# Schwarzschild Black Holes in Extended Spacetime with Two Time Dimensions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. 5D Schwarzschild Equations

## 3. Solution of 5D Equations

#### 3.1. Existence of a Solution

#### 3.2. Solution Classes

- (i)
- ${\alpha}_{<}<{r}_{0}a\left({r}_{0}\right)<{\alpha}_{>}<0$: In this case, it follows from $\left(50\right)$ that $\left(ra\right)\prime <0$ for all $r>0$.

- (ii)
- ${\alpha}_{<}<{\alpha}_{>}<{r}_{0}a\left({r}_{0}\right)<0$: In this case, ${\left(ra\right)}^{\prime}>0$.

- (iii)
- ${r}_{0}a\left({r}_{0}\right)<{\alpha}_{<}<{\alpha}_{>}<0$: In this case, ${\left(ra\right)}^{\prime}<0$.

- (iv)
- ${r}_{0}a\left({r}_{0}\right)>0$: In this last case, $ra>0$ and ${\left(ra\right)}^{\prime}<0$ for all $r>0$. With ${\alpha}_{><}<0$, Equation $\left(51\right)$ can be written as

#### 3.3. Approximate Solution Close to $r=0$

## 4. Physical Interpretation

**Figure 1.**The absolute values of the solutions (

**a**) $A\left(r\right)\text{}\left(96\right)$ and (

**b**) $B\left(r\right)\text{}\left(98\right)$ as a function of distance $r$ from the central point at $r=0$ for Sgr A* (${r}_{S}\approx 1.227\xb7{10}^{10}\text{}\mathrm{m}$ [32]) for different ${C}_{1}$. The crosses indicate the four-dimensional solutions $\left|{A}_{4D}\left(r\right)\right|$ and $\left|{B}_{4D}\left(r\right)\right|\text{}(46,\text{}47)$. The dotted line indicates the Planck length ${l}_{Planck}$.

**Figure 2.**The five-dimensional line element $\left(99\right)$ for the same situation as in Figure 1. The crosses indicate the four-dimensional line element $d{s}_{4D}^{2}={A}_{4D}\left(r\right){c}^{2}d{t}^{2}-{B}_{4D}\left(r\right)d{r}^{2}-{r}^{2}\left(d{\theta}^{2}+{\mathrm{sin}}^{2}\theta d{\phi}^{2}\right)$. As numerical values, we have chosen $\gamma =c$ [25], $dr={l}_{Planck}$, $dt=d\tau ={t}_{Planck}$, the Planck time, and ${K}_{2}={10}^{-40}$.

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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Solution Classes | |
---|---|

(i) | ${\alpha}_{<}<{r}_{0}a\left({r}_{0}\right)<{\alpha}_{>}<0$ |

(ii) | ${\alpha}_{<}<{\alpha}_{>}<{r}_{0}a\left({r}_{0}\right)<0$ |

(iii) | ${r}_{0}a\left({r}_{0}\right)<{\alpha}_{<}<{\alpha}_{>}<0$ |

(iv) | $0<{r}_{0}a\left({r}_{0}\right)$ |

**Table 2.**Asymptotic solution for case (i), ${\alpha}_{<}<{r}_{0}a\left({r}_{0}\right)<{\alpha}_{>}<0$ for $r\to 0,\infty $ and ${C}_{1}\to 0,\infty $.

$\mathit{r}\to 0$ | $\mathit{r}\to \infty $ | ||
---|---|---|---|

$A$ | ${C}_{1}\to 0$ | ${r}^{-1}$ | ${r}^{-\frac{4}{{C}_{1}}}$ |

${C}_{1}\to \infty $ | ${r}^{-\frac{1}{{C}_{1}}}$ | ${r}^{-4}$ | |

$Z$ | ${C}_{1}\to 0$ | ${r}^{-{C}_{1}}$ | ${r}^{-4}$ |

${C}_{1}\to \infty $ | ${r}^{-1}$ | ${r}^{-4{C}_{1}}$ | |

$B$ | ${C}_{1}\to 0$ | $-{C}_{1}-{C}_{1}^{2}$ | $-3$ |

${C}_{1}\to \infty $ | $-\frac{3}{4{C}_{1}}$ | $-3$ |

**Table 3.**Asymptotic solution for case (ii), ${\alpha}_{>}<{r}_{0}a\left({r}_{0}\right)<0$ for $r\to 0,\infty $ and ${C}_{1}\to 0,\infty $.

