# Shadows and precession of orbits in rotating Janis-Newman-Winicour spacetime

Divyesh N. Solanki<EMAIL_ADDRESS>Sardar Vallabhbhai National
Institute of Technology, Surat GUJ 395007, India  Parth Bambhaniya
<EMAIL_ADDRESS>International Center for Cosmology, Charusat University,
Anand, GUJ 388421, India  Dipanjan Dey<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  Kamlesh N. Pathak
<EMAIL_ADDRESS>Department of Physics, S.V National Institute of
Technology, Surat GUJ 395007, India

###### Abstract

In this paper, we construct the rotating Janis-Newman-Winicour (JNW) naked
singularity spacetime using Newman-Janis Algorithm (NJA). We analyse NJA with
and without complexification methods and find that the energy conditions do
satisfied when we skip the complexification step. We study the shadows cast by
rotating JNW naked singularity and compare them with the shadows cast by the
Kerr black hole. We find that the shadow of the rotating naked singularity can
be distinguished from the shadow of the Kerr black hole. While we analyse the
precession of timelike bound orbits in rotating JNW spacetime, we find that it
can have a negative (or opposite) precession, which is not present in the Kerr
black hole case. These novel signatures of the shadow and orbital precession
in rotating JNW naked singularity spacetime could be important in the context
of the recent observation of the shadow of the M87 galactic center and the
stellar dynamics of ‘S-stars’ around Milkyway galactic center.

Key words: Black hole, Naked singularity, Shadow, Timelike orbits, Newman-
Janis Algorithm.

##  I Introduction

In 1965, Newman and Janis showed that the Kerr metric could be obtained from
the Schwarzschild metric using a complex coordinate transformation JNA . For
the static Reissner-Nordström metric, using the same method, one can derive
the Kerr-Newman metric, which represents a spacetime geometry of the
electrically charged and rotating black hole KN . The application of the NJA
for generating interior solutions which match smoothly to the external Kerr
metric is studied in Drake1997 . In this algorithm, authors use two different
types of complexification with no apparent reason. There are many ways to
complexify the coordinates, however, in the above mentioned literature, the
only reason given for choosing those particular complexifications is that it
is successful in generating the Kerr and Kerr-Newman metric. However, the
success of the NJA is limited to the spacetimes which atleast satisfy the
reciprocal condition, $g_{tt}g_{rr}=-1$. Later, it was proved by S.P. Drake,
and P. Szekeres Drake2000  that the only perfect fluid solution generated by
the NJA is the Kerr metric, and the only Petrov typed D solution to the
Einstein-Maxwell equation is the Kerr-Newman metric.

The general theory of relativity predicts that when large enough masses
collapse under the influence of their own gravity, the spacetime singularity
forms necessarily. There is a singularity theorem which shows the
inevitability of the formation of the spacetime singularity under some
circumstances. However, there exists no such theory on the existence of an
event horizon around the singularity. In 1969, Rodger Penrose gave a cosmic
censorship conjecture (CCC) which does not allow horizon-less strong spacetime
singularity penrose . However, there are series of literature on the continual
gravitational collapse of the inhomogeneous matter cloud where it is shown
that spacetime singularities formed during gravitational collapse can be
visible by the outside observerjoshi ; goswami ; mosani1 ; mosani2 ; mosani3 ;
mosani4 . Therefore, the spacetime singularity may or may not be visible by
asymptotic observer, since it depends upon the initial conditions of the
collapsing matter Joshi:2011zm . In the regime of general relativity, such a
scenario can arise where the continual gravitational collapse may lead to an
equilibrium static spacetime which has a central visible or naked singularity
e.g., Joshi-Malafarina-Narayan (JMN), Janis-Newman-Winicour (JNW), and
Bertrand (BST) spacetimes Joshi:2011zm ; JNW ; perlick .

From the above discussion, it is clear now that the naked singularity can be
formed as the end state of continuous gravitational collapse of an
inhomogeneous matter cloud. Therefore, if naked singularity really exits in
nature, it would have distinguishable physical signatures. Since every compact
object in our universe has its intrinsic angular momentum, for more realistic
scenario, one needs to construct rotating counterpart of the naked singularity
spacetime. In this paper, we first construct the rotating version of Janis-
Newman-Winicour (JNW) naked singularity spacetime to investigate its possible
physical signatures in the context of shadow shape and the dynamics of
timelike bound orbits. One can obtain the rotating naked singularity spacetime
by applying the NJA to the static naked singularity metric.

However, when we apply the NJA to other spacetime metrics, such as naked
singularities and other black hole solutions, the final result which is
obtained in the Eddington-Finkelstein coordinates (EFC) may not be transformed
into the BLC due to the complexification procedure as shown in Mustapha2011 .
It is shown by Mustapha Azreg-Aïnou that the rotating spacetime generated by
the NJA and written in the EFC could be transformed into the BLC by dropping
the complexification step Mustapha1 ; Mustapha2 ; Mustapha3 . Recently, the
rotating form of the polytropic black hole spacetime is constructed using the
NJA without complexification polytropic . The use of NJA is described in other
theories of gravity as well, i.e., f(R) gravity, Einstein-Maxwell-Dilaton
gravity, Born-Infeld monopole gravity Canonico . To illustrate this method, in
this paper, we apply the NJA without complexification to the static Janis-
Newman-Winicour (JNW) naked singularity spacetime. Using the procedure given
in Mustapha1 ; Mustapha2 ; Mustapha3 , we obtain the rotating JNW naked
singularity spacetime and it does satisfy all the energy conditions.

As it is stated above, the black hole and naked singularity should have some
different physical signatures, since the geometrical properties of them are
different from each other. Therefore, observational investigations on
different physical properties of ultra compact objects may reveal the nature
of the same. The Event Horizon Telescope (EHT) collaboration recently released
the first-ever image of the shadow of the black hole located at the center of
Messier 87 (M87) galaxy Kazunori_Akiyama . Similar shadows can also be cast by
other compact objects. There are several literature where the shadows cast by
compact objects such as black holes, naked singularities, gravastar and
wormholes are extensively studied Gralla:2019xty ; Abdikamalov:2019ztb ;
Vagnozzi:2019apd ; Gyulchev:2019tvk ; Dey:2013yga ; Dey+15 ; Dey:2020haf ;
atamurotov_2015 ; abdujabbarov_2015b ; ohgami_2015 ; stuchlik_2019 ; Kaur ;
Sakai ; Shaikh:2021 ; Li:2021 ; Hu:2020usx . In Shaikh:2018lcc , authors
discuss the shadow cast by Joshi-Malafarina-Narayan (JMN) naked singularity
spacetime and they compare the results with a shadow cast by the Schwarzschild
black hole. In the first type of JMN (JMN1) naked singularity spacetime, one
can define a range of characteristic parameter ($0<M_{0}<2/3$) for which there
exists no photon sphere and hence no shadow and therefore, an interesting
full-moon image of the central singularity is formed. However, for
$M_{0}>2/3$, JMN1 naked singularity can cast a similar shadow what an equally
massive Schwarzschild black hole can cast. It is generally believed that the
shadows arise due to the presence of a photon sphere. Recently, in Joshi2020 ,
the authors have introduced a new spherically symmetric naked singularity
solution of the Einstein field equation which does not have a photon sphere,
but the singularity still casts a shadow. After that, in Dey:2020bgo , for
null-like and timelike naked singularities, the general conditions for the
shadow formation in the absence of a photon sphere are investigated.

The shape of the shadows and their properties would be different for the
rotating spacetimes. The involvement of a spin parameter can make a spacetime
more realistic, and the shadow cast by that spacetime can give us a more
realistic shadow shape of a compact object. In Shaikh:2019fpu , a general
formula has been derived to obtain the shape of a shadow cast by a compact
object whose gravitational field is described by a rotating spacetime
geometry. Also, this general formula is applied to some known black hole
solutions, and the corresponding results for shadows are reproduced. In our
recent work parth6 , we analyse the shape of the shadows in the deformed Kerr
spacetime. We show that due to the deformation, shadow with oblate and prolate
shape can exist for the negative and positive sign of the deformation
parameter respectively. In last few years, the shadows cast by rotating black
holes are extensively studied in many literature abdujabbarov_2015a ;
abdujabbarov_2017 ; younsi_2016 ; papnoi_2014 ; Afrin ; Ghosh:2020ece ;
Chen:2020aix ; Haroon:2018ryd ; Ovgun:2018tua ; Lee:2021sws ; Konoplya:2021 ;
Shaikh:2021yux . The shadow in the rotating naked singularity may have a
different distinguishable signature. Therefore, in this paper, we study the
shadow shape in rotating JNW spacetime and we show that the shape of the
shadows in rotating JNW spacetime can be distinguished from the shadows cast
by the Kerr black hole.

On the other hand, the GRAVITY, SINFONI, and UCLA galactic center groups are
continuously observing astrometric and spectroscopic data of the S-stars which
are orbiting around our Milky-way galactic center Do:2019txf ; GRAVITY:2018ofz
; Hees:2017aal . These observational data could help to reveal the nature of
the galactic center (Sgr-A*). The relativistic orbital precession in the
Schwarzschild spacetime is discussed in GRAVITY:2020gka . There are large
number of literature where the nature of the orbit precession is extensively
studied in various spacetime geometry Martinez ; Eva ; Eva1 ; Eva2 ; tsirulev
; Joshi:2019rdo ; Bambhaniya:2019pbr ; Dey:2019fpv ; Bam2020 ; Lin:2021noq ;
Deng:2020yfm ; Deng:2020hxw ; Gao:2020wjz . In Joshi:2019rdo ;
Bambhaniya:2019pbr , the periastron precession of particles orbit in static
JMN1, JMN2, JNW, and Bertrand (BST) naked singularity spacetimes are studied,
and the results obtained are compared with that of the Schwarzschild
spacetime. It is concluded that the nature of orbital precession in BST is
similar to that in Schwarzschild spacetime which is in the same direction as
the motion of a particle. On the contrary, an orbit can precess in the
opposite direction of the motion of a particle in JMN1, JMN2, and JNW
spacetimes.

