On the Field Strength of Vacuum Energy and the Emergence of Mass ^†

Abstract

Large inconsistencies in the outcome of precise measurements of Newtonian gravitational ‘constant’ were identified throughout more than three hundred experiments conducted up to date. This paper illustrates the dependency of the Newtonian gravitational parameter on the curvature of the background and the associated field strength of vacuum energy. Additionally, the derived interaction field equations show that boundary interactions and spin-spin correlations of vacuum and conventional energy densities contribute to the emergence of mass. Experimental conditions are recommended to achieve consistent outcomes of the parameter precision measurements, which can directly falsify or provide confirmations to the presented field equations.

1. Introduction

The Newtonian gravitational ‘constant’ $G$ plays a crucial role in theoretical physics, astronomy, geophysics, and engineering. About three hundred experiments attempted to ascertain the value of $G$ up to date. However, the significant inconsistencies in their results have made it unfeasible to reach a consensus on an exact value. Many of them are precision measurements with a relative uncertainty of only 12 to 19 parts per million [1,2,3,4,5].

The achievement of such a low level of uncertainty can indicate that the margin of systematic errors in experiments is narrower than generally anticipated. At the same time, the significant inconsistencies among measurements’ outcomes imply that there could be phenomena that are not yet accounted for in the current framework of physics. This study investigates the impact of the background curvature on the value of $G$, and the influence of boundary interactions and spin-spin correlations on the emergence of mass.

2. Newtonian Gravitational Parameter

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, $\mathcal{R},$ in addition to its energy density and flux. To incorporate the bulk influence, a modulus of spacetime deformation, $E_D$, is utilized. The modulus can be expressed in terms of the bulk resistance to localized curvature or in terms of the field strength of the bulk by using the Lagrangian formulation of energy density existing in the bulk as a manifestation of vacuum energy density as

$$E_D=\frac{T_{\mu \nu }-\frac{1}{2}T{g}_{\mu \nu } }{R_{\mu \nu }/\mathcal{R}}=\frac{-{\mathcal{F}}_{\lambda \rho }{\mathcal{F}}^{\lambda \rho }}{4{\mu }_0}$$

(1)

where ${\mathcal{F}}_{\lambda \rho }$ is the field strength tensor of the bulk and ${\mu }_0$ is vacuum permeability.

By incorporating the bulk influence, the Einstein–Hilbert action can be extended to

$$S=E_D{{\displaystyle \int }}_C^{ }\left[\frac{R}{\mathcal{R}}+\frac{L}{\mathcal{L}}\right]\sqrt{-g} d^4\rho$$

(2)

where $R$ 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 $g_{uv}$ and Lagrangian density$L$, respectively, whereas $\mathcal{R}$ is the scalar curvature of the 4D conformal bulk of metric ${\tilde{g}}_{\mu \nu }$ and Lagrangian density $\mathcal{L}$. Since $E_D$ is constant with regard to the extended action; 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

$$S={{\displaystyle \int }}_B^{ }\left[\frac{-{\mathcal{F}}_{\lambda \rho }{\tilde{g}}^{\lambda \gamma }{\mathcal{F}}_{\gamma \alpha }{\tilde{g}}^{\rho \alpha }}{4{\mu }_0}\right]\sqrt{-\tilde{g}} {{\displaystyle \int }}_C^{ }\left[\frac{R_{\mu \nu }{g}^{\mu \nu }}{{\mathcal{R}}_{\mu \nu }{\tilde{g}}^{\mu \nu }}+\frac{L_{\mu \nu }{g}^{\mu \nu }}{{\mathcal{L}}_{\mu \nu }{\tilde{g}}^{\mu \nu }}\right]\sqrt{-g} d^4\rho d^4\mathsf{\sigma }$$

(3)

