The Boundary Homotopy Retract on the Scalar Hairy Charged Black Hole Spacetime

Abstract

In this paper, we investigate and define the topology of some astrophysical phenomena, like the hairy (scalarized) charged black hole spacetime, to improve our understanding of the kinematics and dynamics of their nature. We use the Lagrangian equation to find different types of geodesic equations. This can be done under some conditions for the variations of the Cosmological constant and Newton’s constant. We show how to induce the two types (null and spacelike) of geodesics as boundary retractions, in order to obtain the boundary homotopy retract of the scalar charged black hole. These types are used the Lagrangian equation in a 4-D scalar charged black hole to explain the event horizon for this black hole.

1. Introduction

In astronomy and astrophysics, white dwarfs, neutron stars, and black holes are thought to be very dense, causing spacetime to be distorted/wraped. Gravitational waves will also be produced in our universe when pairs of these particles are in orbit. Thanks to the LIGO Scientific Collaboration and Virgo Collaboration to directly detect these phenomena in 2015 (LIGO Scientific Collaboration [1]). Up until now, there are more than 100 black hole mergers have been detected since the most significant event from GW150914 [2,3,4,5] and references therein.

One of the important innovations in physics through the last few decades was the accelerated expansion of the universe. General relativity and the quantum field theory fail to justify this phenomenon. However, some pioneer work are under consideration [6,7,8,9]. This role necessitates creative alternate viewpoints capable to provide a satisfactory justification of the Cosmological observations. One of the prospects is to investigate theories of gravity that are more general than general relativity. The scalar-tensor theories constitute one of original Einstein theory’s most natural generalizations [10,11,12]. Black holes are strong-field objects whose characteristics are controlled by Einstein’s theory of gravitation [13]. In mathematical physics, the existence of different types of black holes as well as an explanation of their topological origin has been demonstrated conclusively by the homotopy and retraction theories [14,15,16,17,18,19,20,21,22,23]. The existence of black holes is one of the most exciting predictions of Einstein’s general relativity [24]. On the other hand, the Einstein-scalar-Gauss-Bonnet theories were used to introduce a new kind of instability in a black hole without hair due to the linearized scalar equation. This is called, charged (scalarized) black hole [25,26,27,28]. The motion of the neutron star in the disk galaxy and the key features in studying how physical conditions affect the binary evolution was studied by [29,30,31,32,33,34,35,36,37].

However, the zero temperature limits of holographic fluids and superconductors [38,39] were investigated using the characteristics of the near horizon geometry in black holes. It has been demonstrated that a charged scalar field in this region can condense at a critical temperature, and that this temperature is defined by the horizon’s Anti-de Sitter space-time (AdS) geometry [40]. The effects of applying fractal structure to the horizon geometry were studied in [41,42]. In [43] it has been investigated the geodesic structure of a typical Hayward black hole. The topology of a black hole was discussed in [44]. For simplicity, we denote the scalar hairy charged black hole by $L^4$.

This work is the third in a series dedicated to studying the metric space of a topological black hole with spherical symmetry in AdS space-time. In addition, it discusses the relationship between retraction, homotopy, and folding in the n-dimension. In [45] discussed the spherical topology of black holes in a special case of spherical symmetry. They figured out the minimal deformation, minimal retraction, and equatorial geodesics of the topological black holes to explain the effects on their centers of mass in binary systems. In [45,46,47,48,49,50,51] work the equatorial geodesics on the line element of black holes were determined using a Lagrangian formalism. The space-time morphology, dynamics, and interactions of black holes with other self-gravitating systems were also explored. They specifically looked at the relationship between limit folding and limit retraction. The major object of the current work is to investigate the physical and mathematical of the hairy (scalarized) charged black hole, by adopting some transformations, namely the boundary homotopy retraction.