Similarly, as we discussed above, the rotating spacetimes are more physically
realistic to investigate the orbital precession. In our previous work Bam2020
; parth6 , we have shown that the orbits in deformed Kerr spacetime can
precess in the opposite direction of the motion of a particle when the
deformed Kerr black hole becomes a naked singularity, which is not possible in
Kerr spacetime. We have also shown that the shadow and negative precession of
timelike bound orbits can simultaneously exist in deformed Kerr naked
singularity spacetime. There are some literature where the timelike orbits in
rotating black holes are extensively studied aa4 ; Glampedakis:2002ya ;
Fujita:2009bp ; Pugliese:2013zma ; rana . Here, we analyse the precession of
the timelike bound orbits in rotating JNW spacetime and compare them with the
Kerr black hole case. We find that the rotating JNW spacetime can admit
negative precession for the particular range of scalar field charge $q$ and
spin parameter $a$. These novel features of the shadow shape and the
precession of the timelike bound orbits in rotating JNW spacetime can be
observationally significant to distinguish from the Kerr and other rotating
compact objects.

This paper is constructed in the following way. In section (II), we derive the
rotating JNW naked singularity spacetime using the NJA with and without
complexification. We discuss the energy conditions in both cases. In section
(III), we study the shadow properties and find the shape of the shadows in
rotating JNW spacetime. In addition, we show the comparison of the shape of
the shadow cast by the Kerr black hole and rotating JNW naked singularity. In
section (IV), we obtain the general differential orbit equation for the
timelike bound orbit and compare the orbital precession in the Kerr and
rotating JNW spacetime. Finally, we discuss and conclude our result in (V). We
consider gravitational constant $G$ and speed of light $c$ equal to unity
throughout this paper.

##  II Construction of the rotating JNW spacetime metric

In this section, we determine the rotating JNW naked singularity spacetime by
using the Newman-Janis algorithm (NJA). The final result of the NJA is
obtained in the null coordinates, which must be transformed into the Boyer-
Lindquist coordinates (BLC). The BLCs are more convenient to work with the
rotating spacetime metric. We analyse that it is not possible to perform the
BLC transformation when considering JNW spacetime metric (1) as a seed metric.
To resolve this problem, we drop the complexification step in NJA Mustapha1 ;
Mustapha2 ; Mustapha3 .

###  II.1 JNW metric and the Newman Janis Algorithm

The JNW spacetime metric is given by

$\displaystyle
ds^{2}=-\left(1-\frac{2M}{r\nu}\right)^{\nu}dt^{2}+\left(1-\frac{2M}{r\nu}\right)^{-\nu}dr^{2}$
$\displaystyle+\,r^{2}\left(1-\frac{2M}{r\nu}\right)^{1-\nu}d\Omega^{2}$ (1)

where, $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. It is straight
forward to show that the following expression is the final form of the NJA
with considering JNW metric (1) as a seed metric.

$\displaystyle ds^{2}=\,$
$\displaystyle-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}du^{2}-2dudr$ (2)
$\displaystyle-\,2a\sin^{2}\theta\left[1-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]dud\Phi$
$\displaystyle+\,\rho^{2}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{1-\nu}d\Omega^{2}+2a\sin^{2}\theta
drd\Phi$
$\displaystyle+\,a^{2}\sin^{4}\theta\left[2-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]d\Phi^{2}$

The above form of the spacetime metric seems rather complicated. It can be
made more simple and symmetrical by transforming it into the Boyer-Lindquist
coordinates (BLC). The BLC $(t,r,\theta,\phi)$ represents the rotating black
hole spacetime metrics because all the off-diagonal terms of the metric,
except $dtd\Phi$, vanish; and its axial symmetry becomes apparent. The
transformation into the BLC requires

$\displaystyle du=dt-\xi(r)dr$ (3) $\displaystyle d\Phi=d\phi-\chi(r)dr$ (4)

Here, we note that the transformation functions, i.e., $\xi(r)$ and $\chi(r)$,
strictly depends on coordinate $r$ only; otherwise eq. (3) and (4) will not be
integrable. To write eq. (2) into the BLC like form, the transformation
functions must take the following form.

$\displaystyle\xi(r,\theta)=\frac{1}{\Delta}\left[\rho^{2}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{1-\nu}+a^{2}\sin^{2}\theta\right]$
(5) $\displaystyle\chi(r)=\frac{a}{\Delta}$ (6)
$\displaystyle\Delta=r^{2}-\frac{2Mr}{\nu}+a^{2}$ (7)

It can be seen that the transformation function $\xi$ also depends on
coordinate $\theta$, and therefore the eq. (3) is not integrable. In these
coordinates, eq. (2) is written as

$\displaystyle
ds^{2}=-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}dt^{2}+\rho^{2}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{1-\nu}\left(\frac{dr^{2}}{\Delta}+d\Omega^{2}\right)-2a\sin^{2}\theta\left[1-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]dtd\phi$
$\displaystyle+a^{2}\sin^{4}\theta\left[2-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]d\phi^{2}$
(8)

It can be seen that the above metric reduces to the JNW spacetime metric (1)
when $a=0$. Although the transformation into the BLC was improper in obtaining
eq. (8), we can check the energy conditions because the NJA is just an
algorithm that successfully works for Kerr and Kerr-Newman metric. Thus, we
cannot say whether the final result eq. (8) is a valid solution of the
Einstein field equation. It can only be answered once we know whether the
energy conditions for the metric (8) are satisfied or not. The energy
conditions for the above spacetime metric are discussed in the Appendix, and
we conclude that the rotating JNW spacetime metric (8) obtained using NJA with
complexification violates the energy conditions.
Now, we drop the $\theta$ dependency of the transformation function $\xi$ by
considering a slow rotation approximation in which the second and higher-order
terms of rotation parameter ($a$) are ignored. Thus, eq. (8) reduces to the
following form.

$\displaystyle ds^{2}=-\left(1-\frac{2M}{\nu
r}\right)^{\nu}dt^{2}+\left(1-\frac{2M}{\nu r}\right)^{-\nu}dr^{2}$
$\displaystyle-\,2a\sin^{2}\theta\left[1-\left(1-\frac{2M}{\nu
r}\right)^{\nu}\right]dtd\phi$ $\displaystyle+\,r^{2}\left(1-\frac{2M}{\nu
r}\right)^{1-\nu}d\Omega^{2}$ (9)

And, the transformation functions $\chi$ and $\xi$ become

$\displaystyle\xi(r)=\left(1-\frac{2M}{\nu r}\right)^{-\nu}$
$\displaystyle\chi(r)=\frac{a}{r^{2}}\left(1-\frac{2M}{\nu r}\right)^{-1}$

Eq. (9) is the slow rotation form of the JNW spacetime metric. We note that
the rotating JNW spacetime metric (8) obtained using the NJA with
complexification is not properly transformed into the BLC because of the
complexification of the coordinates. This issue can be solved by skipping the
complexification step Mustapha3 ; polytropic , which is discussed in the next
section.

###  II.2 Newman-Janis Algorithm without Complexification

Consider a general form of the static, and spherically symmetric spacetime
metric.

$ds^{2}=-G(r)dt^{2}+\frac{dr^{2}}{F(r)}+H(r)d\Omega^{2}$ (10)

To apply the algorithm, transform it into the null coordinates
$\\{u,r,\theta,\phi\\}$ using the coordinate transformation

$du=dt-\frac{dr}{\sqrt{F(r)G(r)}}$ (11)

In null coordinates, the spacetime metric (10) takes the form

$ds^{2}=-G(r)du^{2}-2\sqrt{\frac{G(r)}{F(r)}}dudr+H(r)d\Omega^{2}$ (12)

The null tetrad of the above spacetime metric are

$\displaystyle l^{\mu}=\delta^{\mu}_{1}\,,$ (13) $\displaystyle
n^{\mu}=\sqrt{\frac{F(r)}{G(r)}}\delta^{\mu}_{0}-\frac{1}{2}F(r)\delta^{\mu}_{1}\,,$
(14) $\displaystyle
m^{\mu}=\frac{1}{\sqrt{2H(r)}}\left(\delta^{\mu}_{2}+\frac{i}{\sin\theta}\delta^{\mu}_{3}\right)$
(15)

The next step of the NJA is the complexification, which generalize the
function (i.e. $G(r),F(r),H(r)$) as the real function of radial coordinate $r$
and it’s complex conjugate $\bar{r}$. The trick is to skip this step and
proceed directly to the next step which is the complex coordinate
transformation.

$\displaystyle r^{\prime}=r+ia\cos\theta$ (16) $\displaystyle
u^{\prime}=u-ia\cos\theta$ (17)

Under the coordinate transformation, let us consider that the components of
the metric tensor (10) are transformed as

$\\{G(r),F(r),H(r)\\}\to\\{A(r,\theta,a),B(r,\theta,a),\psi(r,\theta,a)\\},$

where the following conditions are satisfied.

$\displaystyle\lim\limits_{a\to 0}A(r,\theta,a)=G(r)\,,$ (18)
$\displaystyle\lim\limits_{a\to 0}B(r,\theta,a)=F(r)\,,$ (19)
$\displaystyle\lim\limits_{a\to 0}\psi(r,\theta,a)=H(r).$ (20)

Now, perform the complex coordinate transformation (16), (17) on the null
tetrad (13), (14), (15).

$\displaystyle l^{\mu}=\delta^{\mu}_{1}\,,$ (21) $\displaystyle
n^{\mu}=\sqrt{\frac{B}{A}}\delta^{\mu}_{0}-\frac{1}{2}B\delta^{\mu}_{1}\,,$
(22) $\displaystyle
m^{\mu}=\frac{1}{\sqrt{2\psi}}\left(\delta^{\mu}_{2}+ia\sin\theta(\delta^{\mu}_{0}-\delta^{\mu}_{1})+\frac{i}{\sin\theta}\delta^{\mu}_{3}\right)$
(23)

The forth null vector $\bar{m}^{\mu}$ is just a complex conjugate of
$m^{\mu}$. These are the components of the null tetrad of the final metric
tensor in null coordinates $(u,r,\theta,\Phi)$.

$\displaystyle
ds^{2}=Adu^{2}+2\sqrt{\frac{A}{B}}dudr+2a\sin^{2}\theta\left(\sqrt{\frac{A}{B}}-A\right)dud\Phi$
$\displaystyle-\,2a\sin^{2}\theta\sqrt{\frac{A}{B}}drd\Phi-\psi d\theta^{2}$
$\displaystyle-\,\sin^{2}\theta\left[\psi+a^{2}\sin^{2}\theta\left(2\sqrt{\frac{A}{B}}-A\right)\right]d\Phi^{2}\,\,\,\,\,\,\,\,\,\,\,$
(24)

The spacetime metric is determined in the null coordinates
$\\{u,r,\theta,\Phi\\}$, which can be transformed into the BLC
$\\{t,r,\theta,\phi\\}$ as

$\displaystyle du=dt-\lambda(r)dr$ (25) $\displaystyle d\Phi=d\phi-\chi(r)dr$
(26)

The transformation functions $\chi$ and $\lambda$ should strictly depend on
the radial coordinate ($r$) only, for eq. (25), (26) to be integrable; which
is not possible in NJA with complexification. As we know, in Boyer-Lindquist
form, the spacetime metric contains only one off-diagonal term which is
$dtd\phi$. Therefore, using the above coordinate transformation (25, 26), and
dropping all off-diagonal terms except $dtd\phi$, we have the following two
expressions.

