Plausible Sources of the Cosmological Constant and Associated Vacuum Properties: Implications for Newton’s G, MOND’s a₀, Vacuum Invariants, and the Tully–Fisher and Faber–Jackson Galactic Relations

1. Introduction

1.1. $\Lambda$CDM Cosmological Constant and QFT Vacuum Catastrophe

The $\Lambda$CDM paradigm has served as the cornerstone of modern cosmology for nearly three decades, offering a robust theoretical framework that aligns with diverse empirical data, including the large-scale clustering of galaxies and the temperature fluctuations observed in the cosmic microwave background (CMB) [1,2]. This model posits a flat spatial geometry governed primarily by cold dark matter (CDM) and a constant vacuum energy density ${\mathit{u}}_0$ characterized by the cosmological constant $\Lambda$ [3,4,5,6]. While measurements from the Planck mission have allowed for the determination of the six $\Lambda$CDM fundamental parameters with remarkable sub-percent accuracy [1], the nature of $\Lambda$ remains one of the most significant unresolved theoretical hurdles in contemporary physics [7,8]. This mystery is now commonly known as the “cosmological constant problem” (CCP).

Several attempts within standard quantum field theory (QFT) to determine $u_0$ by summing the zero-point energy of vacuum oscillations up to the Planck scale have yielded values of approximately ${10}^{115}-{10}^{122}$ GeV ${\mathrm{m}}^{-3}$. This result is about 120 orders of magnitude larger than the observed Planck-2018 value of about 3.3 GeV ${\mathrm{m}}^{-3}$ [1], a discrepancy famously called the “worst theoretical prediction in physics” (and often referred to as the “vacuum catastrophe”) [7,8,9,10]. This catastrophic mismatch continues to drive the search for a more fundamental description of the vacuum manifold.1 In this work, we have obtained a comparable vacuum energy at the Stoney scale and the Planck scale, where our “scaling error” permits a characterization of the assumptions that went wrong in the QFT calculations (see Note 6 and Section 5).

1.2. Cosmological Tensions, Evolving Vacuum Properties, and Varying-G Gravity

As empirical precision has improved over the past decades, the theoretical $\Lambda$CDM framework has encountered increasingly significant internal inconsistencies. In particular, three primary discrepancies (the $H_0$, $S_8$, and $f\sigma {}_8$ tensions [13,14,15]) have intensified to the point of challenging the model’s foundational assumptions. These persistent gaps suggest that our current cosmological paradigm may suffer from either unrecognized systematic biases across independent datasets or a fundamental omission of physics [15]—potentially requiring the modification of gravity at cosmological scales or the introduction of novel undetected fields.

Ultimately, the Hubble tension is the most significant of these challenges, which highlights a stark disagreement regarding the contemporary universal expansion rate $H_0$. Predictions of $H_0$ derived from early-universe physics, calibrated by the CMB using $\Lambda$CDM assumptions, reliably converge to a lower value of $H_0\simeq 67.4$ km ${\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ [1]. Conversely, direct observations of the late universe via the cosmic distance ladder (CDL) consistently indicate a significantly higher expansion rate of $H_0\simeq 73.0$ km ${\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ [15,16,17]. With the statistical significance of this disparity hovering around the $5\sigma$ threshold, the $H_0$ tension cannot be plausibly attributed to merely a statistical fluke [16].

In recent work [18,19,20,21], we were concerned with the major cosmological tensions and (seemingly unrelated) the role of universal constants and dimensionless coupling constants in modern systems of units. In Ref. [18], we addressed the various tensions as resulting from a systematic error ($\gamma =4$–8%) in CDL observations, and we described their resolution in the framework of a slightly modified $\Lambda$CDM model. However, the new “$\gamma \Lambda$CDM model” still inherits the conventional cosmological constant $\Lambda$ in its Friedmann equations, thus it is subject to the CCP inasmuch as the standard $\Lambda$CDM model.

At the same time, the analysis of universal constants and associated systems of units revealed the influence of vacuum constants to several other dimensionful composite constants and the role of dimensionless coupling constants in reformulating systems of units [19,20,21]. One particular composite constant2 [21] alluded to the role of previously neglected quantities characterizing vacuum properties,3 which however evolve in time. These are the vacuum capacitance $\mathcal{C}$ and vacuum inductance $\mathcal{L}$ given by

$$\begin{array}{ccc}\mathcal{C}& =& 4\pi {\epsilon }_0{L}_{\mathrm{S}},\\ \mathcal{L}& =& {\displaystyle \frac{{\mu }_0}{4\pi }}{L}_{\mathrm{S}},\end{array}$$

(1)

where ${\epsilon }_0$ is the vacuum permittivity, ${\mu }_0$ is the vacuum permeability, and $L_{\mathrm{S}}$ is the Stoney length4 [21], which is a small fraction of the Planck length $L_{\mathrm{P}}$ (in fact, $L_{\mathrm{S}}={\mathsf{\alpha }}_{\mathrm{w}}{L}_{\mathrm{P}}$, where ${\mathsf{\alpha }}_{\mathrm{w}}\simeq 0.0341$ is the weak coupling constant [20]).

Since ${\epsilon }_0$ and ${\mu }_0$ are lower limits in nature and $L_{\mathrm{S}}\ll L_{\mathrm{P}}$, the values of $\mathcal{C}$ and $\mathcal{L}$ also represent minima of the corresponding resistive properties set by the vacuum at the Stoney scale. But unlike ${\epsilon }_0$ and ${\mu }_0$, these electromagnetic (EM) quantities of the vacuum are scale-dependent and increase with size as the universe is expanding. At the Planck scale, these quantities already are ${\mathsf{\alpha }}_{\mathrm{w}}^{-1}\approx 30$ times larger.

