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# Positive and Negative Electric Charges and Their Fields are Contracted and Dilated Space

Shlomo Barak

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Positive and Negative Electric Charges and Their Fields are

Contracted and Dilated Space

Shlomo Barak

Taga Innovations 18 Yosef Karo St. Tel Aviv 6701422 Israel 12 Jun 2025

Corresponding author: shlomo@tagapro.com

**Abstract**

In our framework, a positive elementary electric charge and its field is a **contracted** space,
while a negative elementary charge and its field is a **dilated** space. Considering space as a
lattice (a cellular structure), we define the **space density** ρ as the number of space cells per unit
volume, with ρ₀ representing the undistorted density of space. From this, we introduce the
concept of **electric charge density** q, given by:

q = 1/4π · (ρ - ρ₀)/ρ [q]=1. This charge density is positive if ρ > ρ₀ and negative if

ρ < ρ₀. **Electric charge**, in a given zone of space τ, is then: Q = ∫τ dq dτ [Q] = L³. The fields generated by these charges are the spatial extensions of the corresponding deformations, implying no separation between the charge itself and its field. These definitions apply to electric charges and their fields in general, not solely to elementary charges. The above definition of Charge Density q alone, **without any phenomenology**, yields, as we show, the Maxwell equation of Electrostatics ∇ · E = 4πq. Together with the Lorentz Transformation it yields (a known procedure) the entire Maxwell Electromagnetic theory as the Geometrodynamics of Space.

**Keywords:** Electric charge, Electromagnetism, Maxwell equations, Space, Geometrodynamics
# 1. Introduction

## 1.1 General

Until now, no existing theory has succeeded in deriving the fundamental properties of electric charge — namely, its bivalency, stability, quantization, the equality in magnitude of positive and negative charges, the electric field it generates, and the characteristic radii of these charges.

Our model of electric charge and its associated field enables us to derive explicit formulas for the radii and masses of elementary particles such as the electron/positron, muon/antimuon, and quarks/antiquarks. These expressions depend solely on the fundamental constants G, c, $\hbar$, and the fine-structure constant $\alpha$. The calculated values agree with experimental data to high precision, as detailed in Section 1.6.

## 1.2 Our Electromagnetism (EM)

In this paper, we derive the Maxwell equation of electrostatics. The full development of our theory of electromagnetism — as well as the extension of the General Relativity field equations to include the contribution of electric charge to space-time curvature — is presented elsewhere.

Our formulation of electromagnetism (EM) uses only the dimensions of length L (cm) and time T(s). Accordingly, each physical quantity in our EM framework has dimensions of the form $L^x T^y$. By employing General Relativity, we establish a quantitative equivalence between our dimensional system and the conventional system of units.
By relating the field energy density to the charge density, our EM theory becomes intrinsically non-linear — this is its key distinguishing feature (specificity). In the weak-field limit, it reduces to conventional Maxwellian electrodynamics.

Note that Quantum Electrodynamics (QED) is also inherently non-linear.

Our EM framework applies to both a single elementary charge and ensembles of charges.

Moreover, in our model, electromagnetic waves are transverse vibrations of space — analogous to gravitational waves, which are likewise transverse vibrations of space, but composed of gravitons rather than photons.

## 1.3 Space as a Lattice

Our definition of electric charge density is based on the concept of **space density**.

This concept is naturally tied to a cellular (lattice-like) structure of space; although it can be extended to a continuum description, doing so introduces significant challenges. Assigning a cellular structure to space provides explanations for its Hubble expansion, its elasticity (see Section 1.4), and naturally introduces a cutoff in the vacuum-state spectrum of vibrations. Without such a cutoff, the vacuum energy density would diverge to infinity. The necessity of a cutoff has been discussed by Sakharov [1] and Misner *et al.* [2]. The Bekenstein Bound [3] similarly imposes a fundamental limit on the amount of information accessible beyond a black hole's horizon. As Smolin [4] notes: "*There is no way to reconcile this with the view that space is continuous, for that implies that each finite volume can contain an infinite amount of information.*" A relevant review of these ideas can be found in Amelino-Camelia [8].
## 1.4 The Elastic Space

We do not regard space as a passive, static backdrop for fields and particles, but rather as an active, elastic entity. Physicists often hold different — and sometimes conflicting — views about the physical interpretation of the mathematical objects in their models. For example, the mathematical framework of General Relativity employs *n*-dimensional manifolds embedded in higher-dimensional hyperspaces. However, these mathematical objects are not necessarily identical to the physical entities that General Relativity describes. Notably, an *n*-dimensional manifold can be interpreted as equivalent to an *n*-dimensional elastic space.

