r_{\rm ib})+2r_{\rm
ib}(r_{\rm ib}+M)(J+M)\over\sqrt{2}(r_{\rm ib}-M)^{2}(J+M)},$
$\displaystyle\beta$ $\displaystyle={2\sqrt{2}r_{\rm
ib}^{3/2}M(J+M)-\sqrt{2}r_{\rm ib}^{5/2}J\over\sqrt{M}(r_{\rm
ib}-M)^{2}(J+M)},$ $\displaystyle\gamma$ $\displaystyle={3r_{\rm
ib}JM^{1/2}+2r_{\rm ib}M^{3/2}\over\sqrt{2}(r_{\rm ib}-M)(J+M)}.$ (78)

The argument of $\beta\tanh^{-1}$ should be inverted for outspiralling orbits,
where $r>r_{\rm ib}$.

####  VI.2.5 $r_{p}<r_{c}<r_{\rm ib}$ hyperbolic orbits

For $r_{p}<r_{c}<r_{\rm ib}$ we have $\gamma>1,r_{3}<0$, and a test particle
can escape to infinity with non-zero radial velocity. Each of the three
orbital functions must be modified slightly in this limit. They are

$f_{H}(r)={M({J}+2M{\gamma})\over(r_{c}-M)(|r_{3}|+M)\sqrt{{\gamma}^{2}-1}}{\sqrt{r(|r_{3}|+r)}\over
r-M},$ (79)

and

$\tilde{f}(r)={-M^{1/2}({\gamma}^{2}-1)^{-1/2}\over(|r_{3}|+M)^{3/2}(r_{c}-M)^{2}}\tanh^{-1}\left(\sqrt{|r_{3}|/r+1\over|r_{3}|/M+1}\right)\\\
\times\bigg{[}{J}(r_{c}r_{3}-2r_{c}M-3r_{3}M+4M^{2})\\\
-2M{\gamma}(r_{c}r_{3}+Mr_{3}-2M^{2})\bigg{]},$ (80)

and finally

$f_{0}(r)=\left[{2{J}r_{c}^{3/2}-4Mr_{c}^{1/2}(M{\gamma}+{J})\over(r_{c}-M)^{2}\sqrt{r_{c}+|r_{3}|}}\right]\\\
\times{1\over\sqrt{{\gamma}^{2}-1}}\tanh^{-1}\left(\sqrt{|r_{3}|/r+1\over|r_{3}|/r_{c}+1}\right),$
(81)

for the outspiralling orbit $r>r_{c}$. The corresponding inspiralling orbit is
instead described by a function of the same form, but with its $\tanh^{-1}$
argument inverted. This concludes the set of massive particle pseudo-circular
orbits.

##  VII  Radial momentum and orbits of photons

The case of massless particles, which for simplicity we refer to generically
as “photons”, proceeds along similar lines. The trajectories of photon orbits
are well studied in the literature, with the Schwarzschild solutions first
presented in Darwin , and a general discussion of Kerr photon orbits is
presented in Chand . The case of extremal Kerr orbits, which are not simple
limits of the more general $|a|\neq M$ Kerr solutions, do not appear to have
been derived previously, and are presented here. For completeness we also
present the Schwarzschild solutions of Darwin .

We work with the 4-momenta $p^{\mu}={\rm d}x^{\mu}/{\rm d}\sigma$, where
$\sigma$ is a suitable affine parameter. The equations for a circular photon
orbit are now

$g_{\mu\nu}p^{\mu}p^{\nu}=0,\quad p^{\mu}p^{\nu}\partial_{r}g_{\mu\nu}=0,$
(82)

with only $p^{0}$ and $p^{\phi}$ present. The conserved energy and angular
momentum correspond to (minus) $p_{0}$ and $p_{\phi}$. The solution of this
homogeneous system requires the determinant of the coefficients to vanish,
which is the same as solving the following

${\cal D}^{2}=1-3M/r_{p}+2aM^{1/2}/r_{p}^{3/2}=0.$ (83)

Solving this with the standard cubic formula gives the radius of a circular
photon orbit:

$r_{p}=4M\cos^{2}\left[{1\over 3}\cos^{-1}\left(-{a\over M}\right)\right]$
(84)

which varies from $4M$ to $M$ for $-1\leq a/M\leq 1$. With ${\cal D}=0$, we
may now solve for the (conserved) ratio $p_{\phi}/p_{0}$ from either one of
the equations in (82), the first being more convenient. The end result is:

$p_{\phi}=(a-3\sqrt{Mr_{p}})p_{0}.$ (85)

(To arrive at this apparently simple result is non-trivial because it involves
very selectively substituting for $a$ in terms of $r_{p}$ using the original
${\cal D}=0$ equation. See HEL  for an alternative derivation.)

