Four states of matter and centrally symmetric de Broglie particle–wave mechanical systems

The dark issues of cosmological mechanics imply that our accounting for mass and energy at this scale is incorrect. On the other hand, existing theory not only accounts for atomic physics, but does so to a very high degree of accuracy. de Broglie was first to propose a concrete physical picture of the co-existence of both particle and its associated wave. We have previously proposed a Lorentz invariant modification of Newton’s second law that applies when the particle energy itself is of comparable magnitude to the potential energy of the applied external field. A dual particle–wave formulation is developed that allows exceptions to the law that matter cannot be created or destroyed, and accommodates both particle and wave energies. Here we examine this formulation in a centrally or spherically symmetric gravitating environment. A logical analysis of the integrated rate-of-working equation gives rise to four distinct states of matter; positive and negative energies with either non-zero rest mass or zero rest mass. This identification is meaningful only within the extended theory, and it has no meaning within conventional theory. We propose that dark matter and dark energy arise from a particular alignment of the spatial physical force f and the force g in the direction of time, such that the particle and wave energies coincide, and a consistent mathematical framework supports this proposal. Allowable potentials V(r, t) arise as the solutions of certain partial differential equations depending upon the assumed state of matter, and the mathematical framework corresponding to each state is briefly described and, where possible, some of the simpler and physically more interesting solutions are examined. The Newtonian gravitational potential arises from a variety of asymptotic limits, and can be identified as predominantly wave energy, which is entirely consistent with the post-Newtonian approximation.