After the formulation of Einstein’s theory of General Relativity (GR) by utilizing 4D spacetime, Kaluza discovered in 1919 a potential field unification of electromagnetism and gravitation in 5D spacetime. To deal with the fifth dimension, Klein posited that it could be compactified. Nonetheless, these attempts and their expansions to higher dimensions have not culminated in testable predictions nor in the competence to elucidate observations yet. As an alternative to compactification, Gogberashvili, Randall, and Sundrum demonstrated in 1999 that the weak field of gravity could be explained by using a model of 4D spacetime that is embedded in a negatively curved and large fifth dimension; nevertheless, the model demanded massive gravitons [1,2,3].

On the other hand, to achieve an effective action for quantum corrections, several theories were formulated based on alterations of curvature terms and Lagrangian fields that included high-order curvature terms and non-minimally coupled scalar fields [4,5,6].

Moreover, although quantum anomalies can necessitate the use of a non-local Lagrangian, one of the major differences between GR and Quantum Field Theory (QFT) is that GR is background-independent, meaning it does not rely on the global structure, whereas QFT involves a background metric that in turn influences its predictions [7]. Recently, the Planck legacy 2018 (PL18) release confirmed the presence of an enhanced lensing amplitude in cosmic microwave background (CMB) power spectra, which prefers a positively curved early Universe with a confidence level higher than 99% [8,9], which is regarded in this study as curved global structure. In addition, the validity of combining CMB lensing and baryon acoustic oscillation data to recover the spatial flatness was challenged because the curvature parameter tension between these sets of data is 2.5 to 3σ [10]. On the other hand, the closed early Universe provides a physical justification for the anomalous lensing amplitude and is consistent with the low CMB anisotropy observations [8,11,12].

Based on this sign of a primordial global curvature and its feasible evolution over the conformal time into the present Universe spatial flatness, it is obvious that the background-independent theories, such as GR, do not consider the evolution in the background (global or bulk) curvature and regard celestial objects in the early Universe with a preferred curvature on equal footing with their counterparts in the present Universe of a spatially flat background; this shortcoming can be the cause of the dark matter problem.

A desirable gravity theory should consider metrics of both the celestial object and the bulk (background), and reduce to GR in a flat background. Moreover, concerning the wave-particle duality and the quantum interactions, a mutually harmonious geometrical formalism of quantum mechanics should be deemed to maintain the equivalence principle and count for the influence of gravity on the quantum systems. This is because gravity appears to emerge owing to spacetime curvature, imparting it with a geometrical nature, where it cancels out in free fall and cannot be shielded. This study aims to formulate interaction field equations that consider the global (including the background) curvature, signified as the 4D conformal bulk curvature, and its impact on celestial objects that are regarded as 4D relativistic cloud-worlds. It also aims to incorporate the impact of gravity, expressed by 4D conformal bulk curvature, on quantum fields that are regarded as 4D relativistic quantum clouds. This paper is organized as follows. Section 2 presents the electromagnetic-gravitational interaction field equations; Section 3 presents their visualizations; Section 4 incorporates quantum interactions and Section 5 reproduces quantum electrodynamics. Section 6 proposes a theory test. Finally, Section 7 concludes this work.

The Sun flows in a nearly flat background, based on GR, where its induced curvature is proportional to its energy density and flux. Conversely, the Earth flows in a curved background (curved bulk) due to the Sun’s presence, where its own induced curvature is affected by the bulk curvature in addition to its energy density and flux. Moreover, the PL18 release preferred a positively curved early Universe that is a sign of a primordial global curvature or a curved conformal bulk where the evolution of conformal curvature is associated with the Universe scalar factor (Appendix A). To consider the bulk curvature and its evolution over conformal time into present spatial flatness, a modulus of spacetime deformation, $E_D$, in terms of energy density is used based on the elasticity theory [13]. $E_D$ can be expressed in terms of the bulk’s resistance to localized curvature that is induced by celestial objects or in terms of the bulk field strength, ${\mathcal{F}}_{\lambda \rho }$, by using Lagrangian formalism of energy density existing in the bulk as a manifestation of the vacuum energy density as where the stress is signified by the stress-energy tensor $T_{\mu \nu }$ of trace $T$ while the strain is signified by the Ricci curvature tensor $R_{\mu \nu }$ as the change in the curvature divided by the existing curvature $\mathcal{R}$ as the background or bulk conformal curvature.

