# Precession of timelike bound orbits in Kerr spacetime

Parth Bambhaniya<EMAIL_ADDRESS>International Center for Cosmology,
Charusat University, Anand, GUJ 388421, India  Divyesh N. Solanki
<EMAIL_ADDRESS>Sardar Vallabhbhai National Institute of
Technology, Surat GUJ 395007, India  Dipanjan Dey
<EMAIL_ADDRESS>International Center for Cosmology, Charusat
University, Anand, GUJ 388421, India  Ashok B. Joshi<EMAIL_ADDRESS>International Center for Cosmology, Charusat University, Anand, GUJ 388421,
India  Pankaj S. Joshi<EMAIL_ADDRESS>International Center for
Cosmology, Charusat University, Anand, GUJ 388421, India  Vishwa Patel
<EMAIL_ADDRESS>PDPIAS, Charusat University, Anand, GUJ 388421,
India

###### Abstract

Astrometric observations of S-stars provide a unique opportunity to probe the
nature of Sagittarius-A* (Sgr-A*). In view of this, it has become important to
understand the nature and behavior of timelike bound trajectories of particles
around a massive central object. It is known now that whereas the
Schwarzschild black hole does not allow the negative precession for the
S-stars, the naked singularity spacetimes can admit the positive as well as
negative precession for the bound timelike orbits. In this context, we study
the perihelion precession of a test particle in the Kerr spacetime geometry.
Considering some approximations, we investigate whether the timelike bound
orbits of a test particle in Kerr spacetime can have negative precession. In
this paper, we only consider low eccentric timelike equatorial orbits. With
these considerations, we find that in Kerr spacetimes, negative precession of
timelike bound orbits is not allowed.

Key words: Kerr black hole, Naked singularity, Perihelion shift, Galactic
center.

##  I Introduction

In recent days, the study of the nature of Sgr-A* has become a subject of
great interest. Sgr-A* is a highly dense compact object that exists at the
center of our Milky-way galaxy. It is expected that the mass of Sgr-A* is
about $4.3\times 10^{6}M_{\odot}$, which is located at a distance of $8.2$ kpc
($1~{}$kpc $\sim$ $3\times 10^{16}$ km) from the Earth. There are many ‘S’
stars (e.g. S-2, S-38, S-102, etc.) which are orbiting around the compact
radio source Sgr-A*. The observational data of S-stars’ orbits can reveal the
great mystery about the nature of the compact object Sgr-A*. In order to
observe the relativistic corrections in the orbital motion of the S-stars
around Sgr-A*, the GRAVITY and SINFONI collaborations are monitoring their
orbital dynamics continuously.

It is generally believed that super massive black holes (SMBHs) with masses
between $10^{6}-10^{9}M_{\odot}$ exist at the centers of most galaxies
Kormendy . In order to predict the causal nature of the galactic centers, it
is very important to investigate different possible observational features,
such as the shadows of black holes, the relativistic orbits of the S-stars,
accretion disk properties of the compact objects, etc. The Event Horizon
Telescope (EHT) collaboration has released the first-ever image of the shadow
of a black hole located at the center of the Messier 87 (M87) galaxy K.
Akiyama . There are several literature where the detailed analysis of the
shadows cast by spherically symmetric and static black holes and other compact
objects (e.g. different naked singularities, gravastar, etc.) are studied
Shaikh:2018lcc ; Gyulchev:2019tvk ; Shaikh2 ; Nakao1 ; DipanjanRaji ; AJoshi1
. It has been shown that compact objects, like naked singularities can cast
similar type of shadow which is expected to be cast by a black hole
Shaikh:2018lcc ; DipanjanRaji ; AJoshi1 . Recently, GRAVITY collaboration has
shown the possibility of the existence of general relativistic precession of
the orbit of S2 star, where they consider Schwarzschild black hole at the
center Gravity . In our earlier work Dipanjan , we have predicted the
precession angle of the S2 star considering Schwarzschild black hole and naked
singularity at the center. The study of the relativistic orbits of S-stars
near the Milky-way galactic center is very important in astrophysics to verify
the prediction of the general theory of relativity. There are several works
where timelike and lightlike geodesics in different spacetimes are
investigated which are very important in the context of recent observations of
EHT, GRAVITY and SINFONI collaborations parsa ; Banik:2016qvf ; Kovacs:2010xm
; Dey:2013yga ; Dey+15 ; Zhou:2014jja ; Levin:2009sk ; levin ;
Glampedakis:2002ya ; Chu:2017gvt ; Dokuchaev:2015zxe ; Borka:2012tj ;
Martinez:2019nor ; Fujita:2009bp ; Wang:2019rvq ; Suzuki:1997by ;
Zhang:2018eau ; Pugliese:2013zma ; Farina:1993xw ; Dasgupta:2012zf ; Shoom ;
M87 ; Eisenhauer:2005cv ; Eva2 ; Bhatt1 ; Pugliese:2011xn ; Pugliese:2010he ;
Hackmann:2015vla ; Deng:2020yfm ; Gao:2020wjz ; Takamori:2020ntj ;
Collodel:2017end ; Ulbricht:2015vwa ; Bini:2013tia ; Sandoval:2013fua ;
Thornburg:2016msc .

The Schwarzschild black hole is a spherically symmetric, static and vacuum
solution of the Einstein field equation. It describes spacetime geometry of a
non-rotating, uncharged black hole which is characterized by a single
parameter, the Schwarzschild mass M of the black hole. In Parth ;
Joshi:2019rdo , we show that the bound orbits of a freely falling particle in
Schwarzschild spacetime always precess in the positive direction. In positive
precession, the angular distance travelled by a particle between two
successive perihelion points is greater than $2\pi$. On the other hand, the
precession is called negative when the angular distance travelled by a
particle between two successive perihelion points is less than $2\pi$. In
positive precession, it can be shown that the orbit of a particle always
precesses in the direction of the particle motion, whereas in the negative
precession, the orbit precesses in the opposite direction of the particle
motion. However, the negative precession is forbidden in Schwarzschild
spacetime Parth . Therefore, if we consider Sgr-A* to be a Schwarzschild black
hole then the observed precession of ‘S’ stars should always be positive. It
is generally believed that an extended matter distribution (e.g. clusters of
stars, baryonic matter, dark matter, etc. ) can cause or allow for a negative
precession of the stars. In Parth ; Joshi:2019rdo , we study the bound orbits
of a test particle in Schwarzschild, Joshi-Malafarina-Narayan (JMN), and
Janis-Newman-Winicour (JNW) spacetimes. Both the JMN and JNW spacetimes are
naked singularity spacetimes which have spherically symmetric matter
distributions psJoshi1 ; psJoshi2 ; Virbhadra:1998dy ; JNW . The JMN and JNW
naked singularity spacetimes allow both the negative and positive precession
of timelike bound orbits. In DipanjanRaji , it is shown that for some
parameters’ values, when negative precession is allowed in JMN and JNW
spacetimes, these spacetimes would not cast any shadow for the central object.
However, they can cast a shadow when the positive precession is allowed in
these spacetimes. Therefore, in that earlier work DipanjanRaji , we construct
a spacetime configuration in which both the negative precession and shadow can
exist simultaneously. As we know, the Schwarzschild, Kerr and other black hole
solutions can cast a shadow. Therefore, if the negative precession of the
orbit of a test particle in other black holes spacetimes is possible then that
would be another example along with the spacetime configuration of
DipanjanRaji , where a negative precession and shadow both can exist
simultaneously. In this context, in this paper, we investigate the possibility
of a likely negative precession of bound timelike orbits in the Kerr
spacetime, which is physically more realistic solution of a black hole as
compared to the Schwarzschild black hole solution, which allows for no
rotation.