$\mathit{r}\to 0$ | $\mathit{r}\to \mathit{\infty}$ | ||
---|---|---|---|

$A$ | ${C}_{1}\to 0$ | ${r}^{-1}$ | ${e}^{\frac{1}{r}}$ |

${C}_{1}\to \infty $ | ${r}^{-\frac{1}{{C}_{1}}}$ | ${e}^{\frac{1}{r}}$ | |

$Z$ | ${C}_{1}\to 0$ | ${r}^{-{C}_{1}}$ | ${e}^{\frac{{C}_{1}}{r}}$ |

${C}_{1}\to \infty $ | ${r}^{-1}$ | ${e}^{\frac{{C}_{1}}{r}}$ | |

$B$ | ${C}_{1}\to 0$ | $-{C}_{1}-{C}_{1}^{2}$ | $1-\frac{1}{r}-\frac{{C}_{1}}{r}+\frac{{C}_{1}}{4{r}^{2}}$ |

${C}_{1}\to \infty $ | $-\frac{3}{4{C}_{1}}$ | $1-\frac{1}{r}-\frac{{C}_{1}}{r}+\frac{{C}_{1}}{4{r}^{2}}$ |

**Table 4.**Asymptotic solution for case (iii), ${r}_{0}a\left({r}_{0}\right)<{\alpha}_{<}<{\alpha}_{>}<0$ for $r\to 0,\infty $ and ${C}_{1}\to 0,\infty $.

$\mathit{r}\to 0$ | $\mathit{r}\to \mathit{\infty}$ | ||
---|---|---|---|

$A$ | ${C}_{1}\to 0$ | ${r}^{-1}$ | ${e}^{\frac{1}{r}}$ |

${C}_{1}\to \infty $ | ${r}^{-\frac{1}{{C}_{1}}}$ | ${e}^{\frac{1}{r}}$ | |

$Z$ | ${C}_{1}\to 0$ | ${r}^{-{C}_{1}}$ | ${e}^{\frac{{C}_{1}}{r}}$ |

${C}_{1}\to \infty $ | ${r}^{-1}$ | ${e}^{\frac{{C}_{1}}{r}}$ | |

$B$ | ${C}_{1}\to 0$ | $-{C}_{1}-{C}_{1}^{2}$ | $1-\frac{1}{r}-\frac{{C}_{1}}{r}+\frac{{C}_{1}}{4{r}^{2}}$ |

${C}_{1}\to \infty $ | $-\frac{3}{4{C}_{1}}$ | $1-\frac{1}{r}-\frac{{C}_{1}}{r}+\frac{{C}_{1}}{4{r}^{2}}$ |

**Table 5.**Asymptotic solution for case (iv), ${r}_{0}a\left({r}_{0}\right)>0$ for $r\gtrsim {r}_{S,5D},\infty $, and ${C}_{1}\to 0,\infty $.

$\mathit{r}\gtrsim {\mathit{r}}_{\mathit{S},5\mathit{D}}$ | $\mathit{r}\to \mathit{\infty}$ | ||
---|---|---|---|

$A$ | ${C}_{1}\to 0$ | $\frac{4\left(\frac{r}{{r}_{S,5D}}-1\right)}{\sqrt{{C}_{1}}}$ | ${e}^{-\frac{1}{r}}$ |

${C}_{1}\to \infty $ | $1$ | ${e}^{-\frac{1}{r}}$ | |

$Z$ | ${C}_{1}\to 0$ | $1$ | ${e}^{-\frac{{C}_{1}}{r}}$ |

${C}_{1}\to \infty $ | $4\sqrt{{C}_{1}}\text{}\left(\frac{r}{{r}_{S,5D}}-1\right)$ | ${e}^{-\frac{{C}_{1}}{r}}$ | |

$B$ | ${C}_{1}\to 0$ | ${\left(1-\frac{{r}_{S,5D}}{r}\right)}^{-1}$ | $1+\frac{1}{r}+\frac{{C}_{1}}{r}+\frac{{C}_{1}}{4{r}^{2}}$ |

${C}_{1}\to \infty $ | $\frac{2\sqrt{{C}_{1}}}{\sqrt{\frac{r}{{r}_{S,5D}}-1}}$ | $1+\frac{1}{r}+\frac{{C}_{1}}{r}+\frac{{C}_{1}}{4{r}^{2}}$ |

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**MDPI and ACS Style**

Paiman, M.; Cornean, H.; Köhn, C.
Schwarzschild Black Holes in Extended Spacetime with Two Time Dimensions. *Astronomy* **2023**, *2*, 269-285.
https://doi.org/10.3390/astronomy2040018

**AMA Style**

Paiman M, Cornean H, Köhn C.
Schwarzschild Black Holes in Extended Spacetime with Two Time Dimensions. *Astronomy*. 2023; 2(4):269-285.
https://doi.org/10.3390/astronomy2040018

**Chicago/Turabian Style**

Paiman, Mechid, Horia Cornean, and Christoph Köhn.
2023. "Schwarzschild Black Holes in Extended Spacetime with Two Time Dimensions" *Astronomy* 2, no. 4: 269-285.
https://doi.org/10.3390/astronomy2040018