As described in Section 3 below, the evolving $\mathcal{C}$ or $\mathcal{L}$ quantity or, more formally, their geometric mean (G-M) with dimensions of [time] (the “light-crossing time”) applied to a cosmological length scale $L\gg L_{\mathrm{S}}$, viz.

$$\sqrt{\mathcal{L}\mathcal{C}}=\sqrt{{\mu }_0{\epsilon }_0{L}^2}={\displaystyle \frac{L}{c}},$$

(2)

turns out to be instrumental in the determination of the present-day value of the cosmological constant from vacuum properties because any one of them helps define a scale-dependent acceleration.

But there is an alternative pathway that leads to the same outcome (Section 2): assuming that Newton’s G varies in space, we can attribute the dark energy to radial G-gradients ($dG/dr$) that strongly influence the source term in the Poisson equation at cosmological scales [24,25,26,27]. In the theory of varying-G gravity, the calculation of the dark energy density of the universe ${\mathit{u}}_0$ was merely set up but not carried out to completion because, unlike QFT, the dimensional analysis produced the right order of magnitude for ${\mathit{u}}_0$ (Appendix B in Ref. [27]), a satisfactory result at that time.

1.3. Outline

The above two pathways to the present-day dark energy density ${\mathit{u}}_0$ and the associated vacuum properties are the main subjects of this work. The remainder of the paper is organized as follows:

• In Section 2, we calculate ${\mathit{u}}_0$ in varying-G gravity [27].
• In Section 3, we calculate ${\mathit{u}}_0$ from constant and evolving properties of the vacuum [21].
• In Section 4, we compare the results to the corresponding $\Lambda$CDM-inferred Planck values.
• In Section 5, we discuss the vacuum’s invariant and scale-dependent properties, as well as its elasticity and kinematic response to deformations due to the presence of matter or radiation.

2. Dark Energy Density in Varying-$\mathbf{G}$ Gravity

Dimensional analysis shows that energy density (or pressure) has dimensions of field amplitude squared scaled by the corresponding coupling constant. Examples are $\left[{\epsilon }_0\right]{\left[E\right]}^2$ for an electric field E, ${\left[B\right]}^2/\left[{\mu }_0\right]$ for a magnetic field B, and ${\left[a\right]}^2/\left[G\right]$ for a gravitational field producing a typical acceleration a. The latter scaling is more convenient to use, and we do so in the calculations of Section 2 and Section 3. On the other hand, we discuss vacuum EM fields in Section 5.

In varying-G gravity [27], the characteristic acceleration is provided by MOND’s critical acceleration $a_0$ [28,29], and the dark energy density is not constant; in fact, ${\mathit{u}}_0\left(r\right)$ falls off as $r^{-2}$ at large distances r. Thus, we assume that the dark energy density is given by the expression

$${\mathit{u}}_0\left(r\right)={\displaystyle \frac{a_0^2}{G{}_0}}{\left({\displaystyle \frac{r_0}{r}}\right)}^2(r\ge L_{\mathrm{P}}),$$

(3)

where $G{}_0$ is the Newtonian gravitational constant and $r_0=c^2/a_0$ is the MOND length [29]—a scale longer by a factor of $2\times {10}^{61}$ than the Planck length $L_{\mathrm{P}}={\left(hG{}_0/c^3\right)}^{1/2}$ [20]. This profile overestimates the global averages obtained below and should be viewed as a rough approximation in the framework of varying-G gravity. Furthermore, the results do not change if the adopted lower bound $L_{\mathrm{P}}$ is replaced by the Stoney length $L_{\mathrm{S}}\ll L_{\mathrm{P}}$.

The “constant” dark energy density observed at present is obtained from an average over the post-Planckian history of the universe, which is nevertheless heavily weighted toward late times. So, we average ${\mathit{u}}_0\left(r\right)$ over a spherical shell volume $V_0-V_{\mathrm{P}}={\displaystyle \frac{4\pi }{3}}(r_0^3-L_{\mathrm{P}}^3)$ between the radii $r_1=L_{\mathrm{P}}$ and $r_2=r_0\gg L_{\mathrm{P}}$. In the limit of $r_1/r_2\to 0$, we get

$$\langle {\mathit{u}}_0\rangle ={\displaystyle \frac{1}{V_0}}{\int }_0^{r_0}{\mathit{u}}_0\left(r\right)4\pi r^{2}dr=3{\mathit{u}}_0\left(r_0\right)=3{\displaystyle \frac{a_0^2}{G{}_0}}.$$

(4)

Using $a_0=1.112650\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$ and $G{}_0=6.674015\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$ [20], we obtain

$$\langle {\mathit{u}}_0\rangle =3.47\mathrm{GeV}{\mathrm{m}}^{-3},$$

(5)

and an effective mass density of $\langle {\rho }_0\rangle =\langle {\mathit{u}}_0\rangle /c^2=6.19\times {10}^{-27}\mathrm{kg}{\mathrm{m}}^{-3}$.

3. Dark Energy Density from Vacuum Properties

In Newtonian and FLRW cosmologies, Newton’s $G\equiv G{}_0$ is constant in space and time, and there are no G-gradients to mimic an outward pushing dark energy. The formulation of the cosmological constant shifts entirely to vacuum properties, which however do not include a priori a characteristic acceleration scale a or length scale L. An independent scale must then be adopted for the determination of ${\mathit{u}}_0$. Choosing $a=a_0$, as in Section 2, leads to a palatable result, whereas choosing $L=L_{\mathrm{P}}$ results in the vacuum catastrophe [7,8,9,10].