As we know, every celestial body has its own intrinsic spin angular momentum.
Therefore, inclusion of a non-zero spin in a non-rotating spacetime makes the
modified spacetime physically more realistic. There are no restrictions on the
value of the spin angular momentum of a compact object, as long as it is not a
black hole. The spin of a celestial object is typically represented by a
dimensionless spin parameter $\tilde{a}$. This spin parameter can be defined
as $\tilde{a}=\frac{cJ}{GM^{2}}$, where $c,J,G,M$ are the velocity of light,
intrinsic spin angular momentum of the body, Newton’s gravitational constant,
and the mass of the celestial body respectively. Earth, with its spin angular
momentum $J\sim 7.2\times 10^{33}kg~{}m^{2}s^{-1}$ and mass $M=5.972\times
10^{24}kg$ has the spin parameter $\tilde{a}=907$, whereas the sun has spin
parameter $\tilde{a}=0.216$ Nielsen:2016kyw . On the other hand, a rapidly
spinning massive star VFTS102 has $\tilde{a}=75$ Nielsen:2016kyw . Therefore,
one can see that the value of the spin parameter of a celestial object can be
much greater than one. However, if we consider Kerr black hole which is a
rotating generalization of the Schwarzschild black hole, the spin parameter
$\tilde{a}$ cannot be allowed to be greater than unity in order to ensure the
existence of an event horizon. The Kerr black hole is a vacuum axi-symmetric
solution of Einstein field equations, and it is characterized by two
parameters, the total mass of the black hole $M$ and total angular momentum
$J$. Kerr spacetime describes a rotating black hole, if $\tilde{a}\leq 1$,
whereas it describes a rotating naked singularity spacetime if $\tilde{a}>1$.
Astrophysically relevant bound trajectories around a Kerr black hole are
studied in rana . Detailed study of bound nonspacelike geodesics in the Kerr
metric is given in Wilkins ; thorne .

As we mentioned previously, in this paper, our prime focus is to investigate
the nature of perihelion precession of timelike bound orbits in Kerr
spacetime. In Dipanjan , we show that Schwarzschild spacetime does not admit
any negative precession. In this paper, we show that the precession of the
timelike bound orbits in Kerr spacetime is also always positive, when we
consider only low eccentric, equatorial, bound orbits. With the small
eccentricity approximation, one can analyse the nature of the timelike bound
orbits in the strong field region of Kerr spacetime where the relativistic
effects on the bound orbits are likely to be observed.

This paper is organized as follows. In section (II), we review the basic
properties of the Kerr spacetime and derive the analytic solution of the orbit
equation for a test particle in the equatorial plane. We numerically solve the
orbit equation and show the particle trajectories. In section (III), we use an
approximate solution of orbit equation and investigate the nature of
perihelion precession of the timelike bound orbits in Kerr spacetime. We
summarize and conclude our results in section (IV). Throughout the paper, we
use geometrical units with $G=c=1$.

Figure 1:  Above figures show orbits of a test particle in the Kerr spacetime.
Red solid lines indicate particle’s orbits in the Kerr black hole for
$a=0.8,r_{min}=76.15$ and $a=-0.8,r_{min}=75.53$ (first column), in the
extreme Kerr black hole for $a=1,r_{min}=76.22$ and $a=-1,r_{min}=75.45$
(second column) and blue solid lines indicate particle’s orbits in the Kerr
naked singularity for $a=1.1,r_{min}=76.25$ and $a=-1.1,r_{min}=75.41$ (third
column). Black dotted lines indicate particle’s orbits in the Schwarzschild
spacetime (for a=0, $r_{min}=75.85$). Brown circles represent the minimum
approach of the particle towards the center. Here, we considered $M=1$, $L=12$
and total energy $E=-0.001$.

##  II Timelike geodesics in Kerr spacetime

In this section, we study the motion of a test particle in the Kerr spacetime.
The stationary, axisymmetric and rotating Kerr spacetime is given in Boyer-
Lindquist coordinates as,

$ds^{2}=-\left(1-\frac{r_{s}r}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+\left(r^{2}+a^{2}+\frac{r_{s}ra^{2}\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta
d\phi^{2}-\frac{2r_{s}ra\sin^{2}\theta}{\Sigma}dtd\phi\,\,,$ (1)

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$, $\Delta=r^{2}+a^{2}-r_{s}r$ and
$r_{s}=2M$. The spin parameter $a$ with a length dimension is related with the
total angular momentum $J$ as $a=J/M$. Therefore, in terms of dimensionless
spin parameter $\tilde{a}$, we can write $a$ as, $a=M\tilde{a}$. The horizons
are defined by the relation $g_{rr}\rightarrow\infty$ which implies that the
solution of $\Delta=0$ can give us the position of the event horizon.
Therefore,

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}\,\,.$ (2)

There are two horizons, the event horizon at $r=r_{+}$ and Cauchy horizon at
$r=r_{-}$. For $a=M$, both these horizons coincide at $r=M$, and these are
known as extreme Kerr black holes. When $a>M$, there is no horizon and the
Kerr black hole becomes a timelike naked singularity. The study of test
particles motion in Kerr spacetime is important to understand the physical
processes occuring in these spacetimes and their observational consequences.
The Kerr spacetime is independent of $t$ and $\phi$, therefore, the conserved
energy ($e$) and the angular momentum ($L$) per unit rest mass are given by,

$e=g_{tt}U^{t}+g_{t\phi}U^{\phi}\,\,,$ (3)
$L=-g_{t\phi}U^{t}+g_{\phi\phi}U^{\phi}\,\,,$ (4)

where $U^{\mu}$ are the components of the four velocity of a test particle and
$g_{tt}=\left(1-\frac{r_{s}r}{\Sigma}\right)$, $g_{rr}=\frac{\Sigma}{\Delta}$,
$g_{\theta\theta}=\Sigma$,
$g_{\phi\phi}=\left(r^{2}+a^{2}+\frac{r_{s}ra^{2}\sin^{2}\theta}{\Sigma}\right)$,
and $g_{t\phi}=\frac{r_{s}ra\sin^{2}\theta}{\Sigma}$. In a physically
realistic situation, the orbital angular momentum of a test particle and the
spin angular momentum of a central rotating body need not necessarily be
aligned. Here, for simplicity we have restricted our attention to the orbits
in the equatorial plane ($\theta=\frac{\pi}{2}$). Eqs. (3), (4) can be solved
for $U^{t}$ and $U^{\phi}$, and we get,

$U^{t}=\frac{1}{\Delta}\left[\left(r^{2}+a^{2}+\frac{r_{s}a^{2}}{r}\right)e-\left(\frac{r_{s}a}{r}\right)L\right]\,\,,$
(5)
$U^{\phi}=\frac{1}{\Delta}\left[\left(\frac{r_{s}a}{r}\right)e+\left(1-\frac{r_{s}}{r}\right)L\right]\,\,.$
(6)

Using normalization condition $U^{\alpha}U_{\alpha}=-1$ of timelike geodesics
and also using the Eqs. (5), (6), we can derive $r$-component of the four
velocity $U^{r}$ as,

$U^{r}=\pm\sqrt{(e^{2}-1)+\frac{r_{s}}{r}-\frac{L^{2}-a^{2}(e^{2}-1)}{r^{2}}+\frac{r_{s}(L-ae)^{2}}{r^{3}}}\,\,.$
(7)

Figure 2: Above figures show orbits of a test particle in the Kerr spacetime.
Red solid lines indicate the particle’s orbits in the Kerr black hole for
$a=0.8,r_{min}=18.62$ and $a=-0.8,r_{min}=17.03$ (first column). In the
extreme Kerr black hole, for $a=1,r_{min}=18.78$, and for $a=-1$, we have
$r_{min}=16.77$ (second column). The blue solid lines indicate particle’s
orbits in the Kerr naked singularity for $a=1.1,r_{min}=18.86$, and
$a=-1.1,r_{min}=16.64$ (third column). Black dotted lines indicate particle’s
orbits in the Schwarzschild spacetime (for a=0, $r_{min}=17.91$). Brown
circles represent the minimum approach of the particle towards the center.
Here, we have considered $M=1$, $L=6$ and total energy is $E=-0.006$.