This equivalence allows us to apply General Relativity while simultaneously treating our physical space as an elastic three-dimensional lattice. Rindler [6] employs this equivalence to aid the visualization of curved manifolds, while Steane [7] considers it a valid representation of physical reality. Callahan [8] is especially explicit about this point, stating: "...in physics we associate curvature with stretching rather than bending." Indeed, in General Relativity, gravitational waves [9] are literally waves in space itself — implying that attributing elasticity to space is unavoidable.

In our model, the deformation of space is understood as a change in the size of its fundamental cells. We use the terms *positive deformation* and *negative deformation* to indicate that the cells expand or contract, respectively, outward from a given point. Positive deformation corresponds to positive curvature, while negative deformation corresponds to negative curvature [10].

## 1.5 On Our Electric Charge and its Field

In our model, a positive elementary electric charge is a contracted space, whereas a negative elementary electric charge is a dilated space. Relating to space as a lattice (cellular structure),
we define Space Density ρ as the number of space cells per unit volume (denoted ρ₀ for space with no deformations). Based on this we define (postulate, invent) **Electric Charge Density** as:

$q = 1/4π · (ρ - ρ₀)/ρ$ [q] =1. This charge density is positive if ρ > ρ₀ and negative if ρ < ρ₀.

**Electric charge**, in a given zone of space τ, is then: $Q = \int_{\tau}^{\infty} q d\tau$ [Q] = L³.

Our definition of electric charge density alone yields electrostatics, without any phenomenology, and together with the Lorentz Transformation - the entire Maxwell theory [11] [12].

Note that our definition of “charge density” is **axiomatic**. This approach is in the spirit of Einstein [13] that: ... **the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented**...

## 1.6 Attributes of Matter

Our model of the electric charge yields the attributes of matter. Some of which are presented below:

The proton charge radius, rₚ is:

$r_p(\text{calculated}) = 0.877428549 \cdot 10^{-13} \text{ cm. well within the experimental error range [14].}$

$r_p(\text{measured}) = 0.8768(69) \cdot 10^{-13} \text{ cm.}$

Based on rₚ, we calculate the electron radius rₑ:

$r_e = 1.409858587 \cdot 10^{-13} \text{ cm.}$

Based on this rₑ we derive and calculate the mass M of the electron:

$M(\text{calculated}) = 0.910366931 \cdot 10^{-27} \text{gr.}$

A deviation of 0.06% from the measured CODATA 2014 value:

$M(\text{measured}) = 0.910938356(11) \cdot 10^{-27} \text{gr.}$
Based on this M and our model of quarks, we derive and calculate the masses $M_d$ and $M_{\tilde{u}}$ of the d and $\tilde{u}$ quarks:

$M_d = 4.5$ MeV recent experimental value [15]: $M_d=4.8 \pm 0.5$ MeV

$M_{\tilde{u}}=2.25$ MeV recent experimental value [15]: $M_{\tilde{u}}=2.3 \pm 0.8$ MeV

These results speak for themselves, and justify our axiomatic approach.

## 1.7 Recent Papers on Electric Charge

“Nonlinear models of electric charge and magnetic moment” [16] (2015)

“The enigmatic electron” [17] (2013)

“Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation” [18] (2013)

“Duality and ‘particle’ democracy” [19] (2016). Although these papers relate to different aspects of our subject, none presents a similar idea to ours.

# 2. Electric Charge

## 2.1 Electric Charge Density

We define the electric charge density as:

$$q = \frac{1}{4\pi} \frac{\rho - \rho_0}{\rho} \quad [q] = 1 \qquad (1)$$

The factor $1/4\pi$ is introduced for no other reason, than to ensure resemblance to the Gaussian system.

The charge density is **positive** if $\rho > \rho_0$.

The charge density is **negative** if $\rho < \rho_0$.
Necessarily, only two types of electric charge exist, positive and negative. Let n be the number of space cells in a given volume V. Since n = ρ₀V and also n = ρV' we get:

V=n/ρ₀ V'=n/ρ Hence:

$$ (V'-V)/V = -(\rho - \rho_0)/\rho \qquad (2) $$

V'>V is dilation, V'<V is contraction.