$\displaystyle
A\lambda(r)-\sqrt{\frac{A}{B}}+a\sin^{2}\theta\left(\sqrt{\frac{A}{B}}-A\right)\chi(r)=0$
(27) $\displaystyle
a\left(\sqrt{\frac{A}{B}}-A\right)\lambda(r)+a\sqrt{\frac{A}{B}}+\left[\psi-a^{2}\sin^{2}\theta\left(2\sqrt{\frac{A}{B}}-A\right)\chi(r)\right]=0$
(28)

Solving these expressions for $\lambda$ and $\chi$, we obtain

$\displaystyle\chi(r,\theta)=\frac{a}{\psi B+a^{2}\sin^{2}\theta}\,,$ (29)
$\displaystyle\lambda(r,\theta)=\frac{\psi\sqrt{\frac{B}{A}}+a^{2}\sin^{2}\theta}{\psi
B+a^{2}\sin^{2}\theta}$ (30)

Here, again, the BLC transformation functions $\chi$ and $\lambda$ depends on
$\theta$; but this time we have three unknown functions
$A(r,\theta,a),B(r,\theta,a)$ and $\psi(r,\theta,a)$ which play an important
role in dropping the $\theta$ dependency of $\chi$ and $\lambda$. We choose
the following form of the transformation functions, because it successfully
works in Kerr and Kerr-Newman case.

$\displaystyle\chi(r)=\frac{a}{FH+a^{2}}$ (31)
$\displaystyle\lambda(r)=\frac{K+a^{2}}{FH+a^{2}}$ (32)

Where, $K=H\sqrt{\frac{F}{G}}$.
Now, substitute the above expression of $\chi$ and $\lambda$ into the
equations ($\ref{dtdr},\ref{drdphi}$), and solve them for the unknown
functions $A(r,\theta,a)$ and $B(r,\theta,a)$. We obtain

$\displaystyle
A(r,\theta,a)=\frac{(FH+a^{2}\cos^{2}\theta)\psi}{(K+a^{2}\cos^{2}\theta)^{2}}$
(33) $\displaystyle B(r,\theta,a)=\frac{FH+a^{2}\cos^{2}\theta}{\psi}$ (34)

We can verify that the function $A(r,\theta,a)$ and $B(r,\theta,a)$
respectively reduce to the function $G(r)$ and $F(r)$ when $a\to 0$.

$\displaystyle\lim\limits_{a\to 0}A(r,\theta,a)=G(r)$ (35)
$\displaystyle\lim\limits_{a\to 0}B(r,\theta,a)=F(r)$ (36)

Therefore, under the BLC coordinate transformation, the metric (24) becomes

$\displaystyle
ds^{2}=-\frac{(FH+a^{2}\cos^{2}\theta)\psi}{(K+a^{2}\cos^{2}\theta)^{2}}dt^{2}+\frac{\psi}{FH+a^{2}}dr^{2}-2a\sin^{2}\theta\left(\frac{K-FH}{(K+a^{2}\cos^{2}\theta)^{2}}\right)\psi
dtd\phi+\psi d\theta^{2}$ $\displaystyle+\,\psi
sin^{2}\theta\left[1+a^{2}\sin^{2}\theta\left(\frac{2K-FH+a^{2}\cos^{2}\theta}{(K+a^{2}\cos^{2}\theta)^{2}}\right)\right]d\phi^{2}$
(37)

where, $\psi(r,\theta,a)$ is still unknown. It must satisfy the following
constraint which corresponds to the vanishing component of the Einstein
tensor, $G_{r\theta}=0$.

$3a^{2}\sin{2\theta}\psi^{2}K_{,r}+(K+a^{2}\cos^{2}\theta)^{2}(3\psi_{,\theta}\psi_{,r}-2\psi\psi_{,r\theta})=0$
(38)

This non-linear partial differential equation can be solved for $\psi$. One
can verify that a solution of the above partial differential equation is

$\psi(r,\theta,a)=K(r)+a^{2}\cos^{2}\theta$ (39)

For the Kerr and Kerr-Newman metric, $K(r)=r^{2}$, and the above expression
reduces to $\psi=r^{2}+a^{2}\cos^{2}\theta$. After substituting $\psi$ (39)
into the general form of the rotating spacetime metric (37), if the static
spacetime metric (10) can be obtained in the limit $a\to 0$, the rotating
solution is known as normal fluid; otherwise it is known as conformal fluid.
Normal fluid solution requires

$\lim\limits_{a\to 0}\psi(r,\theta,a)=H(r)$ (40)

which implies $F(r)=G(r)$.
The metric (37) can be written in Kerr-like form as

$\displaystyle
ds^{2}=-\left(1-\frac{2f}{\rho^{2}}\right)dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{\Sigma\sin^{2}\theta}{\rho^{2}}d\phi^{2}$
$\displaystyle-\,\frac{4af\sin^{2}\theta}{\rho^{2}}dtd\phi\,\,\,\,\,\,\,\,\,\,$
(41)

Where,

$\displaystyle\rho^{2}=K(r)+a^{2}\cos^{2}\theta\,,$ (42) $\displaystyle
f=\frac{K(r)-F(r)H(r)}{2}\,,$ (43) $\displaystyle\Delta=F(r)H(r)+a^{2}\,,$
(44)
$\displaystyle\Sigma=\big{(}K(r)+a^{2}\big{)}^{2}-a^{2}\Delta\sin^{2}\theta.$
(45)

When $a\to 0$, the rotating solution (95) reduces to the static, and
spherically symmetric spacetime metric if $F(r)=G(r)$.

###  II.3 Rotating JNW spacetime metric

We construct a new rotating spacetime metric which reduces to the static JNW
naked singularity spacetime metric (1) in the limit $a\to 0$. Substitute the
components of the JNW spacetime metric into eq. (95).

$\displaystyle
ds^{2}=-\left(1-\frac{2f}{\rho^{2}}\right)dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+\frac{\Sigma\sin^{2}\theta}{\rho^{2}}d\phi^{2}$
$\displaystyle-\frac{4af\sin^{2}\theta}{\rho^{2}}dtd\phi\,\,\,\,\,\,\,\,\,\,\,\,$
(46)

Where,

$\displaystyle
2f=r^{2}\left(1-\frac{2M}{r\nu}\right)\left[-1+\left(1-\frac{2M}{r\nu}\right)^{-\nu}\right]\,,$
(47)
$\displaystyle\rho^{2}=r^{2}\left(1-\frac{2M}{r\nu}\right)^{1-\nu}+a^{2}\cos^{2}\theta\,,$
(48) $\displaystyle\Delta=r^{2}-\frac{2Mr}{\nu}+a^{2}\,,$ (49)
$\displaystyle\Sigma=(\rho^{2}+a^{2}\sin^{2}\theta)^{2}-a^{2}\Delta\sin^{2}\theta.$
(50)

One can verify that it reduces to the Kerr metric when $\nu=1$. Also it
reduces to eq. (9) under the slow rotation approximation. To check whether the
spacetime metric is physically valid, we must proceed to check the energy
conditions. Consider the inverse form of the rotating spacetime metric (46).

$\displaystyle(\partial_{s})^{2}=-\frac{(K+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta}{\rho^{2}\Delta}(\partial_{t})^{2}+\frac{\Delta}{\rho^{2}}(\partial_{r})^{2}$
$\displaystyle+\,\frac{1}{\rho^{2}}(\partial_{\theta})^{2}+\frac{\Delta-a^{2}\sin^{2}\theta}{\rho^{2}\Delta\sin^{2}\theta}(\partial_{\phi})^{2}-\frac{4aMK}{r\rho^{2}\Delta}\partial_{t}\partial_{\phi}\,\,\,$
(51)

It can also be written as

$\displaystyle(\partial_{s})^{2}=-\left(\frac{K+a^{2}}{\rho\sqrt{\Delta}}\partial_{t}+\frac{a}{\rho\sqrt{\Delta}}\partial_{\phi}\right)^{2}+\frac{\Delta}{\rho^{2}}(\partial_{r})^{2}$
$\displaystyle+\,\frac{1}{\rho^{2}}(\partial_{\theta})^{2}+\left(\frac{a\sin\theta}{\rho}\partial_{t}+\frac{1}{\rho\sin\theta}\partial_{\phi}\right)^{2}\,\,\,$
(52)

Now, we choose a set of orthonormal basis
$\\{u^{\mu},e_{r}^{\mu},e_{\theta}^{\mu},e_{\phi}^{\mu}\\}$, where $u^{\mu}$
can be considered as a four velocity of the fluid.

$\displaystyle
u^{\mu}=\left(\frac{K+a^{2}}{\rho\sqrt{\Delta}},0,0,\frac{a}{\rho\sqrt{\Delta}}\right)\,,$
(53) $\displaystyle
e_{r}^{\mu}=\left(0,\frac{\sqrt{\Delta}}{\rho},0,0\right)\,,$ (54)
$\displaystyle e_{\theta}^{\mu}=\left(0,0,\frac{1}{\rho},0\right)\,,$ (55)
$\displaystyle
e_{\phi}^{\mu}=\left(\frac{a\sin\theta}{\rho},0,0,\frac{1}{\rho\sin\theta}\right).$
(56)

One can verify that $u^{\mu}u_{\mu}=-1$, $e_{i}^{\mu}(e_{i})_{\mu}=1$,
$u^{\mu}(e_{i})_{\mu}=0$; where, $(i\to r,\theta,\phi)$. In terms of these
orthonormal basis, eq. (52) is written as

$(\partial_{s})^{2}=(-u^{\mu}u^{\nu}+e_{r}^{\mu}e_{r}^{\nu}+e_{\theta}^{\mu}e_{\theta}^{\nu}+e_{\phi}^{\mu}e_{\phi}^{\nu})\partial_{\mu}\partial_{\nu}$
(57)

where the metric tensor is

$g^{\mu\nu}=-u^{\mu}u^{\nu}+e_{r}^{\mu}e_{r}^{\nu}+e_{\theta}^{\mu}e_{\theta}^{\nu}+e_{\phi}^{\mu}e_{\phi}^{\nu}$
(58)

Similarly, the energy-momentum tensor can be decomposed as

$T^{\mu\nu}=\rho_{e}u^{\mu}u^{\nu}+P_{r}e_{r}^{\mu}e_{r}^{\nu}+P_{\theta}e_{\theta}^{\mu}e_{\theta}^{\nu}+P_{\phi}e_{\phi}^{\mu}e_{\phi}^{\nu}$
(59)

where, $\rho_{e}$ is the energy density, and $P_{i}$ $(i\to r,\theta,\phi)$
are the principal pressure components. The Einstein field equation in the form
$G_{\mu\nu}=T_{\mu\nu}$ requires

$\displaystyle\rho_{e}=u^{\mu}u^{\nu}G_{\mu\nu}\,,$ (60) $\displaystyle
P_{r}=e_{r}^{\mu}e_{r}^{\nu}G_{\mu\nu}=g^{rr}G_{rr}\,,$ (61) $\displaystyle
P_{\theta}=e_{\theta}^{\mu}e_{\theta}^{\nu}G_{\mu\nu}=g^{\theta\theta}G_{\theta\theta}\,,$
(62) $\displaystyle P_{\phi}=e_{\phi}^{\mu}e_{\phi}^{\nu}G_{\mu\nu}.$ (63)

The components of the energy-momentum tensor of the rotating JNW spacetime are
as follows. We consider $\theta=\pi/2$ for the mathematical simplicity.

