# Static Blackhole with Cosmological Influence: Whittaker Solutions

Santanu Tripathy and K Rajesh Nayak
IISER-Kolkata

(19th October 2023)

###### Abstract

In this article, we investigate the impact of cosmological parameters on black
holes using an exact solution to Einstein’s equations that satisfies the
Whittaker equation of state. We examine a spherically symmetric black hole in
the background of a static Einstein Universe with a perfect fluid source with
cosmological constant. This solution is characterized by two independent
parameters, namely the size of the universe ($R$) and the cosmological
constant ($\Lambda$), which represent the cosmological influences. We explore
phenomena such as periastron precession and the scattering of massless scalar
fields to determine how these cosmological parameters affect the physics
around black holes.

##  1 Introduction

Blackhole physics is an intriguing field within both the framework of general
relativity and alternative theories of gravitation. Extensive investigations
have been conducted on classes of solutions, such as Schwarzschild and Kerr
black holes [1, 2, 3]. These solutions exhibit characteristics such as a flat
spacetime far from the blackhole’s event horizon and represent vacuum
solutions completely devoid of matter. In most studies, the cosmological
influences on astrophysical blackhole are ignored. However, in reality, we
expect the presence of matter surrounding astrophysical blackholes or a
spacetime that may display cosmological properties as one moves away from the
event horizon (EH).

Another crucial property often essential in blackhole physics is the existence
of time symmetry or a timelike Killing-vector. Though concepts such as
isolated horizon and dynamical horizon exist, without time symmetry, it
becomes challenging to define the event horizon at each moment in time [4, 5].
Since the event horizon is a global feature of spacetime, it necessitates
complete knowledge of the solution at all time slices, especially for light
propagation. To investigate blackholes in cosmological backgrounds, it may be
necessary to relax some of the conditions mentioned above. Such models could
provide valuable insights into the properties of blackholes that exhibit non-
flat asymptotic behaviour and time dependent solutions.

When it comes to the physics of static or stationary non-asymptotically flat
blackholes, numerous studies have been conducted on the Schwarzschild-de-
Sitter (SdS) solution [6, 7]. The SdS spacetime is a straightforward extension
of the Schwarzschild solution, incorporating a cosmological constant,
$\Lambda$, but it remains a vacuum solution without any matter field content.
Vaidya introduced a solution to Einstein’s equations with blackholes in the
background of the Static Einstein Universe(SEU) [8]. This solution includes a
matter field generated by a perfect fluid. Another possibility involves a
fully evolving cosmological background, resulting in a time-dependent
solution, such as McVittie’s solution and its generalisation [9, 10, 11, 12].
In these cases, only a Cauchy horizon or apparent horizon can be constructed
at any given time. To determine the location of the event horizon, a complete
solution is required at all times. We focus solely on time-independent
solutions in this article.

In this work, we investigate some of the static solutions to Einstein’s
equations that have a blackhole event horizon surrounded by matter, while the
spacetime is finite or asymptotically non-flat in general. These solutions are
inspired by the solution given by Vaidya, which consists of a Kerr-like event
horizon in the Einstein static universe (Vaidya-Einstein-Kerr or VEK) [8, 13].
A special case of the VEK solution is obtained by setting the rotational
parameter, $a$, to zero, resulting in the static spherically symmetric Vaidya-
Einstein-Schwarzschild(VES) solution, which has been extensively investigated
[14, 15, 16]. Although the VES solution is intriguing, it has one drawback: a
naked singularity at the outer boundary of the spacetime. While this solution
can be matched with the SEU, the first derivative cannot be smoothly matched,
making these discontinuities unphysical [14]. One interesting aspect of the
VES solution is that the density and pressure satisfy the condition
$\rho+3p=0$, which corresponds to a special case of the Whittaker equation of
state [17].

Though the VES solution is very interesting, it has one drawback: a naked
singularity at the outer boundary of the spacetime. While this solution can be
matched with the SEU, though metric coefficient can be matched, the first
derivate can not be matched smoothly, this discontinuities make the matching
unphysical [14]. One interesting point about VES solution is that the density
and pressure satisfy the condition $\rho+3p=0$, which is a special case of the
Whittaker equation of state [17].

The VES solution is a subset of the Whittaker solutions. We extend this idea
further and seek a solution to Einstein’s equations that satisfies the
Whittaker equation, including the cosmological constant $\Lambda$. While
Vaidya’s solution includes $\Lambda$, it appears merely as an additive
constant and does not explicitly appear in the metric coefficients. In this
paper, we extend Vaidya’s solution to include the cosmological constant,
resulting in a more general solution that explicitly depends on $\Lambda$. We
utilise the condition $\rho+3p=0$, which is a special case of the Whittaker
equation of state. This introduces a cosmological horizon (CH) in addition to
the regular blackhole event horizon. The CH blocks the naked singularity at
the outer boundary. This CH is similar to the one present in the de-Sitter
type of solution, although de-Sitter solutions do not have any curvature
singularity beyond the cosmological horizon. With this solution, we explore
the properties of static event horizons in the presence of matter and
asymptotically non-flat spacetime. Now, we have a generalisation of the SdS
solution with matter, a blackhole horizon, and a cosmological horizon.

In the case of VSD, the cosmological influence on the event horizon is
determined by a single parameter, $R$, which can be associated with the size
of the SEU. As the limit $R\rightarrow\infty$, the spacetime solution matches
with the Schwarzschild solution. On the other hand, the Schwarzschild-
Whittaker (SW) solution, as presented here, has two parameters that can
contribute to cosmological influence, namely, $R$ and $\Lambda$. Consequently,
we can now study the effect on the horizon with two additional parameters, in
addition to the mass. In the limiting case of large $R$ and small $\Lambda$,
the SW solution matches with the SdS spacetime.

Using the solution to Einstein’s equations with a blackhole surrounded by
matter and an asymptotically non-flat background, we first examine the
behaviour of the event horizon and its existence with various parameters in
the solutions. Generally, there can be two horizons, as in the case of SdS
spacetime. However, in extreme cases, the inner and outer horizons merge, and
the spacetime become unphysical due to the break down of timelike static fluid
model.

We then investigate the periastron precession of elliptical orbits. The
phenomenon of periastron precession has served as an important test to
understand the behaviour of gravity. Since pure Keplerian orbits do not
exhibit precession, it has provided a robust test for the general theory of
relativity itself. However, in the current context, we use orbital precession
to understand the influence of cosmological parameters on the orbits. We
compare the orbital precession of the SW spacetime with that of Schwarzschild
and SdS spacetime.