Vacuum Properties. The fundamental set of vacuum constants $\{{\epsilon }_0,{\mu }_0,c,Z_0\}$ [19,30] cannot produce an acceleration because of the dimensional structure of the resistive properties, viz.

$$\left[{\mu }_0\right]={\displaystyle \frac{\left[F\right]}{{\left[I\right]}^2}}\mathrm{and}\left[{\epsilon }_0\right]={\displaystyle \frac{{\left[I\right]}^2}{\left[F\right]}}{\displaystyle \frac{1}{{\left[\right]}^2}},$$

where $F=Ma$ is force, I is electric current, and is velocity:

• When I is eliminated between the two relations, then F is eliminated too, and the resulting wave speed cannot define a typical acceleration without a supplementary length L or time T.
• When F is retained, the resulting acceleration is given by $a=Z_0{I}^2/p$, where $Z_0{I}^2$ has dimensions of [power] and $p=M$ represents momentum. But there is presently no bridge to connect charge flow to mechanical momentum in empty space, so this relation is also unable to produce an acceleration scale.

The above impasse can be overcome by including one of the evolving resistive properties of the vacuum: $\mathcal{L}$ or $\mathcal{C}$ introduces a length scale L, and their G-M famously specifies a relaxation timescale $\mathsf{\tau }$ [31]. Below we use the most elegant of the three equivalent descriptions, the scale-dependent G-M $\sqrt{\mathcal{L}\mathcal{C}}$ of vacuum capacitance and vacuum inductance.

Vacuum Dark Energy Density. Using equation (2) for an arbitrary length scale $L\ge L_{\mathrm{S}}$ of the expanding vacuum, we determine first an acceleration scale $a\left(L\right)$, viz.

$$a\left(L\right)={\displaystyle \frac{c}{\sqrt{\mathcal{L}\mathcal{C}}}}={\displaystyle \frac{c^2}{L}},$$

(6)

and then a dark energy density scale ${\mathit{u}}_0\left(L\right)$, viz.

$${\mathit{u}}_0\left(L\right)={\displaystyle \frac{a^2\left(L\right)}{G{}_0}}={\displaystyle \frac{c^4}{G{}_0{L}^2}}={\displaystyle \frac{F_{\mathrm{P}}}{L^2}}(L\ge L_{\mathrm{S}}),$$

(7)

where we have introduced the Planck unit of force $F_{\mathrm{P}}=c^4/G{}_0$ [19] in the last step.5

The interpretation of equation (7) is straighforward: the dark energy content of the universe starts out with enormous values at Stoney/Planck scales, where $L\sim (1$–$30)L_{\mathrm{S}}$, and gradually drops to its present-day value.6 At present, the characteristic length scale L must then be identified with the comoving particle horizon distance $D{}_{\mathrm{ph}}$ [32,33,34].

The particle horizon distance $D{}_{\mathrm{ph}}$ is obtained from the equation

$$D{}_{\mathrm{ph}}={\displaystyle \frac{c}{H_0}}{\int }_0^{\infty }{\displaystyle \frac{dz}{\mathcal{E}\left(z\right)}},$$

(8)

where the normalized Hubble expansion rate $\mathcal{E}\left(z\right)$ is a function of redshift z, viz.

$$\mathcal{E}\left(z\right)=\sqrt{{\Omega }_{\mathrm{m}}{(1+z)}^3+{\Omega }_{\mathrm{r}}{(1+z)}^4+{\Omega }_{\Lambda }},$$

(9)

and the $\Omega$-parameters (for matter, radiation, and the $\Lambda$-field) are provided by Planck-2018 results.

Using Planck-2018 results [1], the dimensionless integral in equation (8) sums up to 3.1807, and the particle horizon distance then is

$$D{}_{\mathrm{ph}}=3.1807\left(c/H_0\right)=4.3681\times {10}^{26}\mathrm{m}=14.156\mathrm{Gpc}.$$

(10)

Finally, equation (7) for $L=D{}_{\mathrm{ph}}$ and $F_{\mathrm{P}}=1.2103\times {10}^{44}$ N [19,20] gives

$${\mathit{u}}_0\left(D{}_{\mathrm{ph}}\right)={\displaystyle \frac{F_{\mathrm{P}}}{{\left(D{}_{\mathrm{ph}}\right)}^2}}=3.96\mathrm{GeV}{\mathrm{m}}^{-3},$$

(11)

which corresponds to an effective mass density of ${\rho }_0={\mathit{u}}_0/c^2=7.06\times {10}^{-27}\mathrm{kg}{\mathrm{m}}^{-3}$.

Vacuum Acceleration. In this framework, the evolving acceleration scale of the vacuum is also physically interesting. Equation (6) for $L=D{}_{\mathrm{ph}}$ specifies a present-day (cosmological) scale of

$$a\left(D{}_{\mathrm{ph}}\right)=2.0575\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}=1.849a_0.$$

Contrary to unsubstantiated claims in the literature (viz. $c{H}_0/\left(2\pi \right)\sim a_0$, that depends crucially on using the 2D geometric tag of $2\pi$ in a 3D kinematic setting), the above comparison indicates that MOND’s critical acceleration is not cosmological in nature (see also Ref. [20]). The perception of equivalence between the two scales will have to be postponed for another 7.74 Gyr, until our particle horizon distance expands out to the formidable MOND length [29]$r_0=c^2/a_0=1.849D{}_{\mathrm{ph}}$ ($\simeq 26$ Gpc).

