Completion and Embedding of the Schwarzschild Solution

An analytic manifold is found, the most important properties of which are that it is complete and that it contains the manifold of the Schwarzschild line element. It is thus the complete analytic extension of the latter. The manifold is represented as a Riemannian surface in a six-dimensional pseudo-Euclidean space. The subspace dϕ=dϑ=0 is visualized as a two-dimensional Riemannian surface in a 3-dimensional hyperplane in the six-dimensional space. Although the manifold admits groups of motion isomorphic to the real 3-dimensional rotation group and the one-dimensional translation group, it is impossible to introduce a global time-coordinate in such a way that the latter is realized as translations in time. Hence in any global set of coordinates the gravitational field is nonstationary, although it can be made stationary for r>1 to any desired approximation. The question of what happens to small test bodies reaching the Schwarzschild critical radius is discussed.