  The work shows that the evolution of the field of the free Klein-Gordon
equation (KGE), in the hydrodynamic representation, can be represented by the
motion of a mass density subject to the Bohm-type quantum potential, whose
equation can be derived by a minimum action principle. Once the quantum
hydrodynamic motion equations have been covariantly extended to the curved
space-time, the gravity equation (GE), determining the geometry of the
space-time, is obtained by minimizing the overall action comprehending the
gravitational field. The derived Einstein-like gravity for the KGE field shows
an energy-impulse tensor density (EITD) that is a function of the field with
the spontaneous emergence of the cosmological pressure tensor density (CPTD)
that in the classical limit leads to the cosmological constant(CC). The
energy-impulse tensor of the theory shows analogies with the modified
Brans-Dick gravity with an effective gravity constant G divided by the field
squared. Even if the classical cosmological constant is set to zero, the model
shows the emergence of a theory-derived quantum CPTD that, in principle, allows
to have a stable quantum vacuum (out of the collapsed branched polymer phase)
without postulating a non-zero classical CC. In the classical macroscopic
limit, the gravity equation of the KGE field leads to the Einstein equation.
Moreover, if the boson field of the photon is considered, the EITD correctly
leads to its electromagnetic energy-impulse tensor density. The outputs of the
theory show that the expectation value of the CPTD is independent by the
zero-point vacuum energy density and that it tends to zero as the space-time
approaches to the flat vacuum, leading to an overall cosmological effect on the
motion of the galaxies that may possibly be compatible with the astronomical
observations.
