A Tachyon Field around the Black Hole
Narges<EMAIL_ADDRESS>(Corresponding Author)
a Department of Theoretical Physics, Faculty of Science, University of
Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN
b ICRANet-Mazandaran, University of Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN

###### Abstract

We study the effects of the presence of the tachyon field around the black
hole. We show that in presence of the tachyon field, unlike the ordinary
canonical scalar field, the time evolution of the black hole mass depends on
the potential of this field. By considering several types of potential, we
study the behavior of the black hole mass and its time evolution and find some
interesting results. We find that the presence of the tachyon field causes the
accretion of the mass into the black hole. We also show that with linear and
hilltop potentials, in some ranges of the parameters space, the mass of the
black hole can decrease even without any Hawking radiation.
Key Words: Tachyon Field, Black Hole, Mass Accretion

##  1 Introduction

Given that in the current universe there are many black holes that have
different masses, the study of black holes, as the thermodynamical systems,
has attracted a lot of attention. As regards some massive black holes have
originated from primordial black holes, one of the interesting aspects in the
study of the black holes is the accretion of the matter into the black hole.
Although the standard argument assumes that the mass of the primordial black
holes remains constant once they are formed [1, 2, 3], the authors of Ref. [4]
have shown that this is not necessarily true. They have studied scenarios in
which the accretion of the black holes’ mass occurs. Satisfying the null
energy condition causes that the mass of the black hole never decreases in a
classical way, but even increases [5]. However, by considering the quantum
processes, there would be the Hawking radiation leading to the mass decrement
of the black hole [6, 7]. By emitting the Hawking radiation, the black holes
can be evaporated completely. So, any black hole in its evolution has been
affected by both decrement (arising from the Hawking radiation) and increment
(due to the accretion of the matter-energy) of the mass. If the Hawking
radiation dominates, the black hole evaporates. If the accretion dominates,
the black hole grows.

In Ref. [4], it is discussed that the primordial black holes might have grown
by absorbing the existing scalar field in the universe, leading to the mass
increment of the black holes. Also, in Refs. [8, 9] it has been shown that by
considering the phantom scalar field, the mass of the black hole effectively
decreases. In Ref. [10], the authors have considered the nonminimally coupled
canonical scalar field and studied the evolution of the black hole mass. They
have shown that the accretion of the nonminimally coupled scalar field causes
the mass to decrease when the black hole mass is smaller than a certain
critical value. Note that, this result came up without exist the phantom
energy in the model.

Another interesting scalar field that can be responsible for both the initial
and late time acceleration phase of the universe is the tachyon field [11, 12,
13, 14, 15, 16, 17, 18]. In this paper, instead of considering the phantom
field or non-minimal coupling, we adopt the tachyon field to study the
accretion of the black hole. We show that when we consider the tachyon field
in the theory, the time derivative of the black hole mass would be
corresponding to the potential of this field. Note that, in the minimal
“simple canonical scalar field”, there is no dependence on the potential in
the time derivative of the black hole mass [4]. In this respect, in our model,
it is possible to study the accretion of the mass in the minimally coupled
scalar field with different potentials. On the other hand, the authors of Refs
[19, 20, 21] have shown that a negative potential can lead to interesting
cosmological results. In this regard, we show that by adopting the suitable
negative potential, there would be a decrease in the black hole mass, at least
in some ranges of the model’s parameter space. This result is obtained without
presenting a nonminimal coupling. Note that, the negative potential for the
tachyon field leads to the negative energy density. Although currently there
is no evidence for the role of the negative energy density in our universe,
some realizations of the relativistic quantum field theories predict this
possibility [22, 23]. Also, some authors have studied the models with negative
energy density and have shown that, at least at a theoretical level, it leads
to interesting cosmological implications [24, 25]. Therefore, it seems
inspiring to consider both negative and positive potentials for the tachyon
field.

With these preliminaries, the paper is organized as follows. In section 2, we
consider the tachyon field around the black hole. By using the energy-momentum
tensor of the tachyon field and also its equation of motion, we seek the
effects of this field on the mass evolution of the black hole. In this
section, we obtain an expression for the time evolution of the black hole mass
in terms of the tachyon field’s potential. In section 3, we study the
evolution of the black hole mass by considering several types of potential. We
show that by considering the tachyon field around the black hole, it is
possible to have accretion of the mass into the black hole. We also show that,
with linear and hilltop potentials, the mass of the black hole can decrease
without considering the Hawking radiation. In section 4, we summarize our
work.

##  2 Tachyon Field around the Black Hole

With a tachyon field, we deal with the following action

$\displaystyle S=\int
d^{4}x\sqrt{-g}\Bigg{[}\frac{1}{2\kappa^{2}}R-V(\phi)\sqrt{1-2X}\Bigg{]},$ (1)

where $R$ is the Ricci scalar, $\kappa$ is the gravitational constant, and
$V(\phi)$ is the potential of the tachyon field. Also, $X$ is defined as
$X=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi$.It should be
noticed that, as it has been demonstrated in Ref. [26], if we consider
$X=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi$ the tachyon
field becomes a phantom. However, in this work we don’t consider this case.
Note that, in Ref. [4], where a canonical scalar field has been considered,
the authors have studied both zero and non-zero potentials. However, in the
case with a tachyon field, considering the zero potential removes the effect
of the tachyon field entirely.