$\displaystyle\rho_{e}=\frac{M^{2}(1-\nu^{2})(\Delta+a^{2})}{r^{6}\nu^{2}}\left(1-\frac{2M}{r\nu}\right)^{-3+\nu}$
(64) $\displaystyle
P_{r}=-P_{\theta}=\frac{M^{2}(1-\nu^{2})}{r^{4}\nu^{2}}\left(1-\frac{2M}{r\nu}\right)^{-2+\nu}$
(65) $\displaystyle
P_{\phi}=P_{\theta}+\frac{2a^{2}}{r^{4}}\left(1-\frac{2M}{r\nu}\right)^{-3+\nu}\bigg{[}\left(1-\frac{2M}{r\nu}\right)^{1+\nu}$
$\displaystyle-\left(1-\frac{M(1+\nu)}{r\nu}\right)^{2}\bigg{]}\,$ (66)

It can be verified that the rotating JNW spacetime metric (46) satisfies the
weak energy condition ($\rho_{e}\geq 0$, $\rho_{e}+P_{i}\geq 0$); null energy
condition ($\rho_{e}+P_{i}\geq 0$); strong energy condition ($\rho_{e}+\sum
P_{i}\geq 0$, $\rho_{e}+P_{i}\geq 0$); and the dominant energy condition
($\rho_{e}>|P_{i}|$). We also note that when $\nu=1$, all the components eq.
(64), (107), (66) vanish, which is obvious as we have already seen that the
rotating JNW metric reduces to the Kerr metric when $\nu=1$. Also, $P_{r}\neq
P_{\theta}\neq P_{\phi}$, therefore, it is an an-isotropic fluid solution.

(a) Shadow of rotating JNW for $q=0.4$ and $a=0.3$.

(b) Shadow of rotating JNW for $q=0.4$ and $a=0.4$.

(c) Shadow of rotating JNW for $q=0.4$ and $a=0.5$.

(d) Shadow of rotating JNW for $q=0.6$ and $a=0.3$.

(e) Shadow of rotating JNW for $q=0.6$ and $a=0.4$.

(f) Shadow of rotating JNW for $q=0.6$ and $a=0.5$.

(g) Shadow of rotating JNW for $q=0.8$ and $a=0.3$.

(h) Shadow of rotating JNW for $q=0.8$ and $a=0.4$.

(i) Shadow of rotating JNW for $q=0.8$ and $a=0.5$.

Figure 1: Figure shows the shape of the shadow cast by the rotating JNW naked
singularity for various values of scalar field charge $q$ and spin parameter
$a$. Here, we consider $M=1$ and $\theta=\pi/2$ for the shape of shadows in
celestial coordinates. The details of this figure are discussed in the text.

##  III Shadows in rotating JNW spacetime

In this section, we review the general formalism of the null geodesic using
the Hamilton-Jacobi separation method and obtained the general expressions for
finding the shape and nature of a shadow. In general relativity, the Hamilton-
Jacobi equation is given by,

$\frac{\partial
S}{\partial\lambda}+\mathcal{H}=0,\;\mathcal{H}=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu},$
(67)

where $\lambda$ is an affine parameter, S is the Jacobi action, $\mathcal{H}$
is the Hamiltonian, and $p_{\mu}$ is the momentum defined by,

$p_{\mu}=\frac{\partial S}{\partial
x^{\mu}}=g_{\mu\nu}\frac{dx^{\nu}}{d\lambda}.$ (68)

In the above equation (67), the Hamiltonian does not depend explicitly on $t$
and $\phi$ coordinates. Therefore we have two constants of motion, the
conserved energy $E=-p_{t}$ and the conserved angular momentum $L=p_{\phi}$
(about the axis of symmetry). Since we have the separable solution of the
above differential equation (67), the Jacobi action can be written in terms of
already known constants of the motion as,

$S=\frac{1}{2}\mu^{2}\lambda-Et+L\phi+S_{r}(r)+S_{\theta}(\theta),$ (69)

where $\mu$ is the rest mass of the test particle. Therefore, for a photon or
null geodesic, we consider $\mu=0$. The metric tensor components of general
rotating spacetime (37) are given as,

$g^{tt}=-\frac{H-a^{2}G\sin^{2}\theta+2a^{2}\sqrt{\frac{G}{F}}\sin^{2}\theta}{P},$
(70) $g^{rr}=\frac{FH+a^{2}\sin^{2}\theta}{H},$ (71)
$g^{\theta\theta}=\frac{1}{H},$ (72)
$g^{\phi\phi}=-\frac{G\csc^{4}\theta}{2a^{2}G^{2}-4a^{2}G\sqrt{\frac{G}{F}}+a^{2}\left(\frac{G}{F}\right)-GH\csc^{2}\theta},$
(73) $g^{t\phi}=g^{\phi t}=\frac{a\left(\sqrt{\frac{G}{F}}-G\right)}{P},$ (74)

where,

$P=-GH+2a^{2}G^{2}\sin^{2}\theta-4a^{2}G\sqrt{\frac{G}{F}}\sin^{2}\theta+a^{2}\left(\frac{G}{F}\right)\sin^{2}\theta$
(75)

Substituting eq. (69) into eq. (67), and using the components of the metric
tensor, we obtain the following expression:

$-\left(FH+a^{2}\sin^{2}\theta\right)\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{\left(\left(\sqrt{\frac{F}{G}}H+a^{2}\sin^{2}\theta\right)E-aL\right)^{2}}{FH+a^{2}\sin^{2}\theta}-\left(L-aE\right)^{2}=\left(\frac{dS_{\theta}}{d\theta}\right)^{2}+L^{2}\cot^{2}\theta-a^{2}E^{2}\cos^{2}\theta,$
(76)

Note that, left and right hand side of above eq. (76) are only functions of
$r$ and $\theta$ respectively, therefore one can write each side equal to a
separation constant since Jacobi action principle is separable solution,

$-\left(FH+a^{2}\sin^{2}\theta\right)\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{\left(\left(\sqrt{\frac{F}{G}}H+a^{2}\sin^{2}\theta\right)E-aL\right)^{2}}{FH+a^{2}\sin^{2}\theta}-\left(L-aE\right)^{2}=\mathcal{K},$
(77)

$\left(\frac{dS_{\theta}}{d\theta}\right)^{2}+L^{2}\cot^{2}\theta-a^{2}E^{2}\cos^{2}\theta=\mathcal{K},$
(78)

where the separation constant $\mathcal{K}$ is known as the Carter constant.
Using above differential equations one can write down the following separated
null geodesic equations for radial ($r$) and tangential ($\theta$) part as,

$H^{2}\left(\frac{dr}{d\lambda}\right)^{2}-R(r)=0\,\,,$ (79)
$H^{2}\left(\frac{d\theta}{d\lambda}\right)^{2}-\Theta(\theta)=0\,\,,$ (80)

where, $R(r)$ and $\Theta(\theta)$ can be considered as radial and tangential
part of effective potentials respectively, which can be defines as,

$R(r)=\left[X(r)-a\xi\right]^{2}-\Delta(r)\left[\eta+(\xi-a)^{2}\right]\,\,,$
(81)
$\Theta(\theta)=\eta+(\xi-a)^{2}-\left(\frac{\xi}{\sin\theta}-a\sin\theta\right)^{2}\,\,$
(82)

where, $\xi=\frac{L}{E}$ and $\eta=\frac{\mathcal{K}}{E^{2}}$. The functions
$X(r)$ and $\Delta(r)$ can be define as,

$X(r)=\sqrt{\frac{F}{G}}H+a^{2}\sin^{2}\theta,$ (83)
$\Delta(r)=FH+a^{2}\sin^{2}\theta.$ (84)

Note that, $R(r)$ and $\Theta(\theta)$ must be non-negative for photon motion.
In a general rotating spacetime, unstable circular orbits exist when the
following conditions hold,

$R(r_{ph})=0,\;\frac{dR(r_{ph})}{dr}=0,\;\frac{d^{2}R(r_{ph})}{dr^{2}}\leq
0\,\,.$ (85)

where, $r=r_{ph}$ is the radius of the unstable photon orbit. Using the above
conditions (85), we obtain the critical impact parameters corresponding to the
maxima of the $R(r)$ as,

