The Sun flows in a spatially flat spacetime background, based on General Relativity, where its induced curvature is proportional to its energy density and flux. On the other hand, the Earth flows in a curved background (curved bulk) due to the Sun’s presence, where its induced curvature is affected by the bulk curvature,
in addition to its energy density and flux. To incorporate the bulk influence, a modulus of spacetime deformation,
, is utilized. The modulus can be expressed in terms of the resistance of the bulk to localized curvature that is induced by celestial objects or in terms of the field strength of the bulk by using the Lagrangian formulation of energy density that exists in the bulk as a manifestation of vacuum energy density as
where
is the field strength tensor of the bulk and
is vacuum permeability.
By incorporating the bulk influence, the Einstein–Hilbert action can be extended to
where
is the Ricci scalar representing a localized curvature, which is induced in the bulk by a celestial object that is regarded as a 4D relativistic cloud-world of metric
and Lagrangian density
, respectively, whereas
is the scalar curvature of the 4D conformal bulk of metric
and Lagrangian density
as its internal stresses and momenta reflecting its curvature. Since
is constant with regard to the extended action under the constant vacuum energy density condition; and by considering the evolution of the bulk owing to the expansion of the Universe, a dual-action concerning the energy conservation on global (bulk) and local (cloud-world) scales can be introduced as follows
Applying the principle of stationary action in [
6] yields
These interaction field equations can be interpreted as indicating that the cloud-world’s induced curvature,
, over the bulk’s conformal (background) curvature,
, equals the ratio of the cloud-world’s imposed energy density and its flux,
, to the bulk’s vacuum energy density and its flux,
, throughout the expanding/contracting Universe. The field equations can describe the interaction and flow of a 4D relativistic cloud-world of intrinsic
and extrinsic
curvatures through a 4D conformal bulk of intrinsic
and extrinsic
curvatures. The boundary term given by the extrinsic curvatures of the cloud-world and bulk is only significant at high energies when the difference between the induced and background curvatures is significant. By transforming intrinsic and extrinsic curvatures of the bulk [
6], comparing Einstein field equations with Equation (1) and then substituting to Equation (4), the interaction field equations can be simplified to
where
, or can be expressed as
because
, is the conformally transformed metric, which takes into account contributions from the cloud-world metric,
as well as the intrinsic and extrinsic curvatures of the bulk based on its metrics,
and
(intrinsic-equivalent metric) respectively, whereas Einstein spaces are a subclass of the conformal space [
7].
is a conformal stress-energy tensor that is defined by including the Lagrangian of the energy density and flux of the cloud-world,
, and the electromagnetic energy flux from its boundary,
, over the conformal time. These interaction field equations could remove the singularities and satisfy a conformal invariance theory. From Equations (5) and (1), the Newtonian gravitational parameter is
where
is the scalar curvature of the bulk. According to Equation (6),
is proportional to
and reflects the field strength of vacuum energy because any changes in the bulk’s metric,
, changes the field strength of the bulk,
, because of the constant modulus,
. In addition, although the ground state of
at the local present Universe appears to be spatially flat, it could have a small temporal curvature reflecting the present value of
. The dependency of
on the curvature of the bulk is discussed and visualized as follows.