# Study of charged gravastar model in $f(\mathcal{Q})$ gravity

Debasmita Mohanty 0009-0006-8118-5327<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Sayantan Ghosh 0000-0002-3875-0849
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

(August 28, 2024)

###### Abstract

In recent days gravastar has been a very lucrative alternative to black holes,
as it does not suffer from the singularity problem as well and it is based on
sound physical grounds. Modified Symmetric teleparallel equivalent of gravity
also has seen quite a few successes in recent years both in cosmology as well
as in astrophysical objects like black holes and wormholes. In this paper, we
have considered the charged gravastar in $f(\mathcal{Q})$ formulation and have
solved it fully analytically and found various physical characteristics like
energy density, entropy and EoS for the gravastar. We have used the Israel
junction condition to make some phenomenological predictions regarding the
potential of the thin shell around the gravastar. We have also studied the
deflection of the angle caused by the gravastar. Finally, we conclude by
noting how future radio telescopes could detect the gravastar shadow and how
can one distinguish it from the black hole event horizon.

Keywords: Charged Gravastar; $f(\mathcal{Q})$ gravity; Junction condition; ADM
mass; Deflection of light.

##  I Introduction

Gravitational Vacuum Condensate Stars or in short just gravastar is a
lucrative alternative to black holes which was first proposed by Mazur and
Mottala [1, 2] in the early twenty-first century. Black holes are the most
fascinating object in general relativity not just because they are the exact
solutions of Einstein’s equation but also the fact that they have entropy
which comes purely from quantum mechanics [3]. It was shown by Roger Penrose
[4] by using topological arguments that even a non-spherical symmetric
collapse could lead to black hole singularity. Black holes or massive compact
objects have been observationally found by Ghez [5] and Gillessen [6]. We note
that Penrose, Ghez, and Gillessen have shared the 2020 Nobel Prize in physics
for their effort. However, there are several problems with taking the final
stage of a massive star collapse to be a black hole. First, as shown by
Penrose under reasonable energy conditions, the black holes or the collapsing
stars would always form a singularity, and in singularity, the metric is not
invertible so the assumptions of general relativity itself fail. We also note
that Hawking radiation of black hole could give Hawking temperature associated
with the black hole as $T_{BH}=\frac{\hbar c^{3}}{8\pi Gk_{B}M}$ also the
black hole entropy could be shown as $S_{BH}=\frac{4\pi k_{B}GM^{2}}{\hbar
c}$. We first note that the immediate problem with such a system in the
context of thermodynamics is that the energy is given as
$dE=T_{BH}dS_{BH}=\frac{\hbar c^{3}}{8\pi Gk_{B}M}d(\frac{4\pi
k_{B}GM^{2}}{\hbar c}$), we note a crucial observation that the reduced plank
constant ($\hbar$) does get cancelled out. So we can claim that the
thermodynamic quantities behave as classical (non-quantum). This creates a
serious paradoxical problem as we noted by Hawking [7] that the energy (or $M$
in natural units) is inversely proportional to temperature. So this leads to
negative heat capacity. This is problematic as we already discussed that even
though the origin of black hole thermodynamics laws is quantum mechanical the
thermodynamic energies behave as a classical theory, and we know from
classical thermodynamics that such negative heat capacity could not make a
system equilibrium. This paradox can be naturally explained by the gravastar
model as shown in [2].
The idea of gravastar comes from the fact that like in a condensed matter
system during the phase transition (when temperature is close to critical
temperature ($T\approx T_{c}$) the entire system goes through a phase
transition like ferromagnet to paramagnet etc.) As the quantum mechanics gets
non-negligible near the event horizon one can expect that the collapsing dust
particles also behave like a quantum many-body system. It is also well known
that a system of bosonic particles in very low temperatures goes to Bose-
Einstein condensation. The idea of Mazur and Motalla was to use this intuition
to explain the dust collapse without the pathological problems encountered by
the black hole. We also note that the use of Bose-Einstein condensation in an
astrophysical context is not at all new it is widely believed that the core or
the neutron star is Bose-Einstein condensate, also there are many studies of
exotic stars like Bose stars [8] which are expected to form due to Bose-
Einstein condensation.
We note that the entropy of an ordinary object with the same radius and mass
as the black hole the entropy is given by $S=\frac{4k+4}{7k+1}S_{BH}$ as shown
in [9] ($k$ is the EoS for the object). Such an expression of entropy is a
unique feature of gravastar as it in the surface matches with Hawking’s semi-
classical treatment of black hole entropy, on the other hand, it is non-
constant like other singularity-free astrophysical objects. As we note, for
gravastar to mimic the entropy formula of a black hole given by Hawking, we
have to take $k=1$, so we take the outer shell of the gravastar as a stiff
matter where $k=1$.
Modified symmetric teleparallel gravity or $f(\mathcal{Q})$ gravity has been
proposed by Jimenez et al. [10]. It has been successfully used in the
cosmological context to explain the late time acceleration [11, 12], also,
there is much literature that has studied the observation data using
$f(\mathcal{Q})$ gravity [13]. $f(\mathcal{Q})$ gravity has also been very
successful in explaining the properties of various astrophysical objects like
black holes [14] and wormholes [15] etc. We also note that the study of
gravastars in modified gravity is not new, there are several articles in the
literature that explore the gravastars in various gravity like $f(R,T)$
gravity [16, 17], $f(T)$ gravity [18], $f(\mathcal{Q})$ gravity [19], Rastall-
Rainbow gravity [20] etc. The property of charged black holes in modified
gravity is of particular interest as the field equation can be most often
solved (in rotating or Kerr black holes in modified gravity background
typically are not exactly solvable due to the high non-linearity of the field
equation). The charged black holes in modified gravity have been also
extensively studied by [21]. We note that due to the black hole no-hair
theorem [22], it can be shown that any black hole can have at most three
variables those are mass ($M$), charge (${Q}$), and angular momentum ($J$)
[23, 24]. We note that even though charged gravastar in various modified
gravity like $f(T),f(R)$ has already been studied, to the best of our
knowledge there has been no study about the charge gravastar in
$f(\mathcal{Q})$ gravity.
In this paper, we have taken the modified symmetric teleparallel gravity in
order to model the gravastar in a charged background. As we know from the
black hole uniqueness theorem a black hole has “no hair” that is a black hole
can be fully classified by its charge ($Q$) mass ($M$) and angular momentum
($J$). As studying rotating gravastar is very difficult in fact as the outside
is a Karr metric we take a charged gravastar and use Reissner-Nordstrom
solutions outside the gravastar to solve the system. We have also used the
Israel junction condition to find the potential as well as the mass of the
thin shell, in this process we have found the deflection of such a gravastar,
giving us the phenomenological tool to distinguish it from black holes. We
have also calculated the ADM mass for the inner and outer regions of the
gravastar in order to incorporate the “effective mass” for gravitational
interaction.
Gravastar is made up of three regions i.e. interior, thin shell and outer
region. We take inner and outer radii of each layer being $r_{1}$ and $r_{2}$,
respectively, and $r_{1}<r_{2}$. The three layers are as follows:

  * •

Interior region: $0\leq r<r_{1}$ with a de Sitter space-time defined equation
of state (EoS), $p=-\rho$.

  * •

Shell region: $r_{1}<r<r_{2}$ with EoS $p=\rho$, an intermediary thin layer
made of an ultra-stiff perfect fluid.

  * •

Exterior region: $r_{2}<r$ with EoS $p=0$ (vacuum), which is described by the
Schwarzschild solution.

We would use the Israel junction condition on this thin shell to study the
behaviour of the gravastar.

Within the context of Einstein’s general relativity, a lot of research on
gravastar models has been published in the literature. Gravastar solutions
with constant pressures and the equation of state have been discussed by
DeBenedictis et al. [25]. In Refs [26, 27, 28], the gravastar model in higher-
dimensional space-time has been discussed.

The format of our paper is as follows: We provided a quick overview of the
gravastar model and the most current research on it in sec I. In sec II, we
shall describe the basic framework of $f(\mathcal{Q})$ gravity. In sec III,
Maxwell’s equation in curved space-time is provided. The summary of the
Einstein-Maxwell field equation in $f(\mathcal{Q})$ gravity is provided in sec
IV. In sec V, we discuss the outline of the conformal killing vectors. In sec
VI, the field equation solution for different regions with varying EoS is
provided. Following that, in sec VII, we looked at the junction condition and
deflection angle. The physical features of the model are described in sec
VIII. In sec IX, we have discussed the ADM mass. Finally, in sec X, we present
the conclusions of our analysis.

##  II $f(\mathcal{Q})$ gravity

The theory of $f(\mathcal{Q})$ gravity deals with a metric-affine space-time,
where the metric tensor $g_{\mu\nu}$ and connection
$\Gamma^{\lambda}_{\mu\nu}$ are treated as independent entities. In this
theory, the non-metricity of the connection is defined by,

$\mathcal{Q}_{\alpha\mu\nu}=\bigtriangledown_{\alpha}g_{\mu\nu}=\partial_{\alpha}g_{\mu\nu}-\Gamma^{\lambda}\,_{\alpha\mu}\,g_{\lambda\nu}-\Gamma^{\lambda}\,_{\alpha\nu}g_{\mu\lambda}.$
(1)

So, basically, the affine connection can be consisted of the following three
independent components,

$\Gamma^{\lambda}\,_{\mu\nu}=\\{{}^{\lambda}\,{}_{\mu\nu}\\}+K^{\lambda}\,_{\mu\nu}+L^{\lambda}\,_{\mu\nu}.$
(2)

The symbols $\\{\lambda{}_{\mu\nu}\\}$ and $K^{\lambda}\,_{\mu\nu}$ represent
the Levi-Civita connection and contortion, respectively and they are defined
as follows,

$\\{{}^{\lambda}\,{}_{\mu\nu}\\}\equiv\frac{1}{2}g^{\lambda\beta}\left(\partial_{\mu}g_{\beta\nu}+\partial_{\nu}g_{\beta\mu}-\partial_{\beta}g_{\mu\nu}\right),$
(3)

$K^{\lambda}\,_{\mu\nu}\equiv\frac{1}{2}T^{\lambda}\,_{\mu\nu}+T_{(}\mu^{\lambda}\,_{\nu}),$
(4)

where $T^{\lambda}\,_{\mu\nu}$ is the antisymmetric part of the affine
connection, which is defined as $T^{\lambda}\,_{\mu\nu}\equiv
2\Gamma^{\lambda}\,_{[\mu\nu]}$.
The disformation $L^{\lambda}\,_{\mu\nu}$ is defined by,

$L^{\lambda}\,_{\mu\nu}\equiv\frac{1}{2}\mathcal{Q}^{\lambda}\,_{\mu\nu}-\mathcal{Q}(_{\mu}\,^{\lambda}\,{}_{\nu}).$
(5)

