The Dilaton Black Hole on a Conformal Invariant Five Dimensional Warped Spacetime: Paradoxes Possibly Resolved?

TL;DR

This paper presents an exact solution for a black hole in a five-dimensional warped conformal dilaton gravity model, addressing paradoxes and exploring geometric and boundary conditions with advanced mathematical tools.

Contribution

It provides a novel exact black hole solution in conformal dilaton gravity on warped 5D spacetime, analyzing its properties and boundary conditions with complex mathematical methods.

Findings

The solution is exact for metric and dilaton fields.

The quintic polynomial has no essential singularities.

The model's boundary conditions involve antipodal identification and Klein surface embedding.

Abstract

A thorough investigation is presented of the exact black hole solution on a warped five-dimensional spacetime in conformal dilaton gravity (CDG), found in earlier work. Summarized, we will prove: The black hole solution in the CDG model on a warped 5D spacetime: 1. It is an exact solution for the metric components as well as for the dilaton field. 2.The quintic polynomial describing the zero's of the model, has no essential singularities. 3.If we write $^{(5)}g_{\mu\nu}=\omega^{4/3}{^{(5)}}{\tilde g_{\mu\nu}}, ^{(5)}\tilde g_{\mu\nu}=^{(4)}\tilde g_{\mu\nu}+n_\mu n_\nu, ^{(4)}\tilde g_{\mu\nu}=\bar\omega^2 {^{(4)}}\bar g{\mu\nu}$ then ${^{(4)}}\bar g{\mu\nu}$ is conformally flat and with $n_\mu$ the normal to the brane. 4. It fits the antipodal boundary condition, i.e., antipodal points in the projected space are identified using the embedding of a Klein surface in $\mathbb{C}^4$. 5. One can apply 't Hooft's back reaction method in constructing the unitary S-matrix and there is no "inside" of the black hole. 6. The contribution from the bulk determines the poles on the effective 4D spacetime. 7. The zeros of the quintic resolvent can analytically described by the icosahedral equation, i.e., in terms of hypergeometric functions and elliptic modular functions. 8. The Hopf fibration of the Klein bottle can be applied.