Applying the principle of stationary action in Ref. [6] yields

$$\frac{R_{\mu \nu }}{\mathcal{R}}-\frac{1}{2}\frac{R}{\mathcal{R}}{g}_{\mu \nu }-\frac{R{\mathcal{R}}_{\mu \nu }}{{\mathcal{R}}^2}+\frac{R\left({\mathcal{K}}_{\mu \nu }-\frac{1}{2}\mathcal{K}{\widehat{q}}_{\mu \nu }\right)-\mathcal{R}\left(K_{\mu \nu }-\frac{1}{2}K{\widehat{q}}_{\mu \nu }\right)}{{\mathcal{R}}^2}=\frac{{\widehat{T}}_{\mu \nu }}{{\mathcal{T}}_{\mu \nu }}$$

(4)

The interaction field equations can be interpreted as indicating that the cloud-world’s induced curvature, $R$, over the bulk’s conformal (background) curvature, $\mathcal{R}$, equals the ratio of the cloud-world’s imposed energy density and its flux, ${\widehat{T}}_{\mu \nu }$, to the bulk’s vacuum energy density and its flux, ${\mathcal{T}}_{\mu \nu }$, throughout the expanding/contracting Universe. The new 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 substituting to Equation (4), the interaction field equations can be simplified to

$$R_{\mu \nu }-\frac{1}{2}R{\widehat{g}}_{\mu \nu }-\left(K_{\mu \nu }-\frac{1}{2}K{\widehat{q}}_{\mu \nu }\right)=\frac{8\pi G_{\mathcal{R}}}{c^4}{\widehat{T}}_{\mu \nu }$$

(5)

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 ${\widehat{g}}_{\mu \nu }=g_{\mu \nu }+2{\tilde{g}}_{\mu \nu }-2{\stackrel{=}{g}}_{\mu \nu }$ because ${\mathcal{R}}_{\mu \nu }/\mathcal{R}={\mathcal{R}}_{\mu \nu }/{\mathcal{R}}_{\mu \nu }{\tilde{g}}^{\mu \nu }={\tilde{g}}_{\mu \nu }$, is the conformally transformed metric, which takes into account contributions from the cloud-world metric, $g_{\mu \nu },$as well as the intrinsic and extrinsic curvatures of the bulk based on its metrics, ${\tilde{g}}_{\mu \nu }$ and ${\stackrel{=}{g}}_{\mu \nu }$ respectively, whereas Einstein spaces are a subclass of the conformal space [7]. ${\widehat{T}}_{\mu \nu }$ is a conformal stress-energy tensor which also includes the electromagnetic energy flux from the boundary of the cloud-world over the conformal time. The interaction field equations could eliminate the singularities and satisfy a conformal invariance theory. From Equations (5) and (1), the Newtonian gravitational parameter is

$$G_{\mathcal{R} }=\frac{c^4}{8\pi E_D}\mathcal{R}$$

(6)

where $\mathcal{R}={\mathcal{R}}_{\mu \nu }{\tilde{g}}^{\mu \nu }$ is the scalar curvature of the bulk. According to Equation (6), $G_{\mathcal{R}}$ is proportional to $\mathcal{R}$ and reflects the field strength of vacuum energy because any changes in the bulk’s metric, ${\tilde{g}}_{\mu \nu } :=\mathcal{R}$, changes the field strength of the bulk, ${\mathcal{F}}_{\lambda \rho }$, owing of the constant bulk modulus, $E_D=-{\mathcal{F}}_{\lambda \rho }{\tilde{g}}^{\lambda \gamma }{\mathcal{F}}_{\gamma \alpha }{\tilde{g}}^{\rho \alpha }/4{\mu }_0$.

Regarding the Earth, the curvature of its background, the curved bulk owing to the Sun’s presence is shown in Figure 1. In this curved background, both Earth and Moon are further inducing different curvature configurations depending on their positions. For instance, at Point A, the Earth’s background curvature is influenced by the Moon’s position as shown by the blue and red-dotted curves. As the background curvature has different values at Point A, $G_{\mathcal{R}}$ is predicted to have different values according to Equation (6). In addition, other nearby planets can influence the background curvature configuration.