Next, we retain the conserved circular $p_{0}$ (which is arbitrary) and
$p_{\phi}$ (from above) values, while also allowing for a $p^{r}$ component to
be present. Our governing equation ($p^{\mu}p_{\mu}=0$) is then:

$r^{2}\left({p^{r}\over p_{0}}\right)^{2}-{p_{\phi}\over p_{0}}\left({2Ma\over
r}-\left(1-{2M\over r}\right){p_{\phi}\over p_{0}}\right)\\\
-\left(\left(r^{2}+a^{2}+{2Ma^{2}\over r}\right)+{2Ma\over r}{p_{\phi}\over
p_{0}}\right)=0.$ (86)

Proceeding as before, this unwieldy equation may be factored in terms of the
roots $r_{p}$ and $r_{3}$ by defining an effective potential. Matters simplify
dramatically:

$\left({p^{r}\over p_{0}}\right)^{2}=\left(1-{r_{p}\over
r}\right)^{2}\left(1-{r_{3}\over r}\right).$ (87)

(The normalisation follows from equation 86 by taking the limit
$r\rightarrow\infty$.) Taking the limit $\lim_{r\to 0}r^{3}(p^{r}/p_{0})^{2}$
gives, from the above equation, $-r_{p}^{2}r_{3}$, and from the governing
equation 86

$-r_{p}^{2}r_{3}=2M\left({p_{\phi}\over
p_{0}}+a\right)^{2}=2M(3\sqrt{Mr_{p}}-2a)^{2}=2r_{p}^{3}$ (88)

where we used (from ${\cal D}=0$)

$2a=3\sqrt{Mr_{p}}-r_{p}^{3/2}M^{-1/2},$ (89)

to arrive at the final equality. Thus, the remarkable solution from the null
geodesic equation that emerges is

$(p^{r})^{2}=(1+2r_{p}/r)(1-r_{p}/r)^{2}(p_{0})^{2}.$ (90)

In common with the case of massive particles inspiralling from the ISCO MB ,
this depends on $a$ only through the photon orbit radius $r_{p}$. This orbital
equation is of the same form as the massive particle hyperbolic orbit.

To determine the photon orbits spiralling from the photon radius, we require
$p^{\phi}$, which is readily evaluated from $p_{\phi}$ and $p_{0}$:

$\displaystyle p^{\phi}$ $\displaystyle={p_{0}\over\Delta}\left(a-{4Ma\over
r}-3({Mr_{p}})^{1/2}+{6M^{3/2}r^{1/2}_{p}\over r}\right),$
$\displaystyle={p_{0}\over\Delta}\left(a-3(Mr_{p})^{1/2}+{2M^{1/2}r^{3/2}_{p}\over
r}\right),$ (91)

where in the final line we have used ${\cal D}=0$ to substitute for $4Ma/r$,
and $\Delta=(r-r_{+})(r-r_{-})$.

###  VII.1 Schwarzschild photon spirals

In the Schwarzschild $a=0$ limit, $p^{\phi}=-3\sqrt{3}Mp_{0}/r^{2}$. Then with
$r_{p}=3M$, $x=r/r_{p}$,

${{\rm d}\phi\over{\rm d}x}=r_{p}{{\rm d}\phi\over{\rm
d}r}=r_{p}{p^{\phi}\over p^{r}}=-{1\over x-1}\sqrt{3\over x(x+2)}$ (92)

Then

$\displaystyle\phi$ $\displaystyle=-\int{{\rm d}x\over x-1}\sqrt{3\over
x(x+2)}$ $\displaystyle=2\tanh^{-1}\sqrt{3x\over x+2},\quad(x<1),$
$\displaystyle=2\tanh^{-1}\sqrt{x+2\over 3x},\quad(x>1).$ (93)

Solving for $r$:

$\displaystyle r$ $\displaystyle={6M\over
3\coth^{2}(\phi/2)-1}\quad{\rm(inspiral)},$ $\displaystyle={6M\over
3\tanh^{2}(\phi/2)-1}\quad{\rm(outspiral)}.$ (94)

These solutions formed part of Darwin’s 1959 analysis Darwin . Note that as in
the case of the massive particle hyperbolic orbits, photons following this
outspiralling orbit reach infinity at a finite angle

$\phi_{\infty}=-2\tanh^{-1}\left(\sqrt{1\over 3}\right).$ (95)

