Point mass cosmological black holes

Javad T. Firouzjaee; Touhid Feghhi

doi:10.1007/s10509-018-3473-0

Astrophysics and Space Science

December 2018, 363:256 | Cite as

Point mass cosmological black holes

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Javad T. Firouzjaee
Touhid Feghhi

Original Article

First Online: 27 November 2018

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Abstract

Real black holes in the universe are located in the expanding accelerating background which is called the cosmological black holes. Hence, to study the background effects, it is necessary to model these black holes in the cosmological background where the dark energy is the dominant energy. In this paper, we argue that most of the dynamical cosmological black holes can be modeled by point mass cosmological black holes. Considering the de Sitter background for the accelerating universe, we present the point mass cosmological background in the cosmological de Sitter spacetime. Our work also includes the point mass black holes which have charge and angular momentum. We study the mass, horizons, redshift structure and geodesics properties for these black holes.

Keywords

Black hole Expanding background Mass Geodesics

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Notes

Acknowledgement

We would like to thank Alireza Allahyari for useful comments.

Appendix: The CBH’s orbits effective potential

Here we want to study the form of effective potential in Schwarzschild-de Sitter spacetime to study the circular orbits. The $\varLambda =0 $ and $9 \varLambda M^{2}=1 $ have been studied in Shapiro et al. (1983), Frolov and Novikov (1998) and Podolsky (1999) respectively. We can use Euler Lagrange Equations with metric (18) which is spherically symmetric and so we can choose $\theta = \frac{\pi }{2}$, hence $d \varOmega ^{2} = d \varphi ^{2} $ (Stuchlik and Slany 2004). The Euler Lagrange equations will result:

$$ \begin{aligned} & \epsilon = -\varPhi \dot{t}^{2} + \varPhi ^{-1} \dot{R}^{2} + R^{2} \dot{\varphi } ^{2} \\ & \dot{t} = \frac{E}{\varPhi } \\ & \dot{ \varphi } = \frac{L}{R^{2}} , \end{aligned} $$

(62)

where dot is partial differential relative to affine parameter and $E$ and $L$ are constants. $\epsilon = -1,0 ,1$ represents timelike, null and spacelike geodesics respectively. From Eq. (62) we can introduce effective potential $V(R)$, where

$$ V(R) = E^{2} - \dot{R}^{2} = \varPhi (R) \biggl( \frac{L^{2}}{R^{2}} - \epsilon \biggr) $$

(63)

We can depict $V(R)$ as a function of $\frac{M}{R}$ for different values of $L$ and $\alpha $ where $\alpha = 9\varLambda M^{2} $ and $0 \leq \alpha \leq 1 $, that $\alpha =1 $ is the extreme case (Podolsky 1999).

If we suppose a massless particle at infinity has the initial velocity $v^{R}_{0} = \sqrt{E^{2} + V(\infty )}$, as the massless particle moves toward central object, it decelerates to $\sqrt{E^{2}-V(R)} $ and if $E^{2} > V_{max}$, the zero mass particle will accelerate toward singularity otherwise $\dot{r}$ will change sign and the particle will be reflected to infinity. According to Figs. 2 and 3 as we decrease $\varLambda $ or increase $L$, it becomes harder for these particles to fall in to singularity. We can also study the circular orbits for mass less particles. We know that if $\frac{\partial V}{\partial R} =0 $ and $\dot{R}=0$, then we will have circular orbits and the stability criterion is $\frac{\partial ^{2} V}{\partial R^{2}} > 0 $. Hence we can write

$$ \begin{aligned} & \frac{\partial V}{\partial R} =0 \quad\rightarrow\quad R= 3M \\ & \dot{R} = 0 \quad\rightarrow\quad E^{2} = V(R), \end{aligned} $$

(64)

which has the same radius ($R= 3M$) as the $\varLambda = 0 $ case (Shapiro et al. 1983; Frolov and Novikov 1998).

