A Class of Eigenvalues of the Fine-Structure Constant and Internal Energy Obtained from a Class of Exact Solutions of the Combined Klein-Gordon-Maxwell-Einstein Field Equations

This paper deals with the combined Klein‐Gordon‐Maxwell‐Einstein field equations, which govern completely and self‐consistently the spinless, charged, gravitating matter distribution. One of the theorems that have been proved here states that, from a static, purely gravitational universe, a class of electrogravitational universes containing a stationary matter field can be constructed, provided a single differential equation is satisfied. The construction of the electrogravitational universe from the Schwarzchild solution hinges on the solubility of the ordinary differential equation $${\mathrm{U″+\alpha }}^2\mathrm{(x\hspace{0.17em}}{\mathrm{csch}}^2{\mathrm{\hspace{0.17em}x)}}^2{U}^3\text{=0}$$ , where the prime denotes differentiation and α² stands for the fine‐structure constant. Next, the following nonlinear eigenvalue problem related to this differential equation has been posed. Are there some positive values of α corresponding to which solutions U(x) exist such that (i) Uis analytic and positive in x ∈ (0, ∞] (⇒ the volume element has one sign), (ii) U(0) = 0 (this condition is physically unpleasant but forced by the differential equation itself), (iii) U′(∞) = 0 (⇒ no force at the center of spherically symmetric mass and charge distributions), (iv) U′(0) = α (⇒ the total charge of the material distribution is α)? The answer is ``yes'' and it has been rigorously proved that there exists a unique solution of the problem. The corresponding value of α comes out to be 1.4343(ℏc)<sup>½</sup>, which, unfortunately, does not agree with the experiment (the discrepancy may be attributed to the neglect of the second quantization). If the restriction in U(x) to be positive is withdrawn, then a countable number of solutions exist with the corresponding eigenvalues for the electronic charge, internal energy, and mass. These solutions give rise to universes which are topologically inequivalent to Euclidean space and contain a finite number of shells. It should be mentioned that the present eigenvalue problem appears as a consequence of the ``Weyl‐Majumdar'' condition on the electrogravitational universe. There may well exist other eigenvalue problems for the fine‐structure constant within the framework of the Klein‐Gordon‐Maxwell‐Einstein field equations without the ``Weyl‐Majumdar'' condition. ``… es kann dann in jedem Punkte das Krümmungsmass in drei Richtungen einen beliebigen Werth haben, wenn nur die ganze Krümmung jedes messbaren Raumtheils nicht merklich von Null verschieden ist …''—Riemann.