Here, $\pm$ signatures are corresponding to the radially outgoing and incoming
timelike geodesics respectively. The expression, in Eq. (7), is equivalent to
the kinetic energy of a test particle. The total relativistic energy is
defined as,

$E=\frac{1}{2}(e^{2}-1)=\frac{1}{2}(U^{r})^{2}+V_{eff}(r)\,\,.$ (8)

Using the expression of $U^{r}$ (Eq. (7)) and the expression of total
relativistic energy (Eq. (8)), we get the following expression of effective
potential,

$V_{eff}(r)=-\frac{r_{s}}{2r}+\frac{L^{2}-a^{2}(e^{2}-1)}{2r^{2}}-\frac{r_{s}(L-ae)^{2}}{2r^{3}}\,\,.$
(9)

Above expression of the effective potential is only applicable for equatorial
timelike geodesics. For bound orbits, the total energy of the particle is
greater than or equal to the minimum effective potential. The minimum
effective potential is determined as,

$\frac{dV_{eff}}{dr}|_{r_{b}}=0;\frac{d^{2}V_{eff}}{dr^{2}}|_{r_{b}}>0\,\,,$
(10)

where the effective potential has a minimum at $r=r_{b}$. Using Eq. (9) and
Eq. (10), we get the following expression of $r_{b}$,

$r_{b}=\frac{1}{2M}\big{(}L^{2}+a^{2}(1-e^{2})+\sqrt{(L^{2}+a^{2}(1-e^{2}))^{2}-12M^{2}(L-ae)^{2}}\big{)}\,\,.$
(11)

The minimum effective potential at $r=r_{b}$ is,

$V_{eff}(r_{b})=\frac{-1}{(L^{2}+a^{2}(1-e^{2})+\sqrt{(L^{2}+a^{2}(1-e^{2}))^{2}-12M^{2}(L-ae)^{2}})^{3}}(2M^{2}(a^{4}(1-e^{2})^{2}+16aLeM^{2}\\\
+L^{2}(L^{2}-8M^{2}+\sqrt{(L^{2}+a^{2}(1-e^{2}))^{2}-12M^{2}(L-ae)^{2}})+a^{2}(2L^{2}(1-e^{2})\\\
-8M^{2}+(1-e^{2})\sqrt{(L^{2}+a^{2}(1-e^{2}))^{2}-12M^{2}(L-ae)^{2}})))\,\,.$
(12)

The bound orbits exist for $V_{min}\leq E<0$. Using the bound orbit
conditions, we can determine the shape of the orbits, that gives how $r$
changes in the equatorial plane with respect to $\phi$,

$\frac{dr}{d\phi}=\frac{\pm\sqrt{(e^{2}-1)+\frac{2M}{r}-\frac{L^{2}-a^{2}(e^{2}-1)}{r^{2}}+\frac{2M(L-ae)^{2}}{r^{3}}}}{\frac{1}{\Delta}\left[\left(\frac{r_{s}a}{r}\right)e+\left(1-\frac{r_{s}}{r}\right)L\right]}\,\,.$
(13)

Using Eq. (13), we can derive second order differential orbit equation of a
massive test particle in Kerr spacetime,

$\frac{d^{2}u}{d\phi^{2}}=\frac{1-2Mu+a^{2}u^{2}}{(L-2Mu(L-ae))^{3}}\left[M(L-2ae(e^{2}-1))+u\left(L(-L^{2}+3a^{2}(e^{2}-1))-2M^{2}(2L+2ae)+uB(\phi)\right)\right]\,\,,$
(14)

where $u=\frac{1}{r}$ and

$B(\phi)=M(7L^{3}-6aeL^{2}+a^{2}L(11-3e^{2})+2a^{3}e(e^{2}-1))+4M^{3}(L-ae)+u[-3a^{2}L^{3}+3a^{4}L(e^{2}-1)\\\
+2M^{2}(L-ae)(-8L^{2}+a(7Le+a(e^{2}-5)))-Mu(L-ae)(a^{2}(-11L^{2}+a(7Le+4a(e^{2}-1)))\\\
-12M^{2}(L-ae)^{2}+10a^{2}Mu(L-ae)^{2})]\,\,.$ (15)

We numerically solve the above orbit equation (Eq. (14)) to investigate the
nature and shape of bound orbits of a test particle which is freely falling in
Kerr spacetime. Fig. (1) shows the bound orbits of a test particle in the Kerr
black hole spacetime and Kerr naked singularity spacetime. In that figure, we
show timelike bound orbits for spin parameters $a=\pm 0.8,\pm 1,\pm 1.1$. As
we know, the values of spin parameter $a=\pm 0.8,\pm 1,\pm 1.1$ correspond to
the Kerr black hole, extreme Kerr black hole and Kerr naked singularity
respectively. In Fig. (1), we consider the particle’s total energy $E=-0.001$,
angular momentum $L=12$ and the mass of the black hole to be $M=1$. The orbit
shown by black dotted lines represents the timelike orbits in Schwarzschild
spacetime (i.e. $a=0$). In Fig.(1), the timelike bound orbits in Kerr black
hole spacetime (i.e. $a<1$), and in Kerr naked singularity spacetime (i.e.
$a>1$) are shown by solid red lines and solid blue lines respectively. It can
be seen that the orbital precession in Kerr spacetime is distinguishable from
the orbital precession in the Schwarzschild spacetime.

All the orbits in Fig. (1) show a positive precession and the non-zero spin
parameter changes the minimum approach ($r_{min}$) and perihelion shift of
those orbits. One can see that for $a>0$, the minimum approach of the particle
(Periastron point) increases as the value of spin parameter increases. On the
other hand, for $a<0$, the minimum approach of the particle decreases as the
value of spin parameter decreases. This effect of spin parameter can also be
seen in Fig. (2), where the particle’s total energy $E=-0.006$ and angular
momentum $L=6$. Since the angular momentum ($L$) considered in Fig.(2) is
smaller than that of the Fig.(1), the minimum approaches of the orbits in the
Fig. (2) are much smaller than the minimum approaches of the orbits in the
Fig. (1). Therefore, the frame-dragging effect of Kerr black hole geometry is
much higher in the second case (i.e. for $L=6$). However, it can be verified
that all the orbits in Fig. (2) always have a positive precession.

There exists a radial limit under which no stable bound orbit is possible in
Kerr spacetime. As we know, in Schwarzschild spacetime, there exists a minimum
value of the radius of the stable circular orbit and it is known as the radius
($r_{ISCO}$) of innermost stable circular orbit (ISCO). For Schwarzschild
spacetime, the ISCO is at $r_{ISCO}=6M$. If we put a condition that the
$r_{b}$ in Eq. (11) should always be real, then the $r_{ISCO}$ for Kerr metric
can be written as,

$\displaystyle r_{ISCO}$ $\displaystyle=$ $\displaystyle
3M-\sqrt{3}ae+\sqrt{9M^{2}-6\sqrt{3}aeM-3a^{2}(1-e^{2})}\,\,,$ (16)
$\displaystyle\approx$ $\displaystyle
6M-2\sqrt{3}a\bigg{(}e+\dfrac{a}{4\sqrt{3}M}\bigg{)}\,\,,$

where the first expression is the exact expression of $r_{ISCO}$ and the
second expression is the approximate one, where we consider small values of
the spin parameter. Using the first expression $r_{ISCO}$, one can verify that
the value of $r_{ISCO}$ is real and finite when $a<M$ (i.e. Kerr black hole).
However, when $a>M$ (i.e. Kerr naked singularity) there exist no real value of
$r_{ISCO}$, which implies that stable circular orbits can extend up to the
singularity. The second expression is useful to understand how much the ISCO
radius differs from $6M$ (i.e. the ISCO radius in Schwarzschild spacetime) due
to the non-zero value of spin parameter. Any bound orbits with a minimum
approach ($r_{min}$) close to $r_{ISCO}$, can have a very large perihelion
shift. One can derive the expression for the smallest possible value of
$r_{min}$ ($r_{min0}$) of a bound timelike orbit in Kerr spacetime by finding
the solution of $V_{eff}|_{r_{min0}}=0$, $\frac{dV_{eff}}{dr}|_{r_{min0}}=0$
and $\frac{d^{2}V_{eff}}{dr^{2}}|_{r_{min0}}<0$. The expression of $r_{min0}$
can be written as,

$r_{min0}\approx 4M-a\left(2e+\frac{a}{4M}\right)\,\,.$ (17)

Timelike bound orbits with the above value of minimum approach have large
amount of perihelion shift. One can verify that though the perihelion shift is
large near $r_{min0}$, the precession is always positive. However, this large
perihelion shift of timelike bound orbits cannot be seen or verified in the
stellar motions of ‘S’ stars. For example, S2 star has a minimum approach
which is $r_{min}=2800M$. Therefore, if we consider the central body (Sgr-A*)
as a Kerr black hole, then the perihelion shift of the orbit of S2 would have
a very small positive value, and that would be very close to the expected
value of perihelion shift of the S2 star in a Schwarzschild background. Hence,
the orbits of S2 star should always have a positive precession if we consider
Schwarzschild or Kerr black hole at the center of our galaxy (Sgr-A*).

