# Quantum Black Holes in Conformal Dilaton–Higgs Gravity on Warped Spacetimes

## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. The Vacuum Case

#### 2.2. Relation with the (2 + 1)-Dimensional Baňadoz–Teitelboim–Zanelli Solution

#### 2.3. Penrose Diagram

#### 2.4. Instanton Solution

## 3. The Non-Vacuum Case

## 4. Application of the Antipodal Boundary Condition

#### 4.1. The Black Hole Paradoxes

- Hawking radiation is in a pure state.
- The information is emitted from the stretched horizon. One can use GRT to describe this phenomenon.
- An infalling observer encounters nothing special when crossing the horizon and is burned at the firewall.

#### 4.2. Antipodicity

## 5. Antipodicity in the Conformal Dilaton–Higgs Model

#### 5.1. Cylindrical Harmonics and the Stress Tensor

#### 5.2. Motivation for the Klein Bottle Surface

#### 5.3. Treatment of the Gravitational Backreaction

#### 5.4. Treatment of the Quantum Fields

## 6. Conclusions

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

**Proposition**

**5.**

## 7. Discussion

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | It is a global symmetry when one considers the spacetime as fixed. |

2 | One can also use the Eddington–Finkelstein coordinates $(U,r,z,\phi ,{y}_{5})$. |

3 | For the time being, we omit the cosmological constant term $\sim \mathsf{\Lambda}{\omega}^{4}$. |

4 | The signs of ${a}_{i}$ and ${C}_{i}$ are not important yet. |

5 | Here, we will not treat the conformal anomalies [30]. |

6 | In this case, the $d{z}^{2}$ term is maintained. |

7 | The sign of ${a}_{2}$ is just a matter of convention. |

8 | This technical aspect is currently under investigation by the author. |

9 | See Slagter [26]. The replacement $t\to i\tau $ has no influence on the solution. |

10 | When we apply the antipodal mapping, ’t Hooft suggests cutting out an ${S}^{3}$ sphere, with topology $\mathbb{R}\times {S}^{3}/{\mathbb{Z}}^{2}$. This is not necessary in our 5D model. The effective 4D spacetime is already present. |

11 | If one allows a Higgs field, then for $d=2$, one calls the configuration a vortex, and for $d=3$, a monopole. |

12 | In Euler angles ($\theta ,\zeta ,\phi $). |

13 | Note that we have the double cover ${\mathbb{C}}^{1}\times {\mathbb{C}}^{1}$ for our original spacetime. Then, $\phi $ indeed runs from $0...2\pi $. |

14 | Remember that $\omega $ contains the gravitational constant $\kappa $ due to the redefinition. |

15 | The scalar-gauge system possesses a quantized magnetic flux $\sim \frac{n}{e}$, which equals the first Chern number of A on ${\mathbb{R}}_{+}^{2}$. This has an important consequence in the expansion of the scalar field in cylindrical harmonics (see Section 5). |

16 | Another interesting method could be derived from the equation of the directrix of the Klein bottle. From the integral curve of the PDEs, $\frac{\partial r}{\partial {t}_{e}}$, one could find, in principle, the elapsed time [37]. |

17 | There is a difference between Maldacena’s method and our model: our method does not need a matter field in the bulk. |

18 | There are no fixed points. |

19 | Also called the “cut-and-paste” procedure or firewall transformation. |

20 | Further, our slice is the Klein surface! |

21 | Remember that we used twice the dilaton separation. |

22 | Remember that our solution for N in Section 4.2 could be written as [25] a meromorphic polynomial $\frac{P(\mathbf{z})}{Q(\mathbf{z})}$, with $Deg(P)=5$. |

23 | With $x=rsin\phi ,y=rcos\phi $. |

24 | Because ${e}^{i(\phi +\pi )}=-{e}^{i\phi}$. |

25 | Let G be the group of self-homeomorphisms of the product space ${S}^{2}\times {S}^{2}$, generated by interchanging the two coordinates of any point and by the antipodal map on either factor. G is then isomorphic to the dihedral group. It contains three subgroups, for example, $K=\{I,(x,y)\to (-x,y),(x,y)\to (x,-y),$ and $(x,y)\to (-x,-y)\}$. It acts freely on ${S}^{2}\times {S}^{2}$. Thus, $({S}^{2}\times {S}^{2})/K=\mathbb{R}{P}^{2}\times {\mathbb{RP}}^{2}$. The most interesting feature is the fact that the twofold symmetric product of $\mathbb{R}{P}^{2}$, $S{P}^{2}(\mathbb{R}{P}^{2})=\mathbb{R}{P}^{4}$. |

