Practical applications of a relativistic theory including tensor and scalar gravitation

A relativistic theory including two types of gravitation, tensor and scalar field, has recently been published by Zhang, providing the origin of the Hubble tension as well as that of the cosmological constant. It is shown here that the combination of scalar and tensor fields enables the relativistic theory to be compatible with Mach's principle (MP). The scalar field can also act as an inflation field at the cosmic inflationary era and convert to dark matter particles during its oscillating era. As the quanta of the excitation states of the scalar field, unlike ordinary matter, this type of dark matter does not couple to the equilibrium state of the scalar field. The masses of the quanta are estimated to be less than 31 meV if we assume that all dark matter is of this type. The most probable mass is 2 meV. Due to the dynamic equilibrium state of the scalar field acting as dark energy, a simulated equation of state is introduced for this dynamical dark energy. Consequently, if the true space of the Universe is closed, observers who are based on the assumption of flat space would discover that the equation of state for dark energy has ever crossed $\ensuremath{-}1$ in the near past. Since Dark Energy Spectroscopic Instrument Data (DESI) imply that the equation of state may cross $\ensuremath{-}1$, then the spatial curvature of our Universe may be positive based on the gravitational scheme. These effects above are analyzed in the frame of general relativity (GR) by canonically subsuming the canonical scalar field into GR. It is also demonstrated that the new interaction between the scalar field and ordinary matter can be independently analyzed in the frame of special relativity (SR). The equations of motion of several given matter fields in the presence of the scalar gravitational field are obtained in SR frame. Unlike the GR effect of the self-interaction potential of the scalar field accelerating the expansion rate of the Universe, the so-called scalar-field-mediating force is caused by the gradient of the scalar field and is entirely an effect of SR, independent of GR. We refer to this new interaction as scalar gravitation to distinguish it from Einstein's tensor gravitation. Since all ordinary matter couples to the scalar field in the same conformal way, the scalar field can result in forces not only between fermions but also bosons. Due to the spontaneous-symmetry-breaking of the coupling, the matter-density-dependent forces are often short ranged. In the current density of the Universe, the range of interaction is estimated to be 2.2 \ensuremath{\mu}m if all dark matter is attributed to the excitation of the scalar field. When the Lorentz invariant effective nonrelativistic energy density ${\ensuremath{\rho}}_{\mathrm{N}}=\ensuremath{\rho}\ensuremath{-}3P$ of the ordinary matter fluid decreases (which does not necessarily mean that the energy density $\ensuremath{\rho}$ of the fluid must be low because the fluid pressure $P$ can be high enough to make ${\ensuremath{\rho}}_{\mathrm{N}}=0$), the range of interaction will increase as a function of $1/\sqrt{{\ensuremath{\rho}}_{\mathrm{N}}}$. Thus, the scalar-mediated force will play a crucial role in the formation of galaxies, especially under relativistic conditions and extremely low energy densities of ordinary matter.