In Fig. (1) and Fig. (2), we show the positive precession of timelike bound
orbits (on $\theta=\frac{\pi}{2}$ plane) in Kerr spacetime for some particular
values of $L$, $E$ and $a$. However, by those figures we cannot or do not
actually claim to prove that the phenomenon of negative precession is always
forbidden or absent in Kerr spacetimes. To prove that for timelike bound
orbits in Kerr spacetime, we need to analytically solve the orbit equation
(Eq. (14)). However, as the Eq. (14) is a fairly complicated non-linear
differential equation, we attempt to solve it here either numerically, or
using a suitable approximation. Therefore, in the next section, we investigate
the nature of perihelion shift of the orbits in Kerr spacetime using an
approximation technique.

(a) $M=1$, $~{}3.8\leq L\leq 5$, $~{}0.895\leq e\leq 0.999$

(b) $M=1$, $~{}5\leq L\leq 10$, $~{}0.895\leq e\leq 0.999$

(c) $M=1$, $~{}10\leq L\leq 20$, $~{}0.895\leq e\leq 0.999$

(d) $M=1$, $~{}20\leq L\leq 35$, $~{}0.895\leq e\leq 0.999$

Figure 3: The above figure shows the solution points for the timelike bound
orbits in $m$ and $a$ coordinates. We have $0<m<1$ and $m>1$, that represent
the positive precession and negative precession of timelike bound orbits
respectively.

##  III An approximate solution of orbit equation

In this section, we investigate whether a negative precession of timelike
orbits of particles is possible for any value of $L,e$ and $a$. For this
purpose, we use an approximate method here where we only consider low
eccentric orbits. Therefore, for the approximate solution of the Eq. (14), we
only consider upto the first order expression in eccentricity $\epsilon$ Parth
; Struck:2005hi ; Struck:2005oi . The approximate solution can be written as,

$u(\phi)=\frac{1}{Mp}(1+\epsilon\cos(m\phi)+\mathcal{O}(\epsilon^{2}))\,\,.$
(18)

where $p$ and $m$ are real positive constants and $m>1$ represents the
precession of the timelike bound orbits in negative direction, while $m<1$
represents the precession in positive direction. When we substitute the above
expression of $u(\phi)$ in the orbit equation (Eq. (14)) and separate the
zeroth order terms and the first order terms of $\epsilon$, we get an
expression of $m$ in terms of $p$. From the zeroth order terms, we get an
equation of fifth order polynomial of $p$,

$\displaystyle g_{5}$
$\displaystyle(L,a,e,M)p^{5}+g_{4}(L,a,e,M)p^{4}+g_{3}(L,a,e,M)p^{3}$
$\displaystyle+$ $\displaystyle
g_{2}(L,a,e,M)p^{2}+g_{1}(L,a,e,M)p+g_{0}(L,a,e,M)=0\,\,,$

where

$\displaystyle g_{5}(L,a,e,M)$ $\displaystyle=$
$\displaystyle-(2ae(1-e^{2})+L)M^{4}\,\,,$ $\displaystyle g_{4}(L,a,e,M)$
$\displaystyle=$
$\displaystyle(3a^{2}(1-e^{2})L+L^{3}+2aeM^{2}+4LM^{2})M^{2}\,\,,$
$\displaystyle g_{3}(L,a,e,M)$ $\displaystyle=$
$\displaystyle(2a^{3}e(1-e^{2})-a^{2}(11-3e^{2})L$ $\displaystyle-$
$\displaystyle 7L^{3}-4LM^{2}+2ae(3L^{2}+2M^{2}))M^{2}\,\,,$ $\displaystyle
g_{2}(L,a,e,M)$ $\displaystyle=$ $\displaystyle
3a^{4}L(1-e^{2})-2ea^{3}M^{2}(5-e^{2})-30aeL^{2}M^{2}$ $\displaystyle+$
$\displaystyle 16L^{3}M^{2}+a^{2}L(3L^{2}+2M^{2}(5+6e^{2}))\,\,,$
$\displaystyle g_{1}(L,a,e,M)$ $\displaystyle=$
$\displaystyle-a^{2}(ae-L)(-4a^{2}(1-e^{2})+7aeL-11L^{2})$ $\displaystyle+$
$\displaystyle 12M^{2}(ae-L)^{3}\,\,,$ $\displaystyle g_{0}(L,a,e,M)$
$\displaystyle=$ $\displaystyle-10a^{2}(ae-L)^{3}\,\,.$

It is very difficult to get an analytical solution of the above fifth order
polynomial equation (Eq. (LABEL:polynomial)). Therefore, we can use numerical
technique and get five solutions of $p$. One can verify that among those five
solutions only one solution has real and positive value. Now, we get the
following expression of $m$ from the first order term of $\epsilon$,

$m^{2}=\dfrac{-1}{M^{4}(2ae+L(p-2))^{4}p^{3}}\big{(}f_{7}(L,a,e,M)p^{7}+f_{6}(L,a,e,M)p^{6}+f_{5}(L,a,e,M)p^{5}+f_{4}(L,a,e,M)p^{4}\\\
+f_{3}(L,a,e,M)p^{3}+f_{2}(L,a,e,M)p^{2}+f_{1}(L,a,e,M)p+f_{0}(L,a,e,M)\big{)}\,\,,$
(20)

where

$\displaystyle
f_{7}(L,a,e,M)=M^{4}(-L^{4}-8ae^{3}LM^{2}+3a^{2}(-1+e^{2})(L^{2}+4e^{2}M^{2}))\,\,,$
$\displaystyle
f_{6}(L,a,e,M)=-2M^{4}(6a^{3}e(-1+e^{2})L-7L^{4}+4aeL(L^{2}-2e^{2}M^{2})+a^{2}(3(-4+e^{2})L^{2}+4e^{2}(-3+2e^{2})M^{2}))\,\,,$
$\displaystyle
f_{5}(L,a,e,M)=-6L^{2}M^{2}(-3a^{4}(-1+e^{2})-16aeLM^{2}+12L^{2}M^{2}+2a^{2}(L^{2}+2(3+e^{2})M^{2}))\,\,,$
$\displaystyle
f_{4}(L,a,e,M)=4LM^{2}(6a^{5}e(-1+e^{2})+3a^{4}(7-2e^{2})L-96aeL^{2}M^{2}+44L^{3}M^{2}-8a^{3}e(3L^{2}+(3+e^{2})M^{2})$
$\displaystyle\hskip 85.35826pt+12a^{2}L(2L^{2}+(2+5e^{2})M^{2}))\,\,,$
$\displaystyle
f_{3}(L,a,e,M)=15a^{6}(-1+e^{2})L^{2}-15a^{4}L^{4}+12a^{2}(ae-L)(a^{3}e(-1+e^{2})-a^{2}(-11+e^{2})L$
$\displaystyle\hskip
85.35826pt-24aeL^{2}+24L^{3})M^{2}-16(-ae+L)^{2}(a^{2}(3+e^{2})-14aeL+13L^{2})M^{4}\,\,,$
$\displaystyle
f_{2}(L,a,e,M)=6a^{4}(ae-L)L(6a^{2}(-1+e^{2})+7aeL-13L^{2})+24a^{2}(-ae+L)^{2}(3a^{2}-16aeL+16L^{2})M^{2}$
$\displaystyle\hskip 85.35826pt+96(-ae+L)^{4}M^{4}\,\,,$ $\displaystyle
f_{1}(L,a,e,M)=8a^{4}(-ae+L)^{2}(3a^{2}(-1+e^{2})+14aeL-17L^{2})-192a^{2}(-ae+L)^{4}M^{2}\,\,,$
$\displaystyle f_{0}(L,a,e,M)=80a^{4}(-ae+L)^{4}\,\,.$

We put the numerical solutions of $p$ in the expression of $m$ given in Eq.
(20) and get the numerical values of $m$. As it is mentioned before, $m>1$
implies negative precession of the timelike orbits in Kerr spacetime.
Therefore, we verify whether there exist any parameter space regions where $m$
is greater than one. To get numerical solution of $p$, we consider some
specific physically realistic parameter spaces for the parameters $a,L$ and
$e$. In our numerical analysis, we always consider $M=1$, and $a$ varies from
$-40$ to $40$, $L$ varies from $3.8$ to $35$ and $e$ varies from $0.895$ to
$0.999$. The spin parameters $a<-40$ and $a>40$ are not physically realistic
as they may represent somewhat extreme spin situations. When specific energy
$e<0.8$, solutions for bound stable orbits becomes hard to find out. Specific
energy cannot have values beyond one, since the total energy of particle
becomes positive, which implies unbound orbits. Our approximate solution
mentioned in Eq. (18) is not a good approximation for the orbits of high
eccentricity. For $L$ less than $3.8$, the eccentricity of the orbits becomes
very high and therefore, using our approximation method, we cannot do
numerical analysis in $L<3.8$ region.