2. The Metric Spacetime of ${\mathit{L}}^{\mathbf{4}}$ and Its Inner and Outer Horizons

The metric spacetime of hair charged (scalarized) black hole is given by

$$\begin{array}{ccc}\hfill d{s}^2& =& -M\left(r\right)d{t}^2+\frac{d{r}^2}{M\left(r\right)}+r^{2}d{\mathsf{\Omega }}_2^2,\hfill \\ \hfill M\left(r\right)& =& 1-\frac{2m}{r}+\frac{Q^2+Q_s^2}{r^2}\hfill \end{array}$$

(1)

where m is the asymptotically measured mass of the black hole, Q electric charge, $Q_s$the scalar charge. Assuming that the black hole’s inner and outer horizons are zeros [52,53] of the metric function $M\left(r\right)$, we can write their respective radii

$$r\pm =m\pm \sqrt{m^2-(Q^2+Q_s^2)}$$

In [12] the spherical symmetric spacetime is described as

$$d{\mathsf{\Omega }}_2^2=d{\theta }^2+si{n}^2\theta d{\varphi }^2$$

(2)

3. Boundary Retraction in Geometric Topology

The continuous cartography of a topological space towards a subspace that preserves the positions of its points is known as a retraction. A retraction for the initial space is then used to describe the subspace. Here, we consider the hair charged black hole and its event horizon with a regular scalar field [28,54,55] to explain the topological aspect.

Definition 1. 

Let $\mathcal{W}$ be an n-dimensional manifold and let ${\mathcal{W}}_0\subseteq \mathcal{W}$ be given the subspace topology. Then we define the boundary retraction from $\mathcal{W}$ onto ${\mathcal{W}}_0$, and ${\mathcal{W}}_0$ is called boundary retract of $\mathcal{W}$, as a continuous map $\lambda :\mathcal{W}\to {\mathcal{W}}_0$. Here we consider the following conditions to be satisfied as boundary retraction

(i) 

$\mathcal{W}$ is open

(ii) 

$\lambda \left(z\right)=z,$$\forall z\in {\mathcal{W}}_0$,

(iii) 

$\lambda (\mathcal{W})={\mathcal{W}}_0$ and

(iv) 

The boundary of the manifold $\lambda (\mathcal{W})$ has a constant curvature.

Definition 2. 

A subset ${\mathcal{W}}_0$ of a manifold $\mathcal{W}$ is called a boundary-homotopy retracts if there is a boundary-retraction $\lambda :\mathcal{W}\to {\mathcal{W}}_0$ and a homotopy $\mathfrak{T}:\mathcal{W}\times [0,1]\to \mathcal{W}$. In which the following criteria must be implemented

(i) 

$\mathfrak{T}\left(w,0\right)=w,\forall w\in \mathcal{W}$

(ii) 

$\mathfrak{T}\left(w,1\right)=\lambda \left(w\right),\forall w\in \mathcal{W},$

(iii) 

$\mathfrak{T}\left(b,s\right)=b,\forall b\in {\mathcal{W}}_0,$and $\forall s\in \left[0,1\right]$.

4. Geodesic in Scalar Hairy Charged Black Hole Metric

In this part we shall elaborate on how to find the geodesic from the Lagrangian equations in ($L^4$)in order to explore the two classes of boundary-retractions, first we deduce the four components $〈f_1,f_2,f_3,f_4〉$ on $L^4$ as:

$$\begin{array}{ccc}\hfill f_1& =& \pm \sqrt{M\left(r\right)}t+b_1,\hfill \\ \hfill f_2& =& \pm \left[m{sinh}^{-1}\left(\frac{r-m}{\sqrt{Q^2+Q_s^2-m^2}}\right)+\sqrt{r^2-2mr+Q^2+Q_s^2}\right]+b_2,\hfill \\ \hfill f_3& =& \pm r\theta +b_3,\hfill \\ \hfill f_4& =& \pm rsin\theta \phi +b_4.\hfill \end{array}$$

(3)

in which $b_1,b_2$, $b_3$, and $b_4$ are constants of integration. The leading Lagrangian equation can be used to determine the geodesic equation from a metric g${}_{\mu \nu }$ as

$$\mathcal{L}=\left(\frac{1}{2}\right)g_{{}_{\mu \nu }}{\dot{x}}^{\mu }{\dot{x}}^{\nu }=\frac{1}{2}\left(-M\left(r\right){\dot{t}}^2+\frac{{\dot{r}}^2}{M\left(r\right)}+r^2\left[({\dot{\theta }}^2+si{n}^2\theta {\dot{\varphi }}^2)\right]\right)$$