By incorporating the bulk influence, a nonlocal action is expressed as where $R$ is the Ricci scalar denoting a localized curvature, which is induced in the bulk by a celestial object that is regarded as a 4D relativistic cloud-world of metric $g_{uv}$ and Lagrangian density$L$ while $\mathcal{R}$ is the scalar curvature of the 4D conformal bulk of metric ${\tilde{g}}_{\mu \nu }$ and Lagrangian density $\mathcal{L}$ as its internal stresses and momenta due to its curvature.

Because of the bulk modulus, $E_D$, is constant with regards to the cloud-world action under the constant vacuum energy density condition; and by considering the expansion of the bulk over conformal time owing to the expansion of the Universe and its implication on the field strength of the bulk, a dual-action concerning the conservation of energy on global (bulk) and local (cloud-world) scales is introduced as follows

This action can be interpreted as describing an interactive flow of a 4D relativistic cloud-world through vacuum energy of field strength that is manifested by the 4D conformal bulk curvature. It implies eight-dimensional degrees of freedom as${ d}^4\rho { d}^4\mathsf{\sigma }={\phi }^2{d}^8\alpha$, where $d^4\rho$ is the standard four dimensions of the Einstein–Hilbert action concerning a celestial object (conventional energy), $d^4\mathsf{\sigma }$ is the four dimensions of the conformal bulk (vacuum energy) regarding the background curvature, and ${\phi }^2$ is a dimension-hierarchy factor. The bulk metric, ${\tilde{g}}_{\mu \nu }$, and the cloud-world metric, $g_{uv}$, are related by Weyl’s conformal transformation as ${\tilde{g}}_{\mu \nu }=g_{\mu \nu }{\mathsf{\Omega }}^2$, where ${\mathsf{\Omega }}^2$ is a conformal function [14]. The global-local action should hold for any variation as follows where $\delta \sqrt{-g}=-\sqrt{-g}{ g}_{\mu \nu }\delta g^{\mu \nu }/2$ [15]. By considering the cloud-world’s boundary term, ${\int }_c^{ }{g}^{\mu \nu }\delta R_{\mu \nu }\sqrt{-g }{d}^4\rho /\mathcal{R}$, the variation in the Ricci curvature tensor, $\delta R_{uv}$, can be expressed in terms of the covariant derivative of the difference between two Levi-Civita connections, the Palatini identity, $\delta R_{\mu \nu }={\nabla }_{\rho }\left(\delta {\Gamma }_{ \nu \mu }^{\rho }\right)-{\nabla }_{\nu }\left(\delta {\Gamma }_{ \rho \mu }^{\rho }\right)$; this variation with respect to the inverse metric, $g^{\mu \nu }$, can be obtained by using the metric compatibility of the covariant derivative, ${\nabla }_{\rho }{g}^{\mu \nu }=0$ [15], as $g^{\mu \nu }\delta R_{\mu \nu }={\nabla }_{\rho }\left(g^{\mu \nu }\delta {\Gamma }_{\nu \mu }^{\rho }-g^{\mu \rho }\delta {\Gamma }_{ \sigma \mu }^{\sigma }\right)$. Thus, the boundary term as a total derivative for any tensor density can be transformed based on Stokes theorem as where the bulk scalar curvature, $\mathcal{R}$, is left outside the integral transformation as it only acts as a scalar. Besides, a second approach can be applied to the bulk’s boundary term as where $-{\tilde{g}}^{\mu \nu }={\tilde{g}}_{\mu \nu }\delta {\tilde{g}}^{\mu \nu }/\delta {\tilde{g}}_{\mu \nu }$ while ${\varnothing }^2={\delta ln\mathcal{R}}_{\mu \nu }/\delta ln{\tilde{g}}_{\mu \nu }$ resembles the Ricci flow in a normalized form, reflecting the conformal evolution in the boundary, which can be expressed as a function

based on Weyl’s transformation as ${\stackrel{̿}{g}}_{\mu \nu }={\tilde{g}}_{\mu \nu }{\varnothing }^2/{\mathsf{\Omega }}^2=g_{\mu \nu }{\varnothing }^2$.