This article is organised as follows: In Section 2, we solve the Einstein’s
equations with a perfect fluid solution and $\Lambda$. We assume that the
fluid satisfies the equation of state $\rho+3p=0$. We are not aware of any
earlier report of such a solution to Einstein’s equations in the literature.
In Section 3, we briefly discuss the properties of the EH and static
observers. The existence of the horizon depends on the values of the
parameters $R$ and $\Lambda$. In Section 4, we discuss the details of geodesic
motion and periastron precession. Our goal here is to compare the results with
the standard Schwarzschild blackhole rather than the actual measurement of
periastron precession. Therefore, we use a numerical method to compute the
precise value of the precession closer to the blackhole. In Section 5, we
study the behaviour of scalar waves. Finally, we close the paper with brief
concluding remarks.

##  2 Schwarzschild-Whittaker Solution

In this section, we extend the VES solution with the Whittaker equation of
state, ensuring it satisfies the Einstein field equations with the
cosmological constant $\Lambda$. We begin with a modified form of the VES
metric, given by,

$ds^{2}=F\left(r\right)dt^{2}-\frac{dr^{2}}{F\left(r\right)}-R^{2}\sin^{2}\left(\frac{r}{R}\right)d\Omega^{2}\,,$
(1)

where,

$d\Omega^{2}\,=\,\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)\,.$

If $F(r)=1$, the metric matches with the SEU, and the parameter $R$ represents
the size of the universe. The cosmological influence here is introduced by the
spherical part of the metric only. Here, the smaller the value of $R$ is, the
larger the cosmological influence is, and as $R\rightarrow\infty$, the
cosmological effect vanishes. Furthermore, we assume the form of $F(r)$,

$\displaystyle F\left(r\right)$ $\displaystyle=$
$\displaystyle\left[1-\frac{2m}{R\tan\left(\frac{r}{R}\right)}-f(r)\right]\,.$
(2)

We obtain a suitable function $f(r)$ that satisfies Einstein’s equations. By
setting $f(r)=0$, we recover the VES solution. The ranges of coordinates are
as follows: $0\leq\frac{r}{R}\leq\pi$, $0\leq\theta\leq\pi$, $0\leq\phi\leq
2\pi$. The Einstein field equations, including the cosmological term
$\Lambda$, are given by:

$G_{\>b}^{a}\>=\>R_{\>b}^{a}-\frac{1}{2}\delta_{\>b}^{a}R\ =\ \kappa
T_{\>b}^{a}+\Lambda\delta_{\>b}^{a},$ (3)

The non-vanishing components of the Einstein tensor for the metric in Eq. (1)
are given by:

$\displaystyle G_{\>0}^{0}$ $\displaystyle=$
$\displaystyle\frac{3}{R^{2}}\left[1-\frac{2m}{R}\cot\left(\frac{r}{R}\right)\right]+\frac{A}{R^{2}}\left\\{R\cot\left(\frac{r}{R}\right)f^{\prime}+\left[\csc^{2}\left(\frac{r}{R}\right)-3\right]f\right\\}\,,$
$\displaystyle G_{\>1}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}\left[1-\frac{2m}{R}\cot\left(\frac{r}{R}\right)\right]+\frac{A}{R^{2}}\cot\left(\frac{r}{R}\right)\left[Rf^{\prime}+\cot\left(\frac{r}{R}\right)f\right]\,,$
$\displaystyle G_{\>2}^{2}=G_{\>3}^{3}$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}\left[1-\frac{2m}{R}\cot\left(\frac{r}{R}\right)\right]+\frac{A}{2R^{2}}\left[R^{2}f^{\prime\prime}+2R\cot\left(\frac{r}{R}\right)f^{\prime}-2f\right]\,.$
(4)

where $f^{\prime}=\frac{\partial f}{\partial r}$. The energy-momentum tensor
$T_{ab}$ is taken to be that of a perfect fluid, takes the standard form,

$T_{\>b}^{a}\ =\ (\rho+p)u^{a}u_{b}\,-\,p\,\delta_{\>b}^{a},$ (5)

here, $u^{a}$ is the four-velocity of the fluid element, which is timelike. In
the case of a static spacetime, the four-velocity is in the direction of the
time-like Killing-vector and is explicitly given by:

$u^{a}\ =\ \frac{1}{\sqrt{g_{00}}}\delta_{0}^{a}\,.$ (6)

The energy momentum tensor simplifies to,

$T_{\>b}^{a}=\mathrm{dia}\left(\rho,\,-p,\,-p,\,-p\right)\,.$ (7)

We substitute the expression for the energy-momentum tensor back into Eq. (4).
For a consistent static solution, we expect $G_{1}^{1}-G_{2}^{2}=0$, which can
be simplified to:

$f^{\prime\prime}=\frac{2f}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\,.$ (8)

In addition, it can be easily shown that,

$\rho+3p=\epsilon\,,$ (9)

where $\epsilon$ is a constant. The Whittaker solutions satisfy the equation
of state given by Eq. (9) [18]. From Eq. (4) and Eq. (8), we obtain,

$\frac{2\Lambda}{\kappa}-\frac{2}{\kappa
R^{2}}\left[\frac{f}{\sin^{2}\left(\frac{r}{R}\right)}\,+\frac{Rf^{\prime}}{\cot\left(\frac{r}{R}\right)}\right]=\epsilon\,,$
(10)

and the solution for $f(r)$ can be readily obtained as,

$f=\frac{\delta}{\tan\left(\frac{r}{R}\right)}-\left(\Lambda-\frac{1}{2}\epsilon\kappa\right)R^{2}\left[1-\frac{r}{R\tan\left(\frac{r}{R}\right)}\right]\,.$
(11)

Here, $\delta$ is an arbitrary integrating constant. Finally, we can solve for
the pressure $p$ and density $\rho$ as,

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{3g_{00}}{\kappa
R^{2}}-\frac{\epsilon}{2}\,,$ $\displaystyle p$ $\displaystyle=$
$\displaystyle-\frac{g_{00}}{\kappa R^{2}}+\frac{\epsilon}{2}\,.$ (12)

The density $\rho$ and pressure $p$ vanish on the Killing-horizon only if we
choose the condition $\epsilon=0$. For non-zero $\epsilon$, the pressure and
density are discontinuous across the horizon and we do not consider in this
article. For spherically symmetric static spacetimes described by the
spacetime metric in Eq. (1), the Killing-horizon is given by the condition
$F(r_{h})=0$. Without loss of generality, in Eq. (11), we set the parameter
$\delta$ to be zero. With these conditions, we obtain,

$F\left(r\right)=1-\frac{2m}{R\tan\left(\frac{r}{R}\right)}-\Lambda
R^{2}\left[1-\frac{r}{R\tan\left(\frac{r}{R}\right)}\right]\,.$ (13)

Although $p<0$, the weak energy condition $\rho+p\geq 0$ is always satisfied
outside the event horizon for this spacetime. This is the form of metric we
use for the rest of our analysis.

