Mass, dual mass, and gravitational entropy Available to Purchase

A mechanism (mathematical transformation) is considered by which a (Schwarzschild) black‐hole singularity can be converted into a (Taub–NUT) (Newman–Unti–Tamburino) wire singularity, or equivalently topological modifications can be induced (e.g., transformation of S²×R² topology into S³×R and conversely). A topological charge—an invariant of the transformation—emerges as a possible candidate for the description of gravitational entropy in the case of source‐free solutions to Einstein’s equation with one Killing vector field. For a Schwarzschild black hole this invariant reduces to the area of the event horizon (or equivalently the Bolt charge) and it reduces to the square of the NUT charge (or equivalently the length of the closed timelike orbits) in the case of a Taub–NUT magnetic monopole. These considerations lead to the proposition that, under extreme conditions, gravitational clumping or entropy increase could be described by a modification in the characteristic classes of the space‐time manifold due to the onset of nontrivial topological features. Further remarks are presented in view of the role of gravitational magnetic monopoles in quantum gravity, and of a possible relation between the notions of gravitational entropy and arrow of time.