In the meantime, $a_0$ should be considered as an additional acceleration scale introduced in modified dynamics, yet also relevant to other vacuum properties because of the localized stress $M{a}_0$ exerted by mass M against the structural integrity of the vacuum manifold. To this end, the connection of $a_0$ to vacuum elasticity is discussed in Section 5.2 below (see “Vacuum Stress and Kinematic Yield”).

Vacuum Gravitational Constant. It is rather evident that the dark energy cannot be assessed without an explicit reference to the Newtonian gravitational constant $G{}_0$, or the gradient $dG/dr$ in varying-G gravity [27]. In varying-G gravity, the minimal set for the determination of ${\mathit{u}}_0$ is $\{a_0,G{}_0\}$, whereas in a constant-G vacuum, the minimal set is $\{{\epsilon }_0,{\mu }_0,G{}_0,\mathsf{\tau }\left(L\right)\}$, where $\mathsf{\tau }\left(L\right)$ is the evolving light-crossing time over a comoving distance L (equation (2)).

The inclusion of $G{}_0$ in the above parametric sets marks the first time that Newton’s gravitational constant is treated as an intrinsic property of the vacuum manifold. The intrinsic properties of the vacuum, including $G{}_0$, are summarized and discussed in detail in Section 5 below. Here, we only review how the gravitational constant may contribute to vacuum properties in the complete absence of interacting matter. In such a setting, we associate $G{}_0$ only with fundamental vacuum invariants, avoiding any throwbacks to Newton’s gravitational law or the gravitational coupling constant ${\mathsf{\alpha }}_{\mathrm{g}}=G{}_0{m}_{\mathrm{e}}^2/\left(hc\right)$ [19], where $m_{\mathrm{e}}$ is the electron mass and h is Planck’s constant.

In this framework, we revisit two composite constants and their two G-Ms constructed from the fundamental set of resisting properties $\{{\epsilon }_0,{\mu }_0,G{}_0\}$ [20,21]:

1.

The gravoelectric constant $G{}_{★}=4\pi {\epsilon }_{0}G{}_0=7.4258\times {10}^{-21}{\mathrm{C}}^2{\mathrm{kg}}^{-2}$ has dimensions ${[Q/M]}^2$, where Q and M represent charge and mass, respectively. Thus, $G{}_{★}$ serves as a bridge between charge and mass, applicable to electrostatic settings.

2.

The gravomagnetic constant $G{}_B=G{}_0{\mu }_0/\left(4\pi \right)=6.6740\times {10}^{-18}{\mathrm{m}}^4{\mathrm{s}}^{-4}{\mathrm{A}}^{-2}$ has dimensions ${[\Phi {}_B/M]}^2$, where $\Phi {}_B$ represents magnetic flux. Thus, $G{}_B$ serves as a bridge between mass and magnetic flux, applicable to EM transport phenomena. However, this vacuum property has profound unforeseen repercussions, as discussed in depth in Section 5.2.

3.

The first G-M, $\sqrt{G{}_{★}G{}_B}=G{}_0/c$, is a composite invariant which is also a lower limit in vacuum. It implies that the gravitational coupling constant $\mathsf{\kappa }=8\pi G{}_0/c^4$ in the Einstein field equations attains a minimum value; thus, Einstein’s ubiquitous coupling of spacetime curvature to the stress-energy tensor is strictly minimal (and independent of the amount of mass present) [8,9,10]. Furthermore, $G{}_0/c$ has dimensions of [velocity]${}^4$[${\mathrm{power]}}^{-1}$, a kinematic scaling that turns out to play an important role in vacuum elasticity in response to EM radiative stresses (Section 5.2).

4.

The second G-M, $\sqrt{G{}_{★}^{-1}G{}_B}={\mathcal{R}}_{\mathrm{P}}$, defaults to an EM vacuum threshold entirely unrelated to $G{}_0$. This behavior has previously appeared in direct comparisons between purely EM Planck units; i.e., in the dual relations ${\mathcal{V}}_{\mathrm{P}}/{\mathcal{I}}_{\mathrm{P}}={\mathcal{R}}_{\mathrm{P}}$ (voltage–current), ${\Phi }_{\mathrm{P}}/Q_{\mathrm{P}}={\mathcal{R}}_{\mathrm{P}}$ (magnetic flux–charge), and $\sqrt{{\mathcal{L}}_{\mathrm{P}}/{\mathcal{C}}_{\mathrm{P}}}={\mathcal{R}}_{\mathrm{P}}$ (inductance–capacitance).

4. Comparisons with Planck–$\Lambda$CDM Results

Table 1 shows a comparison between the two estimates of the dark energy density and the corresponding Planck results over the past ten years, including the older Planck-2015 release [1,36].

Compared to the more recent Planck-2018 results (row 1), the largest upward deviation ($\sim$21%) occurs in the determination of the dark energy from vacuum properties (row 4), whereas varying-G gravity values (row 3) lie higher by only $\sim$6%.

The two independent estimates in rows 3 and 4 differ by $\sim$13%. Given the uncertainties and the approximations involved in these determinations, such differences are not sufficient to distinguish between the two approaches. It is however encouraging that both calculations yield the right order of magnitude, potentially resolving thus the CCP; so, in principle, either one of them could be highlighting the source of the cosmological constant (gradient $dG\left(r\right)/dr$ or vacuum set $\{c,G{}_0,L\}$).

