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

Abstract

We believe that the origin of the universal dark energy may crucially depend on the behavior of the gravitational constant G. If G is constant throughout the universe, then the dark energy density ${\mathit{u}}_0$ is supported by the vacuum via its constant and evolving properties. On the other hand, in varying-G gravity (whose low-acceleration limit is MOND), ${\mathit{u}}_0$ is a manifestation of radial G-gradients in the source of gravity. We estimate the present-day dark energy density of the universe in these two independent cases without using conventional Planck-2018 modeling in deriving the expressions for ${\mathit{u}}_0$. The constant-G derivation uses dimensional analysis, vacuum constants, and a newly discovered evolving bridge between vacuum mechanical and electromagnetic quantities. The varying-G derivation relies on the MOND critical acceleration $a_0$, the source of gravity that falls off as ∼$r^{-2}$ at large distances r, and the assumption that the present-day energy density $u_0$ can be estimated by spatially averaging this radial profile over the MOND volume ${\displaystyle \frac{4\pi }{3}}{r}_0^3$, where $r_0=c^2/a_0$ and c is the speed of light in vacuum. The agreement between these two determinations and the Planck-2018 results from $\Lambda$CDM modeling is at the percent level, so that the results cannot distinguish clearly between constant-G and varying-G gravity. The analysis further indicates that the Newtonian constant $G_0$, the MOND constant $a_0$, and the Planck units of force, power, voltage, and current can be regarded as empirical vacuum constants within the proposed framework, alongside the well-known resistive properties of the vacuum, whereas charge, capacitance, inductance, and various fields are scale-dependent properties evolving in the expanding universe since the Stoney era. Ultimately, the vacuum behaves as a remarkably stiff elastic medium under stress, irrespective of the presence or absence of matter. This property has long been encoded, albeit inconspicuously, in the much discussed Tully–Fisher and Faber–Jackson relations, as well as in the Casimir effect and the field equations of General Relativity.

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 m⁻³. This result is about 120 orders of magnitude larger than the inferred Planck-2018 value of about 3.3 GeV 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 manifold1. 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 7 and Section 5 below).

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 s⁻¹ 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 s⁻¹ 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 to the same extent 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 on 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 properties3, 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, $L_{\mathrm{S}}={\alpha }_{\mathrm{w}}{L}_{\mathrm{P}}$, where ${\alpha }_{\mathrm{w}}\simeq 0.0341$ 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 ${\alpha }_{\mathrm{w}}^{-1}\approx 30$ times larger5.

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 $L\gg L_{\mathrm{S}}$ (similar to that explored in related work by A. Sheykin and S. Manida [25]), 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 dynamical dark energy 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 [26,27,28,29]. 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. [29]), 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 [29].
• 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-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 [29], the characteristic acceleration is provided by MOND’s critical acceleration $a_0$ [30,31], 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 [31]—a scale longer by a factor of $2\times {10}^{61}$ than the Planck length $L_{\mathrm{P}}={\left(h{G}_0/c^3\right)}^{1/2}$ [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 inferred for the present universe is represented by a spatial volume average of the radial profile (3) over the range of physical scales from the Planck length to the MOND length. For the adopted $r^{-2}$ profile, the radial dependence of the density is compensated by the $r^2$ factor in the spherical volume element, so that equal radial intervals $dr$ contribute equally to the integration. Accordingly, 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 obtain

$$\langle {\mathit{u}}_0\rangle ={\displaystyle \frac{1}{V_0}}{\int }_0^{r_0}{\mathit{u}}_0\left(r\right)4\pi r^{2}dr=3{\mathit{u}}_0\left(r_0\right)=3{\displaystyle \frac{a_0^2}{G_0}}.$$

(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 dark energy 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 $\{{\epsilon }_0,{\mu }_0,c,Z_0\}$ [19,32] cannot produce an acceleration because of the dimensional structure of the resistive properties, viz.

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

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

• When I is eliminated between the two dimensional 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 $p=Mv$ 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 $\tau$ [33]. 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 step6.