In Fig. (3), we show the solution points for which bound stable orbits are
possible. We show those points in the $m$ and $a$ coordinates. We know that
$m$ is a function of $a$, $L$ and $e$. Therefore, for a particular value of
$a$, we can get many values for $m$, since at every points other two variables
($L$ and $e$) are varying. We separate $3.8\leq L\leq 35$ into four sections
so that we can get more data points in every section. From Figs. (3(a), 3(b),
3(c), 3(d)), we can see that all the solution points are inside the region
$0\leq m\leq 1$ which indicates that the precession of orbits in Kerr
spacetime is always positive for the classes we have considered. One can see
from Fig. (3) that for any absolute value of the spin parameter, the positive
perihelion shift becomes larger when the sign of the spin changes from
positive to negative. This phenomenon also can be seen from Figs. (1,2).
Therefore, if negative precession is not possible for any positive value of
spin parameter then it would not be possible for any negative value of spin
parameter either. One can consider larger interval of $a$ to verify whether
negative precession of orbits is possible for larger values of $a$ (i.e.
$a>50$), or smaller values of $a$ (i.e. $a<-50$). However, it can be verified
from the Fig. (3) that the solution points are very close to $m=1$ in the
interval $0\leq a\leq 30$ and after that range, those points are diverging
away from $m=1$. Therefore, from the above analysis it can be stated that the
negative precession is forbidden under the given approximation in the Kerr
spacetime.

Till now, in this section, we have done all the analysis considering only the
low eccentricity approximation and we show that negative precession of the
orbits of a test particle is not possible in Kerr spacetime for the equatorial
orbits. We do not consider any weak field approximation for our numerical
analysis. Since eccentricity of a orbit is not directly related with the
perihelion distance, a highly eccentric orbit can be far away from the center,
whereas a low eccentric orbit can be very close to the center. Therefore, in
our analysis, the orbit of the test particle can be very close or far away
from the singularity.

One can consider weak field approximation along with the small eccentricity
approximation to get analytical solutions of $p$ and $m$. From that expression
of $m$, one can get the expression of $m$ in the Schwarzschild limit (i.e.
$a\rightarrow 0$). If we want to consider weak field approximation, we can
neglect third and higher order powers of $u(\phi)$ in the expression of the
orbit equation (Eq. (14)). The approximate orbit equation upto the second
order power of $u(\phi)$ is given by,

$\frac{d^{2}u}{d\phi^{2}}=\frac{1}{L^{5}}[ML^{2}(L+2ae(1-e^{2}))-L(L^{4}+8aLe^{3}M^{2}+3a^{2}(1-e^{2})(L^{2}+4e^{2}M^{2}))u+3M(L^{2}(L^{3}+3a^{2}Le^{2}\\\
+6a^{3}e(1-e^{2}))-4ae^{2}(2eL^{2}+3aL(1-2e^{2})-4a^{2}e(1-e^{2}))M^{2})u^{2}].$
(21)

Substituting the expression of $u(\phi)$ (Eq. (18)) in the above orbit
equation (21), we get the following quadratic equation of $p$ from the
coefficient of zeroth order power of eccentricity ($\epsilon$),

$\displaystyle\tilde{g}_{2}(L,a,e,M)p^{2}+\tilde{g}_{1}(L,a,e,M)p+\tilde{g}_{0}(L,a,e,M)=0\,\,,$
(22)

where

$\displaystyle\tilde{g}_{2}(L,a,e,M)$ $\displaystyle=$ $\displaystyle
2aL^{2}M^{2}e(e^{2}-1)-L^{3}M^{2}\,\,,$ $\displaystyle\tilde{g}_{1}(L,a,e,M)$
$\displaystyle=$ $\displaystyle
L\left[L^{4}+8aLe^{3}M^{2}-3a^{2}(e^{2}-1)(L^{2}+4e^{2}M^{2})\right]\,\,,$
$\displaystyle\tilde{g}_{0}(L,a,e,M)$ $\displaystyle=$
$\displaystyle[-3L^{2}(L^{3}+3Le^{2}a^{2}+6ea^{3}(1-e^{2}))+12ae^{2}M^{2}(2eL^{2}+3aL(1-2e^{2})-4ea^{2}(1-e^{2}))]\,\,.$

Now, we get the following expression of $m$ from the first order term of
eccentricity ($\epsilon$),

$m^{2}=\frac{1}{pL^{5}}[96a^{3}e^{3}M^{2}(e^{2}-1)+L^{5}(p-6)+3a^{2}L^{3}(p-e^{2}(p+6))+4aeL^{2}(9a^{2}(e^{2}-1)+2e^{2}M^{2}(p+6))\\\
-12La^{2}e^{2}M^{2}(-p-6+e^{2}(p+12))]\,\,.$ (23)

(a) $~{}M=1$, $L=6$

(b) $~{}M=1$, $L=12$

Figure 4: This figure shows region plots of $f>0$ and $V_{min}\leq E<0$ for
M=1 and $L=6$ (Fig. 4(a)) and $L=12$ (Fig. 4(b)). Blue region represents
region of negative precession ($f>0$) and orange region represents region of
bound orbits ($V_{min}\leq E<0$).

One can solve the above quadratic equation analytically and get the following
two roots of $p$,

$p_{1}=\frac{1}{2L^{2}M^{2}(L+2ae(1-e^{2}))}[L^{5}+3a^{2}L^{3}(1-e^{2})+8aL^{2}M^{2}e^{3}+12La^{2}e^{2}M^{2}(1-e^{2})\\\
+\\{L^{2}(-12M^{2}(L+2ae(1-e^{2}))(L^{2}(L^{3}+3La^{2}e^{2}+6ea^{3}(1-e^{2}))-4ae^{2}(2eL^{2}+3aL(1-2e^{2})\\\
-4ea^{2}(1-e^{2}))M^{2})+(L^{4}+8aLe^{3}M^{2}+3a^{2}(1-e^{2})(L^{2}+4e^{2}M^{2}))^{2})\\}^{1/2}]$
(24)

$p_{2}=\frac{1}{2L^{2}M^{2}(L+2ae(1-e^{2}))}[L^{5}+3a^{2}L^{3}(1-e^{2})+8aL^{2}M^{2}e^{3}+12La^{2}e^{2}M^{2}(1-e^{2})\\\
-\\{L^{2}(-12M^{2}(L+2ae(1-e^{2}))(L^{2}(L^{3}+3La^{2}e^{2}+6ea^{3}(1-e^{2}))-4ae^{2}(2eL^{2}+3aL(1-2e^{2})\\\
-4ea^{2}(1-e^{2}))M^{2})+(L^{4}+8aLe^{3}M^{2}+3a^{2}(1-e^{2})(L^{2}+4e^{2}M^{2}))^{2})\\}^{1/2}]$
(25)

Now, it can be verified that for $p=p_{2}$, $m$ becomes imaginary and
therefore, the real solution of the quadratic equation of $p$ is $p=p_{1}$. We
get the following expression of $m$ after substituting $p=p_{1}$ in the
expression of $m$,

$m=(1+f(a,M,L,e))^{1/4}\,\,,$ (26)

where

$f(a,M,L,e)=\frac{1}{L^{8}}(-12M^{2}L^{6}+8aeM^{2}L^{3}(L^{2}(5e^{2}-3)+12e^{2}M^{2})+24Lea^{3}M^{2}(e^{2}-1)(L^{2}(e^{2}+3)\\\
+4e^{2}M^{2}(4e^{2}-1))-3a^{4}(e^{2}-1)^{2}(-3L^{4}+24L^{2}e^{2}M^{2}+80e^{4}M^{4})-2a^{2}L^{2}(3L^{4}(e^{2}-1)\\\
+6L^{2}e^{2}M^{2}(1+2e^{2})+8e^{2}M^{4}(8e^{4}+6e^{2}-9)))\,\,.$ (27)

