Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N ) region. The ordinates λn are the positive imaginary parts of the nontrivial zeta zeroes in the critical line : sn = 1 2 + iλn. The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.
We revisit the construction of diffeomorphic but not isometric solutions to the Schwarzschild metric. The solutions relevant to Black Holes are those which require the introduction of non-trivial areal-radial functions that are characterized by the key property that the radial horizon's location is displaced continuously towards the singularity (r=0). In the limiting case scenario the location of the singularity and horizon merges and any infalling observer hits a null singularity at the very moment he/she crosses the horizon. This fact may have important consequences for the resolution of the firewall problem and the complementarity controversy in black holes. It is shown next how modified Newtonian dynamics (MOND) can be obtained from solutions to Finsler gravity, and which in turn, can also be modelled by metrics which are diffeomorphic but not isometric to the Schwarzschild metric. The key point now is that one will have to dispense with the asymptotic flatness condition, by choosing an areal radial function which is finite at = r ∞. Consequently, changing the boundary condition at = r ∞ leads to MONDian dynamics. We conclude with some discussions on the role of scale invariance and Born's Reciprocal Relativity Theory based on the existence of a maximal proper forces.
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