As mentioned earlier, one of the main problems with this spacetime is that, in
the limit as $\frac{r}{R}\rightarrow\pi$, there is a curvature singularity
which is not covered by any event horizon. If the outer solution is matched
with the static Einstein solution, it leads to a surface with a discontinuity
in the first derivatives of the metric. Interestingly, for the spacetime
metric in Eq. (1) with $g_{00}$ given in Eq. (13), it has an outer
cosmological horizon, similar to that of the SdS solution, effectively masking
the singularity. In the present study, we use this metric to investigate the
effect of matter and cosmological influence on the static blackhole.

(a)

(b)

Figure 1:
(a): Plot of location of the horizon as a function of $R$ for different values
of $\Lambda$. The solid line corresponds to the cosmological horizon (CH),
while the dashed line represents the blackhole horizon (BH) for given
$\Lambda$ values. The curves for different $\Lambda$ values are shown with
different colors. The extremal horizons, where CH and BH merge, are marked
with red circles.
(b): Plot of location of the horizon as a function of $\Lambda$ for different
values of $R$. The solid line corresponds to the cosmological horizon (CH),
while the dashed line represents the blackhole horizon (BH) for given $R$
values. The curves for different $R$ values are shown with different colors.
The extremal horizons, where CH and BH merge, are marked with red circles.

###  2.1 Special cases

In this subsection, we examine some of the interesting special cases of the
spacetime solution obtained in the previous section. We begin with the case
when the cosmological constant $\Lambda\rightarrow 0$. In this scenario, we
obtain the VES spacetime, as given by,

$ds^{2}=\left[1-\frac{2m}{R\tan\left(\frac{r}{R}\right)}\right]\,dt^{2}-\left[1-\frac{2m}{R\tan\left(\frac{r}{R}\right)}\right]^{-1}\,dr^{2}-R^{2}\sin^{2}\left(\frac{r}{R}\right)d\Omega^{2}\,.$
(14)

All the additional changes introduced are contributed by the cosmological
constant. As the limit $R\rightarrow\infty$, to the lowest order in $r/R$, we
obtain the Schwarzschild solution. At the second order in $r/R$, we obtain the
SdS spacetime described by the spacetime metric:

$ds^{2}=\left(1-\frac{2m}{r}-\frac{1}{3}\Lambda
r^{2}\right)dt^{2}-\left(1-\frac{2m}{r}-\frac{1}{3}\Lambda
r^{2}\right)^{-1}dr^{2}-r^{2}d\Omega^{2}\,.$ (15)

When $m=0$ in eq. (13), we get another interesting case, where the spacetime
metric is given by,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
F_{e}(r)\,dt^{2}-F_{e}(r)^{-1}\,dr^{2}-R^{2}\sin^{2}\left(\frac{r}{R}\right)d\Omega^{2}\,,$
(16)

Here, $F_{e}(r)=\left[1-\Lambda
R^{2}\left(1-\frac{r}{R\tan\left(\frac{r}{R}\right)}\right)\right]$. This is
not de Sitter spacetime because the curvature scalar $K=R^{ijkl},R_{ijkl}$
diverges at both $r\rightarrow 0$ and $r\rightarrow R\pi$, similar to the case
of VSD spacetime. This interesting special case will be investigated in the
future.

###  2.2 Another Class of Solution

Interestingly, the spatial geometry of the SEU can be changed from positive
($+$ve) to negative ($-$ve) by replacing $\sin\left(\frac{r}{R}\right)$ with
$\sinh\left(\frac{r}{R}\right)$. The metric given in Eq. (1) transforms to,

$ds^{2}=H\left(r\right)dt^{2}-\frac{dr^{2}}{H\left(r\right)}-R^{2}\sinh^{2}\left(\frac{r}{R}\right)d\Omega^{2}\,.$
(17)

It can be easily shown that in this case, the solution for $H(r)$ takes the
form:

$H\left(r\right)=1-\frac{2m}{R\tanh\left(\frac{r}{R}\right)}-\Lambda
R^{2}\left[1-\frac{r}{R\tanh\left(\frac{r}{R}\right)}\right]\,.$ (18)

Proceeding as before, in this case, the expressions for pressure $p$ and
density $\rho$ can be written as,

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle-\frac{3g_{00}}{\kappa
R^{2}}\,,$ $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{g_{00}}{\kappa R^{2}}\,.$ (19)

Although the equation of state is still $\rho+3p=0$, this solution does not
satisfy the energy condition $\rho>0$. In an analogous context, this solution
could represent a blackhole in an anti-de Sitter background. This solution
might be of interest from a cosmological perspective, especially when $m=0$.

Figure 2: Plot of position at which a static observer becomes geodesic.

###  2.3 Radiating Schwarzschild-Whittaker

One way to introduce time dependency is by incorporating radiation, similar to
a Vaidya-like radiating solution, where the blackhole is embedded in incoming
or outgoing radiation [19]. It turns out that, like the VES solution, the SW
solution can be easily extended to incorporate incoming or outgoing radiation
[20, 21].