Figure 1 shows the curvature of the Earth’s background (blue curve), the curved bulk due to the Sun’s presence. In this curved background, the Moon is inducing different curvature configurations depending on its position.

Figure 2 shows six of $G_{\mathcal{R}}$ values by measurements: BIPM-14 [8], BIPM-01 [9], UCI-14 [10], UZur-06 [11], JILA-10 [12] and HUST-05 [13]. These values were among those adopted in the CODATA (Committee on Data for Science and Technology) 2014 recommended value of $\left(6.67408\pm 0.00031\right)\times {10}^{-11}{ \mathrm{m}}^3 {\mathrm{kg}}^{-1}{ \mathrm{s}}^{-2}$ [14].

A one-way ANOVA test was performed on these measurements, resulting in an F-statistic of 302.089 and a p-value of 0.000, which indicates strong evidence against the null hypothesis. This signifies that there is a significant difference in the variances of these measurements. Despite the small relative uncertainty in the measurements, the significant differences in their outcomes that puzzled scientists [1] can be attributed to the differences in the bulk curvature at the time the measurements were conducted, as stated in Equation (6), which can occur owing to varied positions of the Moon and other nearby planets.

3. Emergence of Mass

Analogous to the constant bulk’s modulus, the curvature of the bulk, can be considered constant regarding quantum fields. Consequently, the action can be extended to

$$S={{\displaystyle \int }}_B^{ }\left[\frac{-{\mathcal{F}}_{\lambda \rho }{\tilde{g}}^{\lambda \gamma }{\mathcal{F}}_{\gamma \alpha }{\tilde{g}}^{\rho \alpha }}{4{\mu }_0}\right]\sqrt{-\tilde{g}}{{\displaystyle \int }}_C^{ }\left[\frac{R_{\mu \nu }{g}^{\mu \nu }}{{\mathcal{R}}_{\mu \nu }{\tilde{g}}^{\mu \nu }}\right]\sqrt{-g} {{\displaystyle \int }}_Q^{ }\left[\frac{p_{\mu }{p}_v{q}^{\mu \nu }}{{\pi }_{\mu }{\pi }_v{g}^{\mu \nu }}+\frac{L_{\alpha \beta }{q}^{\alpha \lambda }{L}_{\lambda \gamma }{q}^{\beta \gamma }}{2{\chi }_0{\mathcal{L}}_{\mu \nu }{g}^{\mu \nu }}\right]\sqrt{-q} {\vartheta }^2{d}^{12}\mathsf{\sigma }$$

(7)

where $L_{\alpha \beta }{L}^{\alpha \beta }/2{\chi }_0$ are the Lagrangian densities of two entangled quantum fields that are regarded as 4D relativistic quantum clouds of a metric $q_{\mu \nu }$ and four-momentum $p_{\mu }{p}^{\nu }$, respectively, ${\chi }_0$ is a proportionality constant and ${\vartheta }^2$ is a dimensional-hierarchy factor; while ${\pi }_{\mu }{\pi }^{\nu }$ are the four-momentum of the vacuum energy density of a Lagrangian density ${\mathcal{L}}_{\mu \nu }{g}^{\mu \nu }$. By applying the principle of stationary action, separating the two entangled quantum clouds and utilizing the dimensional analysis, give

$$p_{\mu }-\frac{1}{2}{p}^{\nu }{q}_{\mu \nu }-p^{\nu }{\tilde{q}}_{\mu \nu }-\left(J^{\mu }{A}_{\mu }-\frac{1}{2}{J}^{\mu }{A}^{\nu }{\zeta }_{\mu \nu }\right)+\frac{p^{\nu }}{{\pi }^{\nu }}\left({\mathcal{J}}^{\mu }{\mathcal{A}}_{\mu }-\frac{1}{2}{\mathcal{J}}^{\mu }{\mathcal{A}}^{\nu }{\varsigma }_{\mu \nu }\right)=\frac{\hslash G_{\mathcal{R}}}{2c^2{g}_R}{\mathcal{T}}_{\mu }$$

(8)

where $g_R$ is the gravitational field strength of the parent cloud-world and ${\mathcal{T}}_{\mu }$ is energy density and flux of the quantum cloud of a deformed configuration as shown in Figure 3.