The equation of motion of the tachyon field, obtained from the action (1), is
given by

$\displaystyle\Box\phi-\partial_{\mu}\phi\,\partial^{\mu}\phi\frac{\Box\phi}{1+\partial_{\mu}\phi\,\partial^{\mu}\phi}=\frac{V^{\prime}}{V}\,.$
(2)

Also, we have the following energy-momentum tensor, corresponding to the
tachyon field

$\displaystyle
T_{\mu\nu}=\frac{V}{\sqrt{1+\partial^{\nu}\phi\,\partial_{\nu}\phi}}\partial_{\nu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\,V\,\sqrt{1+\partial^{\nu}\phi\,\partial_{\nu}\phi}\,.$ (3)

The interesting point is that, although our considered tachyon field is not a
usual phantom, depending on the values of the potential, it is possible for
the tachyon field to behave like a phantom. In fact, from equation (3) we have
$\rho+p=-\frac{VX}{\sqrt{1-2X}}$, where $\rho=T_{00}$ and $p=T_{ii}$ with
$i=1,2,3$ (see also Ref. [27], where it has been shown that it is possible to
express density and pressure in terms of Lagrangian and its derivatives).
Considering that the parameter $X$ is given by $X=\frac{1}{2}\dot{\phi}^{2}$,
the tachyon field violates the null energy condition for positive values of
the potential. In this way, although the tachyon field is different from the
usual phantom field, for the positive values of the potential, the tachyon
field has the “phantom-like” behavior. On the other hand, for the negative
values of the potential, we have $\rho+p>0$, meaning the tachyon field doesn’t
violate the null energy condition. While, in this case, the weak energy
condition is violated (since we have $\rho<0$), and in some realizations of
the relativistic quantum field theories this violation is predicted and
possible.

Now, we consider the Schwarzschild coordinates and obtain the spherically
symmetrical version of (2) as

$\displaystyle\ddot{\phi}-\left(1-\frac{2M}{r}\right)\frac{1}{r^{2}}\partial_{r}\left[\left(1-\frac{2M}{r}\right)r^{2}\partial_{r}\phi\right]\hskip
113.81102pt$
$\displaystyle=-\frac{V^{\prime}}{V}\,\left(1-\frac{2M}{r}\right)\,\left[1-\left(1-\frac{2M}{r}\right)^{-1}\dot{\phi}^{2}+\left(1-\frac{2M}{r}\right)(\partial_{r}\phi)^{2}\right]\,,$
(4)

where a dot shows the derivative of the parameter with respect to the time. We
should find spherically symmetric solutions to the equation (2) with the
following boundary condition according to the Bondi accretion process,

$\displaystyle\phi(t,r\rightarrow\infty)=\phi_{c}(t)\,,$ (5)

where $\phi_{c}(t)$ shows the cosmological evolution of the scalar field. In
fact, we assume the black hole lies on a local asymptotically flat space with
boundary condition (5). This is because, in comparison with the cosmological
length or time scales, the black hole is very small [28]. After we find a
solution $\phi(r,t)$ for equation (2), satisfying the boundary condition, we
can obtain its energy flux through the horizon that is completely absorbed by
the black hole. In this regard, in Schwarzschild coordinates, the rate of
accretion of the tachyon field onto the black hole is given by

$\displaystyle\dot{M}=\oint_{r=2M}r^{2}\,\,T_{t}^{\,\,r}\,d\Omega\,.$ (6)

If we adopt Eddington-Finkelstein coordinates $(v,r)$, which is regular on the
horizon, the solution of equation (2) in stationary configuration is given by

$\displaystyle\phi(v,r)=C_{1}+C_{2}\left[v-r+2M\log\Big{(}\frac{2M}{r}\Big{)}\right]\,,$
(7)

where $C_{1}$ and $C_{2}$ are constant. In [29], the authors have found a
specific form of lagrangian leading to the stationary solutions. This means
that they have obtained a form of lagrangian for which the action is invariant
under shift symmetry $\phi\longrightarrow\phi+\lambda$. However, with the
lagrangian form given in (1), we follow a quasi-stationary approach [31]. In
the quasi-stationary approach, we consider a slowly rolling scalar field with
$V^{\prime}(\phi)\approx 0$ and $\ddot{\phi}\approx 0$ which resemble the
slow-roll conditions in inflation. Although the equation (2) is apparently
different from the corresponding ones in Refs. [10, 31], we can consider the
right-hand side of that equation as an effective term and write

$\displaystyle\ddot{\phi}-\left(1-\frac{2M}{r}\right)\frac{1}{r^{2}}\partial_{r}\left[\left(1-\frac{2M}{r}\right)r^{2}\partial_{r}\phi\right]=-V_{eff}^{\prime}\,,$
(8)

where

$\displaystyle
V_{eff}^{\prime}=\frac{V^{\prime}}{V}\,\left(1-\frac{2M}{r}\right)\,\left[1-\left(1-\frac{2M}{r}\right)^{-1}\dot{\phi}^{2}+\left(1-\frac{2M}{r}\right)(\partial_{r}\phi)^{2}\right]\,.$
(9)

Note that, in equation (8) and forthcoming equations, a dot on the parameter
means the derivative over the Eddington-Finkelstein coordinate $v$ which is
equal to derivative over time.