We start with the metric given in Eq. (1), written in the form,

$ds^{2}=g_{00}dv^{2}+2\,dr\,dv-R^{2}\sin^{2}\left(\frac{r}{R}\right)\left[d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right]\,,$ (20)

where

$\displaystyle g_{00}$ $\displaystyle\equiv$ $\displaystyle
F\left(v,r\right)=1-\frac{2\,m\left(v,r\right)}{R\,\tan\left(\frac{r}{R}\right)}-\Lambda
R^{2}\left(1-\frac{r}{R\tan\left(\frac{r}{R}\right)}\right)\,,$ $\displaystyle
v$ $\displaystyle=$ $\displaystyle t-r\,.\ $ (21)

Here, $m\left(v,r\right)$ is a function of time. In this case, the non-
vanishing components of the Einstein tensor are given by,

$\displaystyle G_{\>0}^{0}$ $\displaystyle=$
$\displaystyle\frac{3}{R^{2}}F\left(v,r\right)+\frac{2}{R^{2}}\cot^{2}\left(\frac{r}{R}\right)m^{\prime}\left(v,r\right)+\Lambda\,,$
$\displaystyle G_{\>1}^{0}$ $\displaystyle=$ $\displaystyle\frac{2}{R^{2}}\,,$
$\displaystyle G_{\>0}^{1}$ $\displaystyle=$
$\displaystyle-\frac{2}{R^{2}}\cot^{2}\left(\frac{r}{R}\right)\dot{m}\left(v,r\right)\,,$
$\displaystyle G_{\>1}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}F\left(v,r\right)+\frac{2}{R^{2}}\cot^{2}\left(\frac{r}{R}\right)\frac{\partial}{\partial
r}m\left(v,r\right)+\Lambda\,,$ $\displaystyle G_{\>2}^{2}=G_{\>3}^{3}$
$\displaystyle=$
$\displaystyle\frac{1}{R^{2}}F\left(v,r\right)-\frac{2}{R^{2}}m^{\prime}\left(v,r\right)+\frac{1}{R}\cot\left(\frac{r}{R}\right)m^{\prime\prime}\left(v,r\right)\,+\Lambda.$
(22)

In the above, we have $m^{\prime}=\frac{\partial}{\partial
r}m\left(v,r\right)$ and $\dot{m}=\frac{\partial}{\partial
v}m\left(v,r\right)$.

For our solution, we propose a two-fluid energy-momentum tensor comprising a
timelike static fluid with a timelike four-velocity given by
$u^{a}=\left(\frac{1}{\sqrt{F}},\,0,\,0,\,0\right)$ and a two-component null
fluid with the four-velocities $v^{a}=\left(0,\,1,\,0,\,0\right)$ and
$w^{a}=\left(1,\,-\frac{F}{2},\,0,\,0\right)$. The massive or timelike matter
part of the energy-momentum tensor can be expressed as:

$T_{\>b}^{a}=(\rho_{m}+p_{m})u^{a}u_{b}-p_{m}\delta_{\>b}^{a},$ (23)

With eq. (22), we can solve for $\rho_{m}$ and $p_{m}$ can be written as,

$\displaystyle\rho_{m}$ $\displaystyle=$
$\displaystyle\frac{3}{R^{2}}F\left(v,r\right)\,,$ $\displaystyle p_{m}$
$\displaystyle=$ $\displaystyle-\frac{1}{R^{2}}F\left(v,r\right)\,.$ (24)

We observe that the timelike matter field still satisfies the equation of
state $\rho_{m}+3p_{m}=0,$ indicating special cases of Whittaker solutions. We
use the form of the energy-momentum tensor for the null fluid given by Husain
[22].

$\bar{T}_{\>b}^{a}=-\frac{2}{R^{2}}\dot{m}\left(v,r\right)\,\cot^{2}\left(\frac{r}{R}\right)v^{a}v_{b}+\rho_{r}\left(v^{a}w_{b}+w^{a}v_{b}\right)+p_{r}\left(-\delta_{\>b}^{a}+v^{a}w_{b}+w^{a}v_{b}\right)$
(25)

where,

$\displaystyle\rho_{r}$ $\displaystyle=$
$\displaystyle\frac{2}{R^{2}}m^{\prime}\left(v,r\right)\cot^{2}\left(\frac{r}{R}\right)\,,$
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{2}{R^{2}}m^{\prime}\left(v,r\right)-\frac{1}{R}\cot\left(\frac{r}{R}\right)m^{\prime\prime}\left(v,r\right)$

By selecting a suitable equation of state for $\left(\rho_{r},\,p_{r}\right)$,
one can derive an explicit solution for $m\left(v,r\right)$[22, 23]. We draw
attention to a straightforward special case of a Vaidya-like radiating
solution by setting
$m^{\prime}\left(v,r\right)=m^{\prime\prime}\left(v,r\right)=0$, to obtain the
energy-momentum tensor with

$\bar{T}_{\>b}^{a}=-\frac{2}{R^{2}}\dot{m}\left(v\right)\,\cot^{2}\left(\frac{r}{R}\right)v^{a}v_{b}\,,$
(26)

For the metric given in eq. (20), as $m$ is an arbitrary function of $t-r$,
the geometry lacks time symmetry, i.e., there is no timelike Killing-vector.
The blackhole can shrink or grow depending on the flow of radiation inward or
outward. This demonstrates the possibility of the existence of several
interesting asymptotically non-flat blackhole solutions. These solutions can
play an important role in investigating the properties of blackholes in non-
flat backgrounds.

(a)

(b)

Figure 3:
(a): Plot of the effective potential, $V_{\mathrm{eff}}$, for timelike
geodesics as a function of $r$. We fix the value of $R=5$, while using two
representative values of $\Lambda$, i.e., $\Lambda=0.05$ with dashed curves
and $\Lambda=0.01$ with solid lines.
(b): Plot of the effective potential, $V_{\mathrm{eff}}$, for lightlike
geodesics as a function of $r$. We fix the value of $R=5$, while using two
representative values of $\Lambda$, i.e., $\Lambda=0.05$ with dashed curves
and $\Lambda=0.01$ with solid lines.

##  3 Event Horizon and Static Observer

The Killing-horizon in the static spherically symmetric spacetime is
determined by the condition $g_{00}=0$. For the SW spacetime, this condition
yields the explicit expression

$R\tan\left(\frac{r}{R}\right)\left(1-\Lambda R^{2}\right)+\left(\Lambda
R^{2}r-2m\right)=0\,.$ (27)

In general, there are two horizons: one blackhole horizon closer to the center
and the other cosmological horizon closer to the outer boundary. Since it is
difficult to provide an explicit analytical expression for the position of
these horizons, we numerically determine the horizon’s location using the
condition given by eq. (27). First, we find the horizon’s location by fixing a
range of values of $\Lambda$ from $0.001$ to $0.25$ at equal intervals and
plot them as a function of $R$. In Figure-1a, the solid lines represent the
cosmological horizon, while the dashed lines represent the blackhole event
horizon. For smaller values of $\Lambda$, the solution approaches Vaidya’s
static solution, and for larger values of $R$, it matches with the SdS
solution. Finally, for some combinations of values of $R$ and $\Lambda$, both
horizons meet, and no part of the Universe has a static timelike observer.
This makes the fluid-based source model with the Whittaker equation of state
fail. We refer to this as an extremal condition, as shown in Figure-1a. Such
possibilities are not new; similar situations are observed in SdS and also
blackhole in an expanding Universe [12].