Fig. 3
Effective potential of null geodesics as a function of $\frac{R}{M} $ for different values of $L$. Here we have supposed $\alpha =\frac{1}{10} $

According to Fig. 2, $(\frac{\partial ^{2} V}{\partial R ^{2}})_{| R=3M} < 0 $, that means there is no stable circular orbit for mass less particles except when $E=L =0$ and $R= 3M$. By Eq. (64), if we define critical impact parameter as $b= \frac{L}{E}$, then we can write:

$$ b_{critic} =\frac{L}{E} = \biggl(\sqrt{ \frac{R^{2}}{\varPhi (R)}}\biggr)_{| R=3M} = \frac{3 \sqrt{3} M}{\sqrt{1-\alpha }}. $$

(65)

If the impact parameter of these mass less particles is greater than critical value ($b_{critic}$), then these particles will be captured. This definition leads to a relation for capture cross section for massless particles from infinity:

$$ \sigma _{null} = \pi b^{2} = \frac{27 \pi M^{2}}{1- \alpha }. $$

(66)

We can do the same calculations for the timelike circular orbits. In this case, we have

$$ \frac{\partial V}{\partial R} = 0 \quad\rightarrow\quad L^{2} = \frac{MR^{2} - \frac{ \varLambda }{3}R^{5} }{R - 3M}. $$

(67)

By Eq. (67) and $\dot{R} = 0$ one can easily show:

$$ E^{2}= V (R)=\frac{(1-\frac{\alpha }{27M^{2}}R^{2} -\frac{2M}{R} )^{2}}{R(R - 3M)}. $$

(68)

According to Eq. (68) it’s clear that circular orbits exist for $R > 3M $, but Fig. 4 tells us that effective potential is convex for the given values of cosmological constant, so there is no circular orbit for time like case at these values. The value of $\alpha $ for a black hole with mass $10^{6}$ Sun mass and $\varLambda = 10^{-52}~\mbox{s}^{-1}$ is in order of $10^{-32}$, in this case we can have stable circular orbit and bound orbit (Fig. 8). Figure 4 also represents that for the same value of central mass if we increase the cosmological constant, $\dot{R}$ increases. For the same value of $\varLambda $, Fig. 5 shows that any variation of $L$ in the timelike case is similar to a null one.

Fig. 5
Effective potential of timelike geodesics as a function of $\frac{R}{M} $ for different values of $L$. Here we have supposed $\alpha =\frac{1}{10} $

Figure 6 shows that the spacelike case inside the de Sitter horizon resembles the null and timelike cases, but for outside de sitter horizon as we increase $\varLambda $ (for the same value of central mass), $\dot{R}$ decreases.

Fig. 6
Effective potential of spacelike geodesics as a function of $\frac{R}{M} $ for different values of $\alpha $. Here we have supposed $L=10M $

Fig. 7
Effective potential of spacelike geodesics as a function of $\frac{R}{M} $ for different values of $L$. Here we have supposed $\alpha =\frac{1}{10} $

Figures 8 and 9 represent effective potential for a specific black hole with mass $10^{6}~M_{\odot }$ and $L=5M $ and $3M$. We have also used present value of $\varLambda = 10^{-52}~\mbox{s}^{-1}$. We can see that only for some values of $L$ we can have timelike circular orbits and bound orbits. For the extreme case you can see Podolsky (1999).

Fig. 8
Effective potential of spacelike, null and timelike geodesics as a function of $\frac{R}{M} $ for a black hole with mass $10^{6}~M _{\odot }$ and $\varLambda = 10^{-52}~\mbox{s}^{-1}$. Here we have supposed $L=5M $

Fig. 9
Effective potential of spacelike, null and timelike geodesics as a function of $\frac{R}{M} $ for a black hole with mass $10^{6}~M _{\odot }$ and $\varLambda = 10^{-52}~\mbox{s}^{-1}$. Here we have supposed $L=3M $

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Javad T. Firouzjaee
- 1
- 2
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Touhid Feghhi
- 2

1.Department of PhysicsK.N. Toosi University of TechnologyTehranIran
2.School of AstronomyInstitute for Research in Fundamental Sciences (IPM)TehranIran

Point mass cosmological black holes

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