The interpretation of Equation (7) is straightforward: the dark energy content of the universe starts out with enormous values at Stoney/Planck scales, where L∼$(1$–$30)L_{\mathrm{S}}$, and gradually drops to its present-day value7. At present, the characteristic length scale L could then be identified with the comoving particle horizon distance $D_{\mathrm{ph}}$ [34,35,36] determined in $\Lambda$CDM8 as follows:

The $\Lambda$CDM 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] for evaluation, the dimensionless integral in Equation (8) sums up to 3.1795, and the particle horizon distance then is

$$D_{\mathrm{ph}}=3.1795\left(c/H_0\right)=4.3664\times {10}^{26}\mathrm{m}=14.151\mathrm{Gpc}.$$

(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 ${\rho }_0={\mathit{u}}_0/c^2=7.06\times {10}^{-27}\mathrm{kg}{\mathrm{m}}^{-3}$. Although these numerical values depend on the $\Lambda$CDM particle horizon distance, they still provide a quantitative consistency check of the vacuum elasticity model (7) against the pure $\Lambda$CDM model.

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

$$a\left(D_{\mathrm{ph}}\right)=2.058\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}=1.850a_0.$$

Contrary to unsubstantiated claims in the literature (viz. $c{H}_0/\left(2\pi \right)$∼$a_0$, which 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 7.75 Gyr, until the proper particle horizon distance expands out to the formidable MOND length [31] $r_0=c^2/a_0=1.850D_{\mathrm{ph}}$ ($\simeq 26$ Gpc).

In the meantime, $a_0$ could be considered an additional acceleration scale of the vacuum, yet also relevant to other vacuum properties because of the localized stress $\propto M{a}_0$ 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 [29]. 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 $\{{\epsilon }_0,{\mu }_0,G_0,\tau \left(L\right)\}$, where $\tau \left(L\right)$ is the evolving light-crossing time over a characteristic physical length scale 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 empirical properties of the vacuum, including $G_0$ and $a_0$, 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 ${\alpha }_{\mathrm{g}}=G_0{m}_{\mathrm{e}}^2/\left(hc\right)$ [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 resistive properties $\{{\epsilon }_0,{\mu }_0,G_0\}$ [20,21]:

• The gravoelectric constant $G_{\star }\equiv 4\pi {\epsilon }_0{G}_0=7.4258\times {10}^{-21}{\mathrm{C}}^2{\mathrm{kg}}^{-2}$ has dimensions ${[Q/M]}^2$, where Q and M represent charge and mass, respectively. Thus, $G_{\star }$ serves as a bridge between charge and mass, applicable to electrostatic settings.
• The gravomagnetic constant $G_B\equiv G_0{\mu }_0/\left(4\pi \right)=6.6740\times {10}^{-18}{\mathrm{m}}^4{\mathrm{s}}^{-4}{\mathrm{A}}^{-2}$ 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.
• The first G-M, $\sqrt{G_{\star }{G}_B}=G_0/c$, 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[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).
• The second G-M, $\sqrt{G_{\star }^{-1}{G}_B}={\mathcal{R}}_{\mathrm{P}}$, 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. Present-day dark-energy estimates in five different empirical models.

Model	$\mid —————{\mathit{u}}_{\mathbf{0}}—————\mid$		${{\mathit{\rho }}_{\mathbf{0}}}^{ \left(\mathit{\star }\right)}$	References
	$\left[{\mathbf{10}}^{-\mathbf{10}}\mathbf{Pa}\right]$	$\left[\mathbf{GeV}{\mathbf{m}}^{-\mathbf{3}}\right]$	$\left[{\mathbf{10}}^{-\mathbf{27}}\mathbf{kg}{\mathbf{m}}^{-\mathbf{3}}\right]$
1.  Planck-2018 (⋆⋆)	$5.25$	$3.27$	$5.84$	[1,38]
2.  Planck-2015 (⋆⋆)	$5.35$	$3.34$	$5.96$	[37,39]
3.  Varying-G Gravity       and  MOND	$5.56$	$3.47$	$6.19$	[29,30,31] and this work
4.  Vacuum Properties       +  $\Lambda$CDM’s $D_{\mathrm{ph}}$	$6.35$	$3.96$	$7.06$	[1,19,20,21] and this work
5.  Vacuum Properties       +  Fixed Point ${\Omega }_{\Lambda }$       +  $\Lambda$CDM’s $H_0$	$5.53$	$3.45$	$6.15$	[1] and Note 8 in this work

^(⋆) Here, ${\rho }_0=u_0/c^2,$ and we use SI units to derive the effective mass density ${\rho }_0$. ^(⋆⋆) Relative uncertainties of $\gtrsim$2% and $\gtrsim$3%, respectively.

shows a comparison between three estimates of the dark energy density and the corresponding Planck results over the past ten years, including the older Planck-2015 release [1,37]. 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 for ${\mathit{u}}_0$; so, in principle, either one of them could be highlighting the source of dark energy (the gradient $dG\left(r\right)/dr$ or the vacuum set of properties $\{c,G_0,L\}$). Finally, row 5 comes from the calculations summarized in Note 8 and gives estimates very much comparable to those listed in row 3 for varying-G gravity and MOND.