Next, the non-metricity conjugate is defined by,

$P^{\alpha}\,_{\mu\nu}=-\frac{1}{4}\mathcal{Q}^{\alpha}\,_{\mu\nu}+\frac{1}{2}\mathcal{Q}(_{\mu}\,^{\alpha}\,{}_{\nu})+\frac{1}{4}(\mathcal{Q}^{\alpha}-\tilde{\mathcal{Q}}^{\alpha})g_{\mu\nu}\\\
-\frac{1}{4}\delta^{\alpha}(_{\mu}\mathcal{Q}_{\nu}),$ (6)

where $\mathcal{Q}_{\alpha}\equiv\mathcal{Q}_{\alpha}\,^{\mu}\,{}_{\mu}$ and
$\tilde{\mathcal{Q}}_{\alpha}\equiv\mathcal{Q}^{\mu}\,_{\alpha\mu}$ are the
two independent traces of non-metricity tensor.
The non-metricity scalar is finally defined as follows:

$\mathcal{Q}\equiv-\mathcal{Q}_{\alpha\mu\nu}P^{\alpha\mu\nu}.$ (7)

Along Lagrange multipliers, $f(\mathcal{Q})$ gravity is given by the following
action [10]:

$S=\int\sqrt{-g}d^{4}x\left[\
\frac{1}{2}f(\mathcal{Q})+\lambda_{\alpha}\,^{\beta\mu\nu}R^{\alpha}\,_{\beta\mu\nu}+\lambda_{\alpha}\,^{\mu\nu}T^{\alpha}\,_{\mu\nu}\right.\\\
\left.+\mathcal{L}_{m}+\mathcal{L}_{e}\right]\,,$ (8)

where $g$ is the determinant of the metric $g_{\mu\nu}$, $f(\mathcal{Q})$
represents the arbitrary function of the non-metricity $\mathcal{Q}$.
$\lambda_{\alpha}\,^{\beta\mu\nu},\mathcal{L}_{m},$ and $\mathcal{L}_{e}$
represents the Lagrange multipliers, matter Lagrangian and Lagrangian for the
electromagnetic respectively.

The field equation is obtained by varying action(8) with respect to the metric
[29],

$-T_{\mu\nu}+E_{\mu\nu}=\frac{2}{\sqrt{-g}}\bigtriangledown_{\alpha}\left(\sqrt{-g}f_{\mathcal{Q}}P^{\alpha}\,_{\mu\nu}\right)+\frac{1}{2}g_{\mu\nu}f\\\
+f_{\mathcal{Q}}\left(P_{\mu\alpha\beta}{\mathcal{Q}}_{\nu}\,^{\alpha\beta}-{2\mathcal{Q}}_{\alpha\beta\mu}P^{\alpha\beta}\,_{\nu}\right),$
(9)

where a subscript appears $\mathcal{Q}$ represents a derivative of
$f(\mathcal{Q})$ with respect to $\mathcal{Q}$, namely
$f_{\mathcal{Q}}\equiv\partial_{\mathcal{Q}}f(\mathcal{Q})$. $E_{\mu\nu}$ is
the energy-momentum tensor of the electromagnetic field. The energy-momentum
tensor is defined as follows:

$T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}},$ (10)

We note that $E_{\mu\nu}$, the energy-momentum tensor of the electromagnetic
field can be found as [30],

$E_{\mu\nu}(EM)=\frac{1}{4\pi}\left[-F^{\zeta}_{\mu}F_{\nu\zeta}+\frac{1}{4}F_{\eta\zeta}F^{\eta\zeta}g_{\mu\nu}\right]$
(11)

Now varying equation (8) with respect to the connection we obtain,

$\bigtriangledown_{\rho}\lambda_{\alpha}\,^{\nu\mu\rho}+\lambda_{\alpha}\,^{\mu\nu}=\sqrt{-g}f_{\mathcal{Q}}P^{\alpha}\,_{\mu\nu}+H_{\alpha}\,^{\mu\nu},$
(12)

where $H_{\alpha}\,^{\mu\nu}$ is the hyper momentum tensor density, which is
defined as follows [29]:

$H_{\alpha}\,^{\mu\nu}=\frac{-1}{2}\frac{\delta(\mathcal{L}_{m}+\mathcal{L}_{e})}{\delta\Gamma^{\alpha}\,_{\mu\nu}}.$
(13)

Using the anti-symmetry property of $\mu$ and $\nu$ in the Lagrangian
multiplier coefficients, equation (12) can be simplified to

$\bigtriangledown_{\mu}\bigtriangledown_{\nu}\left(\sqrt{-g}f_{\mathcal{Q}}P^{\mu\nu}\,_{\alpha}+H_{\alpha}\,^{\mu\nu}\right)=0.$
(14)

Taking $\bigtriangledown_{\mu}\bigtriangledown_{\nu}H_{\alpha}\,^{\mu\nu}=0$
(according to Ref. [10] ), the above equation becomes,

$\bigtriangledown_{\mu}\bigtriangledown_{\nu}\left(\sqrt{-g}f_{\mathcal{Q}}P^{\mu\nu}\,_{\alpha}\right)=0.$
(15)

The affine connection has the following form in the absence of curvature and
torsion.

$\Gamma^{\alpha}\,_{\mu\nu}=\left(\frac{\partial
x^{\alpha}}{\partial\theta^{\lambda}}\right)\partial_{\mu}\partial_{\nu}\,\theta^{\lambda}.$
(16)

We can choose a special set of coordinates, the so-called coincident gauge, so
that $\Gamma^{\alpha}\,_{\mu\nu}=0$, in this case. Consequently, non-metricity
reduces to,

${\mathcal{Q}}_{\alpha\mu\nu}=\partial_{\alpha}g_{\mu\nu}.$ (17)

This greatly simplifies the calculation, since only the metric variable is
essential. However, the action is no longer diffeomorphism invariant with the
exception of STGR [31]. One can avoid the issue by employing the covariant
formulation of $f(\mathcal{Q})$ gravity. Since the affine connection in
equation (16) is solely inertial, the covariant formulation could be applied
by first determining the affine connection in the absence of gravity [32].
However, as demonstrated in this work, the off-diagonal component of the field
equations in the coincident gauge would impose stringent constraints on
$f(\mathcal{Q})$ gravity, thereby providing nontrivial functional forms of
$f(\mathcal{Q})$.

##  III Maxwell’s equation in curved space time

We note that all four of Maxwell’s equations can be expressed as,

$\frac{\partial F_{\mu\nu}}{\partial x_{\nu}}=\mu_{0}J_{\mu}.$ (18)

$\frac{\partial F_{\mu\nu}}{\partial x_{\lambda}}+\frac{\partial
F_{\nu\lambda}}{\partial x_{\mu}}+\frac{\partial F_{\lambda\mu}}{\partial
x_{\nu}}=0.$ (19)

For a free electromagnetic field, Lagrangian density is defined as follows,

$L_{EM}=-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu},$ (20)

with

$F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}.$ (21)

We note that if we vary the Lagrangian with respect to $A_{\mu}$ we will get
Maxwell’s equation from the Euler-Lagrange equation.
Consequently, by varying the above Lagrangian with respect to the metric, we
can get its energy-momentum tensor via,

$E_{\mu\nu}=2\frac{\partial\mathcal{L}_{m}}{\partial
g^{\mu\nu}}-\mathcal{L}_{m}g_{\mu\nu},$ (22)

So plugging the EM Lagrangian density we get,

$E_{\mu\nu}(EM)=\frac{1}{4\pi}\left[-F^{\zeta}_{\mu}F_{\nu\zeta}+\frac{1}{4}F_{\eta\zeta}F^{\eta\zeta}g_{\mu\nu}\right]$

We note that for an isolated static charge, $J^{\mu}$ is zero. We also note
that as we are working with spherical polar coordinate, for the field strength
of a static charge only component of interest is the radial potential as,

$A_{0}=\phi\neq 0,\,\,A_{2}=A_{3}=A_{1}=0.$

Hence, out of all the components of $F_{\mu\nu}$ the only nonzero components
will be $F_{01},F_{10}$.
So the electromagnetic field strength tensor can be expressed in the following
manner:

$F_{\mu\nu}=\phi_{,1}\begin{pmatrix}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix}$ (23)

Now from Maxwell’s equation, we get,

$\begin{gathered}\frac{\partial F_{\mu\nu}}{\partial
x_{\nu}}={\mu}_{0}J_{\mu}=0,\\\
\implies(g^{00}g^{11}F_{01}\sqrt{-g})_{,1}=0,\\\
\implies(g^{00}g^{11}\phi\sqrt{-g})_{,1}=0,\\\ \implies
e^{-\frac{\lambda+\nu}{2}}\phi_{,1}r^{2}=Q=constant.\end{gathered}$ (24)

Thus, the energy-momentum tensor of the electromagnetic field is given as,

$E^{\mu}_{\nu}=\frac{1}{8\pi}diag(\frac{Q^{2}}{r^{4}})(1,1,-1,-1).$ (25)

We note that here we use $Q$ as a dimensionless parameter in order to get the
graph. However, the dimension expression for $Q$ is given by
$\frac{Q^{2}G}{4\pi\epsilon_{0}c^{4}}$.

##  IV $f(\mathcal{Q})$ GRAVITY WITH SPHERICAL SYMMETRY

Let’s move on to discussing the spherically symmetric configurations. The
metric follows the standard format in spherical coordinates.

$ds^{2}=-e^{A(r)}dt^{2}+e^{B(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (26)

And the non-metricity scalar $Q$ in terms of $r$ is,

$\mathcal{Q}(r)=-\frac{2e^{-B}}{r}\left(A^{\prime}+\frac{1}{r}\right).$ (27)

The Einstein-Maxwell field equation in $f(\mathcal{Q})$ gravity are given by ,

$\frac{f}{2}+\frac{2}{r}e^{-B}f_{\mathcal{Q}\mathcal{Q}}{\mathcal{Q}}^{\prime}-f_{\mathcal{Q}}\biggl{[}\mathcal{Q}+\frac{e^{-B}}{r}(A^{\prime}+B^{\prime})+\frac{1}{r^{2}}\biggr{]}=\rho+2\pi
E^{2},$ (28)

$-\frac{f}{2}+f_{\mathcal{Q}}\left(\mathcal{Q}+\frac{1}{r^{2}}\right)=p_{r}-2\pi
E^{2},$ (29)
$-\frac{f}{2}-e^{-B}\left(\frac{1}{r}+\frac{A^{\prime}}{2}\right)f_{\mathcal{Q}\mathcal{Q}}{\mathcal{Q}}^{\prime}+f_{\mathcal{Q}}\biggl{[}\frac{\mathcal{Q}}{2}-e^{-B}\left({\frac{A^{\prime\prime}}{2}+\left(\frac{A^{\prime}}{4}+\frac{1}{2r}\right)\left(A^{\prime}-B^{\prime}\right)}\right)\biggr{]}=p_{t}+2\pi
E^{2},$ (30)

and

$\frac{\cot{\theta}}{2}f_{\mathcal{Q}\mathcal{Q}}{\mathcal{Q}}^{\prime}=0.$
(31) $\left(r^{2}E\right)^{\prime}=4\pi r^{2}\sigma(r)e^{\frac{B}{2}},$ (32)

Thus, the invariant energy is,

$E(r)=\frac{1}{r^{2}}\int{4\pi r^{2}\sigma(r)e^{\frac{B}{2}}dr}.$ (33)

Terms $\rho$, $p$, and $E(r)$ denote the matter density, isotropic pressure,
and electric field of a charged fluid sphere, respectively. Also, $\sigma(r)$
denotes the surface energy density of thin shell. The prime notation indicates
differentiation with respect to the radial coordinate, $r$.