On the other hand, this outcome also lends support to varying-G gravity as a viable alternative to constant-G gravitational theories (in particular, Newtonian dynamics, General Relativity (GR), and FLRW cosmologies);7 especially since G-gradients as a source of the cosmological constant appear to fare a little better in comparisons with the Planck results (rows 1–3 in Table 1).

5. The Barely Elastic Vacuum Manifold: Discussion and Conclusions

5.1. Vacuum Properties

The Evolving Vacuum. The calculation of the dark energy density ${\mathit{u}}_0$ carried out in Section 3 is the only one that used vacuum properties and did not miss by 50–120 orders of magnitude [7,8,9,42,43,44]; in fact, this scaling did not miss the order of magnitude at all. The approach was based on a new concept, the evolving properties of the vacuum, that adjust and change in magnitude with the expansion of the universe. Ref. [21] provided the first such evolving quantities, the vacuum capacitance and vacuum inductance (equation (1)), whereas their G-M yielded a prosaic kinematic timescale, the light-crossing time specified by equation (2).

All of these varying quantities start out as minimum resistive properties at the Stoney scale and attain maximum evolving values in the present universe. Their lower limits yield an enormous value for ${\mathit{u}}_0$ (relevant only to the tiny Stoney scale; Note 6), whereas their present-day values yield a reasonable estimate for ${\mathit{u}}_0$ (Section 3). These considerations clearly resolve the problem of the vacuum catastrophe [9] and explain the inability of QFT to obtain the present-day ${\mathit{u}}_0$ value by considering fluctuations up to the Planck scale [42,44]: bulk quantum fluctuations are negligibly small, perhaps even identically zero [11], in the present-day universe.

EM-Field Support. Under these circumstances, we should also consider the vacuum’s ability to support emerging EM fields and their fluctuating sources. Unlike in some unjustifiable man-made definitions of physical constants [19,20,21], the well-known vacuum constants $\{{\epsilon }_0,{\mu }_0,Z_0\}$ consistently carry a 3D geometric tag of $4\pi$ in the following scaling relations:

1.

Magnetic field B.—Letting ${\mathit{u}}_0\to B^2(4\pi /{\mu }_0)$ in equation (7), we obtain $B=\sqrt{F_{\mathrm{P}}{\mu }_0/\left(4\pi \right)}/L$ and $B=8.0\times {10}^{-9}$ T for $L=D{}_{\mathrm{ph}}$. Thus, if ${\mathit{u}}_0$ is assumed to be magnetic pressure, the fluctuating B-field starts out with enormous magnitudes at Planck/Stoney scales and drops immensely at late times.

2.

Electric field E.—Since $E=cB$, it is expected that the evolution of a fluctuating E-field will track closely that of the magnetic field. In this case, we obtain $E=2.4$ V ${\mathrm{m}}^{-1}$ for $L=D{}_{\mathrm{ph}}$, clearly not a substantial field magnitude.

3.

Charge $Q_{\mathrm{vac}}$.—Using Gauss’s law, we find that $Q_{\mathrm{vac}}=\mathcal{C}\sqrt{F_{\mathrm{P}}/\left(4\pi {\epsilon }_0\right)}$, so the evolution of charge in the vacuum follows closely that of the vacuum capacitance $\mathcal{C}\left(L\right)$. For $L=L_{\mathrm{S}}$, the charge $Q_{\mathrm{vac}}=e$ only, and it increases to $Q_{\mathrm{vac}}=e/{\mathsf{\alpha }}_{\mathrm{w}}\approx 30e$ at the Planck scale.8 Thus, some enormous charge densities ($\rho {}_e\sim$10${}^{85-88}\mathrm{C}{\mathrm{m}}^{-3}$) appear at these scales by tiny amounts of charge enclosed within much tinier spherical volumes.

4.

Current $I_{\mathrm{vac}}$.—Using the vacuum’s evolving G-M timescale $\mathsf{\tau }=\sqrt{\mathcal{L}\mathcal{C}}$ (Section 3), we define a current scale by $I_{\mathrm{vac}}=Q_{\mathrm{vac}}/\mathsf{\tau }$, and we obtain

$$I_{\mathrm{vac}}={\displaystyle \frac{1}{{\mathcal{R}}_{\mathrm{P}}}}\sqrt{{\displaystyle \frac{F_{\mathrm{P}}}{4\pi {\epsilon }_0}}}\equiv {\mathcal{I}}_{\mathrm{P}}\simeq 3.479\times {10}^{25}\mathrm{A},$$

(12)

where ${\mathcal{R}}_{\mathrm{P}}$ and ${\mathcal{I}}_{\mathrm{P}}$ are the Planck resistance and current, respectively. Quite unexpectedly, this enormous vacuum quantity turns out to be a universal invariant and implies an enormous invariant voltage as well: $V{}_{\mathrm{vac}}=\sqrt{F_{\mathrm{P}}/\left(4\pi {\epsilon }_0\right)}\equiv {\mathcal{V}}_{\mathrm{P}}$, where ${\mathcal{V}}_{\mathrm{P}}=1.043\times {10}^{27}$ V is the Planck voltage [20].9^,10

In conclusion, it appears that the vacuum cannot generate significant charge fluctuations at the Planck and Stoney scales despite its enormous capacity for supporting currents, voltages, and EM fields—and the absence of charge pairs in significant numbers at such very early epochs renders that enormous underlying EM capacity practically null. Therefore, the invariant vacuum properties are physically important universal constants (item 4), whereas the enormous densities produced by tiny sources in even tinier volumes (such as $\rho {}_e$ in item 3) have no practical meaning.