(4)

$$\frac{d}{ds}\left(\frac{\partial \mathcal{L}}{{\dot{\partial x}}^{\mu }}\right)-\frac{\partial \mathcal{L}}{\partial x^{\mu }}=0,i=1,\dots ,4.$$

(5)

In the case of the Euler-Lagrange equation, relative to the components t and$r$we get:

$$M\left(r\right)\dot{t}=h,$$

(6)

where h is a constant of integration.

$$\frac{d}{ds}\left(\frac{\dot{r}}{M\left(r\right)}\right)+\left(-\frac{m}{r^2}+\frac{Q^2+Q_s^2}{r^3}\right){\dot{t}}^2+\frac{\left(-\frac{m}{r^2}+\frac{Q^2+Q_s^2}{r^3}\right){\dot{r}}^2}{(M\left(r\right){)}^2}-r({\dot{\theta }}^2+si{n}^2\theta {\dot{\varphi }}^2)=0.$$

(7)

Also, the components $\theta$, and $\varphi$ generate.

$$\frac{d}{ds}\left(r^2\dot{\theta }\right)-\frac{r^{2}sin2\theta {\dot{\varphi }}^2}{2}=0$$

(8)

$$r^{2}si{n}^2\theta \dot{\varphi }=k.$$

(9)

for which $k$is a constant of integration.

As we can see in Equation (9), the Euler-Lagrange equation has a physically sound requirement that the geodesic in scalar hairy charged black hole spacetime can be an acceptable solution to the field equations in the limiting insignificant k = 0 case. In this case of $\theta =0$, or $\dot{\varphi }=0$and by using Equation (8) we can obtain

$$\dot{\theta }=\frac{g}{r^2},$$

(10)

where g is a constant

Also, Equation (4) can be written as

$$-M\left(r\right){\dot{t}}^2+\frac{{\dot{r}}^2}{M\left(r\right)}+r^2\left[({\dot{\theta }}^2+si{n}^2\theta {\dot{\varphi }}^2)\right]=H$$

(11)

where $H=2\mathcal{L}$. In fact, $H=0$or 1 for null geodesics and spacelike respectively. Here by using Equations (6), (10) and (11), under condition $\theta =0$, the equation can be simply written as

$${\dot{r}}^2=HM\left(r\right)+h^2-\frac{g^2}{r^2}M\left(r\right)$$

(12)

In a case of radial geodesics motion, we limit ourselves only to $g=0$ and so we obtain,

$${\dot{r}}^2=HM\left(r\right)+h^2$$

(13)

By using the fact, $\frac{dr}{dt}=\frac{dr}{ds}.\frac{ds}{dt}$, and Equation (10), we have

$$\frac{dr}{dt}=\pm M\left(r\right)\sqrt{\frac{HM\left(r\right)}{h^2}+1}$$

(14)

4.1. The Photon-like Particle Phenomena in a Cosmological Function $\left(H=0\right)$

To simplify the calculations we describe the case of null geodesics that corresponds to the gravity $\left(H=0\right)$. Thus we have $\frac{dr}{dt}=\pm M\left(r\right)$. By solving this differential equation, we obtain $t=\pm \left(mln\left|r(r-2m)+Q^2+Q_s^2\right|+\frac{(-Q^2-Q_s^2+2m^2){tan}^{-1}\left(\frac{r-m}{\sqrt{Q^2+Q_s^2-m^2}}\right)}{\sqrt{Q^2+Q_s^2-m^2}}+r+C\right)$. However, the null geodesics could play a significant role in describing the characteristic properties, in particular, the Schwarzschild radius of a black hole in spacetime [43]. For example, if the energy of the photon-like particle has a value that exceeds the effective potential ($V_{eff}$). Then the particle may approach the event horizon with a scalar field from the outside. As a result, it strikes the singularity of the black hole [56]. Several authors investigated [57,58,59] the possible relation between the naked singularity and images of thin accretion disks surrounding in black holes, by using the high-energy spectral fitting and/or the relativistic emission lines profiles. It turns out the naked singularities with photon spheres can also produce images and shadow like in black holes. On the observational side, the null geodesics behavior of gravity can be used to detect gravitational lensing phenomena (Einstein rings) that occur for distant massive stars and galaxies [60,61,62].