##  V CONFORMAL KILLING EQUATION

The use of conformal symmetry under conformal killing vectors (CKV),
represented by the following equation, is an approach that links geometry with
matter systematically [33].

$L_{\xi}g_{ik}=\psi g_{ik},$ (34)

where $L$ stands for the Lie derivative operator, and $\psi$ stands for the
conformal factor. The metric $g$ is conformally translated onto itself along
$\xi$ as a result of the vector $\xi$ producing the conformal symmetry.

Eq.(34) yields the killing vector for $\psi=0$, a homothetic vector for $\psi$
= constant, and conformal vectors for $\psi$ = $\psi(x,t)$. The underlying
space-time is asymptotically flat for $\psi=0$, which means that the Weyl
tensor will also disappear. The conformal killing vectors can therefore be
studied to gain a better understanding of space-time geometry.
So equation (34) becomes,

$L_{\xi}g_{ik}=\xi_{i;k}+\xi_{k;i}=\psi g_{ik}.$ (35)

The conformal killing equations for line element (26) are stated as,

$A^{\prime}{\xi}^{\prime}=\psi,\,\,B^{\prime}{\xi}^{\prime}+2{\xi}^{\prime},_{1}=\psi,\,\,{\xi}^{\prime}=\frac{\psi
r}{2},\,\,{\xi}^{0}=c_{1}.$ (36)

Here the terms ‘prime’ and ‘comma’ refer to the derivative and partial
derivative with respect to ‘$r$’ respectively, and $c_{1}$ is a constant.

The results of the aforementioned equations are,

$e^{A}={c_{2}}^{2}r^{2},$ (37) $e^{B}=\left(\frac{c_{3}}{\psi}\right)^{2},$
(38) ${\xi}^{i}=c_{1}\delta^{i}_{0}+\left(\frac{\psi
r}{2}\right)\delta^{i}_{1}.$ (39)

Here $c_{2}$ and $c_{3}$ are constants of integration. We note that
$\delta^{i}_{j}$ denotes the Kronecker delta, that is $\delta^{i}_{j}=1$ if
$i=j$ else it is $0$.

It is clear that in order to satisfy the off-diagonal terms in the field
equation (31) we can only take $f(\mathcal{Q})$ of the form,

$f(\mathcal{Q})=a{\mathcal{Q}}+b,$ (40)

where $a$ and $b$ are constants. We note that at $a=-1$ and $b=0$, it reduces
to Einstein’s GR.
Substituting equation (40) in equations (28)-(30) we get the following
equations given below.

$\frac{-a{\psi}^{2}}{r^{2}{c_{3}}^{2}}-\frac{2a\psi{\psi}^{\prime}}{r{c_{3}}^{2}}+\frac{b}{2}+\frac{a}{r^{2}}=\rho+2\pi
E^{2}$
$\frac{3a{\psi}^{2}}{r^{2}{c_{3}}^{2}}-\frac{b}{2}-\frac{a}{r^{2}}=p_{r}-2\pi
E^{2}$

$\frac{a{\psi}^{2}}{r^{2}{c_{3}}^{2}}+\frac{2a\psi{\psi}^{\prime}}{r{c_{3}}^{2}}-\frac{b}{2}=p_{t}+2\pi
E^{2}$

From the above equations, we get the following Einstein-Maxwell field
equations (following the assumptions given in [27, 30]),

$-\frac{3a\psi{\psi}^{\prime}}{r{c_{3}}^{2}}+\frac{1}{2}\left(\frac{a}{r^{2}}+b\right)=\rho,$
(41)
$\frac{2a{\psi}^{2}}{r^{2}{c_{3}}^{2}}+\frac{a\psi{\psi}^{\prime}}{r{c_{3}}^{2}}-\frac{1}{2}\left(\frac{a}{r^{2}}+b\right)=p,$
(42)
$\frac{1}{\pi}\biggl{[}\frac{-a{\psi}^{2}}{2r^{2}{c_{3}}^{2}}+\frac{a\psi{\psi}^{\prime}}{2r{c_{3}}^{2}}+\frac{a}{4r^{2}}\biggr{]}=E^{2}.$
(43)

In the next section, we will go over each of the three charged gravastar
regions where we solve the aforementioned field equations (41) - (43)
individually.

##  VI Geometry of gravastar

###  VI.1 Interior region of the Charged Gravastar

Let’s consider the equation of state (EoS) between the matter density $\rho$
and isotropic pressure $p$.

$p=-\rho,$ (44)

as proposed by Mazur and Mottola [1, 2], the metric coefficient for $e^{B}$
can be determined.

By using equation (44), we can obtain a differential equation that is linear
in the conformal factor $\psi$ as follows, from equations (41) and (42).

$\frac{a\psi}{r^{2}{c_{3}}^{2}}\left(\psi-{\psi}^{\prime}r\right)=0.$ (45)

There are two possible values for $\psi$ based on equation (45), $\psi=0$ or
$\psi=\psi_{0}r$, where $\psi_{0}$ represents the constant of integration. We
compute the other physical parameters in the charged gravastar model’s
interior area by using $\psi=\psi_{0}r$ since $\psi=0$ implies asymptotic
flatness.

After obtaining the conformal factor $\psi$ expression, we may write the
metric coefficients expression in terms of the conformal factor as follows:

$e^{-B(r)}={\tilde{\psi_{0}}}^{2}r^{2},$ (46) $e^{A(r)}=c_{2}^{2}r^{2},$ (47)

where $\tilde{\psi_{0}}=\frac{\psi_{0}}{c_{3}}$, and an intriguing feature of
this situation is that $e^{A}$ is directly proportional to $r^{2}$ but $e^{B}$
is inversely proportional to $r^{2}$. One may find the isotropic pressure and
the expression for matter density as,

$\rho=-3a{\tilde{\psi_{0}}}^{2}+\frac{1}{2}\left(\frac{a}{r^{2}}+b\right)=-p.$
(48)

From equation (48), we know that
${\tilde{\psi_{0}}}^{2}<\frac{1}{6a}\left(\frac{a}{r^{2}}+b\right)$ since the
energy density is positive in the interior region of the gravastar which
provides an upper bound for $\tilde{\psi_{0}}$. As we go closer to the
gravastar’s centre, the pressure and density both suffer from central
singularities, which is a normal behaviour of the CKV model. Both variables
are inversely proportional to $r^{2}$ and blow up without bounds.

Again by using $\psi={\psi}_{0}r$ in equation (43), we get the expression for
the electric field as,

$E^{2}=\frac{1}{\pi}\left(\frac{a}{4r^{2}}\right).$ (49)

Using equation (49) in equation (32) we derived charge density as

$\sigma(r)=\frac{\sqrt{a}\tilde{\psi_{0}}}{8{\pi}^{\frac{3}{2}}r}.$ (50)

Here both charge $E$ and charged density $\sigma(r)$ are inversely
proportional to $r^{2}$, while charge does not depend on $\tilde{\psi_{0}}$
but charged density depends on $\tilde{\psi_{0}}$. The active gravitational
mass $M(r)$ can be obtained as the following formula,

$M(r)=\pi\left[\frac{2br^{3}}{3}+4ar(1-\tilde{\psi_{0}}^{2}r^{2})\right].$
(51)

As $r$ approaches zero, the mass function $M(r)$ also approaches zero. This
means the central singularity does not affect the mass function $M(r)$. We
obtain the lower bound for the active gravitational mass, $M(r)>\frac{10\pi
ar}{3}$, using the bound for $\tilde{\psi_{0}}$.

###  VI.2 Thin shell of the charged gravastar

In this region, we assume that the thin shell contains a stiff perfect fluid
that obeys the EoS

$p=\rho.$ (52)

This EoS is a particular case of a barotropic EoS with $p=\omega\rho$ and
$\omega=1$. Barotropic fluids are those in which the pressure only depends on
the density, i.e., $p=p(\rho)$, and vice versa. They are thought to be
implausible, but their simplicity has an educational advantage in showing the
many methods employed to solve various systems and “physically” fascinating
problems. In this regard, we would like to point out that Zel’dovich [34]
first proposed the concept of this type of fluid in relation to a cold
baryonic cosmos and it was described as stiff fluid. The spherical collapse of
an over-density of a barotropic fluid with a linear equation of state in a
cosmic background was examined by Staelens et al. [35] in their study. By
assuming an isotropic pressure with and without the cosmological constant
$\Lambda$, Rahaman and his collaborators [36] were able to create a novel
class of exact interior solutions in $(2+1)$-dimensional space-time using the
barotropic EoS. The stiff fluid model has been used earlier by several
researchers in the field of astrophysics as well as cosmology [37, 38, 39,
40].

We can obtain the following ODE by using equation (52) together with equations
(41) and (42).

$\frac{2r\psi{\psi}^{\prime}}{c_{3}^{2}}+\frac{{\psi}^{2}}{{c_{3}}^{2}}=\frac{1}{2}\left(1+\frac{br^{2}}{a}\right).$
(53)

The aforementioned ODE is a linear equation in $\psi^{2}$, and when
integrated, it yields

$\begin{gathered}\psi(r)=\frac{\sqrt{3ar{c_{3}}^{2}+6ac_{4}+br^{3}{c_{3}}^{2}}}{\sqrt{6}\sqrt{a}\sqrt{r}},\\\
\implies\frac{{\psi}^{2}[r]}{c_{3}^{2}}=\frac{1}{2}+\frac{b}{6a}r^{2}-\frac{\chi}{r}\end{gathered}$
(54)

where $\chi=-\frac{c_{4}}{c_{3}^{2}}$ is a constant of integration.

Utilizing the equation (54) in equations (37) and (38) we achieve the metric
potential as

$e^{-B(r)}=\frac{1}{2}+\frac{br^{2}}{6a}-\frac{\chi}{r},$ (55)

$e^{A(r)}=c_{2}^{2}r^{2},$ (56)

Figures (1) and (2) show the profiles of the metric coefficients inside the
thin shell.
Furthermore, by plugging the EoS $p=\rho$ and using equation (53) into the
equations (41) and (42) we have obtained the expressions for matter density
and isotropic pressure inside the thin shell as,

$\rho=\frac{a}{2r^{2}}-\frac{3a\chi}{2r^{3}}=p.$ (57)

The electric field and charge density of the thin shell are derived as,

$E^{2}=\frac{3a\chi}{4\pi r^{3}},$ (58)

and

$\sigma(r)=\frac{1}{16\pi r^{2}}\sqrt{\frac{3a\chi}{\pi
r}}\sqrt{\frac{1}{2}+\frac{br^{2}}{6a}-\frac{\chi}{r}}.$ (59)

Electric field $E$ depends on $\chi$ and is inversely proportional to
$r^{\frac{3}{2}}$. Similar to the previous instance, the central singularity
affects both pressure and density. From equation (59), we have
$\chi<\frac{(3a+br^{2})r}{6a}$. Thus, we get $0<\chi<\frac{(3a+br^{2})r}{6a}$
from the above two cases.

