A New Self-Dual Gravitational Instanton Solution on a Local Conformal K\"ahlerian Manifold in a Brane World Model

Abstract

An exact gravitational instanton solution on a vacuum Kerr-like warped spacetime in conformal dilaton gravity is found. Remarkably, the metric solution results from a first-order PDE, allowing the connection with self-duality. The singular points are determined by a quintic polynomial. This suggests that this is the highest possible polynomial in describing the singularities of black holes of Petrov type D axially symmetric manifolds and don't fits the Plebanski-Demianski classification of black holes which is determined by a fourth order polynomial. The solution can be described by a locally conformal K\"ahlerian manifold with Euclidean signature and a K\"ahler potential. This is possible for the effective 4D manifold, despite the fact that a K\"ahler manifold in 5D cannot be modelled. We are dealing with an effective 4D self-dual K\"ahler manifold with a recurrent conformal structure. This happened by the projected Weyl tensor of 5D on the brane. The topology of the gravitational instanton would be $S^3\times \mathbb{R}/\mathbb{Z}_2$. The antipodal boundary condition on the hyper-surface of a Klein bottle $\sim \mathbb{C}^1\times\mathbb{C}^1$ is applied to describe the Hawking particles during the evaporation process. We used the Hopf fibration to get $S^2$ as the black hole horizon, where the centrix is not in a torus but in the Klein bottle. The twist fits very well with the antipodal identification of the points on the horizon. No 'cut and past' is necessary, so the Hawing particles remain pure without instantaneous information transport. Finally, we reveal a connection between the description of the interior of our new black hole solution and the similar model proposed by Janis, Newman and Winicour some time ago of the Schwarzschild solution in complex coordinates with a zero rest mass scalar field, which develops an anomalous asymmetry.