As emphasized in Ref. [31], the solution of equation (8) is rather independent
of the form of the potential. In this regard, the field configuration at the
black hole horizon can be approximated as [31, 10, 30]

$\displaystyle\phi(v,r)=\phi_{c}\left(v-r+2M\log\Big{(}\frac{2M}{r}\Big{)}\right)\,,$
(10)

where $\phi_{c}$ has been introduced in boundary condition (5). Now, by
substituting the solution (10) in the equation (8), we get

$\displaystyle\left[1+\frac{2M}{r}+\left(\frac{2M}{r}\right)^{2}+\left(\frac{2M}{r}\right)^{3}\right]\ddot{\phi}=-V_{eff}^{\prime}\,,$
(11)

where we now have the following expression for $V_{eff}^{\prime}$

$\displaystyle
V_{eff}^{\prime}=\frac{V^{\prime}}{V}\,\left[1-\left(1-\frac{16M^{4}}{r^{4}}\right)\dot{\phi}^{2}\right]\,.$
(12)

Note that, since we have considered the slowly varying tachyon field,
$\dot{\phi}$ is almost a constant, and the approximated solution is valid as
long as $\ddot{\phi}$ and $V^{\prime}_{eff}$ are very close to zero.

To study the time evolution of the black hole mass, we should find
$T_{t}^{\,\,r}$. From equations (3) and (10) we obtain

$\displaystyle
T_{t}^{\,\,r}=\frac{\left(\frac{2M}{r}\right)^{2}\,V\,\dot{\phi}_{c}^{2}}{\sqrt{1-\left[1+\frac{2M}{r}+\left(\frac{2M}{r}\right)^{2}+\left(\frac{2M}{r}\right)^{3}\right]\dot{\phi}_{c}^{2}}}\,.$
(13)

The important point about equation (13) is the presence of the potential. When
there is a canonical scalar field, in both minimally or nonminimally coupled
cases, the potential is absent in $T_{t}^{\,\,r}$. However, if we consider the
tachyon field, the potential is presented in $T_{t}^{\,\,r}$. This is an
interesting issue. The reason is that since $T_{t}^{\,\,r}$ is potential
dependent, by considering just the several types of potential we can study the
time evolution of the black hole mass.

In a quasi-stationary approach, it is possible to parameterize the field
absorbed at the black hole horizon, by the field approximated from infinity as
[28, 10, 30]

$\displaystyle\phi_{c}(t)\approx\phi_{{}_{\infty}}+\dot{\phi}_{{}_{\infty}}(t-t_{0})\,,$
(14)

with $\phi_{{}_{\infty}}$ and $\dot{\phi}_{{}_{\infty}}$ to be constant
parameters. In fact, in this regard, we consider $r\gg 2M$ limit and
approximate the field from the infinity to the black hole horizon. Now, from
equations (6), (13) and (14) we get

$\displaystyle\dot{M}=\frac{16\pi\,M^{2}\,V\,\dot{\phi}_{{}_{\infty}}^{2}}{\sqrt{1-\left[1+\frac{2M}{r}+\left(\frac{2M}{r}\right)^{2}+\left(\frac{2M}{r}\right)^{3}\right]\dot{\phi}_{{}_{\infty}}^{2}}}\,.$
(15)

Since we consider $r\gg 2M$ limit, for small values of
$\dot{\phi}_{{}_{\infty}}$, we can rewrite equation (15) as

$\displaystyle\dot{M}=\Bigg{[}16\pi\,M^{2}\,V\,\dot{\phi}_{{}_{\infty}}^{2}\Bigg{]}\Bigg{[}1+\frac{1}{2}\bigg{[}1+\frac{2M}{r}+\left(\frac{2M}{r}\right)^{2}+\left(\frac{2M}{r}\right)^{3}\bigg{]}\dot{\phi}_{{}_{\infty}}^{2}\Bigg{]}\,.$
(16)

Also, in the following, we just keep the terms in $\dot{M}$ which are of the
order of $M^{2}$. This means that we keep the two first terms of the second
bracket of the equation (16). In the following, using equation (16), we study
the evolution of the black hole mass.