Figure 1: Metric coefficient $e^{A}$ inside the thin shell Figure 2: Metric
coefficient $e^{-B(r)}$ inside the thin shell

###  VI.3 Exterior region of charged gravastar

It is assumed that the charged gravastar’s exterior obeys the EoS $p=0$,
proving that the exterior of the shell is totally vacuum-sealed. The exterior
space-time is described by the Reissner-Nordstrom line element given by,

$ds^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (60)

where $M$ and $Q$ represent the mass and charge of the gravastar,
respectively.

###  VI.4 Boundary Condition

In the gravastar configuration, there are two junctions or interfaces. Let’s
call the junction between the intermediate thin shell (at $r=r_{1}$) and
interior space-time junction-$I$ and the junction between the intermediate
thin shell and external space-time (at $r=r_{2}$) junction-$II$. For any
stable arrangement, the metric functions at these interfaces must be
continuous. To determine the values of the unknown constants of our current
study, such as $c_{2}$, $\tilde{\psi_{0}}$, and $\chi$, we matched the metric
functions at these borders.

  * •

Junction-I:

${\tilde{\psi_{0}}}^{2}r_{1}^{2}=\frac{1}{2}+\frac{br_{1}^{2}}{6a}-\frac{\chi}{r_{1}}.$
(61)

  * •

Junction-II:

$c_{2}^{2}r_{2}^{2}=-\left(1-\frac{2M}{r_{2}}+\frac{Q^{2}}{r_{2}^{2}}\right),$
(62)
$\frac{1}{2}+\frac{br_{2}^{2}}{6a}-\frac{\chi}{r_{2}}=-\left(1-\frac{2M}{r_{2}}+\frac{Q^{2}}{r_{2}^{2}}\right).$
(63)

  * •

Obtained Constants :

$c_{2}=\frac{1}{r_{2}}\sqrt{-1+\frac{2M}{r_{2}}-\frac{Q^{2}}{r_{2}^{2}}}.$
(64)

$\chi=\frac{3r_{2}}{2}-2M+\frac{Q^{2}}{r_{2}}+\frac{br_{2}^{3}}{6a}.$ (65)

$\tilde{\psi_{0}}=\frac{1}{r_{1}}\sqrt{\left(\frac{1}{2}+\frac{br_{1}^{2}}{6a}-\frac{\chi}{r_{1}}\right)}.$
(66)

We use the mass of the gravastar, $M=6M_{\odot}$, inner and outer radii,
$r_{1}=10km$, $r_{2}=10.009km$, and $Q=0.002$ (in natural units) to calculate
the values of these constants. Table-I lists the numerical values of
$c_{2},\chi$, and $\tilde{\psi_{0}}$ for various values of model parameter $a$
and $b$ .

Because of the surface tensions at the time-like interfaces $r_{1}$ and
$r_{2}$, the extremely cold radiation fluid in the shell is constrained to the
area-$II$ [1, 2]. It is important to keep in mind that the hollow sphere with
inner and outer radii of $r_{1}$ and $r_{2}=r_{1}+\epsilon$ being considered
here has $\epsilon<<r_{1}$, and according to Mazur and Mottola [1, 2],
$\epsilon$ does not exceed the Planck length when we are matching our interior
space-time to the exterior Reissner-Nordstrom space-time. Our inside region
and the external line element are matched at the boundary. Although it is
clear that the metric coefficients are continuous at $r=a$, this does not
imply that the derivatives of these coefficients are similarly continuous at
the junction surface.

Table 1: Different numerical values of constants assuming $r_{1}=10\,$ km and $r_{2}=10.009$ km, $Q=0.002$. $a$ | $b$ | $c_{2}$ | $\chi$ | $\tilde{\psi_{0}}$
---|---|---|---|---
$-4.2$ | $3.2$ | $0.0445604$ | $0.0445604$ | $0.0482666$
$-3.2$ | $2.2$ | $0.0445604$ | $-111.879$ | $0.0479182$
$-2.5$ | $2.3$ | $0.0445604$ | $-150.734$ | $0.0489987$
$-1.2$ | $1.2$ | $0.0445604$ | $-164.104$ | $0.049365$
$-0.5$ | $1.5$ | $0.0445604$ | $-498.338$ | $0.057773$

##  VII Junction Condition

###  VII.1 Thin shell around gravastar

Even though we have already discussed that the reason to take the EoS of the
thin-shell to be 1 is to match with the black hole entropy formula. We must
note that the thin shell has a nonzero dimension which we have also
calculated. However, in order to get the potential across the thin shell we
note that we have to use the Israel junction condition, which is only valid if
the shell is really thin (mathematically 1 less dimension hypersurface has
been embedded on the manifold). We also note that the following assumption is
justified as beyond shell when we go to de Sitter space inside we expect this
to happen via a quantum phase transition mechanism so the shell is barely a
junction between two phases it can not be ”thick” otherwise it would also be
affected during the phase transition but it definitely exists for the reasons
explained in the introduction.
Study of Junction condition was first done by Sen [41]. Later Lanczos [42]
made several comments about the problem with Den’s approach and how to modify
it. It was Darmois [43] who first pointed out the exact junction condition but
in a rather convoluted way. In this paper, we follow Israel’s prescription
[44, 45] which is far more simple and based on sound physical grounds.
As mentioned earlier we use the Israel junction condition to obtain the
solutions because there are two different metrics across the thin shell that
must match the condition along the boundary (generally, we use the junction
condition because the metric along hypersurfaces must be continuous as well as
differentiable; as a result, we check both the Christoffel symbol and Riemann
curvature tensor to see what the boundary condition leads to, because in thin
shell we have to use the discontinuity across the junction to find the
potential across the thin shell.)

We also observe that $\sum$ would stand for the thin shell or three-manifold.
Reissner-Nordstrom solutions outside of the thin shell are designated by
$\vartheta^{+}$, and inside, there is a $\vartheta^{-}$, and the whole space-
time would be $\vartheta^{+}\cup\sum\cup\vartheta^{-}$. We can also
concentrate on the surface energy density $\varsigma$ and surface pressure
$P$.This can be done as follows:

We have the interior solutions

$ds^{2}=-e^{A(r)}dt^{2}+e^{B(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$

For the exterior solutions, we take the Reissner-Nordstrom solution of the
form

$ds^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)}\\\
+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).$

Here, we note that because of Birkhoff’s theorem outside metric of a static
spherically symmetric gravastar is always the Reissner-Nordstrom metric. We
also note that in both cases, the space-like component is spherically
symmetric, and on the boundary, we can obtain the FLRW metric

$ds^{2}=-d\tau^{2}+\textbf{a}(t)d\Omega^{2}.$ (67)

Now, if we use the first junction condition’s formula, we obtain

$K^{\pm}_{ij}=-n^{\pm}_{\nu}\left(\frac{\partial^{2}x^{\nu}}{\partial\phi^{i}\partial\phi^{j}}+\Gamma^{l}_{km}\frac{\partial
x^{l}}{\partial\phi^{i}}\frac{\partial x^{m}}{\partial\phi^{j}}\right),$ (68)

where $n^{\pm}$ stands for the two-sided unit normal to the surface and $\phi$
stands for the intrinsic co-ordinate in the shell area, which may be
represented as,

$n^{\pm}=\pm\left|g^{lm}\frac{\partial f}{\partial x^{l}}\frac{\partial
f}{\partial x^{m}}\right|^{-1/2}\frac{\partial f}{\partial x^{\nu}},$ (69)

with $n^{\gamma}n_{\gamma}=1$.

We also note that for the second Israel junction condition we can split the
$T_{\mu\nu}$ as
$T_{\mu\nu}=\Theta(l)T_{\mu\nu}^{+}+\Theta(-l)T_{\mu\nu}^{-}+\delta(l)S_{\mu\nu}$
[46], where $S_{\mu\nu}$ denotes the stress-energy tensor for the shell.

Now, to evaluate the surface stress and pressure for the thin shell to sustain
itself, we use the Lanczos equation

$S_{ij}=-\frac{1}{8\pi}(k_{ij}-\delta_{ij}k_{\gamma\gamma}).$ (70)

Where $i,j=0,2,3$, since at the shell, $r$ is constant. The surface energy
tensor can be expressed as $S_{ij}=diag(-\varsigma,P)$. So, the surface energy
density $\varsigma$ and pressure $P$ at the junction surface $r=\textbf{a}$
can be obtained by the following equations:

$\varsigma=-\frac{1}{4\pi\textbf{a}}\left[\sqrt{f}\right]^{+}_{-},$ (71)

and

$P=-\frac{\varsigma}{2}+\frac{1}{16\pi}\left[\frac{f^{\prime}}{\sqrt{f}}\right]^{+}_{-}.$
(72)

Based on the equations (71) and (72), we can derive the expressions for the
aforementioned quantities,

$\varsigma=\left(-\frac{1}{{4\pi\textbf{a}}}\right)\left(\sqrt{-1+\frac{2M}{\textbf{a}}-\frac{Q^{2}}{\textbf{a}^{2}}}-{\textbf{a}\tilde{\psi_{0}}}\right),$
(73)
$P=\frac{1}{8\pi\textbf{a}}\left(\frac{-1+\frac{M}{\textbf{a}}}{\sqrt{-1+\frac{2M}{\textbf{a}}-\frac{Q^{2}}{\textbf{a}^{2}}}}-2\textbf{a}\tilde{\psi_{0}}\right).$
(74)

Figures (3) and (4) respectively show the profiles of surface energy density
and surface pressure inside the thin shell.

The mass of the thin shell can be obtained using the equation for the surface
energy density given by,

$m_{shell}=4\pi\textbf{a}^{2}\varsigma=\textbf{a}^{2}\tilde{\psi_{0}}-\textbf{a}\sqrt{-1+\frac{2M}{\textbf{a}}-\frac{Q^{2}}{\textbf{a}^{2}}}.$
(75)

As a result, the total mass of charged gravastar in terms of $m_{shell}$ takes
the form

$M=\frac{\textbf{a}}{2}\left(1+\frac{Q^{2}}{\textbf{a}^{2}}+\frac{m_{shell}^{2}}{\textbf{a}^{2}}+{\tilde{\psi_{0}}}^{2}\textbf{a}^{2}-2m_{shell}\tilde{\psi_{0}}\right).$
(76)

We note that by following the prescription given by [47, 48]. We can also go
further and calculate the potential $V(r)$ by noting that the energy-momentum
has a conservation relation:

$\frac{d}{d\tau}(\varsigma\phi)=p\frac{d\phi}{d\tau}=0\,,$ (77)

where $\phi=4\pi\textbf{a}^{2}$. From the conservation equation above one can
find

$\varsigma^{\prime}=-\frac{2}{\textbf{a}}(\varsigma+p)\,.$ (78)

Following the prescription given in [49], we note that the last equation has
the form of $\dot{\textbf{a}}^{2}+V(\textbf{a})$, so from the above equation
we can get,

$V(\textbf{a})=\frac{f(\textbf{a})}{2}+\frac{F(\textbf{a})}{2}-\frac{(f(\textbf{a})-F(\textbf{a}))^{2}}{64\textbf{a}^{2}\pi^{2}\varsigma^{2}}-4\textbf{a}^{2}\pi^{2}\varsigma^{2}.$
(79)

We also note that in our case $f(r)$ and $F(r)$ are given by the following,

$f(r)=\tilde{\psi}_{o}^{2}r^{2}$

and

$F(r)=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)$

We note that outside the thin shell, we can take the Reissner-Nordstrom
solution as discussed earlier.

