Narrow Results

In this paper, we investigate the exact time-dependent black hole solution on a warped five-dimensional Randall–Sundrum space–time in conformal dilaton gravity. The zeroes of the model are described by a meromorphic quintic polynomial, which has no essential singularities. The quintic equation can be reduced to the Brioschi form by means of the Weierstrass elliptic curve over $\mathbb{Q}$. The model fits the antipodal boundary condition, i.e. antipodal points in the projected space are identified using the embedding of a Klein surface in ${ℂ}^2$, using the ${\mathbb{Z}}_2$ symmetry on the two sides of the brane. If one writes ${}^{(5)}{g}_{\mu \nu }={\omega }^{4∕3}{}^{(5)}{\tilde{g}}_{\mu \nu },{}^{(5)}{\tilde{g}}_{\mu \nu }={}^{(4)}{\tilde{g}}_{\mu \nu }+n_{\mu }{n}_{\nu }$, ${}^{(4)}{\tilde{g}}_{\mu \nu }={\bar{\omega }}^2{}^{(4)}{ḡ}_{\mu \nu }$, with $n_{\mu }$ the normal to the brane and $\omega$ the dilaton field, then ${}^{(4)}{ḡ}_{\mu \nu }$ is conformally flat. It is the contribution from the bulk which determines the real pole on the effective four-dimensional space–time. There is no objection applying ’t Hooft’s back reaction method in constructing the unitary S-matrix for the Hawking radiation. Again, there is no “inside” of the black hole. The zeroes can also be analyzed by the icosahedron equation and by the Hopf-fibration of the Klein surface.

An exact time-dependent solution of a black hole is found in a conformally invariant gravity model on a warped Randall-Sundrum spacetime, by writing the metric $g_{\mu \nu }={\omega }^{\frac{4}{n-2}}{\tilde{g}}_{\mu \nu }$. Here ${\tilde{g}}_{\mu \nu }$ represents the “un-physical” spacetime and ω the dilaton field, which will be treated on equal footing as any renormalizable scalar field. It is remarkable that the 5D and 4D effective field equations for the metric components and dilaton fields can be written in general dimension n = 4, 5. The location of the horizon(s) are determined by a quintic polynomial. This polynomial is related to the symmetry group of the icosahedron, isomorphic with the Galois group A₅. We applied the antipodal mapping on the axially symmetric black hole spacetime and make some connection with the information and firewall paradoxes. The dilaton field can be used to describe the different notion the in-going and outside observers have of the Hawking radiation by using different conformal gauge freedom. The disagreement about the interior of the black hole is explained by the antipodal map of points on the horizon. The free parameters of the solution can be chosen in such a way that ${\tilde{g}}_{\mu \nu }$ is singular-free and topologically regular, even for ω → 0.