###  VII.2 General Kerr solution $-M<a<M$

The general Kerr photon integral is of the following form

$\phi(r)=\int{2M^{1/2}r_{p}^{3/2}r^{1/2}+(a-3\sqrt{Mr_{p}})r^{3/2}\over(r-r_{+})(r-r_{-})(r-r_{p})\sqrt{r+2r_{p}}}\,{\rm
d}r,$ (96)

whose solution can be found via a partial fraction expansion. Omitting the
lengthy but straightforward details, we find

$\phi(r)=C_{0}\tanh^{-1}\sqrt{3r\over
2r_{p}+r}+C_{+}\tanh^{-1}\sqrt{1+2r_{p}/r\over 1+2r_{p}/r_{+}}\\\
+C_{-}\tanh^{-1}\sqrt{1+2r_{p}/r\over 1+2r_{p}/r_{-}}$ (97)

For an inspiralling $r_{\pm}<r<r_{p}$ solution (the arguments of the final two
$\tanh^{-1}$ functions are inverted for $r<r_{\pm}$). While the outspiraling
solution $r>r_{p}$ is

$\phi(r)=C_{0}\tanh^{-1}\sqrt{2r_{p}+r\over
3r}+C_{+}\tanh^{-1}\sqrt{1+2r_{p}/r\over 1+2r_{p}/r_{+}}\\\
+C_{-}\tanh^{-1}\sqrt{1+2r_{p}/r\over 1+2r_{p}/r_{-}}.$ (98)

The coefficients $C_{0},C_{\pm}$ of these solutions are the following

$C_{0}={-4M^{1/2}r_{p}^{3/2}-2r_{p}(a-3\sqrt{Mr_{p}})\over\sqrt{3}(r_{p}-r_{+})(r_{p}-r_{-})}$
(99)

and

$C_{\pm}=\pm{4M^{1/2}r_{p}^{3/2}r_{\pm}^{1/2}+2r_{\pm}^{3/2}(a-3\sqrt{Mr_{p}})\over(r_{+}-r_{-})(r_{p}-r_{\pm})\sqrt{r_{\pm}+2r_{p}}}$
(100)

Note that for $a=0,r_{p}=3M$, $r_{+}=2M$, $r_{-}=0$, and therefore $C_{0}=2$,
$C_{\pm}=0$.

###  VII.3 Extremal Kerr solutions $a=\pm M$

For an extremal spin $a=\pm M$, the two event horizons of the Kerr black hole
coincide $r_{\pm}=M$, and the partial fractions approach used in the previous
section must be revisited. These solutions appear to have not been previously
discussed in the literature, and display some remarkable properties.

####  VII.3.1 Extremal retrograde spin $a=-M$

For the extremal retrograde spin of $a=-M$, the photon radius is at
$r_{p}=4M$, and the photons orbit has the following shape

$\phi=\int{16M^{2}r^{1/2}-7Mr^{3/2}\over(r-M)^{2}(r-4M)\sqrt{r+8M}}\,{\rm
d}r,$ (101)

which has two possible solutions. For $r<4M$, we have

$\phi(r)={1\over 9}\Bigg{[}{\sqrt{9r(8M+r)}\over
r-M}+8\sqrt{3}\tanh^{-1}\sqrt{{r\over 4M}}\\\
+4\sqrt{3}\ln\left({4+\sqrt{r/M}+\sqrt{24+8r/M}\over
4-\sqrt{r/M}+\sqrt{24+8r/M}}\right)\Bigg{]}$ (102)

while for $r>4M$ we have

$\phi(r)={1\over 9}\Bigg{[}{\sqrt{9r(8M+r)}\over
r-M}+8\sqrt{3}\tanh^{-1}\sqrt{{4M\over r}}\\\
+4\sqrt{3}\ln\left({4+\sqrt{r/M}+\sqrt{24+8r/M}\over
4-\sqrt{r/M}+\sqrt{24+8r/M}}\right)\Bigg{]}.$ (103)

The inspiralling orbit displays pronounced frame dragging, and is displayed in
Fig. 9.

Figure 9: The inspiralling photon trajectory for the extremal retrograde
$a=-M$ Kerr metric. Pronounced frame dragging is clearly visible, with the
trajectory changing direction just outside of the ergo-region $r_{E}=2M$.

####  VII.3.2 Extremal prograde spin $a=+M$

Figure 10: The outspiralling photon trajectory for the extremal prograde
$a=+M$ Kerr metric. This orbit starts formally at Boyer-Lindquist radial
coordinate $r=M$, before escaping to infinity.