Figure 3: Variation of surface energy density inside the thin shell Figure 4:
Variation of surface pressure inside the thin shell Figure 5: Thin-shell
potential across the shell.

We note that the above minimum or the stable circular orbit goes out for
higher values of $Q$ we also note that when $Q$ is very small the potential
profile is very similar to the Schwarzschild solutions as we expect when $Q=0$
Reissner-Nordstrom solution becomes Schwarzschild solution.

###  VII.2 Deflection angle for thin shell around gravastar

It is well known that a massive body acts as a convex lens when light passes
around it. As the path of light follows the null geodesics and around massive
bodies due to nontrivial metric the null geodesic equation does change.
Historically gravitational lensing is the first experimental test of general
relativity (1919). Nowadays weak lensing is an essential tool for studying the
galaxy cluster.
In this paper, we will try to show that if a sufficiently high energy photon
created in the interior region can pass through the thin shell (it is
physically possible as there is no event horizon), one can use a next-
generation Event Horizon like radio telescope to detect such red-shifted
photon. We hope in the future one can find the shadow of the inner region of
the gravastar just like it has been found in black hole. For a given metric
there are several ways to find the deflection angle. One of them is to use the
Hamilton-Jacobi equation for a particle with charge $q$ and mass $m$ given by,

$g^{ij}\left(\frac{\partial S}{\partial
x^{i}}+qA_{i}\right)\left(\frac{\partial S}{\partial
x^{k}}+qA_{k}\right)+m^{2}=0$

We note that one can integrate the equation for $m=0$, and $q=0$ as we are
interested in the photon path. However if one uses the usual null geodesic
equation for the following metric,

$ds^{2}=-C(r)dt^{2}+D(r)dr^{2}+F(r)(d\theta^{2}+\sin^{2}\theta d\phi^{2}).$
(80)

We use the prescription given by Virbhadra et al. [50], to calculated the
deflection angle as,

$\alpha(r_{0})=I(r_{0})-\pi,$ (81)

where

$I(r_{0})=\int^{\infty}_{r_{0}}\frac{2\sqrt{D(r)}dr}{\sqrt{F(r)}\sqrt{\frac{F(r)C(r_{0})}{F(r_{0})C(r)}-1}}.$
(82)

For the interior region,

$C(r)=e^{A(r)}=\frac{r^{2}}{{r_{2}}^{2}}\left(-1+\frac{2M}{r_{2}}-\frac{Q^{2}}{{r_{2}}^{2}}\right),\\\
D(r)=e^{B(r)}=\left(-\frac{Q^{2}}{r_{1}^{3}}+\frac{r^{2}}{2r_{1}^{2}}+\frac{br^{2}}{6\textbf{a}}-\frac{3r^{2}r_{2}}{2r_{1}^{3}}+\frac{2Mr^{2}}{r_{1}^{3}}-\frac{br^{2}r_{2}^{3}}{6\textbf{a}r_{1}^{3}}\right)^{-1},\,\,F(r)=r^{2}$
(83)

Now the above equation becomes,

$I(r_{0})_{inside}=2\bigint\limits^{\infty}_{r_{0}}\frac{\left(-\frac{Q^{2}}{r_{1}^{3}}+\frac{r^{2}}{2r_{1}^{2}}+\frac{br^{2}}{6\textbf{a}}-\frac{3r^{2}r_{2}}{2r_{1}^{3}}+\frac{2Mr^{2}}{r_{1}^{3}}-\frac{br^{2}r_{2}^{3}}{6\textbf{a}r_{1}^{3}}\right)^{-\frac{1}{2}}}{r\sqrt{\left(\frac{r}{r_{0}}\right)^{2}\left[\frac{\left[\left(\frac{r}{r_{2}}\right)^{2}(-1+\frac{2M}{r_{2}}-\frac{Q^{2}}{r_{2}^{2}})\right]_{r=r_{0}}}{\left(\frac{r}{r_{2}}\right)^{2}\left(-1+\frac{2M}{r_{2}}-\frac{Q^{2}}{r_{2}^{2}}\right)}\right]-1}}dr.$
(84)

For the exterior region,

$C(r)=-1+\frac{2M}{r}-\frac{Q^{2}}{r^{2}},\,\,\,D(r)=\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1},\,\,\,F(r)=r^{2}.$
(85)

$\begin{gathered}I(r_{0})_{outside}=2\bigint\limits^{\infty}_{r_{0}}\frac{dr}{r\sqrt{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}\sqrt{(\frac{r}{r_{0}})^{2}\left(\frac{-1+\frac{2M}{r_{0}}-\frac{Q^{2}}{{r_{0}}^{2}}}{-1+\frac{2M}{r}-\frac{Q^{2}}{r^{2}}}-1\right)}}=2\bigint\limits^{\infty}_{r_{0}}\frac{\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-\frac{1}{2}}}{r\sqrt{\left(\frac{r^{4}}{r_{0}^{4}}\right)\left(\frac{-r_{0}^{2}+2Mr_{0}-Q^{2}}{-r^{2}+2Mr-Q^{2}}\right)-1}}dr.\end{gathered}$
(86)

So from the formula of $I(r_{0})_{outside}$, we can see that it is exactly
similar to the Reissner-Nordstrom black hole. However, the inside is
different, as there is no event horizon so if a red-shifted photon can get
deflected from inside through the shell, we can expect future radio telescopes
can give a resolution between the black hole and gravastar.

##  VIII Physical features of the model

In this section, we will discuss some of the physical characteristics of the
constructed structure, including the equation of state, the proper length, the
entropy, and the energy levels within the shell’s region. The stiff perfect
fluid travels along these space-times through the shell region of the
gravastar because the constructed geometry of the gravastar is the matching of
two distinct space-times. It will also examine how the electromagnetic field
affects several physical characteristics of the charged gravastar in the
$f(\mathcal{Q})$ gravity.

###  VIII.1 Proper length of the shell

According to the theories of Mazur and Mottola [1, 2], the stiff fluid of the
shell is situated between the intersection of two space-times. The phase
boundary between the inner space and the intermediate thin shell is at
$r_{1}=d$, and the length of the shell extends up to $r_{2}=d+\epsilon$, which
is the phase border between the outer space and the intermediate thin shell.
Therefore, the required length or appropriate thickness of the shell as well
as the appropriate thickness between these two interfaces can be determined
using the following formula:

$\begin{gathered}\textit{l}=\int^{d+\epsilon}_{d}\sqrt{e^{B(r)}}dr\\\
=\int^{d+\epsilon}_{d}\frac{1}{\sqrt{\frac{1}{2}+\frac{br^{2}}{6a}-\frac{\chi}{r}}}dr=\int^{d+\epsilon}_{d}\frac{1}{f(r)},\end{gathered}$
(87)

where $f(r)=\sqrt{\frac{1}{2}+\frac{br^{2}}{6a}-\frac{\chi}{r}}$.
One can note that the above formula just restates the $ds^{2}$ formula for a
metric so could be used to find any proper length given the metric.

Evaluating the integral given in equation (87) poses a significant challenge
at present. So to solve the above integral, let us take
$\frac{df(r)}{dr}=\frac{1}{f(r)}$. Hence we get,

$\textit{l}=f(d+\epsilon)-f(d).$ (88)

From equation (88), by keeping the linear order of $\epsilon$ and expanding
$f(d+\epsilon)$ in the Taylor series around ‘$d$’ we get

$\textit{l}=\epsilon\frac{df(r)}{dr}\approx\frac{\epsilon}{\sqrt{\frac{1}{2}+\frac{br^{2}}{6a}-\frac{\chi}{r}}}.$
(89)

The higher-order terms of $\epsilon$ can be disregarded since the value of
$\epsilon$ is so small. Fig-(6) displays the proper length variation with
regard to the thin shell radius.

Figure 6: Variation of proper length inside the thin shell

###  VIII.2 Energy

The energy of the shell can be calculated by the formula [33],

$E=\int^{d+\epsilon}_{d}4\pi r^{2}(\rho+2\pi E^{2})dr=2\pi a\epsilon,\ $ (90)

In this, the energy and shell thickness are directly correlated. The unit of
energy is also “km” because, as we can see from equation (90), it is precisely
proportional to the thickness of the shell. Fig-(7) displays the nature of
energy inside the thin shell for different values of $a$.

Figure 7:  Variation of energy inside the thin shell

###  VIII.3 Entropy

Mazur and Mottola [1] studied the inner region and came to the conclusion that
the entropy density is zero, representing the dependability of the condensate
phase. The entropy of a charged gravastar in the intermediate region is
computed as [51]

$\mathcal{S}=\int^{d+\epsilon}_{d}4\pi r^{2}s(r)\sqrt{e^{B(r)}}dr.$ (91)

The entropy density $s(r)$ can be defined as follows using the standard
thermodynamic relation $T\times s=p+\rho$ for a relativistic fluid with zero
chemical potential and at the local temperature $T(r)$:

$s(r)=\frac{\kappa^{2}K_{B}^{2}T(r)}{4\pi\hbar^{2}}=\kappa\left(\frac{K_{B}}{\hbar}\right)\sqrt{\frac{p(r)}{2\pi}},$
(92)

where $\kappa$ is a dimensionless parameter.
Here we followed the treatment and argument given by Mazur and Mottola [1]. We
note that from the expression of $T\times s$ it is clear that in the core
$\rho=-p$ the entropy is zero so the entire contribution of entropy is coming
from the shell, which has been assumed to be a single quantum state (like
Bose-Einstein condensed ground state) which gives contribution to gravastar.
Now equation (91) becomes

$\mathcal{S}=\int^{d+\epsilon}_{d}2\sqrt{2\pi}\kappa\left(\frac{K_{B}}{\hbar}\right)ar\sqrt{\frac{3(r-3\chi)}{3ar+br^{3}-6a\chi}}\,dr.$
(93)

Equation (93) can also be written as

$\mathcal{S}=2\sqrt{2\pi}\kappa\left(\frac{K_{B}}{\hbar}\right)ar\mathcal{J},$
(94)

where

$\mathcal{J}=\int^{d+\epsilon}_{d}\mathcal{D}(r)\,dr,$ (95)

and

$\mathcal{D}(r)=\sqrt{\frac{3(r-3\chi)}{3ar+br^{3}-6a\chi}}.$ (96)

Evaluating the integral given in equation (95) poses a significant challenge
at present, so to solve the above integral let us take $F(r)$ to be the
primitive of $D(r)$. Then equation (95) becomes by using the fundamental
theorem of integral calculus:

$\mathcal{J}=[F(r)]^{d+\epsilon}_{d}=F(d+\epsilon)-F(d).$ (97)

Expanding $F(d+\epsilon)$ in the Taylor series around ‘$d$’ while keeping the
linear order of $\epsilon$ from equation (97), we obtain, from (93), that

$\mathcal{S}=2\sqrt{2\pi}\kappa a\left(\frac{K_{B}}{\hbar}\right)r\times
d\times\epsilon\sqrt{\frac{3(r-3\chi)}{3ar+br^{3}-6a\chi}}.$ (98)

As a result, we were successful in obtaining the entropy expression for the
model we have suggested. One can infer from equation(98) that if the thickness
of the thin shell is $\epsilon<<d$, then
$\mathcal{S}\approx\mathcal{O}(\epsilon)$. Fig-(8) displays the variation of
entropy with regard to the thin shell radius.