However from Equation (13), the relation between r and s can be written as ${\left(\frac{dr}{ds}\right)}^2=h^2$ and the solution is $r=\pm hs$.

4.2. The Dense Particle/Object Phenomena in a Cosmological Function $(H=1)$

In order to drive the equation for a given cosmological timescale $t$in case of spacelike $H=1$, one can solve the equation $\frac{dr}{dt}=\pm M\left(r\right)\sqrt{\frac{M\left(r\right)}{h^2}+1}$, to conclude the following:

$$\begin{array}{ccc}\hfill t& =& \pm (h^{2}r\sqrt{\frac{Q^2+Q_s^2+r(h^{2}r-2m+r)}{{\left(hr\right)}^2}}(-(\left(\right((Q^2+Q_s^2)(\sqrt{m^2-Q^2-Q_s^2}+3m)-\hfill \\ & & \frac{4m^2(\sqrt{m^2-Q^2-Q_s^2}+m)){tan}^{-1}(\frac{m(h^{2}r-\sqrt{m^2-Q^2-Q_s^2}+(h^2+1)r\sqrt{m^2-Q^2-Q_s^2}+Q^2+Q_s^2-m^2}{\left|h\right|\sqrt{Q^2+Q_s^2-2m(\sqrt{m^2-Q^2-Q_s^2}+m)\sqrt{Q^2+Q_s^2+r(h^{2}r-2m+r)}}}))}{(\left|h\right|\sqrt{m^2-Q^2-Q_s^2}\sqrt{Q^2+Q_s^2-2m(\sqrt{m^2-Q^2-Q_s^2}+m)}}-\hfill \\ & & \frac{\begin{array}{c}\left(\right(4m^2(m-\sqrt{m^2-Q^2-Q_s^2})+\\ (Q^2+Q_s^2)(\sqrt{m^2-Q^2-Q_s^2}-3m){tan}^{-1}(\frac{m(\sqrt{m^2-Q^2-Q_s^2}+h^{22}r)-(h^2+1)r\sqrt{m^2-Q^2-Q_s^2}+Q^2+Q_s^2-m^2}{(\left|h\right|(\sqrt{2m(\sqrt{m^2-Q^2-Q_s^2}-m)+(Q^2+Q_s^2)}\sqrt{Q^2+Q_s^2+r(h^{2}r-2m+r)}}))\end{array}}{\left(\left|h\right|\sqrt{m^2-Q^2-Q_s^2}\sqrt{2m(\sqrt{m^2-Q^2-Q_s^2}-m)+Q^2+Q_s^2}\right)}+\hfill \\ & & \frac{2\sqrt{Q^2+Q_s^2+r(h^{2}r-2m+r)}}{h^2+1}+\frac{2m{tanh}^{-1}(\frac{h^{2}r-m+r}{\sqrt{h^2+1}\sqrt{Q^2+Q_s^2+r(h^{2}r-2m+r)}})}{{(h^2+1)}^{\frac{3}{2}}}-\hfill \\ & & \frac{4m{tanh}^{-1}(\frac{m-(h^2+1)r}{\sqrt{h^2+1}\sqrt{Q^2+Q_s^2+r(h^{2}r-2m+r)}})}{\sqrt{h^2+1}}\left)\right)/2\sqrt{Q^2+Q_s^2+r(h^{2}r-2m+r)}+C.\hfill \end{array}$$

Now, we deduce the relation between $r$and s. From Equation (13) we obtain,

$$\frac{dr}{ds}=\pm \sqrt{M\left(r\right)+h^2},$$

(15)

and so

$$s=\pm \int \frac{dr}{\sqrt{1+h^2-\frac{2m}{r}+\frac{Q^2+Q_s^2}{r^2}}}.$$

(16)