On the other hand, the vacuum-based estimates lend support to varying-G gravity and MOND as viable alternatives to constant-G gravitational theories (in particular, Newtonian dynamics, General Relativity (GR), and FLRW cosmologies)9, especially since G-gradients as a source of dark energy appear to be fully consistent with the Planck results (compare 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 [7,8,9,44,45,46]; 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 7), whereas their present-day values yield a reasonable estimate for ${\mathit{u}}_0$ (Section 3). These considerations are relevant to the problem of the vacuum catastrophe [9] and the inability of QFT to obtain the present-day ${\mathit{u}}_0$ value by considering vacuum fluctuations up to the Planck scale [44,46].

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:

• 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.
• 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 m⁻¹ for $L=D_{\mathrm{ph}}$, clearly not a substantial field magnitude.
• 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 $Q_{\mathrm{vac}}=e/{\alpha }_{\mathrm{w}}\approx 30e$ at the Planck scale10. Thus, some enormous charge densities (${\rho }_e\sim$10⁸⁵⁻⁸⁸ C m⁻³) appear at these scales by tiny amounts of charge enclosed within much tinier spherical volumes.
• Current $I_{\mathrm{vac}}$.—Using the vacuum’s evolving G-M timescale $\tau =\sqrt{\mathcal{L}\mathcal{C}}$ (Section 3), we define a current scale by $I_{\mathrm{vac}}=Q_{\mathrm{vac}}/\tau$, 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]11^,12.

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 within 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 11 and 12). Constant and evolving properties are generated from a minuscule fundamental set of only two vacuum invariants, $\{4\pi {\epsilon }_0,{\mu }_0/\left(4\pi \right)\}$, supplemented by Newton’s $G_0$ or a growing length scale $L>L_{\mathrm{S}}$, respectively (see also Ref. [25], where $G_0$ considered as an interaction/nonuniversal constant is rejected, and a cosmological L is instead proposed for adoption).

With help from units of the reformulated Planck system (RPS) [20], some of which turn out to describe vacuum invariants,

Table 2. Constant and evolving vacuum properties.