Figure 8: Variation of entropy inside the shell versus thickness of the shell
of charged gravastar

###  VIII.4 The EoS parameter

The equation of state parameter $\omega$ can be written as [51]

$\omega=\frac{P}{\varsigma}.$ (99)

Now by using equations (73) and (74), we obtain

$\omega=\frac{1}{2}\left[\frac{\frac{-1+\frac{M}{\textbf{a}}}{\sqrt{-1+\frac{2M}{\textbf{a}}-\frac{Q^{2}}{\textbf{a}^{2}}}}-2\textbf{{a}}\tilde{\psi_{0}}}{\left(\textbf{a}\tilde{\psi_{0}}-\sqrt{-1+\frac{2M}{\textbf{a}}-\frac{Q^{2}}{\textbf{a}^{2}}}\right)}\right].$
(100)

The limitation $\frac{2M}{\textbf{a}}-\frac{Q^{2}}{\textbf{a}^{2}}<1$, which
is already satisfied by equation (60), is necessary to maintain the realness
of $\omega$. Now, depending on the signature of the numerator or denominator
of equation(100), $\omega$ may be positive or negative. Fig. (9) displays the
profiles of $\omega$ against ‘a’.

Figure 9: Variation of EoS $\omega$ with respect to a

##  IX ADM mass

We note that mass in general relativity can be a tricky concept as the
gravitation field itself interacts with itself making it subtle to define how
to define for general context. However, after the ADM formulation [52] or the
Hamiltonian formulation of general relativity came it became somewhat clear
that if we take the Hamiltonian and integrate it through space like Cauchy
hypersurface we might be able to define a mass in such a way that incorporates
the gravitational interactions. Note that an asymptotic flatness condition is
needed for such a thing to happen and as we can see at $r\rightarrow\infty$
our Reissner-Nordstrom solution does indeed go to Minkowski metric, so we can
be assured that we can indeed do the ADM mass formula. We note that the ADM
formulation in $f(Q)$ gravity has already been done in [53] and authors have
shown the equivalency of $f(Q)$ gravity and ordinary general relativity.
We have calculated the ADM mass for each of the two components of the
gravastar which is an interior shell and outer region (as a thin shell in
principle has a negligible thickness in this case so we cannot choose a space
like Cauchy surface for $3+1$ splitting). The reason to consider the ADM mass
for gravastar is the fact that we want to distinguish it from the black hole.
As we will show the ADM mass for the charged gravastar is indeed different
from the charged black hole due to its interior being de Sitter space.
We note that the formula for the ADM mass is given by [54],

$M_{ADM}=\frac{1}{16\pi}\lim\limits_{r\rightarrow\infty}\sum_{\mu,\nu=1}^{3}\int_{S}(\partial_{\mu}h_{\mu\nu}-\partial_{\nu}h_{\mu\mu})N^{\nu}dS.$
(101)

where $h_{\mu\nu}$ is the three-dimensional metric from the space like hyper-
surface at $t$ being some constant time. $S$ is the two-sphere on which the
integral has been performed and $N$ is the outward normal.
Now our metric is given in equation (60) is in a 4-dimensional space-time
manifold, over a constant time slice $\Sigma$, the embedded metric takes the
form

$h=ds^{2}_{\Sigma}=\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\theta^{2}\\\
+r^{2}\text{sin}^{2}\theta d\Phi^{2}.$ (102)

Now, in general, the integral in the equation (101) is quite difficult, but
for spherical symmetric metric, it is quite easy as it is done in [55].
The calculation is pretty involved as one has to switch from polar coordinates
to Cartesian coordinates and use the symmetry property to get the answer. Here
we mention the final solution following the prescription given in [55], for a
space-time metric is given by,

$g=\varphi dr^{2}+\chi(r)r^{2}d\Omega^{2}.$ (103)

Also given that $\varphi$ and $\chi$ reach the asymptotic flat space-time
limit as $\varphi-1=o(r^{-\frac{1}{2}})$, $\chi-1=o(r^{-\frac{1}{2}})$ and
$\partial_{r}\varphi=o(r^{-\frac{3}{2}})$,
$\partial_{r}\chi=o(r^{-\frac{3}{2}}).$ Then $M_{ADM}$ is given by,

$M_{ADM}=\lim\limits_{r\to\infty}\frac{1}{2}(-r^{2}\chi^{\prime}+r(\varphi-\chi)).$
(104)

For our case in outside, we take the Reissner-Nordstrom solution so we get
$\varphi(r)=\frac{1}{\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)}$ and
$\chi(r)=1$, so calculating the limit we get,

$\begin{gathered}M_{ADM}=\lim\limits_{r\to\infty}\frac{1}{2}\left[-r^{2}\chi^{\prime}+r(\varphi-\chi)\right]\\\
=\lim\limits_{r\to\infty}\frac{1}{2}r\left[\frac{1}{\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)}-1\right]\\\
=\lim\limits_{r\to\infty}r\left[\frac{\frac{2M}{r}-\frac{Q^{2}}{r^{2}}}{2}\right]\\\
=M-\frac{Q^{2}}{2r}\end{gathered}$ (105)

So, we get the contribution for ADM mass due to the outer region of the thin
shell is given by,

$M_{ADM}=M-\frac{Q^{2}}{2r}.$ (106)

We know that to coincide with the Schwarzschild solution, $Q=0$ as it should
happen. Also, we note that the presence of charge $Q$ can be seen in the
formula for $M_{ADM}$.
Similarly, ADM mass in the interior region can be found by a similar formula
as we note that from equation (83) we get,

$e^{B(r)}=\left(-\frac{Q^{2}}{r_{1}^{3}}+\frac{r^{2}}{2r_{1}^{2}}+\frac{br^{2}}{6\textbf{a}}-\frac{3r^{2}r_{2}}{2r_{1}^{3}}+\frac{2Mr^{2}}{r_{1}^{3}}-\frac{br^{2}r_{2}^{3}}{6\textbf{a}r_{1}^{3}}\right)^{-1}.$

So we can see the radial component of the metric is given in the form
$e^{b(r)}=C(1+Dr^{2})^{-1}$, where $C$ and $D$ are functions made up of the
constants $r_{1}$, $r_{2}$, a, $Q$ etc., as given in the above equation.
For inside the shell if we do the ADM mass calculation by using equation (104)
$\lim\limits_{r\to\infty}\frac{1}{2}r\left[\frac{C}{\left(1+Dr^{2}\right)}-1\right]$.
We can see that the limit diverges for the arbitrary values of $D$, which is
not at all unexpected as the Schwarzschild-de Sitter metric’s radial
coordinate has the form $\frac{1}{1-\frac{2M}{r}-\frac{1}{3}\Lambda r^{2}}$,
here also similarly the ADM mass diverse as this is not asymptotically flat.
This shows that for the interior region, ADM mass does not make sense as the
metric is not asymptotically flat, as we have also taken de Sitter space-time
inside.

##  X Discussion and Conclusion

In this article, we have studied the analytical solution of charged gravastar
in $f(\mathcal{Q})$ gravity. As we have mentioned motivated by the black hole
“no-hair” theorem we have taken charged gravastar as we are modelling
gravastar as an alternative to the black hole after a gravitational collapse.
We have started with the three regions of gravastar which are interior
(EoS=-1), shell (EoS=1), and exterior regions ( Reissner-Nordstrom metric as
there is a non-zero charge). Using two junction conditions we have found the
analytic solution to the metric coefficient which is drawn in fig-(1) and
fig-(2). We also note that as the other region is Reissner-Nordstrom space-
time so it is asymptotically flat (i.e. at $r\rightarrow\infty$ the metric
becomes Minkowski), one needs this to find the ADM mass of a system. As our
solutions come with arbitrary integration constants, and we are keeping the
discussion for generally charged gravastar in table-I we have taken some
typical values of stellar mass black holes for bookkeeping.
In the next part, we have taken the Israel junction condition for the thin
shell (neglecting the thickness) and found out the energy density and pressure
corresponding to the thin shell which we have plotted in fig-(3) and fig- (4)
respectively. We have also calculated the potential across the thin shell and
we note that it can be used as a phenomenological probe to detect not just
gravastar but to distinguish between black hole and gravastar. We can see from
the fig-(5) that the potential does have a varying $r_{min}$ with charge and
after a certain critical charge, the min of $r_{min}$ vanishes so this denotes
the ISCO (innermost stable circular orbit). It is widely known that ISCO has
been a very successful tool for finding black hole accretion disks and
rotation etc. [56]. So one can in principle distinguish between gravastar and
black hole by shifting of Fe (iron) line and measuring the ISCO which is
different from the Reissner-Nordstrom black hole.
We also provide another phenomenological observation that in principle can
distinguish between black holes and gravastars. We have calculated the angle
of deflection for the gravastar interior region and exterior region, even
though the exterior region equation (86) $I(r_{0})$ is the same as that of the
Reissner-Nordstrom black hole, but we can see equation (84)
$I(r_{0})_{inside}$ is much different from that of the Reissner-Nordstrom
black hole. In simple words, as gravastar does not have any event horizon it
is expected that if certain photons can penetrate the shell (which it can as
stiff matter does not violate any causality), we can indeed use a next-
generation event horizon telescope to detect such photons in this way we can
distinguish between the gravastar and black hole.
Finally, in order to investigate the properties of the gravastar we have used
the standard methods developed by [2] to get an overall understanding of the
properties of the gravastar. In fig-(6)the variation of proper length vs
thickness ($\epsilon$) is given as we can see it’s monotonically increasing as
expected. In fig-(7) and fig-(8) we have plotted the functional form of energy
and entropy inside the shell as a function for thickness ($\epsilon$). We have
noted in all three cases that the proper length, energy density, and entropy
have exactly similar features as given by [2]. So we can safely conclude that
our solution of charge gravastar in $f(\mathcal{Q})$ gravity does indeed share
similar characteristics of a gravastar [2].
In fig-(9) we have shown the variation of EoS as a function of a. We see that
for large a it is $-1$ which is expected from the fact we have started with
the de Sitter metric inside. We can also see that as a goes smaller the EoS
state becomes less than $-1$, that is it goes to the phantom region. Even
though in the cosmological context there are problems with dealing with
phantom regions, in this case, it just gets to show that to stabilize the
gravastar we need more than just “Dark Energy”.
We have also calculated the ADM mass sec IX for the gravastar inside and
outside of the shell. We note that the ADM formulation requires $3+1$
splitting of the space-time, as by the assumption the thin shell is really
”thin” to contribute to a $3-$dimensional Cauchy hypersurface. We have noted
that outside the ADM mass is similar to the Reissner-Nordstrom black hole as
expected, but inside it diverges it has to do with the fact inside metric is
de Sitter so it is not asymptotically flat so can not expect the ADM mass to
be finite there.
As far as the observational aspect of the gravastar comes it is a very active
research field where the last words have been said. First, we note that after
detecting the gravitational waveform merging black hole binaries in 2016 [57],
we have officially entered the so-called “multi-messenger era ” of astronomy,
where we can observe our universe using probes very different (like a
neutrino, gravitational wave, etc.) than EM waves (radio telescope, X-ray
telescope, infrared telescope, etc. just to name a few). Definitive
observational proof of gravitational waves indeed gives hope that maybe one
can distinguish gravastar from black holes via GW observation. In the work by
Cardoso et al. [58] it has been shown that to study the event horizon and its
quantum mechanical phenomenon one can use late time ringdown of gravitational
wave. The authors also conclude how the study of late time ringdown can be
used to rule out the alternatives of the black holes like gravastars, bose
stars, etc. The work by Chirenti et al. [59] takes it one step further by
showing it is very difficult to see the difference between the black hole and
gravastar as just like the black hole for gravastar we do not have any results
about the perturbations with respect to some fixed background. The authors
have studied the data from GW150914 to give a heuristic bound on the
confidence by which we can say whether it was a black hole merger or
gravastar.
Apart from gravitational waves, there are other proposals to distinguish
gravastars from black holes as Broderick [60] and Narayan [56] have proposed
that one can do a fallback disk analysis of gravastars just like the
acceleration disk analysis of black hole and the authors also give a bound on
the observational constraints to distinguish gravastars from black holes. We
also note that Uchikata et al. [61] have suggested studying the observational
properties of gravastar from an astrophysical object point of view. As we have
noted earlier gravastar do not have an event horizon so it is natural to
calculate its tidal deformability and I-Love-Q relations for a small
perturbation like a neutron star. The authors have done the calculation for
love number as well as I-Love-Q relation of gravastar under small
perturbations and have given us a phenomenological bound on under what
conditions one can distinguish the gravastar and block hole via observation.
We note that even though this is an exciting era we can ask whether the
gravstars can be distinguished from the black holes by multi-messenger
astronomy. In our work, we propose that one can indeed distinguish them if
some low-energy photons from the interior regions of gravastars can penetrate
the shell and come outside. We can use the angle of deflection formula to see
the difference from the black hole.
We note that after the black hole image formed by the Event Horizon Telescope
[62] it is natural to ask whether such a shadow can be found for gravastar or
not. In our article we have found the deflection angle for the interior region
and hope that such a phenomenological study will help next-generation radio
telescopes to search for more abnormality from black hole shadow and in future
can indeed distinguish between gravastar and black hole.
Finally, we note that also one can test the viability of the various modified
gravity from the shadow of black hole [63], so maybe we can hope future
generation radio telescopes will not just give the shadow of gravastar but
will also provide whether the modification of Einstein gravity by
$f(\mathcal{Q})$ gravity is indeed right choice or not.