For $a=+M$ the photon radius $r_{p}$ coincides with the two event horizon
radii $r_{\pm}$, all occurring at the Boyer-Lindquist co-ordinate $r=M$. For a
photon, there is in fact an orbital solution starting at the Boyer-Lindquist
event horizon $r_{+}=M$, which then escapes to infinity. With
$r_{p}=r_{\pm}=a=M$, this orbital solution satisfies the following integral

$\phi=2M\int{Mr^{1/2}-r^{3/2}\over(r-M)^{3}\sqrt{r+2M}}\,{\rm d}r,$ (104)

which has the following solution

$\phi(r)={2\over 3\sqrt{3}}\Bigg{[}{\sqrt{3r(2M+r)}\over
r-M}+2\tanh^{-1}\sqrt{{M\over r}}\\\
+\ln\left({2+\sqrt{r/M}+\sqrt{6+3r/M}\over
2-\sqrt{r/M}+\sqrt{6+3r/M}}\right)\Bigg{]}.$ (105)

This orbital solution is displayed in Fig. 10.

##  VIII Conclusions

The pseudo-circular orbital solutions of the Schwarzschild, Kerr, and extremal
Kerr, metrics, which are characterised by constants of motion that also
describe a circular orbit, have been explicitly derived. There are eight such
orbital classes, five of which plunge and terminate at the origin, while three
either remain bound (homoclinic orbits), or escape to infinity with zero
(parabolic) or finite (hyperbolic) radial velocities. Each orbital solution
may be written entirely in terms of elementary functions, a result of the
remarkable symmetry imposed on the orbits by the circular constants of motion.
The in- or out-spiralling behaviour of photon trajectories starting at the
unstable photon circular radius have also been discussed, and have
qualitatively similar properties to the hyperbolic class of massive particle
orbits.

The systematic derivation of all eight orbital classes in this paper completes
a line of analysis which dates back to Darwin’s 1959 study of Schwarzschild
orbital motion Darwin . Although the origins of this field are more than half
a century old, these explicit solutions have important modern applications,
e.g. as contextual benchmarks for numerical accretion SK  or photon ray
tracing problems PJ , for the computation of gravitational waveforms arising
from extreme mass ratio inspirals using the Teukolsky formalism T , or in
extending existing theories of black hole accretion Rey .

## Acknowledgments

This work was supported by a Leverhulme Trust International Professorship
grant [number LIP-202-014]. For the purpose of Open Access, AM has applied a
CC BY public copyright licence to any Author Accepted Manuscript version
arising from this submission. This work is partially supported by the Hintze
Family Charitable Trust and STFC grant ST/S000488/1.

## References

  * (1) Bardeen J. M., Press W. H., Teukolsky S. A., 1972, ApJ, 178, 347
  * (2) Chandrasekhar, S. 1983, Mathematical Theory of Black Holes (Oxford: Clarendon Press)
  * (3) Collins N. A., Hughes S. A., 2004, Phys. Rev. D69, 124022
  * (4) Compère G., Liu Y., Long J., 2022, PhRvD, 105, 024075
  * (5) Darwin C., 1959, RSPSA, 249, 180
  * (6) Drasco S., 2006, Class. Quant. Grav. 23, S769
  * (7) Drasco S., Hughes S. A., 2004, Phys. Rev. D 69, 044015
  * (8) Drasco E. F. S., Hughes S. A., 2005, Class. Quant. Grav., 22, 801
  * (9) Drasco S., Hughes S., 2006, Phys. Rev. D 73, 024027
  * (10) Fujita R., Hikida W., 2009, CQGra, 26, 135002
  * (11) Glampedakis K., Hughes S. A., Kennefick D., 2002, Phys. Rev. D 66, 064005.
  * (12) Glampedakis K., 2005, Class. Quant. Grav. 22, S605
  * (13) Hobson, M. P., Efstahiou, G., Lazenby, A. N. 2006, General Relativity: An Introduction for Physicists (Cambridge: Cambridge University Press)
  * (14) Levin J., Perez-Giz G., 2009, PhRvD, 79, 124013
  * (15) Levin J., Perez-Giz G., 2008, PhRvD, 77, 103005
  * (16) Lang R. N., Hughes S. A., 2006, Phys. Rev. D 74, 122001
  * (17) Mummery, A., Balbus, S.A. 2022, PRL, 129, 161101
  * (18) Novikov, I. D., Thorne, K. S. 1973, in Black Holes, ed C. DeWitt & B. DeWitt (New York: Gordon & Breach)
  * (19) Psaltis D., Johannsen T., 2012, ApJ, 745, 1
  * (20) Page D. N., Thorne K. S., 1974, ApJ, 191, 499
  * (21) Reynolds, C. S. Begelman, M. C. 1997, ApJ, 487, 135
  * (22) Schnittman, J. D., Krolik, J. H., Noble, S. C. 2016, ApJ, 819, 48
  * (23) Teukolsky S A, 1973, Astrophys. J. 185 635