Combined GravElectroMagnetic Forces in a Reissner-Nordstr\"om Black-Hole Binary: The Emitted GEM Waves, and the Appearance of the Relativistic Maximum Tension Force and Planck's Constant in the Classical Domain

Two Reissner-Nordström (RN) black holes in a binary, each with mass M and charge Q, are subject to a conservative action-reaction force pair that includes four components (FM→M, FM→Q, FQ→M, and FQ→Q). We combine these four components to obtain the magnitude of the net gravelectromagnetic (GEM) force F as a function of the binary separation R. The resultant force F(R) is especially simple, owing to the conservative nature of the components. The separation R decreases in time, as the binary emits GEM waves. We use an approximation for the dimensionless amplitude h of these waves and the derived force F(R) to develop a formula for h in terms of multiple ratios of the fundamental lengths of the problem: the distance to the binary r, the binary separation R, the Schwarzschild radius RS, and the RN charge radius RQ. For RQ=0, the results reduce to the amplitude of the conventional gravitational radiation produced by the FM→M force component. In the course of these calculations, we have encountered two entirely unexpected properties of the combined GEM field: Newton’s action-reaction principle is not valid for oppositely-charged RN black holes, and the general relativistic maximum force (Fmax=FP/4, where FP is the Planck force) has naturally appeared as the upper limit of classical GEM forces as well. Furthermore, we have serendipitously obtained a remarkable new relation between universal constants of nature that expresses Planck’s constant in terms of classical (non-quantum mechanical) constants αg ∝ 1/NA2, the fine-structure constant α ∝ 1/NA2 as well, and the weak coupling constant αw = √α ∝ 1/NA.