Quaternion algebra on 4D superfluid quantum spacetime: the Lorentz gauge is a gate to the invisible world of dark matter and dark energy

Abstract Quaternion algebra is used for the description of the motion of material objects in 4D space-time. In the same quaternion basis, the energy-momentum density tensor is also written, accurate to arbitrary scalar fields. Quaternionic generating operators defined in this space being applied repeatedly to the energy-momentum density tensor lead to the gravitomagnetic equations that are parent equations for the series of physical equations ranging from the wave equations up to the quantum mechanical ones. Before finally deducing the gravitomagnetic equations, it is necessary to use the Lorentz gauge at a certain stage to cut off the arbitrary scalar fields. We declare that these superfluous scalar fields, inaccessible to direct observation, represent dark matter and dark energy. Dark matter manifesting through the Bose-Einstein condensate looks like a time crystal composed of particle-antiparticle pairs. We show in what way the dark matter can provide a flat profile of the orbital speed of the galaxy's arm rotations.