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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

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

## 3. Boundary Retraction in Geometric Topology

**Definition 1.**

- (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.**

- (i)
- $\mathfrak{T}\left(w,\phantom{\rule{4pt}{0ex}}0\right)=w,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall w\in \mathcal{W}$
- (ii)
- $\mathfrak{T}\left(w,\phantom{\rule{4pt}{0ex}}1\right)=\lambda \left(w\right),\phantom{\rule{4pt}{0ex}}\forall w\in \mathcal{W},$
- (iii)
- $\mathfrak{T}\left(b,\phantom{\rule{4pt}{0ex}}s\right)=b,\phantom{\rule{4pt}{0ex}}\forall b\in {\mathcal{W}}_{0},\phantom{\rule{4pt}{0ex}}$and $\forall s\in \left[0,\phantom{\rule{4pt}{0ex}}1\right]$.

## 4. Geodesic in Scalar Hairy Charged Black Hole Metric

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

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

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

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

## 5. **Boundary Retractions in Scalar Hairy Charged Black Hole**

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

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Abu-Saleem, M.; Taani, A.
The Boundary Homotopy Retract on the Scalar Hairy Charged Black Hole Spacetime. *Axioms* **2022**, *11*, 745.
https://doi.org/10.3390/axioms11120745

**AMA Style**

Abu-Saleem M, Taani A.
The Boundary Homotopy Retract on the Scalar Hairy Charged Black Hole Spacetime. *Axioms*. 2022; 11(12):745.
https://doi.org/10.3390/axioms11120745

**Chicago/Turabian Style**

Abu-Saleem, Mohammed, and Ali Taani.
2022. "The Boundary Homotopy Retract on the Scalar Hairy Charged Black Hole Spacetime" *Axioms* 11, no. 12: 745.
https://doi.org/10.3390/axioms11120745