A New Self-Dual Gravitational Instanton Solution on a Local Conformal Kählerian 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 Kählerian manifold with Euclidean signature and a Kähler potential. This is possible for the effective 4D manifold, despite the fact that a Kähler manifold in 5D cannot be modelled. We are dealing with an effective 4D self-dual Kähler 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 S3× ℝ/ℤ2. The antipodal boundary condition on the hyper-surface of a Klein bottle ∼ ℂ1×ℂ1 is applied to describe the Hawking particles during the evaporation process. We used the Hopf fibration to get S2 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.
Related
- A new toric ALF gravitational instanton
- New gravitational instanton: shadow of an extra dimension
- Study of gravitational radiation at 𝒥 for the general type-D Einstein-Maxwell solution with cosmological constant
- Improving the Sensitivity of Gravitational Wave Detection with Weighted Conformal Prediction
- Gravitational fingerprints of a black hole solution in Palatini Kalb-Ramond gravity
BibTeX
@misc{slagter2026selfdual,
title = {{A New Self-Dual Gravitational Instanton Solution on a Local Conformal Kählerian Manifold in a Brane World Model}},
author = {Reinoud Jan Slagter},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2605.25532},
eprint = {2605.25532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25532},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A New Self-Dual Gravitational Instanton Solution on a Local Conformal Kählerian Manifold in a Brane World Model
AU - Reinoud Jan Slagter
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2605.25532
UR - https://arxiv.org/abs/2605.25532
AB - 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 Kählerian manifold with Euclidean signature and a Kähler potential. This is possible for the effective 4D manifold, despite the fact that a Kähler manifold in 5D cannot be modelled. We are dealing with an effective 4D self-dual Kähler 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× ℝ/ℤ_2. The antipodal boundary condition on the hyper-surface of a Klein bottle ∼ ℂ^1×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.
ER -