Reinstating Schwarzschild’s Original Manifold and Its Singularity

A review of results about this paradigmatic solution to the field equations of Einstein’s theory of general relativity is proposed. Firstly, an introductory note of historical character explains the difference between the original Schwarzschild’s solution and the “Schwarzschild solution” of all the books and the research papers, that is due essentially to Hilbert, as well as the origin of the misnomer. The viability of Hilbert’s solution as a model for the spherically symmetric field of a “Massenpunkt” is then scrutinised. It is proved that Hilbert’s solution contains two main defects. In a fundamental paper written in 1950, J.L. Synge set two postulates that the geodesic paths of a given metric must satisfy in order to comply with our basic ideas on time, namely the postulate of order and the non-circuital postulate. It is shown that neither Hilbert’s solution, nor the equivalent metrics that can be obtained from the latter with a coordinate tranformation that is regular and one-to-one everywhere except on the Schwarzschild surface can obey both Synge’s postulates. Therefore they do not possess a consistent arrow of time, and the only way for obviating this defect is through a change of topology. The true raison d’être of the Kruskal maximal extension with its odd doubling and bifurcate horizon stays just in its capability to produce the needed change of topology, that can be demonstrated through a constructive cut-and-past procedure applied to two Hilbert space-times. The second main defect of Hilbert’s space-time is constituted by the existence of an invariant, local, intrinsic quantity with a simple operational interpretation that diverges when it is calculated at a position closer and closer to Schwarzschild’s surface, i.e. at an internal position in Hilbert’s metric. The diverging quantity is the norm of the four-acceleration of a test particle whose worldline is the unique orbit of absolute rest defined, through a given event, by the unique timelike, hypersurface orthogonal Killing vector. It is an intrinsic quantity, whose local definition only requires the knowledge of the metric and of its derivatives at a given event, just like it happens with the polynomial invariants built with the Riemann tensor and with its covariant derivatives. The regularity of the latter invariants at a given event has been considered by many a relativist like a “rule of thumb” proof of regularity for the manifold at that event, in the persistent lack of a satisfactory definition of local singularity in general relativity. The notion solution includes the topology of space-time in our context. Different solutions may be locally isometric. 1 2 SALVATORE ANTOCI AND DIERCK-EKKEHARD LIEBSCHER The divergence of the above mentioned norm of the four-acceleration, i.e. of the first curvature of the worldline, is a geometric fact. It can be proved however with an exact argument, relying on a two-body solution found by Bach, that a physical quantity, the norm of the force per unit mass exerted on a test particle in order to keep it on the orbit of absolute rest, is equal to the norm of the four-acceleration, hence it diverges too on approaching Schwarzschild’s surface. We claim that the rôle and interpretation of topological differences between partly isometrical manifolds as well as that of the singularities is not really settled, in particular that the Schwarzschild solution and its topological relatives are in more ways singular than the invariants of the Riemann tensor indicate. Due to these facts we assert that the topology chosen by Schwarzschild should be taken as a serious alternative to the commonly used Hilbert or Kruskal topologies.