We also plot the location of horizons by reversing the roles of $\Lambda$ and
$R$. In this case, we fix a set of values of $R$ from $1.58$ to $12.0$ at
equal intervals and plot the location of the horizon as a function of
$\Lambda$, as shown in Figure-1b. From these studies, we conclude that the
larger the values of $R$, the larger both horizons become, and as
$R\rightarrow\infty$, we have the SdS case. However, the blackhole horizon
seems to increase in size with increasing $R$, while the cosmological horizon
appears to shrink with an increase in $\Lambda$.

Because SW spacetime is a static solution, the event horizon and the static
limit match. In usual cases, such as the Schwarzschild solution, a static
observer becomes geodesic only at spatial infinity. However, in SW blackholes,
due to contributions from $M$, $R$, and $\Lambda$, a static observer
experiences both inward and outward pulls, or attractions in the direction of
both horizons. At certain locations, these forces can balance, allowing a
static observer to follow geodesic motion at a finite distance. The
acceleration of a static observer is given by,

$a_{q}=\phi_{,q}\,\,\,\mathrm{where}\,\,e^{-2\phi}=g_{00}\,.$ (28)

The direction of the acceleration is toward the horizons. In between the two
horizons, the acceleration becomes zero, and a static observer follows a
geodesic motion. The spatial location at which the acceleration becomes zero,
allowing a static observer to follow a geodesic motion, is shown in Figure-2.
Here also, we first fix values of $\Lambda$ and solve for the location of the
geodesic as a function of $R$, as shown in Figure-2-(a), we observe that an
increase in $\Lambda$ brings the location of the geodesic closer to the
blackhole horizon. An increase in $R$ pushes the location further from the
blackhole horizon. In Figure-2-(b), we reverse the roles of $\Lambda$ and $R$
and plot the location of a static observer as a function of $\Lambda$ for a
fixed value of $R$. Increasing $\Lambda$ pushes the geodesic position of a
static particle toward the cosmological horizons.

(a) Orit Intergration

(b) Newtonian Orbit

Figure 4: The orbital equation is integrated by combining four sections. We
use $r_{0}=(r_{min}+r_{max})/2$ as the radial starting point, where $r_{min}$
and $r_{max}$ are the radial turning points for a given $Lz$ and $E_{t}$. The
first two sections of orbits are obtained by integrating the orbital equation,
starting from $\\{r_{0},\pi/2\\}$ and proceeding inward ($\epsilon=-1$) and
outward ($\epsilon=1$). Similarly, the other two sections are obtained by
starting from $\\{r_{0},3\pi/2\\}$ and integrating in the inward and outward
directions. We combine all four sections smoothly to obtain the complete
orbit.

##  4 Geodesic motion and Periastron Precession

In this section, we investigate the geodesic motion in SW spacetime. Since the
metric given in eq.(1) is static and spherically symmetric, it is
straightforward to obtain the geodesic equation of motion. Additionally, we
restrict ourselves to the equatorial plane, i.e., $\theta=\frac{\pi}{2}$,
$\dot{\theta}=0$, and $\ddot{\theta}=0$. In this case, the timelike geodesic
equation can be easily separated using constants of motion. With these, we
obtain:

$\displaystyle\dot{t}$ $\displaystyle=$ $\displaystyle\frac{E_{t}}{F(r)}\,,$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{L_{Z}}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\,,$
$\displaystyle\dot{r}^{2}$ $\displaystyle=$ $\displaystyle
E_{t}^{2}-F(r)\left[1+\frac{L_{z}^{2}}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\right]\,.$
(29)

Here, $E_{t}$ and $L_{z}$ are constants of motion, namely, the energy and the
z-component of angular momentum for the timelike particle. For the light or
null geodesic, the corresponding equations are given by:

$\displaystyle\dot{t}$ $\displaystyle=$ $\displaystyle\frac{E_{p}}{F(r)}\,,$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{L_{p}}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\,,$
$\displaystyle\dot{r}^{2}$ $\displaystyle=$ $\displaystyle
E_{p}^{2}-F(r)\frac{L_{p}^{2}}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\,.$ (30)

In this context, we also have $E_{p}$ and $L_{p}$ as conserved quantities.
From the radial equation, we can derive the effective potential for timelike
and null geodesics. The potentials are given by:

$\displaystyle V_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle
F(r)\left[1\,+\,\frac{L_{z}^{2}}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\right]\,,$
$\displaystyle V_{\mathrm{null}}$ $\displaystyle=$ $\displaystyle
F(r)\frac{L_{p}^{2}}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\,.$ (31)

With the function $F(r)$ given by equation (13), the maxima and minima of the
effective potential correspond to unstable and stable circular geodesics. An
important difference in the SdS solution, the position of circular null
geodesics does not depend on $\Lambda$. However, in the case of SW spacetime,
the potential is not just an additive constant, and the position of null
geodesics explicitly depends on $\Lambda$. The effective potentials for both
timelike and null geodesics exhibit similar behaviour, as shown in Figures 3a
and 3b.

For smaller values of $\Lambda$, there exists a single unstable circular
geodesic. In contrast, for larger values of $\Lambda$, the number and nature
of circular geodesics depend significantly on the value of $L_{z}$. As
illustrated in the figures, a low value of $L_{z}$ results in a single
unstable circular geodesic. When $L_{z}=0$, it corresponds to a static
observer, as previously discussed. Increasing the value of $L_{z}$ introduces
a stable circular geodesic in the middle, with two unstable circular geodesics
appearing closer to each horizon. This variation is indeed an intriguing
difference.

Figure 5:  Plots of periastron shift and radial turning points for a massive
particle in Schwarzschild spacetime as functions of $L_{z}$ and ${E_{t}}^{2}$
for $M=1$. The plots in the top row show the behavior of the periastron, while
those in the bottom row display the behavior of the radial turning points
$r_{min}$ and $r_{max}$.  Figure 6: Plots of periastron shift as functions of
the semi-major axis, $a$, and orbital eccentricity, $\epsilon$, in the
Schwarzschild spacetime are presented. The plots in the left column correspond
to variations in $L_{z}$ and depict the precession behavior as a function of
$a$ and $\epsilon$. Similarly, the plots in the right column are obtained by
varying the energy parameter, $E_{t}$.