By using the first approach of the boundary terms’ transformations given in Equation (5), the transformed boundary action, $S_b$, is where $K$ and $\mathcal{K}$ are the traces of the cloud-world and the bulk extrinsic curvatures, respectively; $l$ and $\mathcal{l}$ are the extrinsic traces of the Lagrangian density on the cloud-world and the bulk boundaries, respectively; $q$ and $p$ are the determinants of their induced metrics, respectively; and $ϵ$ equals 1 when the normal ${\widehat{n}}_u$ is a spacelike entity and equals -1 when it is a timelike entity. $f_{\lambda }={\mathcal{F}}_{\lambda \rho }{J}^{\rho }$ is the 4D Lorentz force density. The boundary action should hold for any variation and, by considering the transformed cloud-world’s boundary term, $ϵ{\int }_{\partial C}^{ }K\sqrt{\left|q\right|}{d}^3\varrho /\mathcal{R}$, the variation is where ${K=K}_{\mu \nu }{q}^{\mu \nu }$. By utilizing the Jacobi formula for the determinant differentiation, $\delta \sqrt{\left|q\right|}=-\sqrt{\left|q\right|} q_{\mu \nu }\delta q^{\mu \nu }/2$ and the variation in the metric times the inverse metric, $q^{\mu \nu }\delta q_{\mu \nu }=-q_{\mu \nu }\delta q^{\mu \nu }$; thus, the variation in the boundary term of the cloud-world is, as where ${\delta K}_{\mu \nu }/\delta q_{\mu \nu }K={(\delta K}_{\mu \nu }/K_{\mu \nu }\left)\right(q_{\mu \nu }/\delta q_{\mu \nu })=\delta \mathrm{l}\mathrm{n}{K}_{\mu \nu }/\delta \mathrm{l}\mathrm{n}{q}_{\mu \nu }={\omega }^2$ resembles the Ricci flow in a normalized form, reflecting the conformal distortion in the boundary over conformal time, which can be expressed as a function, according to Weyl’s conformal transformation [16], while the term ${\mathcal{R}}_{\mu \nu }/\mathcal{R}={\mathcal{R}}_{\mu \nu }/{\mathcal{R}}_{\mu \nu }{\tilde{g}}^{\mu \nu }={\tilde{g}}_{\mu \nu }=q_{\mu \nu }{\vartheta }^2$. Thus, the boundary term is $ϵ{\int }_{\partial C}^{ }\left[K_{\mu \nu }-K{\widehat{q}}_{\mu \nu }/2\right]\delta q^{\mu \nu }\sqrt{\left|q\right|} d^3\varrho /\mathcal{R}$, where ${\widehat{q}}_{\mu \nu }=q_{\mu \nu }+2q_{\mu \nu }{\omega }^2-2q_{\mu \nu }{\varnothing }^2$ is the conformally transformed induced metric on the boundary of the cloud-world. The same is applied to the bulk and Lagrangian boundary terms. The variation in the whole action, by renaming the dummy indices, is

While the variation in the inverse metric, $\delta g^{\mu \nu }$, can be omitted from both the curvature and Lagrangian sides of the local action, $C$, (the standard action procedure), this is not the case with regards to the global action, $B$. The outcome of the global action resembles the electromagnetic stress-energy tensor times the variation in inverse metric in addition to the boundary term: ${\mathcal{T}}_{\mu \nu }\delta {\tilde{g}}^{\mu \nu }+{{\tilde{g}}^{\mu \nu }\delta \mathcal{T}}_{\mu \nu }$. By comparing Einstein field equations with Equations (1), (2) and (10); it reveals that ${\mathcal{T}}_{\mu \nu }\delta {\tilde{g}}^{\mu \nu }+{{\tilde{g}}^{\mu \nu }\delta \mathcal{T}}_{\mu \nu } :=E_D=\mathcal{R}{c}^4/8\pi G_{\mathcal{R}}$, which is proportional to the fourth power of the speed of light that in turn is directly proportional to the frequency; this agrees with the frequency cut-off predictions of vacuum energy density in QFT [17,18]. Under the constant vacuum energy condition and by treating the boundary term as in Equation (6), which gives ${\mathcal{T}}_{\mu \nu }\delta {\tilde{g}}^{\mu \nu }=\mathcal{T}{\overline{\tilde{g}}}_{\mu \nu }\delta {\overline{\tilde{g}}}^{\mu \nu }$; this shows changes in vacuum energy density metric in a region changes its boundary, i.e., the global action outcome is a constant representing vacuum energy density and corresponds to $E_D=\mathcal{T}$.

By applying the principle of stationary action for Equation (10), the field equations are

These interaction field equations can be interpreted as indicating that the induced curvature, $R$, by the cloud-world over the bulk (background) curvature, $\mathcal{R}$, equals the ratio of the imposed energy density of the cloud-world and its flux, to the vacuum energy density of the bulk and its flux through the expanding/contracting Universe. The interaction field equations feature the following:

The field equations include four contributions that come from the cloud-world’s intrinsic and extrinsic curvatures and bulk’s intrinsic and extrinsic curvatures. The field equations can be expressed in different forms depending on which contribution is required to be implicit or explicit. By applying the second approach in Equation (6) to the bulk boundary terms, the variation in the action with renaming the dummy indices is

