On Measurable Physical Quantities in the Theory of Double Gravitational Fields

Zhang recently proved that in the sense of flat spacetime, i.e. within the framework of special relativity (SR), there may exist a new type of scalar gravitation in nature. Since the scalar gravitational field couples with any form of field in the standard model (SM) of elementary particles in the same conformal way within the SR frame, all constants of SM are independent of the scalar gravitational interaction. However, the universal coupling property implies the existence of a new universal free fall motion in scalar gravity. This kind of free fall occurs in the SR framework (flat spacetime) because we have not yet considered Einstein's gravity.

The scalar gravitational field can be inserted into curved spacetime by assuming that it couples with Einstein's metric tensor gravitational field in the same way as ordinary matter. Thus, the scalar gravitational interaction does not change any constant in traditional general relativity (GR). Although the geodesic equations for particles are improved to the generalized Newton second law of motion, Einstein's field equations still hold. Consequently, based on the modified GR, the scalar gravitational field can be used to explain not only the origins of dark matter, dark energy and inflation field, but also the Hubble tension in cosmology.

However, in the presence of both scalar and tensor gravitational fields, since any freely falling system is locally neither geodesic nor inertial, it is necessary to carefully re-examine traditional equivalent principles and measurable physical quantities. Based on measurable considerations, it is theoretically demonstrated that the physical quantities measured in experiments, such as energy and three-momentum, correspond to the contravariant four-momentum components of the measured particle. Then, based on the restated principle of equivalence in GR, the expressions for the energy and momentum of a moving particle are obtained in a comoving coordinate system in which all observers are at rest. The correctness of these expressions ultimately requires experimental judgment.