New gravitational instanton: shadow of an extra dimension

Abstract

We present an exact gravitational instanton solution on a five-dimensional, conformally invariant, Kerr-like warped Riemannian brane-world manifold. The geometry can be described as the K\"ahler manifold $\mathbb{C}^1\times\mathbb{C}^1\times \mathbb{R}$. By applying a double cover of $S^3$ through stereographic projection onto $\mathbb{C}P^1\times \mathbb{C}P^1$ of the effective four-dimensional manifold, together with the Klein surface construction, we exploit the underlying $\mathbb{Z}_2$ symmetry. The instanton is then obtained by fibering over the antipodal $S^2$. The metric is determined by a first-order differential equation containing an integer parameter, while the equation governing the angular momentum component decouples from the remaining field equations. Finally, we show that our solution admits an analytic complex transformation to a locally conformally related K\"ahler manifold possessing a K\"ahler potential, thereby making the self-dual structure manifest.