###  4.1 Precession of Periastron

In this section, we examine the precession of the periastron for timelike
elliptical orbits. We begin by utilizing the timelike geodesic equations
provided in eq. (29), and we numerically integrate the $r,\,\phi$ equations to
obtain the complete orbit. In this case, we integrate two first-order
equations, as follows:

$\displaystyle\frac{d\phi}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{L_{z}}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\,,$
$\displaystyle\frac{dr}{d\tau}$ $\displaystyle=$
$\displaystyle\epsilon\left[E_{t}^{2}-F(r)-F(r)\frac{L_{z}^{2}}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\right]^{1/2}\,.$
(32)

In the case of the radial equation, the parameter $\epsilon$ represents the
direction of propagation, i.e., when $\epsilon=+1$, the particle moves along
the outwards or increasing direction of $r$. On the other hand, when
$\epsilon=-1$, the radial coordinate decreases, causing the particle to move
inwards. We integrate the $r$ and $\phi$ equations as functions of proper time
$\tau$, and this allows us to easily construct the spatial orbital equation
$r(\phi)$. For completeness and comparison, we also explicitly provide the
time-like geodesic equation for the Schwarzschild case.

$\displaystyle\frac{d\phi}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{L_{z}}{r^{2}}\,,$ $\displaystyle\frac{dr}{d\tau}$
$\displaystyle=$
$\displaystyle\epsilon\left[E_{t}^{2}-1+\frac{2M}{r}-\frac{{L_{z}}^{2}}{r^{2}}+\frac{2M\,{L_{z}}^{2}}{r^{3}}\right]^{1/2}\,,$
(33)

and in Schwarzschild-de Sitter case,

$\displaystyle\frac{d\phi}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{L_{z}}{r^{2}}\,,$ $\displaystyle\frac{dr}{d\tau}$
$\displaystyle=$
$\displaystyle\epsilon\left[E_{t}^{2}-\left(1-\frac{2M}{r}-\frac{\Lambda}{3}r^{3}\right)\left(1+\frac{{L_{z}}^{2}}{r^{2}}\right)\right]^{1/2}\,.$
(34)

However, the radial turning points correspond to stationary points in the
equation of motion at which the differential equation breaks down and poses
difficulties in the numerical integration, leading to large numerical errors.
To avoid these difficulties, we break the integration of orbital differential
equations into four sections, with each section ending at a stationary point.
We smoothly combine the solutions to obtain the full orbit equation, as shown
in Figure-4a. We use the following simple steps to achieve accurate results
using the Euler method to integrate the $\phi$ and $r$ equations. As a test
case for the procedure, in the case of a Newtonian potential or Kepler
problem, we obtain an exact elliptical orbit without any precession. However,
a deviation from the Newtonian potential leads to a more complicated orbit,
which can be assumed to be instantaneously elliptical, with its semi-major
axis precessing over time. Due to the time-independent or static nature of the
spacetime, $r_{min}$ and $r_{max}$ remain the same as functions of time.
However, they slowly drift as functions of $\phi$, resulting in a slow drift
in the orientation of the semi-major axis. It is this shift or change in the
semi-major axis that we measure and serves as one of the first tests for
general relativity. Our interest is not in testing the theory of gravity;
rather, we aim to evaluate the influence of cosmological parameters, such as
$R$ and $\Lambda$, on the phenomenon of periastron precession. We measure the
periastron precession closer to the event horizon, or in the strong field
limits. For this reason, we do not use the perturbative methods predominantly
used in the similar studies, instead we numerically integrate the orbit
equation. Once again, our interest is not to compare deviations from Newtonian
theory or measure small changes in astrophysical scenarios. We compare
deviations with respect to a Schwarzschild solutions and blackhole influenced
by cosmological parameters.

Figure 7: Plots of periastron shift and radial turning points for a massive
particle in the SdS spacetime as functions of various parameters. The plots in
the top row show the dependence of periastron on the parameters $Lz$,
$E_{t}^{2}$, and $\Lambda$, respectively. Similarly, the bottom row shows the
dependence of radial turning points ($r_{m}in$ and $r_{m}ax$) as functions of
$Lz$, $E_{t}^{2}$, and $Lambda$. The value of mass is fixed at $M=1$.  Figure
8:  Plots of precession of periastron as a function of semi-major axis ($a$)
and eccentricity ($\epsilon$) in the SdS spacetime. In the rightmost column,
variations in ($a$) and ($\epsilon$) are induced by varying $L_{z}$.
Similarly, in the second and third columns, changes in ($a$) and ($\epsilon$)
are induced by varying $E_{t}$ and $\Lambda$, respectively.

Numerical estimating precession involves the following steps:

  1. 1.

For a given $E_{t}$ and $L_{z}$, we determine the radial points $r_{min}$ and
$r_{max}$. In the case of an elliptical orbit, the length of the semi-major
axis $a$ is given by, $a=\frac{1}{2}(r_{min}+r_{max})$. The length of the
semi-minor axis $b$ is calculated as, $b=\sqrt{r_{min}\cdot r_{max}}$. The
eccentricity, denoted as $\epsilon$, can be computed as:
$\epsilon=\sqrt{1-\frac{b^{2}}{a^{2}}}$. Due to stationary points at $r_{min}$
and $r_{max}$, integrating the orbital ordinary differential equations in one
go can result in large errors. To minimize numerical errors, we divide the
orbit into four sections. Starting from the initial coordinates
$\\{r=r_{0},\,\phi=\frac{\pi}{2}\\}$, we first integrate along the increasing
$r$ direction ($\epsilon=+1$) until we reach $r_{max}$. Then, to obtain the
orbit from $\\{r_{0},\,\frac{\pi}{2}\\}$ to $r_{min}$, we integrate along the
decreasing $r$ direction ($\epsilon=-1$) until we reach $r_{min}$. We use a
similar approach to integrate the orbital equations starting from
$\\{r=r_{0},\,\phi=\frac{3\pi}{2}\\}$ and arriving at $r_{max}$ and $r_{min}$.
The details are illustrated in Figure 4a.

  2. 2.

We smoothly combine all four patches to construct the complete orbit. For
testing, we use the Newtonian potential for which the orbit is elliptical with
no periastron precession, as shown in figure 4b. The value of the periastron
precession, denoted as $\phi(\tau_{0})-2\pi$, is calculated, where $\tau_{0}$
is the proper time taken to complete the orbit from $r_{max}\rightarrow
r_{min}\rightarrow r_{max}$.