By transforming intrinsic curvature of the bulk, ${\mathcal{R}}_{\mu \nu }/\mathcal{R}={\mathcal{R}}_{\mu \nu }/{\mathcal{R}}_{\mu \nu }{\tilde{g}}^{\mu \nu }={\tilde{g}}_{\mu \nu }$, where the bulk’s metrics can be incorporated with cloud-world metric into a conformal metric. By applying the principle of stationary action, the simplified interaction field equations are where ${{\widehat{g}}_{\mu \nu }=g}_{\mu \nu }+2{\mathcal{R}}_{\mu \nu }/\mathcal{R}-2{\stackrel{̿}{g}}_{\mu \nu }$, or can be expressed as ${{\widehat{g}}_{\mu \nu }=g}_{\mu \nu }+2{\tilde{g}}_{\mu \nu }-2{\stackrel{̿}{g}}_{\mu \nu }$, is the conformally transformed metric counting for contributions of cloud-world metric, $g_{\mu \nu }$, and bulk’s intrinsic and extrinsic curvatures based on ${\tilde{g}}_{\mu \nu }$ and ${\stackrel{̿}{g}}_{\mu \nu }$(intrinsic-equivalent to reflect external fields1) metrics, respectively, whereas Einstein spaces are a subclass of the conformal space [14]. Similarly, the conformally transformed induced metric on the cloud-world’s boundary is ${\widehat{q}}_{\mu \nu }=q_{\mu \nu }+2{\tilde{q}}_{\mu \nu }-2{\stackrel{̿}{q}}_{\mu \nu }$. The simplified equations feature:

The field equations can remove singularities and satisfy a conformal invariance theory.

This section visualizes the evolution of the 4D relativistic cloud-worlds over the conformal space-time of the 4D bulk. Galaxy formation and evolution as a 4D relativistic cloud-world traveling in a curved 4D conformal bulk, as preferred by the PL18 release, is considered. The entire contribution comes from the boundary term when calculating the black hole entropy using the semiclassical approach [19,20]. This concept can be applied to the field equations in Equation (11) that are multiplied by the bulk curvature, $\mathcal{R}$, as

The field equations can describe the interaction between a 4D relativistic cloud-world of intrinsic $R_{\mu \nu }$ and extrinsic $K_{\mu \nu }$ curvatures with a stress-energy ${\widehat{T}}_{\mu \nu }$ and the 4D bulk of intrinsic ${\mathcal{R}}_{\mu \nu }$ and extrinsic ${\mathcal{K}}_{\mu \nu }$ curvatures with a stress-energy ${\mathcal{T}}_{\mu \nu }$. From Equation (14), the field equations yield where ${\tilde{g}}_{\mu \nu }={\mathcal{R}}_{\mu \nu }/\mathcal{R}={\mathcal{R}}_{\mu \nu }/{\mathcal{R}}_{\mu \nu }{\tilde{g}}^{\mu \nu }$, represents the bulk intrinsic curvature. The conformally transformed metric, ${\widehat{g}}_{\mu \nu }=g_{\mu \nu }+2{\tilde{g}}_{\mu \nu }=g_{\mu \nu }\left(1+2{\mathsf{\Omega }}^2\right)$, can be expressed as where $A$and$B$ are functions of the cloud-world curvature radius $r$, while the conformal function ${\mathsf{\Omega }}^2$ is a function of the bulk curvature radius $\tilde{r}$, which can be influenced by the cloud-world curvature radius. $S^2$ is a dimensionless conformal scale factor and $g_{\omega }=d{\theta }^2+r^2{sin}^2\theta d{\varphi }^2$ is the metric on the two-sphere. The derivations of these functions in Appendix B are where the conformal function ${\mathsf{\Omega }}^2$ relies on the gravitational potential of the bulk, while its influence is inversely proportional to the cloud-world potential.

In the case of PI18′s preferred early Universe positive curvature, the gravitational potential of the bulk can be expressed in terms of the early Universe plasma of mass, $M_p$, while $\tilde{r}$denotes the radius of the curvature of the bulk, where the bulk’s potential decreases with the Universe expansion and vanishes in the flat spacetime background ($\tilde{r}\to \infty )$. The minus sign of ${\mathsf{\Omega }}^2$ reveals a spatial shrinking by evolving in the conformal time. Consequently, the conformally transformed metric tensor, ${\widehat{g}}_{\mu \nu }=g_{\mu \nu }+2{\tilde{g}}_{\mu \nu }=g_{\mu \nu }\left(1+2{\mathsf{\Omega }}^2\right)$, is

This nonlocal metric reduces to the Schwarzschild metric in a flat background ($\tilde{r}\to \infty )$. The conformally transformed metric can be visualized by evolving in the conformal time by using Flamm’s approach as where $C$ is a constant, and $\mathcal{O}$ denotes less significant terms. Figure 1 shows this scenario, which reveals that a galaxy forms as a dual forced vortices due to the curvature of the background.