This implies that

$$s=\pm \left(\begin{array}{c}\frac{r\sqrt{r((h^2+1)r-2m)+h^2}}{(h^2+1)\left|r\right|}-\\ \frac{2\sqrt{Q^2+Q_s^2}\sqrt{m}{tan}^{-1}\left(\frac{\sqrt{(h^2+1)(Q^2+Q_s^2)-m^2}\left[\frac{mr\sqrt{\frac{=(Q^2+Q_s^2)(r((h^2+1)r-2m)+Q^2+Q_s^2)}{m^2-(h^2+1)(Q^2+Q_s^2)}}}{\left|r\right|}-(Q^2+Q_s^2)\right]}{\sqrt{-\frac{(h^2+1)(Q^2+Q_s^2)}{m}}\sqrt{m}(mr-(Q^2+Q_s^2))}\right)}{\sqrt{-\frac{(h^2+1)(Q^2+Q_s^2)}{m}}(h^2+1)}\end{array}\right)+C$$

(17)

Now, Equation (12) can be written as

$${\dot{r}}^2={\left(\frac{dr}{ds}\right)}^2=HM\left(r\right)+h^2-\frac{g^2}{r^2}M\left(r\right).$$

(18)

By comparing Equation (18) to the motion equation

$$\frac{{\dot{r}}^2}{2}+{\mathcal{V}}_{eff}=0,$$

(19)

we get

$${\mathcal{V}}_{eff}=-\frac{1}{2}\left[HM\left(r\right)+h^2-\frac{g^2}{r^2}M\left(r\right)\right].$$

(20)

In fact, in case of $H=0$ we have

$${\mathcal{V}}_{eff}=-\frac{1}{2}\left[h^2-\frac{g^2}{r^2}M\left(r\right)\right].$$

(21)

4.3. For Photon-like Particle $(H=0)$

Let us consider the problem of the radial geodesic motion ($g=0$) for a photon in a charged black hole. Then, we have

$${\mathcal{V}}_{eff}=-\frac{1}{2}{h}^2.$$

(22)

This indicates that these particles will behave like free particles i.e., its ${\mathcal{V}}_{eff}=0$ at $h=0$. As a result, they will move in the circular orbits around the Schwarzschild radius, and they should also be stable under perturbations.

In case of a circular geodesic, $g\ne 0$, the effective potential can be written as,

$${\mathcal{V}}_{eff}=-\frac{1}{2}{h}^2+\frac{g^2}{2r^2}\left[1-\frac{2m}{r}+\frac{Q^2+Q_s^2}{r^2}\right].$$

(23)

Within a certain limit $r\to 0,{\mathcal{V}}_{eff}$ achieves a large value and when $r\to \infty ,{\mathcal{V}}_{eff}\to -\frac{h^2}{2}$. These radial position values r would be defined as knowing the horizon, where $M\left(r\right)=0$. By using Equation (23), we have ${\mathcal{V}}_{eff}=-\frac{1}{2}{h}^2<0$, for every h. In this case, these particles can only move inside the inner and outer horizons due to their real velocity. We should note that the geometry of the near horizon of the metric space has scaling invariance at zero temperature. This is due to dynamical critical exponent $\dot{\varphi }$ [1].

4.4. For Massive Particle $(H=1)$

The effective potential in this situation is

$${\mathcal{V}}_{eff}=-\frac{1}{2}\left[M\left(r\right)+h^2-\frac{g^2}{r^2}M\left(r\right)\right].$$

(24)

The motion of the massive particle for $h\ge 0$. When $h=0$ and $g=0$, we can obtain

$${\mathcal{V}}_{eff}=-\frac{1}{2}\left[1-\frac{2m}{r}+\frac{Q^2+Q_s^2}{r^2}\right].$$

(25)

Inside the black hole, particles can move based on the values of the parameters. For $h\ne 0$ and $g=0$, the effective potential can be calculated as

$${\mathcal{V}}_{eff}=-\frac{1}{2}\left[M\left(r\right)+h^2\right].$$

(26)

This is interpreted the same way as when photons are moving toward the horizon, which is $M\left(r\right)=0$ at horizon and ${\mathcal{V}}_{eff}<0$. In this approach, particles between the horizons move with real velocity. If $r\to \infty$, we get ${\mathcal{V}}_{eff}\to \frac{-1}{2}(1+h^2)$. This could help us to study the concept of “holographic superconductors [32,63]” based on the dynamics of a charged scalar field outside the horizon.