We study the SW, SdS, and Schwarzschild solutions separately, with the
Schwarzschild case serving as a reference. We investigate the influence of
$\Lambda$ on the de-Sitter case and the influence of $\Lambda$ and $R$ on SW,
considering the variations in orbital parameters such as $E_{t}$ and $L_{z}$,
which in turn affect $r_{min}$ and $r_{max}$.

Figure 9:  Plots of periastron shift and radial turning points for a massive
particle in SW spacetime are presented as functions of various parameters. In
this case, we fix the value of the mass, $M=1$. The top row of plots depicts
the dependence of periastron on various parameters, including
$L_{z},\,{E_{t}}^{2},\,\Lambda,\,\rm{and},\,R$. Similarly, the bottom row
shows the dependence of radial turning points, $r_{min}$ and $r_{max}$, as
functions of $L_{z},\,{E_{t}}^{2},\,\Lambda,\,\rm{and},\,R,$ respectively.
Figure 10: Plots of precession of periastron as a function of semi-major axis,
$a$, and eccentricity, $\epsilon$, in SW spacetime are presented. In the
rightmost column, the changes in $a$ and $\epsilon$ are induced by varying
$L_{z}$. Similarly, in the next three columns, the changes in $a$ and
$\epsilon$ are induced by varying $E_{t}$, $\Lambda$, and $R$, respectively.

####  4.1.1 Schwarzschild Spacetime

First, we look at the precession of periastron in the case of the
Schwarzschild blackhole as our reference. We arbitrarily choose a range of
$E_{t}$ and $L_{z}$ such that $r_{min}$ is sufficiently close to the event
horizon but not too close. We start by fixing the value of $E_{t}^{2}$ and
find ${r_{min},\,r_{max}}$ by varying $L_{z}$. Once we have obtained the
turning points, we integrate the orbital equation to calculate the periastron
precession. The results are summarized in figure-5. It is easy to see that the
precession is sensitive to $L_{z}$, as it determines the shape of the
effective potential. However, the precession weakly depends on ${E_{t}}^{2}$.
The orbital eccentricity changes considerably within the range of $L_{z}$
values considered. In Figure 5, we observe a linear dependence of periastron
precession on the semi-major axis, while its relationship with orbital
eccentricity is more complex. Figures 5 and 6 reveal a weak dependence on
$E_{t}$. Although eccentricity increases with higher $E_{t}$ values, leading
to an increase in the semi-major axis, the combined effect results in only a
marginal change in periastron precession.

####  4.1.2 Schwarzschild de-Sitter Spacetime

In this section, we explore the influence of the cosmological constant on the
precession of the periastron using the SdS solution. Our aim is not to
constrain the cosmological constant in weaker gravitational fields such as the
Sun but to examine the qualitative differences in the strong field due to the
presence of the cosmological constant and understand the dependence of
precession on the value of $\Lambda$. In this case, we have three parameters:
$L_{z}^{2}$, $E_{t}$, and $\Lambda$. As before, we fix any two parameters
within the given range and vary the third parameter to determine the turning
points. With the turning points, we can integrate the orbital equation to
determine the periastron shift. We repeat this procedure to obtain the
periastron as a function of all parameters. The results are shown in Figure 7.
Although $\Lambda$ has to be reasonably small, it has a considerable influence
on the periastron precession. Similar to the case of the Schwarzschild
blackhole, in this case, the precession weakly depends on the energy of the
particle. In Figure 8, we observe that in the case of SdS spacetime, the
dependence on the semi-major axis obtained by changing $L_{z}$ is stronger
than in the Schwarzschild case. Meanwhile, the dependency with eccentricity is
very similar. In the case of SdS spacetime, the dependency of periastron
precession on orbital energy is weak.

Figure 11: Plots of the effective potential for a scalar field as a function
of $\frac{r}{R}$ for different values of $l$ ranging from 0 to 6. We have
fixed the values of $R=10$, $M=1$, and $\Lambda=0.001$.

####  4.1.3 Schwarzschild-Whittaker Spacetime

In this section, we investigate the periastron precession of elliptical orbits
in the SW blackhole, which is influenced by two parameters, $R$ and $\Lambda$,
introducing cosmological effects. This results in a total of four parameters:
$L_{z}$, $E_{t}$, $R$, and $\Lambda$. We systematically study the precession
of the periastron, similar to our approach in the previous case. The results
are summarized in Figure-9. Interestingly, it appears that the contribution to
precession from $R$ tends to suppress the effects of $\Lambda$. However, this
effect may be attributed to the fact that $R$ has a larger impact on the
orbital parameters $r_{min}$ and $r_{max}$. While precession is directly
dependent on $r_{min}$ and $r_{max}$, we have yet to determine the precise
independent relationship between the two.

(a) $V_{scalar}(r)$ for $l=0$

(b) $V_{scalar}(r)$ for $l=1$

Figure 12: Plots of effective potential for scalar field as a function of
$\frac{r}{R}$, for $M=1$ and $\Lambda=0.001$ with various values of $R$ from
$5.0$ to $14.0$.

##  5 Scalar Waves in Schwarzschild-Whittaker

In this section, we investigate a massless scalar field within the SW
spacetime background. The evolution of the massless scalar field in a curved
geometry is described by the Klein-Gordon equation, which can be written as
follows:

$\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{a}}\left[\sqrt{-g}\>g^{ab}\frac{\partial\psi}{\partial
x^{b}}\right]\,=\,0\,.$ (35)

Where $g$ is the determinant of the metric tensor $g_{ab}$, and for the
spacetime metric given by eq. (1), $g$ takes the expression
$|g|=R^{4}\sin^{4}\left(\frac{r}{R}\right)\sin^{2}\theta$. The Klein-Gordon
equation can be simplified to the following form:

$\frac{1}{R^{2}\sin^{2}\left(\frac{r}{R}\right)\sin\theta}\frac{\partial}{\partial
x^{a}}\left[R^{2}\sin^{2}\left(\frac{r}{R}\right)\sin\theta\,g^{ab}\frac{\partial\psi}{\partial
x^{b}}\right]=0\,.$ (36)

Following the standard procedure, we decompose the $\left(\theta,\phi\right)$
part of $\psi(t,r,\theta,\phi)$ in terms of spherical harmonics, and the
temporal part in terms of the Fourier frequency $\omega$, resulting in the
form:

$\psi\left(t,r,\theta,\phi\right)=e^{i\omega
t}\,\psi_{r}\left(r\right)\,Y_{l}^{m}\left(\theta,\phi\right)\,,$

where $Y_{l}^{m}\left(\theta,\phi\right)$ are spherical harmonics. With this
decomposition, we can separate the radial equation, which can be written as:

$\frac{F(r)}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\frac{d}{dr}\left[R^{2}\sin^{2}\left(\frac{r}{R}\right)F\left(r\right)\frac{d\psi_{r}\left(r\right)}{dr}\right]+\left[\omega^{2}\,-\,\frac{l\left(l+1\right)F(r)}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\right]\psi_{r}\left(r\right)=0\,.$
(37)

We further simplify the equation by changing our analysis to a tortoise-like
coordinate system. We define a new variable $y$ such that,

$dy=F\left(r\right)^{-1}dr\,,$ (38)

and changing dependent variable from $\psi_{r}$ to $u(r)$, which is given by,

$u\left(r\right)=R\sin\left(\frac{r}{R}\right)\psi_{r}\left(r\right)\,.$ (39)

With the above substitution, the radial equation takes the form of a time-
independent Schrödinger equation,

$\frac{d^{2}u}{dy^{2}}\,+\,\left[\omega^{2}-V_{scalar}\left(r\right)\right]u=0\,,$
(40)

with the potential $V_{scalar}(r)$ given by,

$V_{scalar}\left(r\right)=F(r)\left[\frac{1}{R\,\tan\left(\frac{r}{R}\right)}\frac{dF\left(r\right)}{dr}-\frac{1}{R^{2}}F\left(r\right)+\frac{l\left(l+1\right)}{R^{2}\sin^{2}\left(\frac{r}{R}\right)}\right]\,.$
(41)

Note that the potential, $V_{scalar}(r)$, vanishes on the event horizon, which
is given by the condition $F\left(r_{h}\right)=0$. Plots of the scalar
potential, $V_{scalar}(r)$, as functions of various parameters are shown in
Figures 11, 12, and 13. One important difference from the Schwarzschild case
is that the potential takes negative values. The negative value of the
potential on the boundary implies an exponential growth in the scalar field,
leading to instability. However, at a finite radial point with negative
potential, the field can have bound states, provided the fields have energies
in a suitable range. If the energy level is kept suitably high, one can have
the scattering of scalar field by the horizon. In addition, if the potential
is negative beyond the horizon, this region is outside our domain.

(a) $V_{scalar}(r)$ for $l=0$

(b) $V_{scalar}(r)$ for $l=1$

Figure 13: Plots of effective potential for scalar field as a function of
$\frac{r}{R}$, for $M=1$ and $R=10$. with various values of $\Lambda$ from $0$
to $0.0008$.

In Figure 11, we display the potential for different values of $l$ ranging
from $0$ to $6$ as a function of $\frac{r}{R}$, with $R=10$, $M=1$, and
$\Lambda=0.001$ held constant. In this case, the potential is well-behaved,
approaching zero on the boundary and beyond. For the lowest value of $l=0$,
the minimum value inside the domain falls below zero, indicating the
possibility of bound states between the blackhole and cosmological horizon.
This could be attributed to a radial mode or spherical accretion of the field,
influenced by the presence of two horizons, suggesting that the free fall of
matter may be affected by both horizons.

Another interesting observation is the presence of two peaks, each closer to
the blackhole and cosmological horizon. This configuration allows the field to
become trapped between these peaks. However, the behaviour of the potential is
strongly dependent on other parameter ranges due to the metric’s dependence on
multiple parameters.

Next, we investigate the influence of $R$ on the effective potential
$V_{scalar}(r)$. The results are depicted in Figure 12 for two values of $l$
($0$ and $1$). The metric coefficient $g_{00}$, denoted as $F(r)$, undergoes
significant changes when altering the values of $R$ or accounting for
cosmological influence. In these plots, $g_{00}=F(r)$ is also shown with a
suitable scaling factor, $\alpha$, to bring the plots within the range. This
is done to emphasize the regions where $F(r)>0$ and $V_{scalar}$.

For lower values of $R=\\{4,\,6,\,\rm{and}\,8\\}$ (as shown in Figure 12a),
only the blackhole horizon exists, and there is no cosmological horizon. In
all these cases, $V_{scalar}$ approaches negative values in the outer region,
while $F(r)>0$. These parameters indicate a potentially unstable spacetime and
require more in-depth study.

Conversely, for larger values of $R=\\{10,\,12,\,\rm{and}\,14\\}$ (Figure
12a), the potential exhibits dual peaks. In these cases, both blackhole and
cosmological horizons coexist, where the potential vanishes. Here, bound
states are possible as the potential takes on negative values.

We also provide plots for $l=1$ in Figure 12b. In Figure 13, we present plots
of $V_{scalar}$ for different values of $\Lambda$ while keeping other
parameters fixed. For $l=0$ and $\Lambda=\\{0.0007,\,0.0008\\}$, the potential
blows up at the cosmological horizon. In these scenarios, the scalar field
might be able to evolve and necessitates further detailed investigation.
Figure 13a illustrates that these two cases feature both horizons.

However, for the remaining values of $\Lambda$, only the blackhole horizon is
present, and the potential assumes negative values at the outer boundary,
potentially indicating an unstable spacetime. In contrast, for $l=1$, the
potential becomes zero at the cosmological horizon (Figure 13b). It is evident
that the behaviour of the massless scalar field in the SW solution is complex
and warrants a more detailed analysis of stability, a subject we intend to
address in the future.

##  6 Conclusions

In this article, we explore solutions to Einstein’s equations with an event
horizon surrounded by matter and asymptotically not flat. This is a
generalization of the solution proposed by Vaidya [8] and falls under a
special case of the Whittaker equation of state, which we refer to as the
Schwarzschild-Whittaker solution. In this case, we have two independent
parameters that induce cosmological effects: the size of the universe, denoted
as $R$, and the cosmological constant $\Lambda$. We also extend the solution
to a radiating Vaidya-like solution, incorporating a timelike perfect fluid
with ingoing/outgoing radiation fields. We investigate various physical
phenomena, such as the precession of periastron and the scattering of a
massless scalar field, to understand the effects of cosmological parameters.
Our preliminary studies suggest that the size of the universe and the
cosmological parameters have roughly opposite contributions to the physics of
blackholes in a cosmological background. Further detailed investigations are
necessary for a proper understanding of cosmological influences on blackholes.

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