Vacuum Invariants. The new EM invariants advance our understanding of vacuum properties considerably; especially those properties that do not at all depend on Planck’s constant h (Notes 9 and 10). Constant and evolving properties are generated from a minuscule fundamental set of only two invariants, $\{4\pi {\epsilon }_0,{\mu }_0/\left(4\pi \right)\}$, supplemented by Newton’s $G{}_0$ or a length L, respectively.

With help from units of the reformulated Planck system (RPS) [20], some of which turn out to describe vacuum invariants, Table 2 summarizes fundamental and derived vacuum properties. It is rather evident, yet crucially important, that there is no bridge in any of the tabulated sets of quantities which would produce an energy or a mass scale. The main obstacle is that power is defined only in Set 2 that includes $G{}_0$, whereas time is defined only in Set 3 that includes a length scale L.

Vacuum Resistance $\mathbf{G}{}_{\mathbf{0}}$. The inability to define mass in any of the sets of Table 2 strengthens the argument that Newton’s $G{}_0$ may be a vacuum resistance irrespective of mass, as featured in Set 2 of the table. In constant-G gravity, $G{}_0$ is considered to be an invariant, whereas in varying-G gravity, $G\left(L\right)$ is assumed to be an evolving property. In either case, the origin of the gravitational constant remains open-ended.11

Another hint comes from the dimensionless (man-made) constants, the fine-structure constant (FSC) $\mathsf{\alpha }={\left(4\pi {\epsilon }_0\right)}^{-1}{e}^2/\left(hc\right)$ and the gravitational coupling constant ${\mathsf{\alpha }}_{\mathrm{g}}=G{}_0{m}_{\mathrm{e}}^2/\left(hc\right)$ [19]. Besides the enlisted electronic properties e (charge) and $m_{\mathrm{e}}$ (mass), the other parameters are apparently associated with vacuum properties, so that $G{}_0$ may not be an exception after all. The vacuum’s quantum stiffness (represented here by $hc$) has been previously discussed in various contexts [51,52,53,54,55], all related to the CCP and the scaling of quantum fluctuations in the expanding vacuum. But we also see that ${\left(4\pi {\epsilon }_0\right)}^{-1}$ mediates $e^2$ and, quite in line, $G{}_0$ mediates $m_{\mathrm{e}}^2$. The correspondence is complete only if the gravitational constant is also a vacuum property; otherwise, ${\mathsf{\alpha }}_{\mathrm{g}}$ is oddly more complex and packs more information than the FSC, an indefensible position in our opinion.

Naturally, the above interpretation is open to debate. For instance, G. E. Volonik [54] writes that gravity is an emergent property of the vacuum at low energies, and that the dark energy background does not gravitate. This could be interpreted in two different ways: (a) Particle excitations and $G{}_0$ emerge together at some energy threshold; or (b) $G{}_0$ has always existed, but it could not mediate vacuum energy in the absence of matter. We argue in favor of option (b) because this is what we read in the coupling constants $\mathsf{\alpha }$ and ${\mathsf{\alpha }}_{\mathrm{g}}$—vacuum invariant factors of ${\left(4\pi {\epsilon }_0\right)}^{-1}$ and $G{}_0$ each coupling to a tangible property of the electron, respectively, where they effectively mediate the corresponding field interactions (see also Refs. [50,51,55,56] for related investigations). Furthermore, we show below that, irrespective of the presence or absence of matter, $G{}_0$ is the singular coupling constant that mediates all types of stress forces responsible for vacuum deformations (curvature strain).

5.2. Vacuum Strain Under Stress

Vacuum Stress and Kinematic Yield. The gravomagnetic constant $G{}_B$ with dimensions ${[\Phi {}_B/M]}^2$ was presented in Section 3 as a bridge between mass and magnetic flux. Surprisingly, this constant is also key to understanding (a) the roles of $G{}_0$ and $a_0$ as vacuum constants, (b) the origin of the Tully–Fisher (TF) [57,58,59] and Faber–Jackson (FJ) [60,61,62] galactic relations, and (c) the vacuum’s kinematic yield in response to material and/or radiative stresses.

The dimensions of $G{}_B=G{}_0{\mu }_0/\left(4\pi \right)$ can be recast in the equivalent form ${\left[\right]}^4/{\left[I\right]}^2$, where I represents electric current [21]. This form leads to the dimensional relation ${}^4=G{}_B{I}^2$. In a vacuum devoid of matter, we assume the presence of EM radiation with power $\mathcal{P}{}_{\mathrm{em}}$ such that $\mathcal{P}{}_{\mathrm{em}}=I^2{\mathcal{R}}_{\mathrm{P}}$. Eliminating $I^2$ between the two equations, we find that

$$v^4=G{}_0\left({\displaystyle \frac{\mathcal{P}{}_{\mathrm{em}}}{c}}\right),$$

(13)

where the term in parentheses has dimensions of [force] (i.e., $F{}_{\mathrm{vac}}=\mathcal{P}{}_{\mathrm{em}}/c$) and leads to a generalized stress-strain relation at each point of the vacuum manifold, viz.

$$v^4=G{}_{0}F{}_{\mathrm{vac}},$$

(14)

a dead ringer for the famous TF/FJ relations of galactic dynamics, in which $F{}_{\mathrm{vac}}=M{a}_0$ [27,28,29]. The same “stress–strain” elastic relation has also been obtained in the absence of matter when the force comes from an electrostatic potential difference or voltage $\mathcal{V}$, in which case $F{}_{\mathrm{vac}}=\left(4\pi {\epsilon }_0\right){\mathcal{V}}^2$ [20]. Various related stress–strain cases are summarized in Table 3 and discussed below.