The visualization of Equation (19) as the scenario of the galaxy formation as a dual forced vortices due to the curvature of the background is shown in Figure 1, the evolution of the 4D cloud-world of metric $g_{\mu \nu }$ through its travel and spin in the conformal space-time of the 4D bulk of metric ${\tilde{g}}_{\mu \nu }$.

Regarding the wave-particle duality and analogous to the constant modulus of the bulk, ${-\mathcal{F}}_{\lambda \rho }{\mathcal{F}}^{\lambda \rho }/4{\mu }_0$, the curvature of the bulk, including both that which is conformal, ${\mathcal{R}}_{\mu \nu }{\tilde{g}}^{\mu \nu }$, and induced by a celestial object, $R_{\mu \nu }{g}^{\mu \nu }$, can be considered constant regarding quantum fields, $L_{\alpha \beta }{L}^{\alpha \beta }/2{\chi }_0$. Accordingly, the dual action in Equation (3) can be extended in terms of quantum waves, as follows

This action can be interpreted as conceptualizing that a 4D conformal bulk, as a manifestation of vacuum energy of a conformal time flow, embeds a 4D relativistic cloud-world representing a celestial object of a conventional time flow that in turns encapsulates 4D relativistic quantum clouds of a quantum time flow and so forth, where $L_{\alpha \beta }{L}^{\alpha \beta }/2{\chi }_0$ are Lagrangian densities of two entangled quantum fields of a metric tensor $q_{\mu \nu }$ and four-momentum $p_{\mu }{p}^{\mu }$ respectively, while, ${\pi }_{\mu }{\pi }^{\mu }$ are the four-momentum of vacuum energy density of a Lagrangian density ${\mathcal{L}}_{\mu \nu }{g}^{\mu \nu }$ and ${\chi }_0$ is a proportionality constant.

By considering the induced curvature by the cloud-world in the bulk, the conceptual picture can be simplified to an interactive flow of 4D relativistic quantum clouds with the vacuum energy of a field strength that relies on the 4D conformal bulk curvature, which harmonizes the dual-action concept. The action should hold for any variation, as follows

By considering the boundary term of the quantum cloud$,{\int }_Q^{ }{q}^{\mu \nu }\delta \left(p_{\mu }{p}_v\right)\sqrt{-q}{d}^4\alpha /{\pi }_{\mu }{\pi }^{\mu }$, the variation in the four-momentum $\delta p_{\mu }$, i.e., the change in the total energy of the charged fields enclosed within the quantum cloud boundary $Q$, can represent the flow of the four-current $J_{\mu }$ through the cloud boundary $\partial Q$, where multiplying this current by the four-potential that is generated by the current itself, $A_{\mu }$, and that which is externally applied, $B_{\mu }$, gives the following scalar: $\delta p_{\mu }\equiv \delta \surd (E/c-p)\equiv (A_{\mu }+B_{\mu })J^{\mu }$. This deduction is based on the gauge theory. However, the cloud’s volume and its boundary surface should be taken into consideration. The boundary term signifies the variation of two entangled quantum clouds, $q^{\mu \nu }\delta \left(p_{\mu }{p}_v\right)$; thus,

The variation in this transformed boundary term with relabeling $A_{\rho }+B_{\rho }$ as $A_{\rho }$, yields where $e_{\mu \nu }$ is the induced metric on the boundary of the quantum cloud. The last two terms resemble the Ricci flow in a normalized form, reflecting the conformal distortion in the boundary, which can be expressed as a conformal function as where $e^{\lambda \gamma }=-e_{\lambda \gamma }\delta e^{\lambda \gamma }/\delta e_{\lambda \gamma }$, and ${\varnothing }^2=\delta ln\left(J_{\lambda }{A}_{\rho }{J}^{\rho }{A}_{\gamma }\right)/\delta ln{e}_{\lambda \gamma }$ is the conformal function.

On the other hand, the cloud-world’s boundary term can be transformed as where $g^{\mu \nu }=-g_{\mu \nu }\delta g^{\mu \nu }/\delta g_{\mu \nu }$ while ${\varnothing }^2=\delta ln\left({\pi }_{\mu }{\pi }_v\right)/\delta ln{g}_{\mu \nu }$, which resembles normalized Ricci flow and ${\stackrel{̿}{q}}_{\mu \nu }$ represents the metric of the bulk regarding its extrinsic curvature. Accordingly, the variation in the whole action is