In the above expressions of $m$, $f>0$ implies $m>1$ which is the necessary
condition for the negative precession and $f<0$ implies $m<1$ which is the
necessary condition for the positive precession. As $m$ must be a real and
positive number, $f\geq-1$. Therefore, for the positive precession, we must
have $-1<f<0$. In the Schwarzschild limit or with the approximation of
negligibly small value of $a$, the expression of $m$ written above reduces to,

$m=\left(1-\dfrac{12M^{2}}{L^{2}}\right)^{1/4}\,\,,$ (28)

where $f=\dfrac{-12M^{2}}{L^{2}}$. Therefore, to satisfy the condition
$-1<f<0$ we need,

$\dfrac{L}{M}>\sqrt{12}\,\,.$ (29)

From the expression of $m$ in Eq. (28), one can verify that in Schwarzschild
spacetime, $m$ is always less than one which implies positive precession of
timelike bound orbits Parth ; Joshi:2019rdo . If we consider $L>>M$, we can
write down an approximate expression of $m$ for the Schwarzschild spacetime,

$m=\left(1-\dfrac{3M^{2}}{L^{2}}\right)\,\,.$ (30)

From the above expression of $m$, we can get the following positive perihelion
shift of timelike bound orbits in Schwarzschild spacetime Parth ;
Joshi:2019rdo ,

$\delta\phi_{prec}=\frac{6\pi M^{2}}{L^{2}}\,\,.$ (31)

In Kerr spacetime, we can also get the minimum value of $\frac{L}{M}$ for
bound orbits,

$\dfrac{L}{M}\approx\sqrt{12}-\dfrac{a}{M}\bigg{(}e+\dfrac{a}{\sqrt{12}M}\bigg{)}\,\,,$
(32)

where we consider only upto the second order power of $a$.

Previously, we have shown that the negative precession does not occur in Kerr
spacetime, where we do not consider any weak field approximation. Now, with
the weak field approximation, it should be obvious that we would get the same
result. To verify that, in Fig. (4), we show two region plots for fixed mass
(M) and angular momentum (L), where we show the regions of the negative
precession $f>0$ (i.e. the blue region) and the bound orbits $V_{min}\leq E<0$
(i.e. the orange region) for different values of $a$ and $e$. Existence of any
common region between these two regions implies that in Kerr spacetime,
negative precession of bound timelike orbit is possible. However, we do not
find any of such overlapping regions. Therefore, again we get the same result
that we have verified previously without considering any weak field
approximation.

##  IV Discussion and Conclusion

In this paper, we have studied the nature of timelike bound orbits of a test
particle in Kerr spacetime geometry. In order to find the particle
trajectories in Kerr spacetime, we derive an orbit equation and solve the same
numerically. In Fig. (1), we show the trajectories of a test particle in Kerr
spacetime for the spin parameter values $a=\pm 0.8,\pm 1,\pm 1.1$, where we
take black hole mass $M=1$, specific angular momentum of the test particle
$L=12$ and the total energy of the test particle $E=-0.001$. In Fig. (2), we
show particle trajectories for the same spin parameter values where $M=1,L=6$
and $E=-0.006$. In both the cases we get positive precession of the bound
orbits. We have investigated the particle trajectory only on the equatorial
plane ($\theta=\pi/2$).

Then in section (III), we consider an approximation solution of the orbit
equation (Eq. (14)), in order to understand the nature of perihelion shift of
timelike bound orbits in Kerr spacetime. In that approximation, we consider
only small values of eccentricity ($\epsilon$), and therefore, we neglect
second and higher order terms of $\epsilon$. With this approximation, we show
that in Kerr spacetime, negative precession of the timelike bound orbits is
forbidden, no matter how much far or how much close the orbit is from the
center.

Finally, we take a weak field approximation and we show that the solutions of
Kerr spactime reduce to the solution of Schwarzschild with the approximation
$a\rightarrow 0$. In this paper and in Parth ; Joshi:2019rdo , we show that
negative precession of timelike orbits is not possible in Kerr and
Schwarzschild spacetimes respectively. In Parth ; Joshi:2019rdo , we also
showed that naked singularity models, such as JMN, JNW spacetimes admit both
negative and positive precession of timelike orbits.

As we know, GRAVITY and SINFONI collaborations are continuously observing the
stellar motions of ‘S’ stars around the Milky-way galaxy center Sgr-A*. Hence,
any evidence of negative precession of any ‘S’ star can raise big question on
the existence of Kerr black hole at the Milky-way galaxy center.

Of course, we have not scanned here the full space of bound orbits around the
Kerr black hole or the Kerr naked singularity. Our analysis, however, clearly
points that in the classes of bound orbits we analysed, using the
approximation and numerical techniques as we have stated, we have not found
any negative precession for the bound orbits in both these cases. In
particular, the case of Kerr black hole needs to be analysed in more detail to
make sure if it forbids the negative precession for the bound timelike
trajectories always. If that turns out to be the case, that will support the
conjecture that black holes never allow for the negative precession, however,
naked singularities allow the same as shown by some of the naked singularity
spacetimes investigated as we pointed out here.

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# ePose: Let’s Make EfficientPose More Generally Applicable

Austin Lally
Oregon State University
<EMAIL_ADDRESS>Robert Bain
Oregon State University
<EMAIL_ADDRESS>Mazen Alotaibi
Oregon State University
<EMAIL_ADDRESS>
###### Abstract

EfficientPose is an impressive 3D object detection model. It has been
demonstrated to be quick, scalable, and accurate, especially when considering
that it uses only RGB inputs. In this paper we try to improve on EfficientPose
by giving it the ability to infer an object’s size, and by simplifying both
the data collection and loss calculations. We evaluated ePose using the
Linemod dataset and a new subset of it called ”Occlusion 1-class”. We also
outline our current progress and thoughts about using ePose with the NuScenes
and the 2017 KITTI 3D Object Detection datasets. The source code is available
at https://github.com/tbd-clip/EfficientPose.

##  1 Introduction

Object detection is one of the core problems in computer vision. Given an
input image and a predefined object class, the task is to locate an instance
of the desired object type in the image. Typically this is accomplished by
reporting a bounding box. There are several variants of the problem. In multi-
object detection, all instances of the object type are located. In semantic
segmentation, each pixel of the image that belongs to an object of the class
is labeled. Instance segmentation further distinguishes between each separate
occurrence. Multi-class object detection, as the name suggests, requires
locating instances of each of a set of predefined object classes instead of
just one.

A natural extension of the object detection problem is to define three-
dimensional bounding cubes in world coordinates instead of just drawing two-
dimensional bounding boxes on the image plane. This problem, called 3D object
detection, is especially important in autonomous driving and other robotics
applications as agents need to predict the future positions of obstacles in 3D
space. It has received a lot of attention in recent research. LiDAR-based
point cloud-based methods [7, 1] have driven considerable progress. However,
LiDAR sensors are not always available due to their high cost, so there is
also some research into 3D object detection from RGB images [15, 2, 14].

In section 2 we discuss related work in 3D object detection. Section 3 covers
the primary evaluation metric and our high-level technical approach. We
describe our experiments in section 4, and finally we summarize our thoughts
and opportunities for future work in section 5.

##  2 Related Work

We now review existing work on 3D Object Detection that uses RGB images as an
input. We divide existing detectors into two categories. We first cover
traditional detectors, which require no prior information about the target
object’s shape, and then discuss detectors that leverage detailed 3D-models of
the target object (note: we use the term “3d-models” to distinguish mesh-style
object models from deep learning models).

###  2.1 Traditional Detectors

Traditional detectors don’t require 3D-models of the target objects to
estimate 3D bounding cubes. Some models, like MonoPSR [8], use a mature 2D
object detector to produce accurate 2D bounding boxes first, and then regress
a 3D bounding cube from an initial 3D proposal from there. Similarly, [11]
predicts a 3D bounding cube from a 2D bounding box, but in this case they
first predict the object’s orientation using a novel hybrid discrete-
continuous loss.

PerspectiveNet [6] instead defines “perspective points” as the 2D projections
of 3D keypoints. Using the geometric constraints imposed by the perspective
projection, it reconstructs the 3D position of the target object using
detected keypoints without prior knowledge about the object’s 3D shape.

###  2.2 Detectors Based on 3D-models

These detectors require 3D-models of the target object to estimate the 3D
bounding cubes. There are two main approaches: (i) viewpoint-based methods and
(ii) keypoint-based methods.