##  3 Evolution of the Black Hole Mass

To find $M$ from equation (16), we follow Ref. [10] and consider $\dot{M}$ as

$\displaystyle\dot{M}=f(t)\,M^{2}\,,$ (17)

where, in our case,

$\displaystyle
f(t)=16\pi\,V\,\dot{\phi}_{{}_{\infty}}^{2}\bigg{(}1+\frac{1}{2}\dot{\phi}_{{}_{\infty}}^{2}\bigg{)}\,.$
(18)

Note that, $f(t)$ in equation (18) depends on the potential. However, by
considering the canonical scalar field, even the non-minimally coupled one,
there is no dependence on the potential [4, 10]. From equations (17) and (18)
we see that a negative potential can cause decreasing in the black hole mass
even in the absence of Hawking radiation. This is an interesting issue and we
discuss it in the next subsections.

Now, the solution of equation (17) is given by

$\displaystyle M(t)=\frac{M_{0}}{1-M_{0}\,{\cal{F}}(t)}\,,$ (19)

where $M_{0}$ is the mass at time $t_{0}$ and ${\cal{F}}$ is defined as

$\displaystyle{\cal{F}}(t)=\int_{t_{0}}^{t}f(t^{\prime})\,dt^{\prime}\,.$ (20)

Since the mass is positive, so there is always a constraint on equation (19)
as $M_{0}\,{\cal{F}}(t)<1$. In this step, we should specify the form of the
potential. We consider four types of potential as linear, quadratic, natural,
and hilltop potentials. In the following, we study each case separately.

###  3.1 Linear Potential

The first potential which we consider is the linear potential as $V\sim\phi$
[32]. By this potential, we find the function $f(t)$ as

$\displaystyle
f(t)=8\,\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\pi\left(\dot{\phi}_{{}_{\infty}}\,\left(t-{\it
t_{0}}\right)+\phi_{{}_{\infty}}\right)\,,$ (21)

and ${\cal{F}}(t)$ as

$\displaystyle{\cal{F}}(t)=16\,\pi\dot{\phi}^{2}_{{}_{\infty}}\left(1+\frac{1}{2}\,\dot{\phi}^{2}_{{}_{\infty}}\right)\left(\dot{\phi}_{{}_{\infty}}\,\left(\frac{1}{2}\,{t}^{2}-t_{0}\,t\right)+\phi_{{}_{\infty}}\,t\right)\,.$
(22)

By substituting equations (21) and (22) in equation (17) and (19), we can
obtain the black hole mass and its time evolution as follows

$\displaystyle
M=\frac{M_{0}}{-16\,\pi\dot{\phi}_{\infty}^{2}\left(1+\frac{1}{2}\,{\dot{\phi}}_{\infty}^{2}\right)\left(\dot{\phi}_{\infty}\,\left(\frac{1}{2}\,{t}^{2}-t_{0}\,t\right)+\phi_{\infty}\,t\right)M_{0}+1}\,,$

$\displaystyle\dot{M}=\frac{16\,\pi\left(\dot{\phi}_{\infty}\,\left(t-t_{0}\right)+\phi_{\infty}\right){\dot{\phi}}_{\infty}^{2}\left(1+\frac{1}{2}\,{\dot{\phi}}_{\infty}^{2}\right)M_{0}^{2}}{\left(-16\pi{\dot{\phi}}_{\infty}^{2}\left(1+\frac{1}{2}{\dot{\phi}}_{\infty}^{2}\right)\left(\dot{\phi}_{\infty}\left(\frac{1}{2}\,{t}^{2}-t_{0}\,t\right)+\phi_{\infty}\,t\right)M_{0}+1\right)^{2}}\,.$

Equation (3.1) sets the following constraint on the model’s parameter space in
the linear potential case

$\displaystyle
16\,\pi\dot{\phi}_{\infty}^{2}\left(1+\frac{1}{2}\,{\dot{\phi}}_{\infty}^{2}\right)\left(\dot{\phi}_{\infty}\,\left(\frac{1}{2}\,{t}^{2}-t_{0}\,t\right)+\phi_{\infty}\,t\right)M_{0}<1\,.$
(25)

Figure 1: Presentation of the range of parameters $\phi$ and $t$, for the
linear potential, satisfying the constraint (25). The upper panel is
corresponding to the case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
the lower panel is corresponding to the case with
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. Also, the parameter $\chi$
represents the left-hand side of equation (25).