$\xi=\frac{X_{ph}\Delta^{\prime}_{ph}-2\Delta_{ph}X^{\prime}_{ph}}{a\Delta^{\prime}_{ph}},$
(86)
$\eta=\frac{4a^{2}X^{\prime}_{ph}\Delta_{ph}-\left[(X_{ph}-a^{2})\Delta^{\prime}_{ph}-2X^{\prime}_{ph}\Delta_{ph}\right]^{2}}{a^{2}\Delta^{\prime
2}_{ph}},$ (87)

where prime denoted for derivative concerning $r$ and subscript “ph” denoted
for the quantities which are evaluated at $r=r_{ph}$. The above expressions
are the general forms of the critical impact parameters $\xi$ and $\eta$ for
the unstable photon orbits, which would be important to find out the shape of
the shadows. However, one can obtain the apparent shape of the shadow by using
the celestial coordinates $\alpha$ and $\beta$ which lie in the line of sight
of the observer direction. The expressions of the celestial coordinates
$\alpha$ and $\beta$ can be written in the following way,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\lim_{r_{0}\to\infty}\left(-r_{0}^{2}\sin{\theta_{0}}\frac{d\phi}{dr}\right)\,\,,$
(88) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\lim_{r_{0}\to\infty}\left(r_{0}^{2}\frac{d\theta}{dr}\right)\,\,,$
(89)

where $r_{0}$ and $\theta_{0}$ are the coordinates of the asymptotic observer.
Note that, if we consider the general rotating metric to be an asymptotically
flat in the limit of $r\to\infty$, then we can consider $G\to 1$, $F\to 1$,
$H\to r^{2}$, $X\to r^{2}$ and $\Delta\to r^{2}$. Therefore, the above
expressions of the celestial coordinates $\alpha$ and $\beta$ becomes,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\frac{\xi}{\sin{\theta_{0}}}\,\,,$ (90) $\displaystyle\beta$
$\displaystyle=$
$\displaystyle\pm\sqrt{\eta+a^{2}\cos^{2}{\theta_{0}}-\xi^{2}\cot^{2}{\theta_{0}}}\,\,,$
(91)

from the above expressions (86) and (87), one can obtain the celestial
coordinates $\alpha$ and $\beta$ for the rotating JNW spacetime by using its
metric tensor components. Note that, we used JNW metric tensor components
without adopting any complexification method since the general rotating metric
we obtained by skipping the complexification method. The metric tensor
components of the JNW spacetime are given as,

$G=F=\left(1-\frac{2M}{r\nu}\right)^{\nu}\,\,;\
H=r^{2}\left(1-\frac{2M}{r\nu}\right)^{1-\nu}$ (92)

using above equations we can find that the expressions for $X(r)$ and
$\Delta(r)$,

$X(r)=r^{2}\left(1-\frac{2M}{r\nu}\right)^{1-\nu}+a^{2}\sin^{2}\theta,$ (93)
$\Delta(r)=r^{2}\left(1-\frac{2M}{r\nu}\right)+a^{2}\sin^{2}\theta.$ (94)

therefore, using the above expressions, one can calculate the critical impact
parameters and celestial coordinates for the rotating JNW spacetime. We have
constructed the shape of the shadows using the expressions of celestial
coordinates for rotating JNW spacetime, where we consider the inclination
angle of the observer $\theta_{0}=\pi/2$.

(a) Timelike bound orbit in rotating JNW metric for $q=0.4$ and $a=0.3$.

(b) Timelike bound orbit in rotating JNW metric for $q=0.4$ and $a=0.5$.

(c) Timelike bound orbit in rotating JNW metric for $q=12$ and $a=0.3$.

(d) Timelike bound orbit in rotating JNW metric for $q=12$ and $a=0.5$.

Figure 2: The figure shows the timelike bound orbits in the rotating JNW naked
singularity spacetime for scalar field charge $q=0.4,12$ and spin parameter
$a=0.3,0.5$ respectively. Here, we consider mass (ADM mass for JNW) $M=1$,
conserved angular momentum $L=6$ and total energy $E=-0.006$. Note that we
consider the timelike bound orbits in the equatorial plane (i.e.
$\theta=\pi/2$). The black dotted lines represent the bound orbits in the Kerr
spacetime, and the solid green lines represent the bound orbits in the JNW
spacetime. The corresponding minimum approach of a test particle towards the
center is shown by the blue (in the Kerr metric) and the red (in the JNW
metric) circles. The details of this figure are discussed in the text.

In fig.(1), we have shown that the shape of the shadows for different values
of spin parameter $a$ and scalar field charge $q$, where the blue line
indicate the shadows in rotating JNW spacetime and the red dotted line
indicate the shadows in the Kerr spacetime. It is well known that the shadow
shapes in the Kerr black hole spacetime always be in contour shape, and as we
increase the spin parameter, it will become more and more prolate. One can see
in fig.(1), the horizontal radius of the red dotted line will decreases from
the left side as we increase the value of a spin parameter. On the other hand,
we found that the shadows in rotating JNW spacetime changes from contour shape
to arc shape as we increase the values of spin parameter and scalar field
charge or one of them.

##  IV Timelike geodesics in rotating JNW spacetime

In this section, we derive the effective potential and the orbit equation for
a test particle in rotating JNW spacetime. Solving the orbit equation
numerically, we determine the nature of the orbit precession. We proceed by
considering a general form of the spacetime metric, i.e.,

$ds^{2}=-g_{tt}dt^{2}+g_{rr}dr^{2}-2g_{t\phi}dtd\phi+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$
(95)

which is written in the Boyer-Lindquist Coordinates (BLC). For the above
spacetime metric, the effective potential takes the form

$V_{eff}=\frac{1}{2g_{rr}}+\frac{e^{2}-1}{2}+\frac{L^{2}g_{tt}+2eLg_{t\phi}-e^{2}g_{\phi\phi}}{2g_{rr}(g_{t\phi}^{2}+g_{tt}g_{\phi\phi})}$
(96)

where, we have considered a case of equatorial orbits for the mathematical
simplicity. Substituting the metric components and taking
$\theta=\frac{\pi}{2}$ into eq. (96), we obtain the effective potential for
the orbits in equatorial plane of rotating JNW spacetime.

$\displaystyle
V_{eff}=\frac{(L-ae)^{2}}{2r^{2}}\left(1-\frac{2M}{r\nu}\right)^{2\nu-1}-\frac{1}{2}$
$\displaystyle+\,\frac{\Delta+2ae\left(L-ae\right)}{2r^{2}}\left(1-\frac{2M}{r\nu}\right)^{\nu-1}$
(97)

Now, we proceed to derive an orbit equation. For the general rotating
spacetime metric (95), we obtain parth6

$\displaystyle\left(\frac{dr}{d\phi}\right)^{2}=-\frac{(g_{t\phi})^{2}+g_{tt}g_{\phi\phi}}{g_{rr}(eg_{t\phi}+Lg_{tt})^{2}}[(g_{t\phi})^{2}$
$\displaystyle-$ $\displaystyle e^{2}g_{\phi\phi}+2eLg_{t\phi}$
$\displaystyle+$ $\displaystyle g_{tt}$
$\displaystyle(L^{2}+g_{\phi\phi})]\,\,\,$ (98)

where, $\theta=\frac{\pi}{2}$. Now, using the differential relation
$\frac{d^{2}r}{d\phi^{2}}=\frac{1}{2\frac{dr}{d\phi}}\frac{d}{d\phi}\left(\frac{dr}{d\phi}\right)^{2}.$,
we can determine the orbit equation as,

$\frac{d^{2}u(\phi)}{d\phi^{2}}=-\frac{1}{r^{2}}\frac{d^{2}r}{d\phi^{2}}+\frac{2}{r^{3}}\left(\frac{dr}{d\phi}\right)^{2}$
(99)

where, $u=\frac{1}{r}$. The above eq. (99) is the non-linear second order
differential equation, thus it is very difficult to find the complete analytic
solution of this equation. Hence, we can solve this orbit equation numerically
for the rotating JNW metric.

In figure (2), we show the timelike bound orbits in rotating JNW spacetime and
compare them with the orbits in Kerr spacetime. Note that, for simplicity, we
define the particle motion in the equatorial plane (i.e. $\theta=\pi/2$). In
fig. (2(a)),(2(b)), (2(c)) and (2(d)), we show the bound orbits for scalar
field charge $q=0.4,12$ corresponding to the spin parameter $a=0.3,0.5$. The
green and black dotted lines represent the orbits of the particles in the
rotating JNW and Kerr spacetimes respectively. The red and blue circles denote
the periastron points (minimum approach towards the center) for rotating JNW
and Kerr spacetime respectively. The parameters we have consider the mass (ADM
mass for JNW) $M=1$, conserved angular momentum $L=6$, total energy
$E=-0.006$.

One can see from the figure (2), the timelike bound orbital motion in the JNW
spacetime is significantly different from the orbital motion in Kerr
spacetime. We find that the timelike bound orbits can have negative (or
opposite) precession in rotating JNW spacetime. In other words, if the angular
distance traveled by a particle between two successive periastron points is
less than $2\pi$, then the next orbit would shift in the opposite direction of
the particle’s orbiting direction. Hence, it is known as negative (or
opposite) precession. On the other hand, in Bam2020 , it is shown that the
negative precession is not present in the Kerr spacetime. The Kerr spacetime
exhibits positive precession since the angular distance traveled by a particle
between two successive periastron points is always greater than $2\pi$.

The spherically symmetric and static JNW spacetime can cast the shadow due to
the existence of the photon sphere Dey:2020haf . The photon sphere exist for
the range of parameter $0.5\leq\nu<1$ and absent for the range of $0<\nu<0.5$.
On the other hand, the negative precession of timelike bound orbits is present
Bambhaniya:2019pbr  only within the range $0<\nu<0.1$. Therefore, in
Dey:2020haf , it is shown that the negative precession of the timelike bound
orbits and shadow formation can not occur simultaneously. Similarly in the
rotating JNW spacetime, the parametric range slightly differs as we increase
the spin parameter $a$.

##  V Discussions and conclusions

The conclusions of this paper are in the following way.

  * •

We construct the rotating JNW spacetime using the NJA without employing the
complexification of coordinates. We have shown that the rotating JNW spacetime
obtained using the NJA with considering the complexification step cannot be
written into the BLC. If we anyhow write that metric into the BLC-like form by
performing an improper coordinate transformation, we find that it does not
satisfy the energy conditions. To resolve this problem, we drop the
complexification step in the NJA so that the rotating solution can properly be
transformed into the BLC. As a result, we obtain the rotating JNW spacetime
into the BLC, which does satisfy the energy conditions.

  * •

As we know, the rotating Kerr black hole can cast prolate contour shape shadow
(see dotted red line in fig.(1)). While we increase the spin parameter up to
$M\geq a$, it will become more and more prolate contour, where $M$ is the mass
of the black hole. But, as we reach the $M<a$ limit, the Kerr black hole
becomes the Kerr naked singularity and it would not cast any shadow. To get
more insight, in this paper, we obtain the shape of the shadows cast by
rotating JNW naked singularity and compare them with the shape of shadows cast
by the Kerr black hole. We find that the shape of the shadows in rotating JNW
spacetime becomes prolate arc shape from the prolate contour shape as we
increase the scalar field charge $q$ and/or spin parameter $a$ (see blue line
in fig. (1)).