Viewpoint-based methods predict 3D bounding cubes with textured models. [10]
proposed predicting several 3D bounding cubes for each object instance to
estimate the bounding cubes distribution generated by symmetries and
repetitive textures. Each predicted hypothesis corresponded to a single 3D
translation and rotation, and estimated hypotheses collapsed onto the same
valid bounding cube when the object appearance was unique.

Keypoint-based methods predict 3D bounding cubes by detecting specified
keypoints and then solving Perspective-n-Point (PnP) problem to estimate the
3D bounding cubes. [2] proposed a 6D object pose estimation method that
predicts 2D bounding boxes using a 2D object detector, uses network heads to
estimate the missing depth information to lift 2D bounding boxes to 3D space,
then uses PnP to refine the final pose.

The advantage of using detectors based on 3D-models is they are robust and
more accurate compared to traditional detectors that don’t use 3D-models to
estimate the 3D bounding cubes. However, detectors based on 3D-models will
require 3D-models to be used during training and evaluation of the detectors
to calculate the loss rather than using the 3D bounding cube. Since ground-
truth 3D-models are not always available (e.g., pedestrian detection), this is
a rather limiting requirement. The other drawback to 3D-model-based detection
is that it requires the detector to predict a full 3D-model, which is
computationally expensive compared to simple bounding cube estimation.

##  3 Technical Approach

###  3.1 Evaluation Metric

We evaluate our models with the ADD(-S) metric [5]. Symmetric and asymmetric
objects are evaluated differently (ADD and ADD-S respectively). This is to
ensure that the symmetric object labels are not unduly punished by a higher
loss for predictions that might be semantically the same but just rotated
relative to the ground truth label. This is the effect of the minimization in
the for loop, which allows the comparison of object points to any other object
points in $\mathcal{M}$ (originally a set of surface points from a given
object of interest). We adopt the same (rather arbitrary) threshold of 10% of
the object diameter to count an object as being correctly detected or not.
Said another way, if the average point pair distance between the model
prediction and the ground truth is larger than 1/10th of the object’s
diameter, then the object was detected incorrectly.

$\mathrm{ADD}=\frac{1}{m}\sum_{\mathbf{x}\in\mathcal{M}}\|(\mathbf{R}\mathbf{x}+\mathbf{t})-(\tilde{\mathbf{R}}\mathbf{x}+\tilde{\mathbf{t}})\|_{2}$

$\mathrm{ADD}$-$\mathrm{S}=\frac{1}{m}\sum_{\mathbf{x}_{1}\in\mathcal{M}}\min_{\mathbf{x}_{2}\in\mathcal{M}}\|(\mathbf{R}\mathbf{x}+\mathbf{t})-(\tilde{\mathbf{R}}\mathbf{x}+\tilde{\mathbf{t}})\|_{2}$

###  3.2 Training with Bounding Cubes

While EfficientPose [2] uses a 3D-model with many surface points $\mathcal{M}$
to model the target object, we wanted to reduce the requirement for prior
information. One of our main contributions is to train the model instead on a
3D bounding cube. In this way, the loss function is computed against only
eight points (instead of an average of 500 points) per object, which reduces
the number of input data points by many times.

###  3.3 Improving the loss

The other limiting assumption made by [2] is that the precise size of the
target object is known ahead of time. We aim to remove this requirement and
instead let the network learn the object’s size. We do this by adding an
additional module, identical to EfficientPose’s rotation subnet, to learn a
3-dimensional scale value for the object’s bounding cube. We additionally add
a scaling term to the loss function in order to train the scaling subnet.

##  4 Experiments

Our experiments do not focus much on the speed of training. EfficientPose [2],
which our work is built upon, also defined epochs in an odd way, which makes
it difficult to compare results. Their definition is a hard-coded number of
training steps per epoch, thus changing the mini-batch size or including less
or more data-augmentation does not change the number of training steps in an
epoch. Given more time, or if we were developing a product to be used in
production, we would want to make this more in line with the standard
definitions. For these reasons we will loosely talk about the time it took to
train these models. The models in this report were trained on an RTX 2080 and
RTX 8000 GPU with 8GB and 48GB of VRAM respectively. The latter proved useful
as even small batch sizes of the smaller models ($\phi$ = 0) required a
tremendous amount of VRAM to train (e.g. a batch size of 12 took 11GB). We
used exclusively the smaller $\phi$ = 0 models, which took roughly 3 days each
to train to completion. This included transfer learning EfficientDet [13]
weights pretrained on COCO [9]. The rotation and translation subnetworks were
always initialized from untrained weights though, as was the scaling
subnetwork that will be introduced in section 4.2.

The authors of [2] implemented group normalization [16] in order to train
large $\phi$ = 3 models, which require large amounts of VRAM to perform
backprop. The experiments after and including scaling Linemod cat detector are
done using batch normalization instead. Various batch sizes were used
throughout training, between 1 and 50. Our intuition after tuning these models
is that group norm did help at lower batch sizes, but that a batch size of 16
with the small $\phi$ = 0 models works well too. This large of a batch size
was enabled by the larger memory GPU (the RTX 8000).

Table 1 summarizes the best scores achieved in each of our experiments.

###  4.1 Simplifying the Loss

Figure 1: Results of detecting the camera (bottom left) after training a model
with the bounding cube for the clothes iron (top left). Figure 2: ADD(-S)
score as a function of training epochs while training a camera detection model
on a clothes iron mesh. The highest score achieved was 0.828, after 320
epochs.

EfficientPose [2] requires a detailed set of surface points $\mathcal{M}$ to
model each potential object of interest in the image. This is a rather
restricting assumption that limits the applicability to mostly toy problems.
We were interested in calculating the loss without the need for this set of
surface points. The intuition that motivated this experiment was that the
ADD(-S) loss function would seem to be fairly agnostic to the choice of
$\mathcal{M}$ (see figure 1 for an example). A quick test of this hypothesis
was conducted by swapping the Linemod `.ply` model files used to train the
various single object detectors. The camera and clothes iron $\mathcal{M}$
were swapped and good results were still obtained. The ADD(-S) score was 0.83.
This could have undoubtedly been improved upon, as training takes quite a
while and we were convinced by the results seen in figure 2 that this method
was working to train a detector. Thus, training was stopped before convergence
so we could move on to the next experiment into simplifying the loss.

Figure 3: Our scaling subnetwork architecture. The diagram is an updated
version from [2].

We build on the previous experiment to further demonstrate that the ADD(-S)
loss of [2] creates a good learning signal essentially regardless of
$\mathcal{M}$. Since many 3D object data sets already have bounding cubes as
part of the ground truth labels, we sought to use these in the loss function
as a replacement for $\mathcal{M}$. This set of 3D points would be of length
8, reducing the number used by roughly 60x compared to [2]. This also produced
good results, even though the model was trained with the 3D bounding cube
vertices of the wrong `.ply` model again. To reiterate: the camera detection
model seen in figure 1 was trained with the bounding cube vertices obtained
from the clothes iron `.ply` model file. This model was trained until
convergence and achieved 0.91 ADD(-S) (it actually did as well as 0.95 but the
checkpoint directory has been lost to time).

Experiment | ADD(-S)
---|---
Camera detector w/ iron model | 0.83
Camera detector w/ iron bounding cube | 0.91
Kitty detector w/scaling | 0.83
Occlusion 1-class w/bounding cube prior | 0.06
Occlusion 1-class w/o symmetric objects | 0.13
Table 1: Best ADD(-S) achieved for each experiment.

###  4.2 Scaling Subnetwork

Another disappointing assumption made by [2] is that oracle sizing information
is known about each object of interest. This too, like the need for an
accurate and detailed set of surface points, limits the applicability of this
otherwise powerful method. If the scaling can instead be made part of
inference, this method has a chance of working on interesting problem
formulations.

