Exact vacuum solution with Hopf structure in general relativity

Abstract

An exact solution to the vacuum Einstein equations is presented, whose structure is based on the Hopf fibration. The solution employs a geodesic null vector field that defines a twisting congruence and appears in the metric in Kerr-Schild form. This solution is of Petrov type D and involves two parameters. Remarkably, the resulting spacetime is regular, with no curvature singularities. Both the Kretschmann scalar and the Chern-Pontryagin scalar are nonzero and remain finite throughout the spacetime. In addition, the Newman-Penrose Weyl scalar $\hat{I}{¨}_2$ possesses both nonzero real and imaginary parts, reflecting the topologically nontrivial nature of the gravitational field. The spacetime also admits two Killing vector fields and a Killing-Yano tensor, which induces an associated Killing tensor, revealing its hidden symmetry. The derivation is simple and self-contained, offering a transparent and geometrically guided approach to finding new exact solutions in general relativity.

5 pages, 1 figure, 1 table; accepted for publication in Phys.Rev.D