Therefore, for the numerical study of the model, we should consider the above
constraint. In this regard, we perform numerical analysis on this constraint
to find the suitable ranges of the parameters. To perform this analysis, we
should specify the parameters $\phi_{\infty}$ and $\dot{\phi}_{\infty}$. Given
that these parameters are arbitrary constant, we consider two cases as
$\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. We also adopt $M_{0}=1$ and
$t_{0}=0$. By these adoptions of the parameters, we perform numerical analysis
on the parameter space of $\phi$ and $t$, satisfying the constraint (25). The
results are shown in figure 1, where the parameter $\chi$ represents the left-
hand side of the constraint (25). The region of $\phi_{\infty}$ and $t$
leading to $\chi<1$ are viable. This figure helps us to adopt appropriate
values of parameter $\phi_{\infty}$ and therefore $\dot{\phi}_{\infty}$, to
study the black hole mass and its time evolution numerically. In figure 2, we
plot the evolution of the black hole masse for
$\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$, for some sample values of
$\phi_{\infty}$ as $0.3$, $0.1$, $-0.1$ and $-0.3$. These adopted values of
$\phi_{\infty}$ satisfy the constraint (25). From figure 2, we see that by
adopting $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$, $\phi_{\infty}=0.1$ and
$\phi_{\infty}=0.3$, we get accretion of the tachyon field into the black hole
and the black hole mass increases. By adopting
$\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$, $\phi_{\infty}=-0.1$ and
$\phi_{\infty}=-0.3$, the black hole mass decreases even without considering
the Hawking radiation. In the case with
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$, for all adopted values of
$\phi_{\infty}$, the tachyon field at first causes increasing of the black
hole mass, then it decreases the mass. This interesting result is obtained
because of the fact that with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$, for
$\phi_{\infty}=-0.1$ and $\phi_{\infty}=-0.3$, and with
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$, for all adopted values of $\phi$,
the potential can be negative leading to negative values of $\dot{M}$. The
evolution of $\dot{M}$ versus time is shown in figure 3, for both cases with
$\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. We conclude that, by considering
the tachyon field with the linear potential, it is possible to have a decrease
in the black hole mass without considering the Hawking radiation. Note that,
in the ranges of the parameters space leading to black hole mass decrease, we
have $\rho+p>0$. This means that, in these ranges, the null energy condition
is not violated. Therefore, in these ranges, the tachyon field is not a
phantom. In this case, we face the mass decrease of the black hole because of
the negative energy density of the tachyon field.

Figure 2: Time evolution of the black hole mass for different values of
$\phi_{\infty}$, with linear potential. The left panel is corresponding to the
case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and the right panel is
corresponding to the case with $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$.

Figure 3: Time evolution of $\dot{M}$ for different values of $\phi_{\infty}$,
with linear potential. The left panel is corresponding to the case with
$\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and the right panel is corresponding
to the case with $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$.

###  3.2 Quadratic Potential

The next potential which we study is the quadratic potential as
$V\sim\phi^{2}$. With the quadratic potential, we find the following
expressions

$\displaystyle
f(t)=8\,\pi\left(\dot{\phi}_{{}_{\infty}}^{2}+2\right)\left(\dot{\phi}_{{}_{\infty}}\,\left(t-t_{0}\right)+\phi_{{}_{\infty}}\right)^{2}\,\dot{\phi}_{{}_{\infty}}^{2}\,,$
(26)

and

$\displaystyle{\cal{F}}=\frac{16}{3}\,\pi\dot{\phi}_{{}_{\infty}}\,\left(1+\frac{1}{2}\,\dot{\phi}_{{}_{\infty}}^{2}\right)\bigg{(}\dot{\phi}_{{}_{\infty}}\,t-\dot{\phi}_{{}_{\infty}}\,t_{0}+\phi_{{}_{\infty}}\bigg{)}^{3}\,.$
(27)

With these expressions and by using equations (17) and (19), we get

$\displaystyle
M=\frac{\,M_{0}}{3-8\,\pi\,\left(\left(t-t_{0}\right)\dot{\phi}_{{}_{\infty}}+\phi_{{}_{\infty}}\right)^{3}\dot{\phi}\,\left({\dot{\phi}}_{{}_{\infty}}^{2}+2\right)M_{0}}\,,$
(28)

$\displaystyle\dot{M}=\frac{72\left({\dot{\phi}}_{{}_{\infty}}^{2}+2\right){\dot{\phi}}_{{}_{\infty}}^{2}\pi\left(\left(t-t_{0}\right)\dot{\phi}_{{}_{\infty}}+\phi_{{}_{\infty}}\right)^{2}{M_{0}}^{2}}{\left(-3+8\,\pi\,\left(\left(t-t_{0}\right)\dot{\phi}_{{}_{\infty}}+\phi_{{}_{\infty}}\right)^{3}\dot{\phi}_{{}_{\infty}}\,\left({\dot{\phi}}_{{}_{\infty}}^{2}+2\right)M_{0}\right)^{2}}\,.$
(29)

To have the positive black hole mass, from equation (28), the following
constraint should be satisfied

$\displaystyle
8\,\pi\,\left(\left(t-t_{0}\right)\dot{\phi}_{{}_{\infty}}+\phi_{{}_{\infty}}\right)^{3}\dot{\phi}_{{}_{\infty}}\,\left({\dot{\phi}}_{{}_{\infty}}^{2}+2\right)M_{0}<3\,.$
(30)

Figure 4: Presentation of the range of parameter $\phi_{\infty}$ and $t$, for
the quadratic potential, satisfying the constraint (30). The left panel is
corresponding to the case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
the right panel is corresponding to the case with
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. Also, the parameter $\chi$
represents the left hand side of equation (30).