In Equation (14), the force $F{}_{\mathrm{vac}}$ represents the localized stress exerted against the vacuum, while ${}^4$ represents the kinematic response of the metric to local curvature change. The coupling constant is $G{}_0$, even in the absence of gravitational interactions (pure EM case; equation (13)), which again signifies a vacuum origin of the gravitational constant. The vacuum’s leverage is also evident in the definitions of $F{}_{\mathrm{vac}}$, which involve scalings by $a_0$, $1/c$, and $4\pi {\epsilon }_0$ (Table 3). Furthermore, in all cases, $\ll c$ and can be thought as the intrinsic propagation or relaxation velocity at which local structural adjustments of the metric proceed in response to the applied (matter, voltage, or EM) stress field.

A small velocity $\Delta \ll c$ whose vacuum kinematics scales as ${\left(\Delta \right)}^4$ can also be obtained from a scalar form of the Einstein field equations, $R=(8\pi G{}_0/c^4)P$, where R is geometric scalar curvature and P is an effective isotropic pressure representing the trace of the stress–energy tensor. To map vacuum strain to local kinematics, we relate curvature to an effective acceleration $\Delta a$ associated with the local metric deformation, viz. $R={\left(\Delta a\right)}^2/c^4$, and we recast pressure as $P=F{}_{\mathrm{vac}}/\left(4\pi r^2\right)$, where $F{}_{\mathrm{vac}}$ is the effective force exerted over a spherical surface of radius r. Then, we find that

$${\left(\Delta a\right)}^2={\displaystyle \frac{2G{}_{0}F{}_{\mathrm{vac}}}{r^2}},$$

(15)

and, substituting $\Delta a={\left(\Delta \right)}^2/r$, we finally obtain a stress–strain elastic relation of the form (14) (Table 3), viz.

$${\left(\Delta \right)}^4=2G_{0}F{}_{\mathrm{vac}}.$$

(16)

The extra factor of 2 is a signature of GR, likely analogous to that factor of 2 separating the relativistic and Newtonian predictions for the gravitational deflection of light [63,64], and the one appearing in the horizon radius $R_{\mathrm{S}}=2G_{0}M/c^2$ of the Schwarzschild metric [33].

Vacuum Stress–Strain Law. The coupling between vacuum stress and kinematic strain in the above equations admits a unified, field-theoretic formulation. First, by recognizing that a continuous EM radiant flux of energy E through a surface satisfies $\mathcal{P}{}_{\mathrm{em}}=dE/dt=c(dE/dr)$, the radiative stress is reduced to a radial energy gradient. Second, to encompass a static mass defect M, where $M{a}_0$ acts as an equivalent radial energy variation required to sustain the metric deformation, we adopt the relation $M{a}_0=dE/dr$. Substituting these stress terms into the $v^4$ stress–strain relation yields the generalized local expression

$$v^4=G{}_0\left({\displaystyle \frac{dE}{dr}}\right).$$

(17)

In a field theory, this isotropic localized stress must be distributed throughout the surrounding vacuum manifold. We define ${\rho }_{\mathrm{s}}$ as the stress density (with dimensions of [force]/[volume]), and $■$ as the vacuum stress flux vector field (with dimensions of [pressure]), such that

$${\rho }_{\mathrm{s}}=\nabla ·■.$$

(18)

Integrating this density over a bounded spatial volume V and applying the divergence theorem, the total stress force is expressed as the flux of $■$ through the closed boundary surface $\partial V$, viz.

$${\int }_V{\rho }_{\mathrm{s}}dV={\oint }_{\partial V}■·d\mathbf{A}\left(\mathrm{total}\mathrm{stress}\mathrm{force}\right).$$

(19)

Thus, the localized expression (17) generalizes to a global field equation of the 2D integral form

$$v^4=G{}_0{\oint }_{\partial V}■·d\mathbf{A}(\mathrm{vacuum}\mathrm{stress}--\mathrm{strain}\mathrm{law}),$$

(20)

where the stress vector is $■\equiv (dE/dV)\widehat{\mathbf{n}}$, and $\widehat{\mathbf{n}}$ is the outward normal to the boundary $\partial V$.

Equation (20) has the form of Gauss’s law [25], although it works backwards: the source of stress mediated by $G{}_0$ on the right-hand side gives rise to a kinematic response of the vacuum fabric described by ${}^4$ on the left-hand side. The effective Young’s modulus $Y{}_{\mathrm{vac}}$ of this constitutive relation [65,66] then is

$$Y{}_{\mathrm{vac}}\propto {\displaystyle \frac{1}{G{}_0}}\propto F_{\mathrm{P}},$$

(21)

a remarkable result concerning vacuum elasticity. Evidently, Newton’s constant $G{}_0$ is exceedingly small because the vacuum’s resistance against deformation is immense (and vice versa).

The surface integral in equation (20) represents the vacuum force $F{}_{\mathrm{vac}}$ over the boundary $\partial V$, and it is the field-theoretic analogue of the vacuum force in equation (14) above. These equations can be recast in a dimensionless form that readily provides additional quantitative information: Using the shorthand $F{}_{\mathrm{vac}}$ for the surface integral, dividing both sides by $c^4$, and introducing the Planck force $F_{\mathrm{P}}=c^4/G{}_0$ into the resulting proportion, we obtain the elegant relation

$${\displaystyle \frac{F{}_{\mathrm{vac}}}{F_{\mathrm{P}}}}={\left({\displaystyle \frac{v}{c}}\right)}^4.$$

(22)

This proportion signifies that the dimensionless vacuum strain in response to the exerted stress is a fourth-order effect in the kinematic ratio $/c\ll 1$, leading again to the conclusion that the classical vacuum behaves as a remarkably stiff elastic medium.12^,13^,14 Related works and conclusions on vacuum rigidity can be found in Refs. [51,55,71,72,73,74] (although equation (22) has not been heretofore explicitly considered).