26 | Remember, $SO(3)\cong {\mathcal{M}}_{0}(\mathbb{C})=SU(2)/\{\pm I\}$. |

27 | For a more extensive description, see t’ Hooft [10]. |

28 | Another issue is the proposition that the entropy is proportional to the horizon area and not the volume. There is no inside in the antipodal map. In the context of the Klein surface, does the area extend to the Klein surface? In the RSII model, this is not an issue because the two mirror surfaces cancel each other. |

29 | From the black hole, where we have an FLRW spacetime, the warp factor is ${y}_{5}$-dependent. One then Fourier expands the perturbative 5D graviton amplitude as $f(\mathbf{x},{y}_{5})={\sum}_{m}{e}^{im{y}_{5}}{f}_{m}(\mathbf{x})$ The KG equation for f is replaced with our dilaton field. |

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**Figure 1.**Penrose diagram. The antipodal points $P(X)$ and $P(\overline{X})$ are identified. Particles going in will generate waves that approach the horizon from the outside. Those passing through the horizon will reappear from “the other side” of the black hole. Note that ${t}^{*}=log(U/V)\sim log(t-{t}_{H})$. In this approach the regions I and II are CPT invariant, implying that time runs backward in region II.

**Figure 2.**Stable solutions of $X,\omega $, and N for the case of a constant gauge field. The potential is $V={\beta}_{1}{X}^{{\beta}_{2}}{\omega}^{\frac{2}{3}-\frac{\eta {\beta}_{2}}{{y}_{0}}}$. We used a cosine function for the initial values of the scalar field and a sine function for the dilaton field. For N, we used the vacuum solutions. Further, we applied Neumann boundary conditions. Note that N develops a singularity and approaches a constant value with increasing time. The solution critically depends on the parameters of the potential, for example, the scale ${y}_{0}$ of the extra dimension.

**Figure 3.**Same as in Figure 2, but now we use a tanh function as the initial value for $\omega $. We observe that the polynomial behavior in the metric component N is induced by the dilaton. We also plotted the total metric component ${\omega}^{4}{N}^{2}$.

**Figure 4.**Left: Stereographic projection of ${S}^{2}\to \mathbb{R}{P}^{2}\to \mathbb{C}{P}^{1}$. To ensure a one-to-one mapping, the ${\mathbb{Z}}_{2}$ symmetry identification is applied: the two antipodal planes are identified, $z\to -\frac{1}{\overline{z}}$. Right: Stereographic projection of ${S}^{3}\subset {\mathbb{R}}^{4}\simeq \mathbb{C}\times \mathbb{C}$ also with antipodal identification, representing the projection of an embedded Klein bottle.

**Figure 5.**In the RS2 model (1 brane), there is a mirror symmetry ${\mathbb{Z}}_{2}$, i.e., ${y}_{5}\to -{y}_{5}$. So, we have a contribution from the antipode when considering Hawking radiation (see text). The antipodal map on the brane is suppressed for clarity. Right: the ingoing and outgoing Hawking particles.

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**MDPI and ACS Style**

Slagter, R.J.
Quantum Black Holes in Conformal Dilaton–Higgs Gravity on Warped Spacetimes. *Universe* **2023**, *9*, 383.
https://doi.org/10.3390/universe9090383

**AMA Style**

Slagter RJ.
Quantum Black Holes in Conformal Dilaton–Higgs Gravity on Warped Spacetimes. *Universe*. 2023; 9(9):383.
https://doi.org/10.3390/universe9090383

**Chicago/Turabian Style**

Slagter, Reinoud Jan.
2023. "Quantum Black Holes in Conformal Dilaton–Higgs Gravity on Warped Spacetimes" *Universe* 9, no. 9: 383.
https://doi.org/10.3390/universe9090383