Figure 5: Time evolution of the black hole mass for different values of
$\phi_{\infty}$, with quadratic potential. The left panel is corresponding to
the case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and the right panel is
corresponding to the case with $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$.

Figure 6: Time evolution of $\dot{M}$ for different values of $\phi_{\infty}$,
with quadratic potential. The left panel is corresponding to the case with
$\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and the right panel is corresponding
to the case with $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$.

This means that, to the numerical study of the model, we should consider the
parameters’ values satisfying the constraint (30). Here also, we perform a
numerical analysis which is shown in figure 4. As the linear case, we have
considered $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$ and also adopted $M_{0}=1$ and
$t_{0}=0$. As figure shows, with $\phi=0.1\,\dot{\phi}$, the constraint (30)
implies some constraint on the parameter $\phi$. However, with
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$, there is no constraint on
$\phi_{\infty}$ and all considered values of $\phi_{\infty}$ are viable. Now,
we can study the evolution of the black hole mass in the quadratic potential
case. In this regard, we analyze the time evolution of the black hole mass by
using equation (28). The results are shown in figure 5. As figure shows, for
both cases with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$ and with quadratic potential, the
black hole mass increases continually. Correspondingly, the value of $\dot{M}$
in every time is positive. This issue has been shown in figure 6. As a result,
the accretion of the tachyon field with quadratic potential into the black
hole leads to an increase in the black hole mass. Note that, with a quadratic
potential, we have $\rho+p<0$, leading to violation of both null and weak
energy conditions. This means that a tachyon field with quadratic potential
behaves like a phantom (despite it is not a usual phantom field).

###  3.3 Natural Potential

With the natural potential
$V\sim\Lambda^{4}\big{[}1+\cos\big{(}\frac{\phi}{l}\big{)}\big{]}$ [33, 34,
35], we find the function $f(t)$ as

$\displaystyle
f(t)=8\,\pi\,\Lambda^{4}\left[1+\cos\left({\frac{\dot{\phi}_{{}_{\infty}}\,\left(t-t_{0}\right)+\phi_{{}_{\infty}}}{l}}\right)\right]\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\,,$
(31)

and ${\cal{F}}$ as

$\displaystyle{\cal{F}}=8\,\pi\,\Lambda^{4}\,\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\Bigg{[}t+\frac{l}{\dot{\phi}_{{}_{\infty}}}\sin\left({\frac{\dot{\phi}_{{}_{\infty}}\,t}{l}}-{\frac{\dot{\phi}_{{}_{\infty}}\,t_{0}-\phi_{{}_{\infty}}}{l}}\right)\Bigg{]}\,.$
(32)

Now, from equations (17), (19), (31) and (32), we find the black hole mass and
its time evolution in the natural potential case as follows

$\displaystyle
M=M_{0}\Bigg{[}-8\,\pi{\Lambda}^{4}\,\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\bigg{[}t+\frac{l}{\dot{\phi}_{{}_{\infty}}}\sin\left({\frac{t\,\dot{\phi}_{{}_{\infty}}}{l}}-{\frac{\dot{\phi}_{{}_{\infty}}\,t_{0}-\phi_{{}_{\infty}}}{l}}\right)\bigg{]}M_{0}+1\Bigg{]}\,,$
(33)

$\displaystyle\dot{M}=\Bigg{[}8\,\pi\,{\Lambda}^{4}\,\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right){M_{0}}^{2}\bigg{(}1+\cos\left({\frac{\dot{\phi}_{{}_{\infty}}\,\left(t-t_{0}\right)+\phi_{{}_{\infty}}}{l}}\right)\bigg{)}\Bigg{]}\hskip
42.67912pt$
$\displaystyle\Bigg{[}\bigg{[}-8\,\pi{\Lambda}^{4}\,\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\bigg{(}t+\frac{l}{\dot{\phi}_{{}_{\infty}}}\sin\left({\frac{t\,\dot{\phi}{{}_{\infty}}}{l}}-{\frac{\dot{\phi}_{{}_{\infty}}\,t_{0}-\phi_{{}_{\infty}}}{l}}\right)\bigg{)}M_{0}+1\bigg{]}^{2}\Bigg{]}\,.$
(34)

For the natural potential case, equation (33) gives the following constraint
on the model’s parameters

$\displaystyle
8\pi{\Lambda}^{4}\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\Bigg{[}t+\frac{l}{\dot{\phi}_{{}_{\infty}}}\sin\left({\frac{t\dot{\phi}_{{}_{\infty}}}{l}}-{\frac{\dot{\phi}_{{}_{\infty}}t_{0}-\phi_{{}_{\infty}}}{l}}\right)\Bigg{]}M_{0}<1.$
(35)

Figure 7: Presentation of the range of parameter $\phi_{\infty}$ and $t$, for
the natural potential, satisfying the constraint (35). The left panel is
corresponding to the case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
the right panel is corresponding to the case with
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. Also, the parameter $\chi$
represents the left hand side of equation (35).