Spherical Symmetry. For a spherically symmetric energy distribution, $\mathit{u}\left(r\right)=dE/dV$ and the vacuum stress flux field is radial with outward-pointing unit vector $\widehat{\mathbf{r}}$, viz.

$$■\left(r\right)=\mathit{u}\left(r\right)\widehat{\mathbf{r}}={\displaystyle \frac{1}{4\pi r^2}}\left({\displaystyle \frac{dE}{dr}}\right)\widehat{\mathbf{r}};$$

(23)

then, the stress density (18) turns out to be

$${\rho }_{\mathrm{s}}\left(r\right)={\displaystyle \frac{1}{r^2}}{\displaystyle \frac{d}{dr}}\left(r^{2}u\left(r\right)\right)={\displaystyle \frac{1}{4\pi r^2}}\left({\displaystyle \frac{d^{2}E}{d{r}^2}}\right).$$

(24)

In spherical symmetry, we can distinguish the types of vacuum fields and their source densities in the presence of matter or radiation15^,16 by reducing equations (23) and (24) as follows:

• Matter present and the TF/FJ relations.—Substituting the static mass defect relation $dE/dr=M\left(r\right)a_0$, the stress flux vector $■\left(r\right)$ becomes proportional to the surface mass density $\sigma \left(r\right)\equiv M\left(r\right)/\left(4\pi r^2\right)$ [27], viz.

  $$■\left(r\right)=a_0\sigma \left(r\right)\widehat{\mathbf{r}},$$

  (25)

  and the stress–strain law (20) takes the TF/FJ form

  $$v^4={\mathcal{A}}_{0}M\left(r\right),$$

  (26)

  where ${\mathcal{A}}_0\equiv a_{0}G{}_0$ is MOND’s universal constant and $M\left(r\right)$ is the enclosed mass. The divergence of $■\left(r\right)$ reveals that the vacuum stress density ${\rho }_{\mathrm{s}}\left(r\right)$ is directly proportional to the localized volumetric mass density ${\rho }_{\mathrm{m}}\left(r\right)$, yielding the localized source relation

  $${\rho }_{\mathrm{s}}\left(r\right)=a_0{\rho }_{\mathrm{m}}\left(r\right).$$

  (27)
• Matter absent and a purely EM vacuum stress.—Substituting the radiative power relation $dE/dr={\mathcal{P}}_{\mathrm{em}}\left(r\right)/c$, the stress flux $■\left(r\right)$ becomes EM momentum flux $\mathbf{S}\left(r\right)/c$, viz.

  $$■\left(r\right)={\displaystyle \frac{\mathbf{S}\left(r\right)}{c}},$$

  (28)

  and the stress–strain law (20) takes the form

  $$v^4=\left({\displaystyle \frac{G{}_0}{c}}\right)\mathcal{P}{}_{\mathrm{em}}\left(r\right),$$

  (29)

  where $\mathcal{P}{}_{\mathrm{em}}\left(r\right)=\left(4\pi r^2\right)\left|\mathbf{S}\left(r\right)\right|$ and the Poynting vector $\mathbf{S}$ due to vacuum EM fields $\mathbf{E}$ and $\mathbf{B}$ is defined by $\mathbf{S}\equiv (\mathbf{E}\times \mathbf{B})/{\mu }_0$ in SI units [31]—or $\mathbf{S}\equiv {\displaystyle \frac{c}{4\pi }}(\mathbf{E}\times \mathbf{B})$ in the commonly used Gaussian–CGS units [30]. Then, in the absence of matter, the stress density ${\rho }_{\mathrm{s}}\left(r\right)$ corresponds to the localized temporal variation of the EM energy density ${\mathit{u}}_{\mathrm{em}}\left(r\right)$, viz.

  $${\rho }_{\mathrm{s}}\left(r\right)={\displaystyle \frac{1}{c}}\nabla -0.25ex·\mathbf{S}\left(r\right)=-{\displaystyle \frac{1}{c}}\left({\displaystyle \frac{\partial {\mathit{u}}_{\mathrm{em}}}{\partial t}}\right),$$

  (30)

  implying that a localized vacuum stress density is generated by the dynamical time-dependent accumulation or depletion of field energy within the vacuum fabric. In the final step, Poynting’s theorem was applied to a pure vacuum devoid of matter, where there are no charge carriers to sustain a conduction current; thus, the localized current density vanishes identically ($\mathbf{J}\equiv \mathbf{0}$). Consequently, the Joule heating (or mechanical work) term $\mathbf{J}-0.25ex·\mathbf{E}$ drops out of the energy conservation equation [30,31], even in the presence of a time-varying electric field $\mathbf{E}$.

Equations (27) and (30) highlight a profound property of field behavior in vacuum: whereas the mere presence of mass is sufficient to permanently stress the vacuum fabric, EM energy can do the same only if it is dynamically varying over time. This dichotomy fully justifies the perpetual GR perspective—i.e., the treatment of gravitational fields as static elastic deformations of the vacuum manifold—while distinguishing gravitation from the treatment of transient stresses exerted onto the vacuum by propagating EM waves. Some additional elastic properties of the vacuum manifold are described briefly in Notes 12 and 14–16.