1. Fundamental Set $\{4\mathit{\pi }{\mathit{\epsilon }}_0,{\mathit{\mu }}_0/\left(4\mathit{\pi }\right)\}$
${\epsilon }_0$: Permittivity    ${\mu }_0$: Permeability			Lower limits
Derived G-M Constants
Speed of Light:	$c$	$=1/\sqrt{{\mu }_0{\epsilon }_0}$	Upper limit
Impedance of Free Space:	$Z_0$	$=\sqrt{{\mu }_0/{\epsilon }_0}$	Matching threshold
Planck Resistance:	${\mathcal{R}}_{\mathrm{P}}$	$=Z_0/\left(4\pi \right)$	Matching threshold
MOND Critical Acceleration:	$\mathcal{N}\left(a_0\right)$	$=\mathcal{N}{\left(4\pi {\epsilon }_0\right)}^{(\star )}$	Threshold, SI values only
2. Constant Set $\{\mathbf{4}\pi {\epsilon }_{\mathbf{0}},{\mu }_{\mathbf{0}}/\left(\mathbf{4}\pi \right),{\mathit{G}}_{\mathbf{0}}\}$
${\epsilon }_0$: Permittivity    ${\mu }_0$: Permeability    $G_0$: Newton’s Constant			Lower limits
Derived Invariants (⋆⋆)
Planck Force:	$F_{\mathrm{P}}$	$=c^4/G_0$	Upper limit
Planck Voltage:	${\mathcal{V}}_{\mathrm{P}}$	$=\sqrt{F_{\mathrm{P}}/\left(4\pi {\epsilon }_0\right)}$	Upper limit
Planck Current:	${\mathcal{I}}_{\mathrm{P}}$	$={\mathcal{V}}_{\mathrm{P}}/{\mathcal{R}}_{\mathrm{P}}$	Upper limit
Planck Power:	${\mathcal{P}}_{\mathrm{P}}$	$={\mathcal{V}}_{\mathrm{P}}{\mathcal{I}}_{\mathrm{P}}=c^5/G_0$	Upper limit
3. Evolving Set $\{\mathbf{4}\pi {\epsilon }_{\mathbf{0}},{\mu }_{\mathbf{0}}/\left(\mathbf{4}\pi \right),\mathbf{L}\}$
${\epsilon }_0$: Permittivity    ${\mu }_0$: Permeability    L: Length Scale			Lower limits ($L=L_{\mathrm{S}}$)
Derived Quantities $(L>L_{\mathrm{S}})$
Capacitance:	$\mathcal{C}$	$=4\pi {\epsilon }_{0}L$	Increasing
Inductance:	$\mathcal{L}$	$={\displaystyle \frac{{\mu }_0}{4\pi }}L$	Increasing
Light-Crossing Time:	$\tau$	$=\sqrt{\mathcal{L}\mathcal{C}}=L/c$	Increasing
Electric Charge:	$Q$	$=\mathcal{C}{\mathcal{V}}_{\mathrm{P}}=4\pi {\epsilon }_0{\mathcal{V}}_{\mathrm{P}}L$	Increasing
Acceleration:	$a$	$=c/\tau =c^2/L$	Decreasing
Electric Field:	$E$	$={\mathcal{V}}_{\mathrm{P}}/L$	Decreasing
Magnetic Field:	$B$	$=E/c={\mathcal{V}}_{\mathrm{P}}/\left(cL\right)$	Decreasing
Magnetic Flux:	${\Phi }_B$	$=B{L}^2=({\mathcal{V}}_{\mathrm{P}}/c)L$	Increasing

^(⋆) Equal SI magnitudes $\mathcal{N}\left(·\right)$. In SI units, ${\displaystyle \frac{a_0}{4\pi {\epsilon }_0}}=1{\mathrm{V}}^2{\mathrm{kg}}^{-1}$ is a scale of the energy density of a capacitor [21]. ^(⋆⋆) Also note the vacuum invariants ${\mu }_{\mathrm{P}}=c^2/G_0$ (linear mass density) and ${\stackrel{•}{M}}_{\mathrm{P}}=c^3/G_0$ (mass flow rate) [21], as well as ${\mathcal{A}}_0=a_0{G}_0$ (MOND universal constant) [20].

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 ${\mathit{G}}_{\mathbf{0}}$. 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-ended13.

Another hint comes from the dimensionless (man-made) constants, the fine-structure constant (FSC) $\alpha ={\left(4\pi {\epsilon }_0\right)}^{-1}{e}^2/\left(hc\right)$ and the gravitational coupling constant ${\alpha }_{\mathrm{g}}=G_0{m}_{\mathrm{e}}^2/\left(hc\right)$ [19]. Besides the listed 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 [53,54,55,56,57], 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, ${\alpha }_{\mathrm{g}}$ 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 [56] 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 $\alpha$ and ${\alpha }_{\mathrm{g}}$—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. [52,53,57,58] 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 elastic stress forces responsible for vacuum deformation (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) [59,60,61] and Faber–Jackson (FJ) [62,63,64] 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 ${\left[v\right]}^4/{\left[I\right]}^2$, where I represents electric current [21]. This form leads to the dimensional scaling $v^4=G_B{I}^2$, which we adopt without introducing an additional numerical factor between the two sides. 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 get

$$v^4=G_0\left({\displaystyle \frac{{\mathcal{P}}_{\mathrm{em}}}{c}}\right),$$

(13)

where the term in parentheses has dimensions of [force] (i.e., $F_{\mathrm{vac}}={\mathcal{P}}_{\mathrm{em}}/c$). Here, we identify this quantity with the vacuum deformation force and adopt the resulting relation as a stress–strain constitutive relation at each point of the vacuum manifold, viz.

$$v^4=G_0{F}_{\mathrm{vac}},$$

(14)

a close analogue of the famous TF/FJ relations of galactic dynamics [29,30,31], provided that $F_{\mathrm{vac}}=M{a}_0$ in the elasticity framework. 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 $\mathcal{V}$ [20], in which case $F_{\mathrm{vac}}=\left(4\pi {\epsilon }_0\right){\mathcal{V}}^2$. Four individual stress–strain cases are summarized in

Table 3. Vacuum kinematic yield caused by various types of stress $\propto F_{\mathrm{vac}}$.