Figure 8: Time evolution of the black hole mass for different values of
$\phi_{\infty}$, for the natural potential. The left panel is corresponding to
the case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and the right panel is
corresponding to the case with $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$.

Figure 9: Time evolution of $\dot{M}$ for different values of $\phi_{\infty}$,
for the natural potential. The left panel is corresponding to the case with
$\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and the right panel is corresponding
to the case with $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$.

The parameter space satisfying the constraint (35) is shown in figure 7, for
two cases with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. To plot this figure, we have
considered $\Lambda=1$, $l=1$, $M_{0}=1$ and $t_{0}=0$. As the figure shows,
for both cases, there are some values of the parameters which satisfy the
constraint (35) and can be used in numerical analysis of $M$ and $\dot{M}$.
Here, we consider $\phi_{\infty}=-0.3$ , $-0.1$, $0.1$ and $0.3$. With these
adopted values of $\phi_{\infty}$, we study the evolution of the black hole
mass versus time for both cases with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$
and $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. The results are shown in figure
8. This figure shows that, as time goes, the mass of the black hole increases.
This means that with natural potential we always have the accretion of the
mass into the black hole and there is no evaporation or decreasing of the
black hole mass. This result is verified by figure 9, where we have plotted
the time evolution of $\dot{M}$. When we adopt a natural potential for the
tachyon field, we find $\rho+p<0$. Therefore, in this case also, both null and
weak energy conditions are violated. In this regard, it seems that a tachyon
field with a natural potential behaves like a phantom.

###  3.4 Hilltop Potential

With the hilltop potential which is defined as $V\sim
V_{0}\left[1-\left(\frac{\phi}{\mu}\right)^{q}\right]$ [33, 34, 35], we find
the following expression for $f(t)$

$\displaystyle
f(t)=16\,\pi\,V_{0}\left[1-\left({\frac{\dot{\phi}_{{}_{\infty}}\,\left(t-t_{0}\right)+\phi_{\infty}}{\mu}}\right)^{q}\right]\dot{\phi}^{2}_{{}_{\infty}}\left(1+\frac{1}{2}\,\dot{\phi}^{2}_{{}_{\infty}}\right).$
(36)

Also, the hilltop potential gives ${\cal{F}}$ as

$\displaystyle{\cal{F}}=8\,\pi\,V_{0}\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\Bigg{(}t-\frac{l}{\dot{\phi}_{{}_{\infty}}\,\left(1+q\right)}\left({\frac{\dot{\phi}_{{}_{\infty}}\,t}{\mu}}+{\frac{-\dot{\phi}_{{}_{\infty}}\,t_{0}+\phi_{\infty}}{\mu}}\right)^{1+q}\Bigg{)}\,.$
(37)

By having the above equations and using equations (17) and (19), we find the
black hole mass as follows

$\displaystyle M=M_{0}\Bigg{[}\Bigg{(}1-8\pi
V_{0}\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\bigg{(}t-\frac{\mu}{\dot{\phi}_{{}_{\infty}}\,\left(1+q\right)}$
$\displaystyle\left({\frac{\dot{\phi}_{{}_{\infty}}\,t}{\mu}}-{\frac{\dot{\phi}_{{}_{\infty}}\,t_{0}-\phi_{\infty}}{\mu}}\right)^{1+q}\bigg{)}M_{0}\Bigg{)}\Bigg{]}\,,$
(38)

and its time evolution as

$\displaystyle\dot{M}=\Bigg{[}16\pi
V_{0}\dot{\phi}^{2}_{{}_{\infty}}\left(1+\frac{1}{2}\dot{\phi}^{2}_{{}_{\infty}}\right)M_{0}^{2}\bigg{(}1-\left({\frac{\dot{\phi}_{{}_{\infty}}\
\left(t-t_{0}\right)+\phi_{\infty}}{\mu}}\right)^{q}\bigg{)}\Bigg{]}\hskip
42.67912pt$ $\displaystyle\Bigg{[}\Bigg{(}1-8\pi
V_{0}\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\bigg{(}t-\frac{\mu}{\dot{\phi}_{{}_{\infty}}\left(1+q\right)}\left({\frac{\dot{\phi}^{2}_{{}_{\infty}}t}{\mu}}-{\frac{\dot{\phi}_{{}_{\infty}}t_{0}-\phi_{\infty}}{l}}\right)^{1+q}\bigg{)}M_{0}\Bigg{)}^{2}\Bigg{]}.$

For the hilltop potential case, equation (3.4) gives the following constraint
on the model’s parameters

$\displaystyle 8\pi
V_{0}\dot{\phi}^{2}_{{}_{\infty}}\left(\dot{\phi}^{2}_{{}_{\infty}}+2\right)\Bigg{(}t-\frac{\mu}{\dot{\phi}_{{}_{\infty}}\left(1+q\right)}\left({\frac{\dot{\phi}_{{}_{\infty}}t}{\mu}}-{\frac{\dot{\phi}_{{}_{\infty}}t_{0}-\phi_{\infty}}{\mu}}\right)^{1+q}\Bigg{)}M_{0}<1\,.$
(40)

Figure 10: Presentation of the range of parameter $\phi_{\infty}$ and $t$,
with hilltop potential, satisfying the constraint (40). The left panel is
corresponding to the case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
the right panel is corresponding to the case with
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. Also, the parameter $\chi$
represents the left hand side of equation (40).