Data availability There are no new data associated with this article.

###### Acknowledgements.

DM expresses gratitude to the BITS-Pilani, Hyderabad campus, India, for the
financial support. SG acknowledges Council of Scientific and Industrial
Research (CSIR), Government of India, New Delhi, for junior research
fellowship (File no.09/1026(13105)/2022-EMR-I). PKS acknowledges the National
Board for Higher Mathematics (NBHM) under the Department of Atomic Energy
(DAE), Govt. of India for financial support to carry out the Research project
No.: 02011/3/2022 NBHM(R.P.)/R & D II/2152 Dt.14.02.2022 and IUCAA, Pune,
India for providing support through the visiting Associateship program. We are
very much grateful to the honorable referees and to the editor for the
illuminating suggestions that have significantly improved our work in terms of
research quality, and presentation.

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# Large-scale ASR Domain Adaptation Using Self- and Semi-supervised Learning

###### Abstract

Self- and semi-supervised learning methods have been actively investigated to
reduce labeled training data or enhance model performance. However, these
approaches mostly focus on in-domain performance for public datasets. In this
study, we utilize the combination of self- and semi-supervised learning
methods to solve unseen domain adaptation problems in a large-scale production
setting for online ASR model. This approach demonstrates that using the source
domain data with a small fraction of the target domain data (3%) can recover
the performance gap compared to a full data baseline: 13.5% relative WER
improvement for target domain data.

Index Terms—  speech recognition, domain adaptation, self-supervised learning,
semi-supervised learning, RNN-T

##  1 Introduction

It is natural to have poor speech recognition accuracy with small labeled
dataset. There are 2 major approaches to resolving this problem: self-
supervised learning (self-sup) and semi-supervised learning (semi-sup). Self-
sup pre-trains the audio encoder to learn meaningful representations.
Contrastive Predictive Coding (CPC) [1], Wav2vec [2] and Wav2vec2.0 [3] show
that contrastive loss can be good pre-training objectives. Autoregressive
Predictive Coding (APC) [4] shows that mean squared error (MSE) also can be a
good objective. Semi-sup trains the ASR model using the pseudo labels
generated by a teacher model. Noisy student training (NST) [5] is a popular
semi-supervised learning approach, which also works for ASR [6, 7]. Recently,
it is also shown that combining self- and semi-sup methods improves the
performance of automatic speech recognition (ASR) [8] significantly, leading
to the state-of-art (SOTA) Librispeech Word Error Rate (WER). However, the
current methods focused on maximizing the performance for in-domain data. The
largest dataset of the studies is labeled Librispeech data (1k hours) [9] and
unlabeled LibriLight data (60k hours) [10].

In large-scale production ASR systems, there is often the challenge of
mismatched domain between the training data (source domain) and the real world
data (target domain) [9296327]. It is also common to have an order of
magnitude bigger dataset with various contexts (e.g. command, chat, caption)
than the public dataset like Librispeech. To the best of our knowledge, there
are no extensive studies to tackle large-scale domain adaptation using both
self- and semi-supervised learning.

In this paper, we propose a combined self- and semi-sup approach for domain
adaptation. Our method completely recovers target domain accuracy using only a
small fraction of labeled target data. In addition, the improved
generalization power enhances source domain accuracy as well. The other main
contributions of our work is that we analyze how much self- and semi-sup
contribute on domain adaptation. Self-supervised learning improves overall
model generalization and semi-sup directly closes out-of-domain generalization
gap, which means both are complementary.

##  2 Related works

We use the three popular self-supervised learning methods in this work:
Wav2vec [1, 2], and Wav2vec2.0 [3], APC [4].

The Wav2vec loss maximizes the mutual information between the context latent
vector and the future inputs [2]. Instead of the convolution layers, we use
conformer blocks [DBLP:conf/interspeech/GulatiQCPZYHWZW20] for the context
network. As we directly use the log-mel features instead of the waveform, we
exclude the feature network. The output of the audio encoder directly predicts
the future log-mel features.

The Wav2vec2.0 loss maximizes the mutual information between the context
latent vector and the masked input features [3]. We exclude the feature
network for the same reason. We do not quantize the input features as we do
not observe any performance difference in our experiments.

The APC loss minimizes the MSE between the input log-mel feature and the
predicted log-mel feature by the audio encoder [4]. APC paper uses LSTM or
transformer blocks but we use conformer blocks.

Noisy student training (NST) [6] is one of the most popular semi-sup methods.
It trains a model with both labeled data and unlabeled data. The teacher model
produces pseudo labels of unlabeled data from non-augmented input feature. The
method trains the student model with augmented input feature and the pseudo
labels as the ground truth. The current Librispeech SOTA paper [8] uses NST
with Librispeech data (labeled data) and LibriLight data (unlabeled data).
Unlike image domain, it’s hard to use soft label for sequence model. NST in
speech domain [6, 8] uses hard label.

##  3 Methods

###  3.1 Self-Supervised learning

All of the self-supervised methods are used to pre-train the audio encoder of
the RNN-T model [12] using all the source and target domain data. This is
followed by supervised training of the entire model using only the labeled
source domain data. However, with this 2-stage training approach, it is
difficult to determine the optimal pre-training hyper parameters and
checkpoints because the frame accuracy metrics in self-sup are not always a
good indicator of the final WER performance. To resolve this issue, we propose
a joint train for both RNN-T and self-sup loss:

$\centering{\cal L}={\cal L}_{\tt RNN-T}+\lambda{\cal L}_{\tt self-
sup}\@add@centering$ (1)

where $\lambda$ is a hyper-parameter. We find $\lambda=0.5$ works well.

Wav2vec2.0 is a bi-directional method. When we train it with a causal audio
encoder, it does not converge. Our workaround is to pre-train with right
context and subsequently fine-tune with left context only. Joint Wav2vec2.0
and RNN-T train does not have the issue. The joint train works on a causal
audio encoder, because RNN-T loss can guide Wav2vec2.0 objective. As our RNN-T
model is online, we use a causal audio encoder for all joint experiments. We
add an additional MLP layer between the audio encoder and RNN-T joint layer,
and between the audio encoder and self-supervised loss. The representation of
the last audio encoder layer is specialized by 2 MLP layers for RNN-T loss and
self-supervised loss respectively.

###  3.2 Semi-supervised learning

After self-supervised pre-training, we train the ASR model using RNN-T loss
with both source domain data (labeled data) and target domain data (unlabeled
data) [13]. NST produces pseudo labels for target domain data.

The teacher model is trained with source domain data (same for the student
model). As a result, the pseudo labels generated for the target domain data is
error-prone, which is harmful for domain adaptation. When pseudo labels are
generated, the teacher model filters out low confidence utterances by
Confidence Estimation Module (CEM) [14]. When the teacher model is trained by
RNN-T loss, we add CEM whose inputs are the audio encoding and the beam search
labels from the RNN-T model. The CEM is trained to minimise the binary cross
entropy between the estimated confidence $p$ and the binary target sequence
$c$. The target sequence $c$ contains a 1 when the prediction word is correct
and 0 otherwise. The average word-level confidence is used to filter
utterances. The teacher’s multi-task training objective consists of both the
RNN-T loss and confidence loss.

Fig. 1: Domain adaptation: Self-sup pre-trains the audio encoder. Supervised
and semi-sup train all the modules.

###  3.3 Domain adaptation approach

Figure 1 visualizes the proposed domain adaptation method. First of all, self-
sup trains the audio encoder with source and target domain data. Then, RNN-T
model is trained by RNN-T loss with source and target domain data. NST
produces pseudo label for target domain data.

##  4 Experimental Setup

###  4.1 Model Architecture Details

We use a Conformer RNN-T [12] model in the experiments. The audio encoder has
$17$ Conformer blocks with model dimension $512$. As the model is online ASR,
we restrict the model from using any future information [15]. Each conformer
block [DBLP:conf/interspeech/GulatiQCPZYHWZW20, 15] uses causal convolution
and left-context attention layers. The convolution kernel size is $15$ and the
self-attention layer consists of $8$ heads with $65$ left context length. The
RNN-T decoder consists of a label encoder with $2$ LSTM layers with $2048$
units projected down to $640$ output units, and a joint network with a single
feed-forward layer with $640$ units, as shown in Fig 1. The architecture and
training procedure is similar to the LibriSpeech SOTA WERs model [8]. The
total number of weights is 137M, while that of the SOTA model is 1B.