The non-vanishing angular momentum case is now being considered. If $g\ne 0$ with $h=0$. As a result, the effective potential is

$${\mathcal{V}}_{eff}=-\frac{1}{2}\left(1-\frac{2m}{r}+\frac{Q^2+Q_s^2}{r^2}\right)\left(1-\frac{g^2}{r^2}\right).$$

(27)

In this instance the shape of potential overlaps with the roots the $M\left(r\right)$ functions, and it means that particles tracing out the time-like trajectories are captured by the black hole. This causes these particles to move in s fixed radius of circular orbits. However, both photons and massive particles have a small deflection in their distance to the horizon geometry during the interactions at radial motion. These relationships depend only on the characteristics of geodesic radial and are independent of the black hole parameters such as mass and energy. The effective potential for the ability of photon-like particles to move radially suggests that if their energy is zero, they can act as free particles [43].

On the other hand, in a special case of Equation (6), where h$=0$, this means that the thermodynamics are in AdS. As a result, the hairy charged black hole will be thermodynamically preferred at low temperatures [64]. In this approach, we can get the solution $t=k$where k is constant of integration. Additionally, in this case if $M\left(r\right)=0$yields the surface $r^{+}=m+\sqrt{m^2-\left(Q^2+Q_s^2\right)}$represents the outer event horizon and the surface$r^{-}=m-\sqrt{m^2-\left(Q^2+Q_s^2\right)}$represents the inner event horizon. It is noteworthy to mention here that this can be explained by the presence of the photon-like particle in quantum aspects and quantum gravity effects on microscopic scales (see i.e., [41]) and also on the macroscopic horizon scales, similar to the singularity of black holes or the information paradox (see i.e., [65,66,67,68]) due to their gravitational effect.

5. Boundary Retractions in Scalar Hairy Charged Black Hole

In this section, we present the two classes of boundary retractions that illustrate the geodesic as the boundary retractions of the event horizon scalar hairy charged black hole $L^4$. In our equations, using Equation (9), we set $\dot{\varphi }=0$. let’s first consider $\varphi =0$, and the components are:

$$\begin{array}{cc}\hfill \underset{(\varphi =0)}{f_1}& =\pm \sqrt{M\left(r\right)}t+b_1,\hfill \\ \hfill \underset{(\varphi =0)}{f_2}& =\pm \left[m{sinh}^{-1}\left(\frac{r-m}{\sqrt{Q^2+Q_s^2-m^2}}\right)+\sqrt{r^2-2mr+Q^2+Q_s^2}\right]+b_2,\hfill \\ \hfill \underset{(\varphi =0)}{f_3}& =\pm r\theta +b_3,\hfill \\ \hfill \underset{(\varphi =0)}{f_4}& =b_4.\hfill \end{array}$$

(28)

This is a result of a subspace geodesic${\mathcal{C}}_{(\varphi =0)}\subseteq$$L^4$.Thus, we obtain the first type ofboundary retraction $\lambda$${}_{(\varphi =0)}:$$L^4\to {\mathcal{C}}_{(\varphi =0)}$. If $sin\theta =0$, we then consider a new subspace ${\mathcal{C}}_{(\theta =0)}\subseteq L^4$whose given components are as follows:

$$\begin{array}{cc}\hfill \underset{(\theta =0)}{f_1}& =\pm \sqrt{M\left(r\right)}t+b_1,\hfill \\ \hfill \underset{(\theta =0)}{f_2}& =\pm \left[m{sinh}^{-1}\left(\frac{r-m}{\sqrt{Q^2+Q_s^2-m^2}}\right)+\sqrt{r^2-2mr+Q^2+Q_s^2}\right]+b_2,\hfill \\ \hfill \underset{(\theta =0)}{f_3}& =b_3,\hfill \\ \hfill \underset{(\theta =0)}{f_4}& =b_4.\hfill \end{array}$$

(29)

Furthermore, these forms of boundary retractions explain the event horizon of $L^4$.