Stress	Exerted	Kinematic	Stress–Strain	Composite	Intrinsic (⋆⋆)
Source	Force ${\mathit{F}}_{\mathbf{vac}}$	Yield (⋆)	Relation	Coupling	Dimensions
1.  Mass M	$M{a}_0$	$v^4$	$v^4={\mathcal{A}}_{0}M$	${\mathcal{A}}_0=a_0{G}_0$	$a_0$: ${\left[v\right]}^2/\left[L\right]$
2.  EM Power ${\mathcal{P}}_{\mathrm{em}}$	${\mathcal{P}}_{\mathrm{em}}/c$	$v^4$	$v^4={\displaystyle \frac{G_0}{c}}{\mathcal{P}}_{\mathrm{em}}^{(▸)}$	${\displaystyle \frac{G_0}{c}}=c^{-1}{G}_0$	$c^{-1}$: $\left[T\right]/\left[L\right]$
3.  Electrostatic Potential $\mathcal{V}$	$\left(4\pi {\epsilon }_0\right){\mathcal{V}}^2$	$v^4$	$v^4=G_{\star }{\mathcal{V}}^2$	$G_{\star }=4\pi {\epsilon }_0{G}_0$	$4\pi {\epsilon }_0$: $\left[\mathcal{C}\right]/\left[L\right]$
4.  Isotropic Pressure P (GR)	$\left(4\pi r^2\right)P$	${(\Delta v)}^4$	${(\Delta v)}^4=G_{R}P$	$G_R=8\pi r^2{G}_0$	$8\pi r^2$: $\left[V\right]/\left[L\right]$

^(⋆) Derived from the kinematic yield, the dimensionless vacuum strain is $ϵ={(v/c)}^4$ or $\Delta ϵ={(\Delta v/c)}^4$ (GR). ^(⋆⋆) Those per unit length L coupling to $G_0$—$a_0$, $c^{-1}$, $4\pi {\epsilon }_0$, and $8\pi r^2\propto 1/R$—define the structural compliance of the vacuum, representing its intrinsic capacity to accommodate kinematic, temporal, electrostatic, and pressure gradients across space. Coupled to $G_0$, these compliances map disparate forms of localized stresses onto a singular geometric strain ${(v/c)}^4$ that commonly appears as a local kinematic yield $v^4$. The elastic Young’s modulus (Equation (21)) of this mapping is discussed in Note 14 and applied to BHs in Note 16. The scalar curvature R is discussed in Note 18 in relation to the Casimir effect. ^(▸) The coupling $G_0/c$ is produced by the first G-M of $G_{\star }$ and $G_B$, so that $G_0/c=\sqrt{G_{\star }{G}_B}$ (item 3 near the end of Section 3).

and discussed below.

In Equation (14), the force $F_{\mathrm{vac}}$ represents the localized stress exerted against the vacuum, while $v^4$ 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 suggests a vacuum origin of the gravitational constant within this framework. The vacuum’s leverage is also evident in the definitions of $F_{\mathrm{vac}}$, which involve scalings by $c^{-1}$ and $4\pi {\epsilon }_0$ (Table 3). Furthermore, in all cases, $v\ll c$ and v can be thought of as the intrinsic propagation or relaxation velocity at which local structural adjustments of the metric proceed in response to the applied stress field (matter, EM, or voltage).

A small velocity $\Delta v\ll c$ whose vacuum kinematics scales as ${(\Delta v)}^4$ can also be motivated via a heuristic analogy assuming a scalar curvature of the form $R=(8\pi G_0/c^4)P$, where P is an effective isotropic pressure. To map vacuum strain to local kinematics, we relate this curvature to an effective strain acceleration $\Delta a$ denoting the mechanical rate of metric deformation (rather than particle acceleration along a GR geodesic), viz. $R={(\Delta a)}^2/c^4$—a dimensional kinematic equation adopted for the purposes of the elastic vacuum model. Then, 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, and we find that

$${(\Delta a)}^2={\displaystyle \frac{2G_0{F}_{\mathrm{vac}}}{r^2}},$$

(15)

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

$${(\Delta v)}^4=2G_0{F}_{\mathrm{vac}}.$$

(16)

The extra factor of 2 in Equation (16) is reminiscent of the factor of 2 in gravitational light deflection [65,66] or the Schwarzschild horizon radius $R_{\mathrm{S}}=2G_{0}M/c^2$ [35]. Its origin is the particular form $R\propto 8\pi G_0$ that was adopted for the scalar curvature.