The scaling subnetwork we introduce is an exact copy of the rotation
subnetwork from the original EfficientPose architecture. The architecture is
shown in figure 3. The Refinement Module is just a sequence of $D_{iter}$ 3x3
convolutional layers. $N_{iter}$, $D_{iter}$, and $D_{rot}$ are defined as:

$N_{iter}(\phi)=1+\lfloor\phi/3\rfloor$ (1)

$D_{iter}(\phi)=D_{rot}(\phi)=2+\lfloor\phi/3\rfloor$ (2)

For this initial proof of concept the scaling subnet only had to learn to
scale all the points in $\mathcal{M}$ by 1. In later experiments we use a
bounding cube prior that requires more intricate scaling to be learned. The
results can be seen in figure 4. The kitty detector was also trained using the
correct bounding cube vertices (not a swapped model’s vertices as done in the
previous subsection), and got an ADD(-S) of 0.83. It intuitively makes sense
that the old performance of 0.98 ADD(-S) reported by the authors of
EfficientPose should act as an upper bound on our model’s performance given
that they use ground truth scaling information, where instead our method must
guess at that information.

Figure 4: Sample detection results for the kitty detector with the initial
proof-of-concept scaling subnet.

Given that when [2] added the translation and rotation subnetworks to
EfficientDet it only reduced inference from 35 to 27 frames per second, our
addition of an exact copy of only the rotation subnetwork should have a small
impact on performance.

###  4.3 Occlusion 1-Class

The occlusion dataset is a subset of the Linemod dataset. It includes a single
scene with 8 annotated objects. Some of these are heavily occluded, as
indicated by the name of the dataset. We made the task even more difficult by
mapping all of the objects to a single class, where originally each object had
its own class. This new dataset allows us to benchmark our EfficientPose
additions against use cases where there are multiple objects of interest in a
given image, and the use cases where there is intra-class size variance (i.e.
when objects in the same class have different scalings).

Unlike the scaling in the previous section, here we use a bounding cube prior
that is an average bounding cube over all 8 objects in the class. The prior is
then directly scaled using the output of the scaling subnetwork.

Figure 5: Example inference from our new scaling subnet addition. Note that
the two worse object predictions are both symmetric objects. This highlights
how adding scaling makes the ADD-S metric inappropriate for training. Figure
6: Results from the same experimental setup as 5, but with the symmetric
objects removed from training. The ADD(-S) score then jumps from 0.06 to 0.13.
This demonstrates that training our method on symmetric objects will not only
fail, but also degrades the performance on other, non-symmetric objects.

Using a class average object diameter of 170 mm led to an ADD(-S) score of
only 0.06 for this difficult dataset. Upon inspecting the results, one quickly
realizes that the symmetric objects are failing to be learned. The model
always predicts scalings that result in small volumes. This effect can be seen
in figure 5. The ADD-S metric and loss for symmetric objects was discovered to
have become inappropriate for a network that includes a scaling inference,
because the inner-minimization allows the most favorable points to be compared
in order to avoid over-penalizing model predictions. The model then learns to
make a very small volume guess that is close to just 1 or 2 points.

We took the two symmetric objects out of the dataset and re-ran with the same
hyper-parameters (6). The ADD(-S) results more than doubled to 0.13 ADD(-S).
This indicates that not only will we need to change the loss for symmetric
objects, but currently including symmetric objects will hinder the performance
w.r.t. the other asymmetric objects in the dataset.

##  5 Discussion

###  5.1 Challenges

####  5.1.1 Symmetry Problem

We tried to solve the symmetry problem where our detector fails to estimate 3D
bounding cubes for symmetrical objects. To solve this problem, we introduced
an additional loss to the translation loss to estimate the volume of the
object. We have tried two approaches to define the additional loss: a simple
volume estimation and a rotation-robust IoU (RIoU) [17]. The simple volume
estimation compares the volume of predicted 3D bounding cube and the target 3D
bounding cube. Rotation-robust IoU [17] computes the Intersection over Union
(IoU) for the 3D bounding cube by taking rotation into consideration, as
defined in equation 3.

$\begin{split}I_{R\operatorname{Io}U}&=\min(I1,I2)\cdot\left|\cos\left(2\cdot\left(r_{g}-r_{p}\right)\right)\right|\\\
U_{R\operatorname{Io}t}&=\max\left(I_{RI\mathrm{o}U},l_{g}-w_{g}+l_{p}-w_{p}-I_{RI\mathrm{o}U}\right)\\\
R\mathrm{IoU}&=\frac{I_{\mathrm{RIO}U}}{U_{\mathrm{RToU}}}\end{split}$ (3)

In these equations (as described in [17]), $l_{g}$,$w_{g}$, and $r_{g}$
represent length, width, and rotation around z-axis respectively of the ground
truth 3D bounding cube. $l_{p}$,$w_{p}$, and $r_{p}$ represent length, width,
and rotation around z-axis respectively of the predicted 3D bounding cube.
$I1$ represents the intersection of the predicted and ground truth bounding
cubes in the coordinate space of the ground truth, and $I2$ represents the
intersection of the predicted and ground truth bounding cubes in the
coordinate space of the prediction.

Neither definition of the loss yielded good results when training on symmetric
objects without using 3D-models of the objects compared to training on
symmetric objects with 3D-models of objects on Linemod dataset. This means
that detection of symmetric objects remains an unsolved problem in our
approach, but we hope that with further research an appropriate loss function
can be found to resolve the issue. We suggest an alternative approach in
section 5.2.

####  5.1.2 NuScenes

NuScenes [3] offers an autonomous driving dataset that looks promising for our
use case. It consists of 1000 20-second video scenes captured in Boston and
Singapore. In each keyframe, 23 object classes are annotated with ground-truth
3D bounding cubes. In each keyframe, several sensor images are captured along
with the ground truth annotations. There are six cameras, one LiDAR, five
RADAR, one GPS, and an IMU. We ignore the other sensors and use only the RGB
camera images for training and evaluation.

The ground-truth bounding cubes are given as an 8x3 matrix of coordinates,
with the four front points listed first in clockwise order, followed by the
back four. This fits well with the loss model described in section 4.1.
Unfortunately, after building a data loader for this dataset and incorporating
it into the EfficientPose codebase, the model was entirely unsuccessful and
failed to produce almost any detections. Debugging this behavior led to the
discovery that the dataset includes many fully occluded ground-truth bounding
cubes, as seen in figure 7.

Figure 7: NuScenes images with heavily occluded ground-truth bounding cubes.

Two reasons are given for this in the NuScenes issue tracker. First, the LiDAR
hat is mounted much higher on the vehicle than the cameras, so although
objects are hidden from camera view they may be visible by LiDAR in the same
keyframe. Second, NuScenes propagates temporarily occluded bounding cubes
through intermediate frames (although it’s unclear under what conditions they
do this, and how they define temporarily in this case). This obviously harms
training when looking only at camera data, because the model is trying to
learn to detect what isn’t visible.

###  5.2 Future Work

Future researchers might be interested in a solution to the volume collapse
that we came up with. Each scaling subnet output element could be put through
an exponential and then offset by some small amount (e.g. 0.2). [12] scales
its anchor bounding boxes in a similar way, but without the offset. The offset
would essentially enforce a minimum volume of the predicted bounding cube,
preventing the undesirable effects of the current ADD-S loss that unknowingly
encourages the scaling net to predict very small volume bounding cubes.

We are still interested in applying ePose to a large dataset like NuScenes,
even if NuScenes in particular is not applicable to a model like ours that
does not leverage LiDAR information. We are currently in the process of
evaluating ePose on the 2017 KITTI dataset for 3D Object Detection [4]. This
dataset has multiple levels of difficulty, classified by the area of each
annotation’s bounding box, and only includes 1 degree of rotation about the
y-axis. The 2017 KITTI dataset annotates the pose and rotation of bounding
cubes centered around pedestrians, cyclists, and cars. The volume overlap
threshold used for the KITTI leaderboards is 70% for cars, and only 50% for
the other 2 classes. It would be interesting to see how our model ranks on the
leaderboard, against solutions that use LiDAR or depth information.

###  5.3 Conclusion

EfficientPose [2] is a promising method for 3D object pose estimation, but it
has real limitations when it comes to applying it to some real-world
situations. We endeavored to improve the flexibility of the approach by
reducing the amount of detailed object information necessary to train the
network. We found that it is in fact possible to train the network using 3D
bounding cubes instead of high-fidelity 3D-models. We also found that a
scaling subnetwork successfully learns object sizes to a reasonable degree. We
found that symmetric objects pose a real challenge, and that additional
research is needed to get the scaling module working well in those cases. We
also proposed possible future extensions to our work. We consider our work to
be a substantial contribution to the field, and hope that continued work in
this direction can lead to highly successful real-time 3D object detection
from RGB images.

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