  * •

In Bambhaniya:2019pbr ; Bam2020 , we have shown that the Schwarzschild and
Kerr black holes admit only positive orbital precession. In other words,
timelike bound orbits only precess in the direction of particle’s motion.
Here, we have examined the precession of timelike bound orbits in rotating JNW
spacetime. We find that the rotating JNW spacetime can have positive as well
as negative (or opposite) precession of the timelike bound orbits.

In the context of the observations by the EHT, GRAVITY, SINFONI, and UCLA
galactic center groups, these studies on shadows and precession of timelike
bound orbits are very useful. These novel features of the shadow shape
(prolate arc) and precession of timelike bound orbits (negative precession) in
rotating JNW spacetime could be observationally significant to differentiate
the naked singularities from the black holes.

##  VI Appendix

We discuss about the energy conditions of the metric (8) obtained using the
original NJA. Consider the inverse form of the spacetime metric.

$\displaystyle\partial_{s}^{2}=-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{-\nu}\left\\{1-\frac{a^{2}\sin^{2}\theta}{\Delta}\left[1-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]^{2}\right\\}\partial_{t}^{2}+\frac{1}{\rho^{2}}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu-1}\big{\\{}\Delta\partial_{r}^{2}+\partial_{\theta}^{2}\big{\\}}$
$\displaystyle-\,\frac{2a}{\Delta}\left[1-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\right]\partial_{t}\partial_{\phi}+\frac{1}{\Delta\sin^{2}\theta}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\partial_{\phi}^{2}$
(100)

It can be written as

$\displaystyle\partial_{s}^{2}=-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{-\nu}\partial_{t}^{2}+\frac{1}{\rho^{2}}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu-1}\big{[}\Delta\partial_{r}^{2}+\partial_{\theta}^{2}\big{]}+\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu}\bigg{[}\frac{a\sin\theta}{\sqrt{\Delta}}\left\\{1-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{-\nu}\right\\}\partial_{t}$
$\displaystyle+\,\frac{1}{\sqrt{\Delta}\sin\theta}\partial_{\phi}\bigg{]}^{2}\,\,\,$
(101)

Now, we choose a set of orthonormal basis
$\\{u^{\mu},e_{r}^{\mu},e_{\theta}^{\mu},e_{\phi}^{\mu}\\}$, which satisfies
$u^{\mu}u_{\mu}=-1$, $e_{i}^{\mu}(e_{i})_{\mu}=1$, $u^{\mu}(e_{i})_{\mu}=0$;
where, $(i\to r,\theta,\phi)$.

$\displaystyle
u^{\mu}=\left(\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{-\nu/2},0,0,0\right)$
(102) $\displaystyle
e_{r}^{\mu}=\left(0,\frac{\sqrt{\Delta}}{\rho}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\frac{\nu-1}{2}},0,0\right)$
(103) $\displaystyle
e_{\theta}^{\mu}=\left(0,0,\frac{1}{\rho}\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\frac{\nu-1}{2}},0\right)$
(104) $\displaystyle
e_{\phi}^{\mu}=\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{\nu/2}\left(\frac{a\sin\theta}{\sqrt{\Delta}}\left\\{1-\left(1-\frac{2Mr}{\nu\rho^{2}}\right)^{-\nu}\right\\},0,0,\frac{1}{\sqrt{\Delta}\sin\theta}\right)$
(105)

Using eq. (60), (61), (62), (63), we determine the components of the energy
momentum tensor. For mathematical simplicity we take $\theta=\pi/2$.

$\displaystyle\rho_{e}=P_{r}+\frac{2a^{2}M^{2}(1-\nu)^{2}}{r^{6}\nu^{2}}\left(1-\frac{2M}{r\nu}\right)^{-3+\nu}$
(106) $\displaystyle
P_{r}=-P_{\theta}=\frac{M^{2}(1-\nu^{2})}{r^{4}\nu^{2}}\left(1-\frac{2M}{r\nu}\right)^{-2+\nu}$
(107) $\displaystyle
P_{\phi}=P_{\theta}-\frac{2a^{2}M^{2}(1-\nu^{2})}{r^{6}\nu^{2}}\left(1-\frac{2M}{r\nu}\right)^{-3+\nu}$
(108)

We see that

$\rho_{e}+P_{r}+P_{\theta}+P_{\phi}=\rho_{e}+P_{\phi}=-\frac{4a^{2}M^{2}(1-\nu)}{r^{6}\nu}\left(1-\frac{2M}{r\nu}\right)^{-3+\nu}<0$
(109)

which implies that the spacetime metric (8) does not obey the weak energy
condition, null energy condition, and the strong energy condition. We note
that the dominant energy condition is also violated. Thus, the rotating
spacetime metric obtained using the Newman-Janis algorithm with
complexification is not physically a valid solution of the Einstein field
equation.

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[table]capposition=top

# Monaural Singing Voice Separation with Skip-Filtering Connections and
Recurrent Inference of Time-Frequency Mask

Stylianos Ioannis Mimilakis∗, Konstantinos Drossos†, João F. Santos§‡,
Gerald Schuller∗, Tuomas Virtanen†, Yoshua Bengio§¶

∗Fraunhofer IDMT, Ilmenau, Germany
†Tampere University of Technology, Tampere, Finland
§Université de Montréal, Montreal, Canada
‡INRS-EMT, Montreal, Canada
¶CIFAR Fellow

## I. Introduction

The problem of music source separation has received a lot attention in the
fields of audio signal processing and deep learning [1]. The most adopted
solution is the estimation of a time-varying and source-dependent filter,
which is applied to the mixture [2]. Performing the filtering operation is
done by treating audio signals as wide-sense stationary. This involves
transforming the mixture signal using the short-time Fourier transform (STFT).
Then, the source-dependent filtering operation is applied to the complex-
valued coefficients of the mixture signal. More formally, let $\mathbf{x}$ be
the time-domain mixture signal vector of $J$ sources.
${\mathbf{Y}}\in\mathbb{C}^{M\times N}$ is the complex-valued STFT
representation of ${\mathbf{x}}$, comprising of $M$ overlapping time frames
and $N$ frequency sub-bands. The estimation of the $j$-th target source
(${\mathbf{\hat{Y}}^{j}}\in\mathbb{C}^{M\times N}$) is achieved through:

${\mathbf{\hat{Y}}^{j}}={\mathbf{Y}}\odot{\mathbf{M}^{j}}\text{,}$ (1)

where $\odot$ is element-wise product and ${\mathbf{M}^{j}}\in\mathbb{R}_{\geq
0}^{M\times N}$ is the $j$-th source-dependent filter, henceforth denoted as
mask. In [2] was shown that a preferred way for estimating the $j$-th source
is to derive the mask through the generalized Wiener filtering using
$\alpha$-power magnitude spectrograms as:

${\mathbf{M}^{j}}=\frac{|{\mathbf{\hat{Y}}^{j}}|^{\circ\alpha}}{\underset{j}{\sum}|{\mathbf{\hat{Y}}^{j}}|^{\circ\alpha}}\text{,}$
(2)

where, $|\cdot|$ and $\circ$ denote the entry-wise absolute and exponentiation
operators respectively, and $\alpha$ is an exponent chosen based on the
assumed distributions that the sources follow. Finding $\alpha$ (and thus an
optimal ${\mathbf{M}^{j}}$ for the source estimation process [2]) is an open
optimization problem [2, 3].

Deep learning methods for music source separation are trained using
synthetically created mixtures ${\mathbf{Y}}$ (adding signals
${\mathbf{Y}}^{j}$ together, i.e., knowing the target decomposition). They can
be divided into two categories. In the first category, the methods try to
predict the mask directly from the mixture magnitude spectrum [4] (i.e.
$f_{1}:|{\mathbf{Y}}|\rightarrow{\mathbf{M}^{j}}$). This requires that an
optimal ${\mathbf{M}^{j}}$ is given (e.g. all the non-linear mixing parameters
of the target source are known) during training as a target. However, such
information for the ${\mathbf{M}^{j}}$ is unknown, and an approximation of
${\mathbf{M}^{j}}$ is computed from the training data using Eq. (2) and
empirically chosen $\alpha$ values, under the hypothesis that the source
magnitude spectra are additive, which is not true for realistic audio signals
[2, 3]. This implies that such models are optimized to predict non-optimal
masks. The methods in the second category try to estimate all sources from the
mixture(i.e.
$f_{2}:|{\mathbf{Y}}|\rightarrow|{\mathbf{\hat{Y}}^{j}}|^{\circ\alpha}\;\forall\,j\in
J$) and then use these estimates to compute a mask [5, 6, 7, 8, 9]. This
approach is widely adopted, since it is straightforward by employing denoising
autoencoders [10], with noise corresponding to the addition of other sources.
However, the masks are dependent on the initial $\alpha$-power magnitude
estimates of the sources ($|{\mathbf{\hat{Y}}^{j}}|^{\circ\alpha}$), and the
mask computation is not a learned function. Instead, the mask computation uses
a deterministic function which takes as inputs the outcomes
($|{\mathbf{\hat{Y}}^{j}}|^{\circ\alpha}\forall j\in J$) of deep neural
networks, e.g. as in [8].

An exception to the above are the works presented in [11] and [12], where
these methods jointly learned and optimized the masking processes described by
Eq. (1) and (2). In [11], highway networks [13] were shown to be able to
approximate a masking process for monaural solo source separation and in [12],
a more robust alternative to [11] is presented. The approach in [12] uses a
recurrent encoder-decoder with skip-filtering connections, which allow a
source-dependent mask generation process, applicable to monaural singing voice
separation. However, the generated masks are not robust against interferences
from other music sources, thus requiring a post-processing step using the
generalized Wiener filtering [12].

In this work we present a method for source separation that learns to generate
a source-dependent mask which does not require the generalized Wiener
filtering as a post-processing step. To do so, we introduce a novel recurrent
inference algorithm inspired by [14] and a sparsifying transform [15] for
generating the mask ${\mathbf{M}^{j}}$. The recurrent inference allows the
proposed method to have a stochastic depth of RNNs during the mask generation
process, computing hidden, latent representations which are presumably better
for generating the mask. The sparsifying transform is used to approximate the
mask using the output of the recurrent inference. In this method the mask
prediction is not based on the above mentioned assumptions about the
additivity of the magnitude spectrogram of the sources, is part of an
optimization process, and is not based on a deterministic function.
Additionally, the method incorporates RNNs instead of feed-forward or
convolutional layers for the mask prediction. This allows the method to
exploit the memory of the RNNs (compared to CNNs) and their efficiency for
modeling longer time dependencies of the input data. The rest of the paper is
organized as follows: Section II presents the proposed method, followed by
Section III which provides information about the followed experimental
procedure. Section IV presents the obtained results from the experimental
procedure and Section V concludes this work.