Vacuum Stress–Strain Relation. 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 ${\mathcal{P}}_{\mathrm{em}}\equiv dE/dt=c(dE/dr)$, the radiative stress is reduced to a radial energy gradient. Second, to encompass a static mass defect M, for which $M{a}_0$ 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 (14) yields the generalized scaling 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 $\mathbf{\Pi }$ as the vacuum stress flux vector field (with dimensions of [pressure]), such that

$${\rho }_{\mathrm{s}}=\nabla ·\mathbf{\Pi }.$$

(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 $\mathbf{\Pi }$ through the closed boundary surface $\partial V$, viz.

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

(19)

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

$$v^4=G_0{\oint }_{\partial V}\mathbf{\Pi }·d\mathbf{A}\left(\mathrm{vacuum}\mathrm{stress-strain}\mathrm{relation}\right),$$

(20)

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

Equation (20) has the form of Gauss’s law [27], 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 $v^4$ on the left-hand side. The effective Young’s modulus $Y_{\mathrm{vac}}$ of this constitutive relation [67,68] therefore scales as

$$Y_{\mathrm{vac}}\propto {\displaystyle \frac{1}{G_0}}\propto F_{\mathrm{P}},$$

(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 this 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 stress–strain constitutive 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 $v/c\ll 1$, leading again to the conclusion that the classical vacuum behaves as a remarkably stiff elastic medium14^,15^,16. Related works and conclusions on vacuum rigidity can be found in Refs. [53,57,73,74,75,76] (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 $\hat{\mathbf{r}}$, viz.

$$\mathbf{\Pi }\left(r\right)=\mathit{u}\left(r\right)\hat{\mathbf{r}}={\displaystyle \frac{1}{4\pi r^2}}\left({\displaystyle \frac{dE}{dr}}\right)\hat{\mathbf{r}},\mathrm{where}[\mathbf{\Pi }]=\left[\mathrm{pressure}\right];$$

(23)

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

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

(24)

In spherical symmetry, we can distinguish the types of vacuum fields and their source densities in the presence of matter or radiation17^,18 by reducing Equations (23) and (24) as follows:

➀

Matter present, the TF/FJ galactic relations.—Substituting the static mass defect relation $dE/dr=M\left(r\right)a_0$, the stress flux vector $\mathbf{\Pi }\left(r\right)$ becomes proportional to the surface mass density $\sigma \left(r\right)\equiv M\left(r\right)/\left(4\pi r^2\right)$ [29], viz.

$$\mathbf{\Pi }\left(r\right)=a_0\sigma \left(r\right)\hat{\mathbf{r}},$$

(25)

and the stress–strain relation (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 $\mathbf{\Pi }\left(r\right)$ 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)

➁

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

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

(28)

and the stress–strain relation (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 [33]—or $\mathbf{S}\equiv {\displaystyle \frac{c}{4\pi }}(\mathbf{E}\times \mathbf{B})$ in the commonly used Gaussian–CGS units [32]. 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.

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

(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 $\mathbf{J}·\mathbf{E}$ drops out of the energy conservation equation [32,33], even in the presence of a time-varying electric field $\mathbf{E}$.

Equations (27) and (30) highlight a specific property of the dynamic source density ${\rho }_{\mathrm{s}}$ within this elastic framework: whereas the mere presence of mass is sufficient to permanently source this term, EM energy contributes to ${\rho }_{\mathrm{s}}\left(r\right)$ only if ${\mathit{u}}_{\mathrm{em}}$ is varying over time. We emphasize that this dynamic source term does not replace the standard EM stress–energy tensor in GR, which also accounts correctly for the gravitational effects of static EM fields. Rather, this distinction justifies treating gravitational fields as static elastic deformations of the vacuum manifold, while cleanly separating them from transient dynamic stresses exerted onto the vacuum by propagating EM waves. Some additional properties of the elastic vacuum model are described briefly in Notes 14 and 16–18, whereas BH thermodynamics from vacuum elasticity will be covered in a future publication.