Citation: Krupka, D.; Brajerˇ cík, J. Schwarzschild Spacetimes: Topology. Axioms 2022, 11, 693. https:// doi.org/10.3390/axioms11120693 Academic Editor: Sidney A. Morris Received: 14 November 2022 Accepted: 2 December 2022 Published: 4 December 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). axioms Article Schwarzschild Spacetimes: Topology Demeter Krupka 1,2,†,‡ and Ján Brajerˇ cík 3, * ,‡ 1 Lepage Research Institute, 081 16 Presov, Slovakia 2 Faculty of Mathematics and Computer Science, Transilvania University in Brasov, 500091 Brasov, Romania 3 Faculty of Humanities and Natural Sciences, University of Presov in Presov, 081 16 Presov, Slovakia * Correspondence: jan.brajercik@unipo.sk † Current address: Faculty of Humanities and Natural Sciences, University of Presov in Presov, 17. Novembra 1, 081 16 Presov, Slovakia. ‡ These authors contributed equally to this work. Abstract: This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, R × (R 3 \{(0, 0, 0)}) and S 1 × (R 3 \{(0, 0, 0)}). The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R × (R 3 \{(0, 0, 0)}) and a winding mapping of the real line R onto the circle S 1 . The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to S 1 × (R 3 \{(0, 0, 0)}) by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom. Keywords: manifold topology; Einstein equations; spherical symmetry; Schwarzschild spacetime; special orthogonal group; SO(3)-action; invariant metric 1. Introduction In this paper, a Schwarzschild spacetime, or a spherically symmetric spacetime, is a smooth 4-dimensional manifold X endowed with a left action of the special orthogonal group SO(3) and a non-singular, symmetric (0, 2)-tensor field g, satisfying the following two conditions: (1) g is SO(3)-invariant. (2) g solves the Einstein vacuum equations. where g is a Schwarzschild metric on X. Standard topological properties are required: X is Hausdorff, second countable, and connected. As g can be understood as an extremal of an integral variational functional, the Hilbert variational functional, no a priori restrictions of the signature of g are imposed. In this paper, we revisit and extend several constructions of classical general relativ- ity theory, especially the theory of spherically symmetric spacetimes (Einstein 1915 [1], Hilbert 1915 [2], Schwarzschild 1916 [3]). Since Schwarzschild, spherically symmetric mod- els became a principal application of the theory, stimulating extensive research on the basis of classical differential geometry on Riemannian spaces (see Hawking, Ellis 1973 [4] and, for a more comprehensive contemporary discussion De Felice, Clarke 1990 [5], and Kriele 1999 [6]). Less is known, however, on the effort focused on a deeper understanding of what is going on from the topological point of view. For first steps in this direction, we refer to Axioms 2022, 11, 693. https://doi.org/10.3390/axioms11120693 https://www.mdpi.com/journal/axioms