The Schwarzchild manifold of general relativity theory is unsatisfactory as a particle model because the singularity at the origin makes it geodesically incomplete. A coupling of the geometry of space‐time to a scalar field φ produces in its stead a static, spherically symmetric, geodesically complete, horizonless space‐time manifold with a topological hole, termed a drainhole, in its center. The coupling is Rμν=2φ,μφ,ν; its polarity is reversed from the usual to allow both the negative curvatures found in the drainhole and the completeness of the geodesics. The scalar field satisfies the scalar wave equation □φ=0 and has finite total energy whose magnitude, expressed as a length, is comparable to the drainhole radius. On one side of the drainhole the manifold is asymptotic to a Schwarzschild manifold with positive mass parameter m, on the other to a Schwarzschild manifold with negative mass parameter , and − > m. The two‐sided particle thus modeled attracts matter on the one side and, with greater strength, repels it on the other. If m is one proton mass, then − m̄/m ≈ 1+10−19 or 1+10−39, according as the drainhole radius is close to 10−33cm or close to 10−13 cm; the ratios of total scalar field energy to m in these instances are 1019 and 1039. A radially directed vector field which presents itself is interpreted, for purposes of conceptualization, as the velocity of a flowing ``substantial ether'' whose nonrigid motions manifest themselves as gravitational phenomena. When the ether is at rest, the two‐sided particle has no mass on either side, but the drainhole remains open and is able to trap test particles for any finite length of time, then release them without ever accelerating them; some it can trap for all time without accelerating them. This massless, chargeless, spinless particle can, if disturbed, dematerialize into a scalar‐field wave propagating at the wave speed characteristic of the space‐time manifold.

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The symbol t is now overloaded, representing both coordinate function and constant. Such ambiguities are to be resolved by appeal to context.
8.
Here the argument ρ is suppressed for convenience’s sake. It will be suppressed again on occasion.
9.
Einstein was convinced that the luminiferous ether, driven out of physicists’ thoughts by the special theory of relativity, had returned as an essential feature of the general theory, intimately involved with gravity and only secondarily if at all connected with electromagnetism. He did not, however, recognize in it any degree of substantiality or of motility. In his essay “Relativity and the Ether” he wrote: “According to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, an ether exists. In accordance with the general theory of relativity space without an ether is inconceivable…. But this ether must not be thought of as endowed with the properties characteristic of ponderable media, as composed of particles the motion of which can be followed; nor may the concept of motion be applied to it.” [A. Einstein, Essays in Science, translated by A. Harris (Philosophical Library, New York, 1934)].
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Anyone willing to consider the ether‐flow hypothesis as plausible might wish to ask whether, if the earth is conceivably a conglomeration of ether sinks and sources, the conventional interpretation of the Michelson‐Morley experiment ought not to be revised.
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A. Trautman has found before now that the Schwarzschild and the de Sitter line elements can be cast in the form (1). He has also applied the term “the ether” in these instances, but in a way that precludes any attribution of substantiality to the ether. [Perspectives in Geometry and Relativity, Essays in Honor of Václav Hlavatý, edited by B. Hoffman (Indiana U.P., Bloomington, 1966), p. 413].
13.
One could raise the point that the Schwarzschild manifold, which satisfies Eq. (24) no matter what the coupling constant K (just let φ be a constant), already has a central hole, the “wormhole” of the Kruskal‐Fronsdal extension. This hole, however, is not a permanent feature of the spatial cross sections; it develops into the Schwarzschild singularity when pursued in either temporal direction. Although at this point I have not said it in so many words, I have in mind that the particle models I seek shall be static, which alone would rule out the Schwarzschild manifold, even without the completeness requirement. One might also wish to say that a Schwarzschild interior solution, properly matched to an exterior solution, would provide a model of just the kind I want, without even introducing the scalar field φ. I would have to reply that such a manifold can only represent a mass body, not a particle. To electromagnetic geons as particle models (J. A. Wheeler, Ref. 5) my foremost objection would be that they are not static.
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H. Yilmaz, Introduction to the Theory of Relativity and the Principles of Modern Physics (Blaisdell, New York, 1965), p. 176.
The Rκλ used by Yilmaz are the negatives of those appearing here. This has the consequence that his Eq. (18.2), although seemingly equivalent to the present Eq. (26), actually involves the coupling of opposite polarity and therefore is not satisfied by his line element (18.3) unless φ is constant.
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These concepts are also combined, in a somewhat analogous fashion, in the Kerr solution of the Einstein vacuum field equations. [
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23.
There is an earlier instance in which to meet a new criterion the polarity of the coupling in Eq. (60) was partially reversed. Einstein and Rosen reversed it for the coupling of space‐time geometry to the electromagnetic field in order to represent an elementary charged mass particle as one of their “bridges” (A. Einstein and N. Rosen, Ref. 2).
24.
At the Relativity Conference in the Midwest, Cincinnati, Ohio, June 2–6, 1969, István Ozsváth was good enough to raise and to discuss with me this issue. Also, the referee has pressed it upon me in a friendly manner. The referee’s argument, based upon energy conservation, is persuasive but not, to my mind, conclusive. For this reason I shall not recapitulate it here; neither shall I at the end profess to have resolved the issue.
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É. Cartan, Leçons sur la géométrie des espaces de Riemann (Gauthier‐Villars, Paris, 1946), 2nd ed.
See also H. Flanders, Differential Forms with Applications to the Physical Sciences (Academic, New York, 1963),
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