As the gravitational field strength of the cloud-world of mass $M_{ }$ and at curvature radius $R$ is $g_R=M{G}_{\mathcal{R}}/R^2$, a plane wavefunction, $\psi =A{e}^{-i\left(\omega t-kx\right)}$, can be expressed by utilizing Equation (8) as $\psi =A{e}^{-i\left(R^2/2M{c}^2\right) T_{\mu }{x}^{\mu }}$, consequently, the quantized field equations are

$$i\hslash {\gamma }^{\mu }{\partial }_{\mu }\psi -\frac{1}{2}i\hslash {\gamma }^{\mu }{\partial }^{\nu }\left(q_{\mu \nu }+2{\tilde{q}}_{\mu \nu }\right)\psi -\left(J^{\mu }{A}_{\mu }-\frac{1}{2}{J}^{\mu }{A}^{\nu }{\zeta }_{\mu \nu }\right){\psi }^{ }+\left({\mathcal{J}}^{\mu }{\mathcal{A}}_{\mu }-\frac{1}{2}{\mathcal{J}}^{\mu }{\mathcal{A}}^{\nu }{\varsigma }_{\mu \nu }\right)i\hslash {\gamma }^{\mu }{\partial }^{\nu }\psi /{\pi }^{\nu }=\frac{1}{2}\frac{\hslash }{x^{\mu }}R{\partial }_R\psi$$

(9)

where ${\gamma }^{\mu }{\partial }^{\nu }\psi /{\pi }^{\nu }$ signifies the spin-spin correlation of conventional, ${\gamma }^{\mu }{\partial }^{\nu }\psi$, and vacuum energy fields, ${\pi }^{\nu }$. Although ${\pi }_{\mu }{\pi }^{\nu }$ of the two entangled fields signifying the momentum of vacuum energy density that could be of a total zero spin, ${\pi }^{\nu }$ signifies a single field of vacuum energy of a possible spin, which can be conjectured as an analogue of a part of the singlet Cooper pair of a total zero spin. Equation (9) reveals that the spin-spin correlation and the bulk’s boundary interactions based on its field strength, ${\mathcal{J}}^{\mu }{\mathcal{A}}_{\mu }$, contribute to the emergence of mass. Equation (9) reduces to quantum electrodynamics in Ref. [6] as

$$i\hslash {\gamma }^{\mu }{\partial }_{\mu }\psi -mc\psi =e{\gamma }^{\mu }{A}_{\mu }\psi$$

(10)

This resembles the Dirac equation and the interaction with the electromagnetic field.

4. Conclusions and Future Experiment Recommendations

To date, about three hundred experiments have attempted to determine the value of $G_{\mathcal{R}}$, with many of them being precision measurements. However, the significant inconsistencies in their outcomes have made it unfeasible to reach a consensus on an exact value, which puzzled scientists. The interaction field equations showed the dependency of $G_{\mathcal{R}}$ on the background curvature and the associated field strength of vacuum energy. In addition, the equations revealed that the boundary interactions of conventional and vacuum energy densities and their spin-spin correlations contribute to the emergence of mass.

To achieve consistent $G_{\mathcal{R}}$ measurements, it is necessary to consider the positions of the Moon and other nearby planets, as they can influence the curvature of the background. Future experiments should aim to address this issue of inconsistent $G_{\mathcal{R}}$ measurements by accounting for the influence of these celestial bodies. One simple approach could be to conduct measurements twice, with one set taken when the Moon is on the horizon and another set taken when it is on the opposite side of the Earth. Finally, to ensure higher consistency, the positions of nearby planets can be also considered as well as the influence of variations in the background curvature on the field strength of vacuum energy which can impact both the value of the experiment’s emergent masses and the distance between the masses owing to the dependency of the Planck length on the value of $G_{\mathcal{R}}$.