The model input is a vector of size $528$, consisting of $4$ contiguous frames
of $128$-dimension log-mel features [16] sub-sampled by a factor of $3$ and
one-hot domain-id vector of size $16$. The model label uses $4,096$ word
pieces [17] as sub-word label.

###  4.2 Data Sets

We use large multi-domain (MD) data sets [18] in English. MD utterances
include multi domain data such as search, farfield, telephony and YouTube. All
datasets are anonymized and hand-transcribed except for YT which uses YouTube
video transcription with confidence filter [19]. We divide MD data into source
domain data and target domain data. We pre-train the model using source domain
data by self-sup and transfer it to target domain data by semi-sup. As shown
in Table 1, we use Medium-form (MF) utterances as target domain and MD-MF as
the source domain. MF is average 10.4 secs length in natural conversation
domain. Short-form (SF) is average 4.6 secs length in voice command domain.

For evaluation metrics, we calculate the WER of MF to measure the performance
on target domain and the WER of SF for the performance on the source domain.
The goal of domain adaptation is to minimize the amount of required
transcriptions of MF while maintaining WERs for both MF and SF. In the final
experiments, we use 3% MF as labeled data, because 800 hours data is
manageable amount for hand-transcription. NST produces pseudo labels for rest
of 97% MF data.

Table 1: Overview of training data sets. MF denotes the target domain utterances. MD denotes all domain utterances including MF, SF and YT. MDsrc (MD-MF) denotes the source domain utterances.  Data set | Domain | Hours
---|---|---
Medium-form (MF) | Target |  $26$k
Short-form (SF) | Source |  $27$k
Youtube (YT) | Source |  $226$k
Multi-domain (MD) | Source $\&$ Target |  $400$k
MD-MF (MDsrc) | Source |  $374$k
3$\%$ MF \+ MDsrc (MD3p) | Source $\&$ Target |  $375$k

##  5 Experimental results

In this section, we conduct extensive domain adaptation experiments using
self- and semi-supervised learning.

###  5.1 Self-supervised Learning

In Table 2, we compare the WERs of supervised and self-supervised learning.
The first block lists supervised experiments with MD, MDsrc, and MD3p data. MD
has full MF data. MDsrc doesn’t have any MF data. MD3p has 3% MF data. As
expected, SF WERs are the same but MF WERs are better when more MF data is
used.

The second block lists three different self-sup experiments. One of 3 self-sup
methods pre-trains the audio encoder with MD, and RNN-T loss trains the RNN-T
model with MDsrc. Wav2vec and APC have better WERs on both MF and SF than
Wav2vec2.0, unlike what Wav2vec2.0 paper reported [3]. The downstream ASR
model is online RNN-T, which is a causal model. Wav2vec and APC are causal
models like GPT-3, but Wav2vec2.0 is full context (non-causal) model like
BERT. It shows causal self-sup has better performance for causal downstream
task. Even though Wav2vec and APC have the same WERs, we use Wav2vec for rest
of experiments. In our experience, APC is more sensitive to checkpoint
fluctuations. When we choose a pre-trained checkpoint, Wav2vec works between
50k and 1.2M steps, but APC works only near 100k steps. In addition, APC
requires total variation auxiliary loss to stabilize it [20].

In the third block of Table 2, Wav2vec MD3p enhances WERs for both source
domain (SF, 6.0 to 5.8) and target domain (MF, 3.7 to 3.6), compared to
Supervised MD3p. However, there is huge gap between Supervised MD and Wav2vec
MD3p. Self-sup enhances overall model generalization, but cannot reduce gap of
out-of-domain (OOD) generalization.

Table 2: Comparisons of Word Error Rate (WER) between supervised baselines, three self-sup algorithms. Algorithms | Data | Word Error Rate ($\%$)
---|---|---
|  | MF (Target) | SF (Source)
Supervised | MD | 3.2 | 6.0
Supervised | MDsrc | $6.2$ | $6.0$
Supervised | MD3p | $3.7$ | $6.0$
Wav2vec | MDsrc | ${4.5}$ | $5.8$
Wav2vec2.0 | MDsrc | ${4.6}$ | $5.9$
APC | MDsrc | ${4.5}$ | $5.8$
Wav2vec | MD | 3.2 | 5.8
Wav2vec | MD3p | 3.6 | 5.8

###  5.2 Semi-Supervised Learning

The first block in Table 3 shows semi-sup results. Semi-sup MDsrc denotes
RNN-T training using MDsrc for labeled data and MF for unlabeled data. Semi-
sup MD3p denotes RNN-T training using MD3p for labeled data and 97% MF for
unlabeled data. NST produces the pseudo labels for unlabeled data. Compared to
Supervised MDsrc, Semi-sup MDsrc improves MF (target domain) WER from 6.2 to
3.4. There is only 0.2 % gap between Supervised MD. Semi-sup can reduce most
of gap of OOD generalization, even without target domain data. Semi-sup MD3p
closes the all the gap, whose MF WER is same to the full data baseline: 3.2%.
All semi-supervised experiments have the same SF WER to the baseline as all of
them uses same amount of source domain data.

The teacher model for semi-sup has almost the same architecture as the student
model. One difference for the teacher model is that it is non-causal. The
self-attention right context is $64$ and convolution layer is not causal. The
teacher model uses the same amount of data as the student model. The teacher
model is trained with MDsrc for Semi-sup MDsrc and MD3p for Semi-sup MD3p.

Table 3: WERs for semi-supervised learning. Algorithms | Data | Word Error Rate ($\%$)
---|---|---
|  | MF (Target) | SF (Source)
Semi-sup | MDsrc | ${3.4}$ | $6.0$
Semi-sup | MD3p | 3.2 | $6.0$
Self + Semi-sup | MD3p | 3.1 | 5.7

###  5.3 Self + Semi-Supervised Learning

The second block in Table 3 shows self + semi-sup WERs. In Table 2, Supervised
MD uses all the labels of both source domain (MDsrc) and target domain (MF)
data. Wav2vec MD3p has better SF (source domain) WER than the supervised
baseline because self-sup improves overall generalization as mentioned in
Section 5.1. Semi-sup MD3p has the same MF (target domain) WER to the baseline
because semi-sup resolves OOD generalization as mentioned in Section 5.2.
Combined both self- and semi-sup are complementary. In Table 3, Self + Semi-
sup MD3p show even better WERs for both MF (target domain) and SF (source
domain). We are actually surprised that that Self + Semi-sup with ${3\%}$
target domain has better WERs than supervised learning with ${100\%}$ target
domain. Our domain adaptation method not only reduces all the OOD
generalization gap but also improves source domain performance. Semi-sup plays
a much more critical role to close the OOD gap.

###  5.4 Confidence filter for Semi-Supervised Learning

In Table 3, all semi-supervised experiments use a confidence filter [14] that
filters out target domain (MF) data when the utterance-level confidence score
is less than $0.9$. This results in dropping $24$% of MF utterances and
improves the teacher WER on the target domain by $54$%. Without confidence
filter, both MF and SF WERs are slightly worse.

Using only high-confidence utterances results in higher quality pseudo
labeling, which enhances MF (target domain) WER from 3.2 to 3.1. As the
teacher is trained with MD3p, it tends to be more confident to source-domain-
like SF data. This results in additional SF (source domain) WER improvement
from 5.8 to 5.7. It indicates that confidence score threshold should be tuned
per fraction of target label data.

Table 4: WERs of combined self- and semi-sup with/without confidence filter on MD3p. Algorithms | Word Error Rate ($\%$)
---|---
| MF (Target) | SF (Source)
Self + Semi-sup w/o filter | ${3.2}$ | $5.8$
Self + Semi-sup w filter | ${3.1}$ | $5.7$

###  5.5 Joint RNNT and Self-Sup train

In Fig 3, Joint W2V2 (Green line) reaches WER 4.6 at 200k step, while W2V2
requires 900k steps for both pre-training and supervised learning. Total
training steps is reduced significantly. In addition, Joint W2V2 reaches WER
4.3, which W2V2 never reach. It’s because Joint W2V2 utilizes Wav2vec2.0 loss
on the causal encoder, so there is no transition gap from non-causal to causal
unlike the Wav2vec2.0 experiment. It brings out the full potential of
Wav2vec2.0 objective. However, Joint W2V (Purple line) requires the same steps
to reach the converging point to W2V (Blue line), as both utilize a causal
audio encoder.

In Table 5, Joint W2V + NST and Joint W2V2 + NST has additional semi-sup
stage. They have the same MF (target domain) WER to Self + Semi-sup MD3p in
Table 3. However, their SF (source domain) WER is worse than Self + Semi-sup
MD3p (5.7%). We leave the weaker in-domain generalization power of Joint train
for future study.

Fig. 2: Comparison of WER convergence with and without Joint train for wav2vec and wav2vec2.0.  Table 5: WERs of Joint train with MD3p. Algorithms | Word Error Rate ($\%$)
---|---
| MF (Target) | SF (Source)
Joint W2V | ${4.5}$ | $5.8$
Joint W2V2 | ${4.3}$ | $5.8$
Joint W2V + NST | 3.1 | $5.9$
Joint W2V2 + NST | 3.1 | $5.8$

##  6 Conclusion

In this paper, we investigated a domain adaptation method by combining self-
and semi-supervised learning. Self-sup improves overall generalization and
semi-sup closes out-of-domain generalization gap. Both methods complement each
other. Extensive experimental results demonstrate that the proposed methods
using only 3% of the labeled target domain data obtain better WERs on both the
target and source domains than the baseline.

##  7 Acknowledgements

We thank Hasim Sak, Anshuman Tripathi, Jaeyoung Kim, Qian Zhang, Yonghui Wu,
Ruoming Pang, Yu Zhang, Arun Narayanan, Tony Bruguier, Lillian Zhou, Petr
Zadrazil, Zhehuai Chen, Andrew Rosenberg for helpful discussions.

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##  Appendix A Appendix

###  A.1 Self-sup with X$\%$ MF data

Self-sup enhances WERs when labeled data is less than ${10\%}$ (low resource
regime, less than 2.6k hours). However, self-sup has diminishing returns in
high resource regime.

Fig. 3: Comparison of WER on X$\%$ MF data between supervised baseline and
Wav2vec.

###  A.2 Semi-sup with full MD data

In semi-sup training, ${3\%}$ label (i.e. ${97\%}$ pseudo label) and ${0\%}$
label don’t make any difference. Pseudo label is as good as human
transcription. It means we need only ${3\%}$ label with semi-sup.

Table 6: WERs for semi-supervised learning. Algorithms | Data | Word Error Rate ($\%$)
---|---|---
|  | MF (Target) | SF (Source)
Semi-sup | MDsrc | ${3.4}$ | $6.0$
Semi-sup | MD3p | 3.2 | $6.0$
Semi-sup | MD | 3.2 | $6.0$
Self + Semi-sup | MD3p | 3.1 | 5.7
Self + Semi-sup | MD | 3.1 | 5.7