## II. Proposed Method

Our proposed method takes as an input the time domain samples of the mixture,
and outputs time domain samples of the targeted source. The model consists of
four parts. The first part implements the analysis and pre-processing of the
input. The second part generates and applies a mask, thus creating the first
estimate of the magnitude spectrogram of the targeted source. The third part
enhances this estimate by learning and applying a denoising filter, and the
fourth part constructs the time domain samples of the target source. We call
the second part the “Masker” and the third the “Denoiser”. We differentiate
between the Masker and the Denoiser because the Masker is optimized to predict
a time-frequency mask, whereas the Denoiser enhances the result obtained by
time-frequency masking. We implement the Masker using a single layer bi-
directional RNN encoder ($\text{RNN}_{\text{enc}}$), a single layer RNN
decoder ($\text{RNN}_{\text{dec}}$), a feed-forward layer (FFN), and skip-
filtering connections between the magnitude spectrogram of the mixture and the
output of the FFN. We implement the Denoiser using one FFN encoder
($\text{FFN}_{\text{enc}}$), one FFN decoder ($\text{FFN}_{\text{dec}}$), and
skip-filtering connections between the input to the Denoiser and the output of
the $\text{FFN}_{\text{dec}}$. We jointly train the Masker and the Denoiser
using two criteria based on the generalized Kullback-Leibler divergence
($D_{KL}$), as it is shown in [3, 16] to be a robust criterion for matching
magnitude spectrograms. All RNNs are gated recurrent units (GRU). The proposed
method is illustrated in Figure 1.

### i. Input preprocessing

Let ${\mathbf{x}}$ be the vector containing the time-domain samples of a
monaural mixture from $J$ sources, sampled at $44.1$kHz. We compute the STFT
of ${\mathbf{x}}$ from time frames of $N=2049$ samples, segmented with Hamming
window and a hop size of 384 samples. Each time frame is zero-padded to
$N^{\prime}=4096$. Subsequent to the STFT we retain only the positive
frequencies, corresponding to the first $N=2049$ frequency sub-bands. This
yields the complex-valued time-frequency representation of ${\mathbf{x}}$,
${\mathbf{Y}}\in\mathbb{C}^{M\times N}$, and the corresponding magnitude
$|{\mathbf{Y}}|\in\mathbb{R}_{\geq 0}^{M\times N}$. We split $|{\mathbf{Y}}|$
in $B=\lceil M/T\rceil$ subsequences, with $T$ being the length of the
subsequence, and $\lceil\cdot\rceil$ is the ceiling function. Each subsequence
$b$ overlaps with the preceding one by an empirical factor of $L\times 2$, in
order to use some context information for the encoding stage. We use each
subsequence $b$ in $|{\mathbf{Y}}|$, denoted as $|{\mathbf{Y}}_{\text{in}}|$
as an input to the skip-filtering connections (presented later). Furthermore
we produce a low-bandwidth version of $|{\mathbf{Y}}|$, which is used for
encoding, by preserving only the first $F=744$ frequency sub-bands at each
frame yielding $|{\mathbf{Y}}_{\text{tr}}|\in\mathbb{R}_{\geq 0}^{T\times F}$
$|{\mathbf{Y}}_{\text{tr}}|$. This operation retains information up to $8$
kHz, in order to reduce the number of trainable parameters but preserving the
most relevant information of the singing voice source.

Figure 1: Illustration of our proposed method.

### ii. The Masker

RNN encoder We use $|{\mathbf{Y}}_{\text{tr}}|$ as an input to the
$\text{RNN}_{\text{enc}}$. The forward GRU of the $\text{RNN}_{\text{enc}}$
takes $|{\mathbf{Y}}_{\text{tr}}|$ as an input. The backward one takes as an
input the
${|\overleftarrow{{\mathbf{Y}}}_{\text{tr}}}|=[|\mathbf{y}_{\text{tr}_{T}}|,\ldots,|\mathbf{y}_{\text{tr}_{t}}|,\ldots,$
$|\mathbf{y}_{\text{tr}_{1}}|]\text{,}$ where
$|\mathbf{y}_{\text{tr}_{t}}|\in\mathbb{R}_{\geq 0}^{F}$ is a vector in
$|{\mathbf{Y}}_{\text{tr}}|$ at time frame $t$, and $\overleftarrow{}$
indicates the direction of the sequence. The outputs from the Bi-GRU
$\mathbf{h}_{t}$ and $\overleftarrow{\mathbf{h}}_{t}$ are updated at each time
frame $t$ using residual connections [17] and then concatenated as

${\mathbf{h}_{\text{enc}_{t}}}=\big{[}({\mathbf{h}}_{t}+|{{\mathbf{y}}_{\text{tr}_{t}}}|)^{\text{T}},({\overleftarrow{\mathbf{h}}_{t}}+{\overleftarrow{|\mathbf{y}}_{\text{tr}_{t}}|})^{\text{T}}\big{]}^{\text{T}}\text{.}$
(3)

The output of the $\text{RNN}_{\text{enc}}$ for all $t\in T$ is denoted as
$\mathbf{H}_{\text{enc}}\in\mathbb{R}^{T\times(2\times F)}$ and is followed by
the context information removal defined as:

$\tilde{\mathbf{H}}_{\text{enc}}=[{\mathbf{h}}_{\text{enc}_{1+L}},{\mathbf{h}}_{\text{enc}_{2+L}},\ldots,{\mathbf{h}}_{\text{enc}_{T-L}}]\text{,
}$ (4)

yielding
$\tilde{\mathbf{H}}_{\text{enc}}\in\mathbb{R}^{T^{\prime}\times(2\times F)}$
for $T^{\prime}=T-(2\times L)$. Residual connections are used to ease the RNN
training [17].

Recurrent inference and mask prediction Inspired by recent optimization
methods employing stochastic depth [14], we propose a recurrent inference
algorithm that processes the latent variables of the $\text{RNN}_{\text{dec}}$
which affect the mask generation. We use this algorithm in order to employ a
stochastic depth for the network parts responsible for predicting the mask,
increasing the performance of our method. The recurrent inference is an
iterative process and consists in reevaluating the latent variables
$\mathbf{H}^{j}_{\text{dec}}$, produced by the $\text{RNN}_{\text{dec}}$,
until a convergence criterion is reached, thus avoiding the need to specify a
fixed number of applications of the $\text{RNN}_{\text{dec}}$. The stopping
criterion is a threshold on the mean-squared-error
($\mathcal{L}_{\text{MSE}}$) between the consecutive estimates of
$\mathbf{H}^{j}_{\text{dec}}$, with a $\mathcal{L}_{\text{MSE}}$ threshold
$\tau_{\text{term}}$. A maximum number of iterations ($iter$) is used to avoid
having infinite iterations for convergence between the above mentioned
consecutive estimates. $\mathbf{H}^{j}_{\text{dec}}$ is used only for the
singing voice, i.e. $j=1$. Let $\mathcal{G}^{j}_{\text{dec}}$ be the _source-
dependent_ and trainable function of the $\text{RNN}_{\text{dec}}$. The
recurrent inference is performed using Algorithm 1.

1:$\mathbf{S}^{j}_{0}\leftarrow\mathcal{G}^{j}_{\text{dec}}(\tilde{\mathbf{H}}_{\text{enc}})$

2:for $i\in\\{1,\dots,iter\\}$ do

3:
$\mathbf{H}^{j}_{\text{dec}}\leftarrow\mathcal{G}^{j}_{\text{dec}}(\mathbf{S}^{j}_{i-1})$

4: if
$\mathcal{L}_{\text{MSE}}(\mathbf{S}^{j}_{i-1},\mathbf{H}^{j}_{\text{dec}})<\tau_{\text{term}}$
then

5: Terminate the process

6: $\mathbf{S}^{j}_{i}\leftarrow\mathbf{H}^{j}_{\text{dec}}$ return
$\mathbf{H}^{j}_{\text{dec}}$

Algorithm 1 Recurrent Inference

$\mathbf{H}^{j}_{\text{dec}}$ is then given to the FFN layer with shared
weights through time frames, in order to approximate the $j$-th source-
dependent mask as:

$\tilde{\mathbf{M}}^{j}=\text{\small{ReLU}}(\mathbf{H}^{j}_{\text{dec}}\mathbf{W}_{\text{mask}}+\mathbf{b}_{\text{mask}})\text{,
}$ (5)

where ReLU is the element-wise rectified linear unit function producing a
sparse [15] approximation of the target source mask
$\tilde{\mathbf{M}}^{j}\in\mathbb{R}_{\geq 0}^{T^{\prime}\times N}$. The
sparsification is performed in order to improve the interference reduction of
[12]. The ReLU function can produce high positive values inducing distortions
to the audio signal. However, the reconstruction loss (see Eq. (9)) will
alleviate that. $\mathbf{W}_{\text{mask}}\in\mathbb{R}^{(2\times F)\times N}$
is the weight matrix of the FFN comprising a dimensionality expansion up to
$N$, in order to recover the original dimensionality of the data.
$\mathbf{b}_{\text{mask}}\in\mathbb{R}^{N}$ is the corresponding bias term.

Skip filtering connections and first estimate of the targeted source We obtain
an estimate of the magnitude spectrum of the target source
$|\hat{\mathbf{Y}}_{\text{filt}}^{j}|\in\mathbb{R}_{\geq 0}^{T^{\prime}\times
N}$ through the skip-filtering connections as:

$\displaystyle|\hat{{\mathbf{Y}}}_{\text{filt}}^{j}|=$
$\displaystyle|\tilde{{\mathbf{Y}}}_{\text{in}}|\odot\tilde{\mathbf{M}}^{j}\text{,
where}$ (6) $\displaystyle|{\mathbf{Y}}_{\text{in}}|=$
$\displaystyle[|\mathbf{y}_{\text{in}_{L}}|,\cdots,|\mathbf{y}_{\text{in}_{T-L}}|]\text{.}$
(7)

### iii. The Denoiser

The output of the Masker is likely to contain interferences from other sources
[12]. The Denoiser aims to learn a denoising filter for enhancing the
magnitude spectrogram estimated by this masking procedure. This denoising
filter is implemented by an encoder-decoder architecture with the
$\text{FFN}_{\text{enc}}$ and $\text{FFN}_{\text{dec}}$ of Fig. 1.
$\text{FFN}_{\text{enc}}$ and $\text{FFN}_{\text{dec}}$ have shared weights
through time frames. The final enhanced magnitude spectrogram estimate of the
target source $|\hat{{\mathbf{Y}}}^{j}|$ is computed using

$|\hat{{\mathbf{Y}}}^{j}|=\text{\small{ReLU}}(\text{\small{ReLU}}(|\hat{{\mathbf{Y}}}_{\text{filt}}^{j}|\mathbf{W}_{\text{enc}}+\mathbf{b}_{\text{enc}})\mathbf{W}_{\text{dec}}+\mathbf{b}_{\text{dec}})\odot|\hat{{\mathbf{Y}}}_{\text{filt}}^{j}|\text{,}$
(8)

where $\mathbf{W}_{\text{enc}}\in\mathbb{R}^{N\times\lfloor N/2\rfloor}$ and
$\mathbf{W}_{\text{dec}}\in\mathbb{R}^{\lfloor N/2\rfloor\times N}$ are the
weight matrices of the $\text{FFN}_{\text{enc}}$ and
$\text{FFN}_{\text{dec}}$, with the corresponding biases
$\mathbf{b}_{\text{enc}}\in\mathbb{R}^{\lfloor N/2\rfloor}$,
$\mathbf{b}_{\text{dec}}\in\mathbb{R}^{N}$, respectively.
$\lfloor\cdot\rfloor$ denotes the floor function.

