# The (A)symmetry between the Exterior and Interior of a Schwarzschild Black Hole

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## Abstract

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## 1. Introduction

## 2. Systems of Co-Ordinates in Space-Time with a Horizon

- (a)
- t-observers$${U}_{t}=\frac{1}{\sqrt{{g}_{tt}}}{\partial}_{t}$$
- (b)
- T-observers$${U}_{T}=-\frac{1}{\sqrt{{h}_{TT\text{}}}}{\partial}_{T}$$

## 3. Uniformly Accelerated Motion along Straight Line

- (a)
- Outside the horizon, ${\xi}^{\alpha}\equiv {x}^{\alpha}$ and $f={g}_{tt}$ is a function of spatial coordinate ${\xi}^{1}$
- (b)
- Inside the horizon, ${\xi}^{\alpha}\equiv {X}^{\alpha}$ and $f={h}_{TT}$ is a function of temporal coordinate ${\xi}^{0}$.

- (a)
- outside the horizon$${a}^{t}=-{g}_{rr}{u}^{r}\frac{d}{dr}({g}_{tt}{u}^{t})$$$${a}^{r}={g}_{tt}{u}^{t}\frac{d}{dr}({g}_{tt}{u}^{t})$$
- (b)
- inside the horizon$${a}^{T}={h}_{RR}{u}^{R}\frac{d}{dT}({h}_{RR}{u}^{R})$$$${a}^{R}=-{h}_{TT}{u}^{T}\frac{d}{dT}({h}_{RR}{u}^{R})$$

- (a)
- Outside the horizon$$\frac{d}{dr}({g}_{tt}{u}^{t})=\pm \alpha .$$

- (b)
- Inside the horizon$$\frac{d}{dT}({h}_{RR}{u}^{R})=\pm \alpha .$$

#### 3.1. Black Hole Exterior

#### 3.2. Black Hole Interior

## 4. Uniform Acceleration on a Photon Sphere and on Its Analogue inside the Horizon

#### 4.1. Black Hole Exterior

#### 4.2. Black Hole Interior

## 5. Discussion and Final Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Squared speed ${v}^{2}$ (Equation (35)) of a uniformly accelerated test particle initially at rest at ${r}_{0}=2{r}_{g}\equiv 4$ (in this case ${r}_{g}\equiv 2$) escaping radially from the gravitational field for different values of α = 0.1 (red), α = 0.5 (green), α = 1 (black).

**Figure 2.**Squared speed ${\tilde{v}}^{2}$ (Equation (37)) of a test particle initially, ${T}_{0}=0.9\text{}{r}_{g}$ at rest (in this case ${r}_{g}\equiv 10$) uniformly accelerated along homogeneity axis $R$(=t) for different values of α = 0.05 (blue), α = 0.1 (red), α = 0.2 (green), α = 1 (black).

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**MDPI and ACS Style**

Gusin, P.; Augousti, A.; Formalik, F.; Radosz, A.
The (A)symmetry between the Exterior and Interior of a Schwarzschild Black Hole. *Symmetry* **2018**, *10*, 366.
https://doi.org/10.3390/sym10090366

**AMA Style**

Gusin P, Augousti A, Formalik F, Radosz A.
The (A)symmetry between the Exterior and Interior of a Schwarzschild Black Hole. *Symmetry*. 2018; 10(9):366.
https://doi.org/10.3390/sym10090366

**Chicago/Turabian Style**

Gusin, Pawel, Andy Augousti, Filip Formalik, and Andrzej Radosz.
2018. "The (A)symmetry between the Exterior and Interior of a Schwarzschild Black Hole" *Symmetry* 10, no. 9: 366.
https://doi.org/10.3390/sym10090366