The so-called ‘Schwarzschild solution’ is not Schwarzschild’s solution, but a corruption, due to David Hilbert (December 1916), of the Schwarzschild/Droste solution, wherein m is allegedly the mass of the source of an associated gravitational field and the quantity r is said to be able to go down to zero (although no proof of this claim has ever been advanced), so that there are two alleged ‘singularities’, one at r = 2m and another at r = 0. It is routinely claimed that r = 2m is a ‘coordinate’ or ‘removable’ singularity which denotes the so-called ‘Schwarzschild radius’ (event horizon) and that a ‘physical’ singularity is at r = 0. The quantity r in the so-called ‘Schwarzschild solution’ has never been rightly identified by the physicists, who, although proposing many and varied concepts for what r denotes, effectively treat it as a radial distance from a source of the gravitational field at the origin of coordinates. The consequence of this is that the intrinsic geometry of the metric manifold has been violated. It is easily proven that the said quantity r is in fact the inverse square root of the Gaussian curvature of a spherically symmetric geodesic surface in the spatial section of the ‘Schwarzschild solution’ and so does not in itself define any distance whatsoever in the that manifold. With the correct identification of the associated Gaussian curvature it is also easily proven that there is only one singularity associated with all Schwarzschild metrics, of which there is an infinite number that are equivalent. Thus, the standard removal of the singularity at r = 2m is, in a very real sense, removal of the wrong singularity, very simply demonstrated herein. In addition, the ‘field equations’ Rμν = 0 define a spacetime that contains no matter, and since the ‘Principle of Superposition’ does not apply in General Relativity, it is impossible for Schwarzschild black holes to persist and mutually interact in a mutual spacetime that by construction contains no matter. Consequently, there are no black holes associated with the equations Rμν = 0 and therefore no related gravitational waves.