### iv. Training details and post-processing

We train our method to minimize the objective consisting of a reconstruction
and a regularization part as:

$\begin{split}\mathcal{L}=D_{KL}(|{{\mathbf{Y}}}^{j}|\;||\;|\hat{{\mathbf{Y}}}^{j}|)+\lambda_{\text{rec}}D_{KL}(|{{\mathbf{Y}}}^{j}|\;||\;|\hat{{\mathbf{Y}}}_{\text{filt}}^{j}|)\\\
+\lambda_{mask}|\text{diag}\\{\mathbf{W}_{\text{mask}}\\}|_{1}+\lambda_{\text{dec}}||\mathbf{W}_{\text{dec}}||_{2}^{2}\text{,}\end{split}$
(9)

where $|{{\mathbf{Y}}}^{j}|$ is the magnitude spectrogram of the true source,
diag$\\{\cdot\\}$ denotes the elements on the main diagonal of a matrix,
$|\cdot|_{1}$, and $||\cdot||_{2}^{2}$ are the $\ell_{1}$ vector norm and the
squared matrix $L_{2}$ norm respectively, and $\lambda_{\text{mask}}$, and
$\lambda_{\text{dec}}$ are scalars. For $\lambda_{\text{rec}}$ the following
condition applies:

$\lambda_{\text{rec}}=\begin{cases}1,&\text{if}\
D_{KL}(|{{\mathbf{Y}}}^{j}|\;||\;|\hat{{\mathbf{Y}}}_{\text{filt}}^{j}|)\geq\tau_{\text{rec}}\\\
&\text{and
}D_{KL}(|{{\mathbf{Y}}}^{j}|\;||\;|\hat{{\mathbf{Y}}}^{j}|)\geq\tau_{\text{min}}\\\
0,&\text{otherwise}\end{cases}\text{,}$ (10)

where $\tau_{\text{rec}}$ and $\tau_{\text{min}}$ are hyper-parameters
penalizing the mask generation process, allowing a collaborative minimization
of the overall objective. The usage of
$\lambda_{\text{rec}}D_{KL}(|{{\mathbf{Y}}}^{j}|\;||\;|\hat{{\mathbf{Y}}}_{\text{filt}}^{j}|)$
will ensure that ${\mathbf{M}^{j}}$ can be used to initially estimate the
target source, which is then improved by employing the Denoiser and
$D_{KL}(|{{\mathbf{Y}}}^{j}|\;||\;|\hat{{\mathbf{Y}}}^{j}|)$. The penalization
of the elements in the main diagonal of $\mathbf{W}_{\text{mask}}$ will ensure
that the generated mask is not something trivial (e.g. a voice activity
detector), while the reconstruction losses using the $D_{KL}$ will ensure that
a source-dependent mask is generated, that minimizes the aforementioned
distance. The squared matrix $L_{2}$ norm is employed to improve the
generalization of the model.

By processing each subsequence using the proposed method, the estimates are
concatenated together to form $|\hat{{\mathbf{Y}}}^{j}|\in\mathbb{R}_{\geq
0}^{M\times N}$. For the singing voice we retrieve the complex-valued STFT
$\hat{Y}^{j=1}$ by means of 10 iterations of the Griffin-Lim algorithm (least
squares error estimation from modified STFT magnitude) [18] initialized with
the mixture’s phase and using $|\hat{{\mathbf{Y}}}^{j}|$. The time-domain
samples $\mathbf{\hat{x}}^{j=1}$ are obtained using inverse STFT.

## III. Experimental Procedure

We use the development subset of Demixing Secret Dataset
(DSD$100$)111http://www.sisec17.audiolabs-erlangen.de and the non-
bleeding/non-instrumental stems of MedleydB [19] for the training and
validation of the proposed method. The evaluation subset of DSD$100$ is used
for testing the objective performance of our method. For each multi-track
contained in the audio corpus, a monaural version of each of the four sources
is generated by averaging the two available channels. For training, the true
source $|\mathbf{Y}^{j}|$ is the outcome of the ideal ratio masking process
[20], element-wise multiplied by a factor of $2$. This is performed to avoid
the inconsistencies in time delays and mixing gains between the mixture signal
and the singing voice (apparent in MedleydB dataset). The length of the
sequences is set to $T=60$, modeling approximately $0.5$ seconds, and $L=10$.
The thresholds for the minimization of Eq.(9) are $\tau_{\text{rec}}=1.5$ and
$\tau_{\text{min}}=0.25$ and the corresponding scalars are
$\lambda_{\text{mask}}=1e^{-2}$, and $\lambda_{\text{dec}}=1e^{-4}$. The
hidden to hidden matrices of all RNNs were initialized using orthogonal
initialization [21] and all other matrices using Glorot normal [22]. All
parameters are jointly optimized using the Adam algorithm [23], with a
learning rate of $1e^{-4}$, over batches of $16$, an $L_{2}$ based gradient
norm clipping equal to $0.5$ and a total number of $100$ epochs. All of the
reported parameters were chosen experimentally with two random audio files
drawn from the development subset of DSD$100$. The implementation is based on
PyTorch222http://pytorch.org/.

We compared our method with other state-of-the-art approaches dealing with
monaural singing voice separation, following the standard metrics, namely
signal to noise ratio (SIR) and signal to distortion ratio (SDR) expressed in
dB, and the rules proposed in the music source separation evaluation campaign
[1] (e.g. using the proposed toolbox for SIR and SDR calculation). The
compared methods are: i) GRA: Deep FFNs [4] for predicting both binary and
soft masks [20] which are then combined to provide source estimates, ii) CHA:
A convolutional encoder-decoder for magnitude source estimation, without a
trainable mask approximation [6] iii) MIM-HW: Deep highway networks for music
source separation [11] approximating the filtering process of Eq.(1),
retrained using the development subset of DSD$100$, and iv) MIM-DWF, MIM-DWF+:
The two GRU encoder-decoder models combined with generalized Wiener filtering
[12], trained on the development subset of DSD$100$ (MIM-GRUDWF) and the
additional stems of MedleydB (MIM-DWF+). The methods denoted as MIM-HW, MIM-
DWF, and MIM-DWF+ were re-implemented for the purposes of this work. For the
rest of the methods we used their reported evaluation results obtained from
[1]. Our proposed methods are denoted as GRU-NRI, which does not include the
recurrent inference algorithm, and two methods using different hyper-
parameters for the recurrent inference algorithm: GRU-RIS${}^{\text{s}}$,
parametrized using a maximum number of iterations $iter=3$, and
$\tau_{\text{term}}=1e^{-2}$, and GRU-RIS${}^{\text{l}}$ parametrized using a
maximum number of iterations $iter=10$ and $\tau_{\text{term}}=1e^{-3}$, which
where selected according to their performance in minimizing Eq. (9).

## IV. Results & Discussion

Table 1 summarizes the results of the objective evaluation for the
aforementioned methods by showing the median values obtained from the SDR and
SIR metrics. The proposed method based on recurrent inference and sparsifying
transform is able to provide state-of-the-art results for monaural singing
voice separation, without the necessity of post-processing steps such as
generalized Wiener filtering, and/or additionally trained deep neural
networks. Compared to methods that approximate the masking processes (GRA,
MIM-HW, MIM-DWF, and MIM-DWF+) there are significant improvements in overall
median performance of both the SDR and SIR metrics, especially when the masks
are not a learned function, such as in the case of CHA.

Table 1: Median SDR and SIR values in dB for the investigated approaches.
Proposed approaches are underlined. Higher is better.

Method | SDR | SIR | Method | SDR | SIR
---|---|---|---|---|---
GRA[4] | -1.75 | 1.28 | MIM-DWF+[12] | 3.71 | 8.01
MIM-HW[11] | 1.49 | 7.73 | GRU-NRI | 3.62 | 7.06
CHA[6] | 1.59 | 5.20 | GRU-RISs | 3.41 | 8.32
MIM-DWF[12] | 3.66 | 8.02 | GRU-RISl | 4.20 | 7.94

Using the proposed method, a gain of $0.49$ dB for the SDR is observed between
MIM-DWF+ and GRU-RIS${}^{\text{l}}$ and $0.30$ dB for the SIR between the MIM-
DWF and GRU-RIS${}^{\text{s}}$. Finally, by allowing a larger number of
iterations during the recursive inference the mask generation performance and
using skip-filtering connections we see an increase in SDR which outperforms
the previous methods MIM-DWF and MIM-DWF+, but at the cost of a loss in SIR. A
demo for the proposed method is available at https://js-
mim.github.io/mss_pytorch/.

## V. Conclusion

In this work we presented an approach for singing voice separation that does
not require post-processing using generalized Wiener filtering. We introduced
to the skip-filtering connections [12] a sparsifying transform yielding
comparable results to approaches that rely on generalized Wiener filtering.
Furthermore, the introduced recurrent inference algorithm was shown to provide
state-of-the-art results in monaural singing voice separation. Experimental
results show that these extensions outperform previous deep learning based
approaches for singing voice separation.

## VI. Acknowledgements

The research leading to these results has received funding from the European
Union’s H2020 Framework Programme (H2020-MSCA-ITN-2014) under grant agreement
no 642685 MacSeNet. Part of the computations leading to these results were
performed on a TITAN-X GPU donated to K. Drossos from NVIDIA. The authors
would like to thank Paul Magron for the precious feedback.

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