6. Boundary Homotopy Retracts on (${\mathit{L}}^{\mathbf{4}}$)

In this section, we use the two classes of boundary retracts as stated earlier to construct the boundary homotopy retract of $L^4$.

The boundary homotopy retracts on $L^4$ is defined as $\mathfrak{T}:$$L^4\times [0,1]\to$$L^4$.

It follows from the retractions:

$\lambda$${}_{〈\varphi =0〉}:$$L^4$$\to {\mathcal{C}}_{〈\varphi =0〉},$$\lambda$${}_{〈\theta =0〉}:$$L^4$$\to {\mathcal{C}}_{〈\theta =0〉}$that the boundary homotopy retracts of $L^4$into ${\mathcal{C}}_{〈\varphi =0〉}\subseteq$$L^4$ can be written as

$${\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\varphi =0〉}}}:L^4\times [0,1]\to L^4$$

where ${\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\varphi =0〉}}}(f,s)=(1+e^{-s})\frac{cos\frac{\pi s}{2}}{2}{\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\varphi =0〉}}}(f,0)+(1+e^{-s})\frac{sin\frac{\pi s}{2}}{2}$$\mathfrak{T}$${}_{{}_{{\mathcal{C}}_{〈\varphi =0〉}}}($f$,1),\forall$f∈$L^4$, ∀s$\in [0,1]$, in which ${\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\varphi =0〉}}}(f,0)=\left\{f_1,f_2,f_3,f_4\right\}$and

$${\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\varphi =0〉}}}(f,1)=\left\{\underset{〈\varphi =0〉}{f_1},\underset{〈\varphi =0〉}{f_2},\underset{〈\varphi =0〉}{f_3},\underset{〈\varphi =0〉}{f_4}\right\}.$$

Also, the boundary homotopy retract of $L^4$ into ${\mathcal{C}}_{〈\theta =0〉}\subseteq$ $L^4$ can be written as ${\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\theta =0〉}}}:$ $L^4\times [0,1]\to$ $L^4$ where ${\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\theta =0〉}}}(f,s)=(1+e^{-s})\frac{cos\frac{\pi s}{2}}{2}{\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\theta =0〉}}}(f,0)+(1+e^{-s})\frac{sin\frac{\pi s}{2}}{2}$$\mathfrak{T}$${}_{{}_{{\mathcal{C}}_{〈\theta =0〉}}}$ $(f,1)$, $\forall f$∈ $L^4$$,\forall$s$\in [0,1]$ in which ${\mathfrak{T}}_{{}_{{\mathcal{C}}_{〈\theta =0〉}}}(f,0)=\left\{f_1,f_2,f_3,f_4\right\}$and${\mathfrak{T}}_{{}_{{\mathcal{C}}_{(\theta =0)}}}(f,1)=\left\{\underset{〈\theta =0〉}{f_1},\underset{〈\theta =0〉}{f_2},\underset{〈\theta =0〉}{f_3},\underset{〈\theta =0〉}{f_4}\right\}$. wherethe components$f_i,$$\underset{〈\varphi =0〉}{f_i},\underset{〈\theta =0〉}{f_i},i=1,2,\dots ,4$are given in Equations (3), (28) and (29) respectively. From the above discussion, we can show that $L^4$can be deformed continuously into spaces or subspaces.

7. Conclusions

In this contribution, we have considered boundary homotopy and boundary retraction of scalar hairy black holes ($L^4$). We focus on the geometrical topology of the metric space and its application to the event horizon. The structure of $L^4$ whether they can be null geodesics (if $H=0$) and/or space-like (if $H=1$), depends on the geodesic and the retractions in geometric topology. The thermodynamics of photon-like particles play a crucial role in approaching the geometry (thickness and structure) of the event horizon with an external scalar field from the outside. This may occur when the energy exceeds the maximum value of the effective potential with a specific angular momentum. On the other side, in the case of the massive particle model $M\left(r\right)=0$ at the horizon and ${\mathcal{V}}_{eff}<0$, as a consequence, the particles move with a real velocity between the inner and outer horizons.