Figure 11: Time evolution of the black hole mass for different values of
$\phi_{\infty}$, with hilltop potential. The left panel is corresponding to
the case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and the right panel is
corresponding to the case with $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$.

Figure 12: Time evolution of $\dot{M}$ for different values of
$\phi_{\infty}$, with hilltop potential. The left panel is corresponding to
the case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and the right panel is
corresponding to the case with $\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$.

The values of the model’s parameter which is used to perform numerical
analysis on the model with hilltop potential should satisfy the constraint
(40). The acceptable ranges of the parameters are shown in figure 10, for two
cases with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$. In this figure, the parameter
$\chi$ is the left-hand side of the constraint (40). The viable regions of
$\phi_{\infty}$ and $t$ are those that lead to $\chi<1$. By using these viable
values, we study the evolution of the black hole mass versus time numerically.
The results are shown in figure 11. Our numerical analysis shows that, in the
case with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and for $\phi_{\infty}=0.1$
and $\phi_{\infty}=0.3$, the presence of the tachyon field causes the black
hole mass first increases and then decreases by time. Therefore, with hilltop
potential, it is possible to have a decreasing black hole mass without
considering the Hawking radiation. In the case with
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$, if we consider $\phi_{\infty}=-0.1$
and $\phi_{\infty}=-0.3$, it is possible to have decrease in the mass of the
black hole. However, for $\phi_{\infty}=-0.3$, the evolution of the black hole
mass follows a strange pattern and is not favor. The time evolution of
$\dot{M}$ for both cases with $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$ and
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$ is shown in figure 12. This figure
also confirms that, for $\phi_{\infty}=0.1\,\dot{\phi}_{\infty}$
,$\phi_{\infty}=0.1$ and $\phi_{\infty}=0.3$ and also for
$\phi_{\infty}=-0.1\,\dot{\phi}_{\infty}$, $\phi_{\infty}=-0.1$ and
$\phi_{\infty}=-0.3$, the mass of the black hole decreases even there is no
Hawking radiation.

Here also, in the ranges of the parameters space leading to black hole mass
decrease, we have $\rho+p>0$. Therefore, in these ranges, the null energy
condition is not violated. This means that, in these ranges, the tachyon field
is not a phantom. As the linear potential case, the mass decrease of the black
hole occurs due to the negative energy density of the tachyon field.

##  4 Summary

In this paper, we have studied the accretion of the black hole mass in the
presence of the tachyon field. Given that the previous works on the black hole
mass accretion were with canonical scalar fields, in both the minimally and
non-minimally coupled cases, we have considered a non-canonical scalar field
on this issue. As regards the energy-momentum tensor of the tachyon field has
non-diagonal components, the potential of the field is involved in the
equations corresponding to the mass and its time evolution. In this regard,
there is no need for the non-minimal coupling between the tachyon field and
gravity. We have used four types of potential as linear, quadratic, natural,
and hilltop potentials. To study the black hole mass, we have used the fact
that some interesting cosmological solutions are corresponding to the negative
potentials. In this regard, we have used the parameter space leading to both
the negative and positive potentials. We have restricted the values of the
model’s parameters by implying the constraint $M\geq 0$, where $M$ is the
black hole mass. To show those values of the model’s parameters satisfying the
constraint $M\geq 0$, we have plotted some figures which gave us the viable
ranges of the parameters. We have used these viable ranges to study the time
evolution of the parameters $M$ and $\dot{M}$. By numerical analysis of $M$
and $\dot{M}$, we were wonder whether black hole mass decreases or not. Our
numerical study has shown that if we consider the linear potential, for the
cases with $\phi_{\infty}=\pm 0.1\,\dot{\phi}_{\infty}$, $\phi_{\infty}=0.1$
and $\phi_{\infty}=0.3$ we have accretion of the mass into the black hole.
However, with $\phi_{\infty}=\pm 0.1\,\dot{\phi}_{\infty}$,
$\phi_{\infty}=-0.1$ and $\phi_{\infty}=-0.3$, the black hole decreases
without considering the Hawking radiation.

Another interesting point is that the tachyon field with linear and hilltop
potentials, in some ranges of the parameter space leading to the black hole
mass decrease, satisfies the null energy condition. In the cases with
quadratic and natural potentials, the null energy condition is violated and
the tachyon field has “phantom-like” behavior.

Acknowledgement
I thank the referee for the very insightful comments that have improved the
quality of the paper considerably.

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