### 3. Electric Charge Q in a Given Volume τ

We define Electric Charge in a given zone of space τ as:

$$ Q = \int_{\tau}^{r} q d\tau $$

Electric charge has the dimensions of volume [Q] = L³

For clarity, in this section alone, we omit the factor 1/4π in (1). For the spherical symmetric case where dτ = 4πr²dr, for a given r, the radius of the charge Q, we get the result:

$$ Q = \int_{0}^{r} q 4\pi r^2 dr = 4\pi/3 \cdot r^3 (1 - \rho_0/\rho). \quad \text{Thus } \rho > \rho_0 \text{ gives } Q > 0 \text{ whereas } \rho < \rho_0 \text{ gives } Q < 0. $$

For us, outside observers, positive charge in a given spherical zone of space with radius r, means more space cells in the zone than in an un-deformed space (contraction), whereas negative charge means less space cells in the zone than in an un-deformed space (dilation). In [20] using our Geometry of Deformed Cellular Spaces, we relate curvature to this space deformation and open the way to the application of GR in issues related to charge.

Note that the equality $|Q_+| = |Q_-|$, of the absolute values of the bivalent elementary charges means, according to the integral above, $(1 - \rho_0/\rho_+)$ = $-(1 - \rho_0/\rho_-)$ and hence $2/\rho_0 = 1/\rho_+ + 1/\rho_-$.

Note that both $\rho_+$ and $\rho_-$ are, probably, functions of r and not just constants.

Fig. (1) suggests simplistic models of positive and negative charges, both as spheres of “radius” $r_0+$ and $r_0^-$. The contracted space in the sphere with radius $r_0+$, our $Q_+$, contracts space around it
(its field) whereas the dilated space in the sphere with radius $r_0$, our $Q_-$, dilates space around it (its field). In this model, of charge and its field, **there is no physical separation between the particle and its field** (Einstein Vision), and the integral of $\rho$ over the entire space, for two bivalent elementary charges together, is zero. Note that in the field of a positive charge space is also curved positively. Similarly, in the field of a negative charge, space is curved negatively. Hence the field equation is non-linear, as is the field equation of gravitation.

<figure><img src="image_2.png" /><figcaption>Fig. (1) A Charge and its Field</figcaption></figure>

## 4. On the “border” between the charge and the Field

In our model there is no separation between the charge and its field; they are a continuous deformation of space – Einstein’s vision. This model solves also the known issue: Is the elementary charge Electrostatic Energy in the charge or in the Field?

This long-standing open issue, addressed by Feynman [21] and others, is: The energy needed to create a charge $Q$ of radius $r$, by bringing in from infinity infinitesimal amounts of charge, despite repulsion, is:

$$U = Q^2/2r$$

The energy density of the electrostatic field $E = Q/r^2$ is:

$$\epsilon = 1/4\pi E^2$$
and the entire energy in the field is also: $U = \int_{r}^{\infty} \epsilon d\tau = Q^{2}/2r$

Thus, we ask where is the energy, in the field or in the charge?

Our model of the elementary electric charge solves this issue.

In our model there is no separation between the charge and its field; they are a continuous deformation of space. The above calculated energy, in both cases, is the same **total energy** of the charge and its field.

Thus, r can only be considered as an artificial “border” between the two. We **define** - freely invent (Einstein’s expression) - this r to be a virtual border for which on both of its sides resides the same **half** of the above U.

Hence, for r calculation we take $r = Q^{2}/4U$.

Note that U is the space energy of strain, be it contraction or dilation.

## 5. The Elastic Spatial Vector u and the Electric Field E

By relating to space as an elastic medium we can use the theory of elasticity and its Elastic Displacement Vector $u = r' - r$. In Appendix A we show that:

$$ \nabla \cdot \mathbf{u} = - \frac{\rho - \rho_0}{\rho} \qquad (A2) $$

Thus, according to (1):

$$ \nabla \cdot \mathbf{u} = -4\pi q \qquad (3) $$

By defining the Electric field vector E as:

$$ \mathbf{E} = -\mathbf{H}\mathbf{u} \quad H = 1 \quad [\mathbf{H}] = \mathbf{T}^{-2} \quad [\mathbf{E}] = \mathbf{LT}^{-2} \qquad (4) $$

equation (3) becomes the **Maxwell known equation of Electrostatics**:

$$ \nabla \cdot \mathbf{E} = 4\pi Hq \qquad (5) $$
Note that any deformation (strain) in space is related to a stress; hence the introduction of H.

E expresses, therefore, the tension in space due to a deformation in it.

For a positively charged particle, E points outwards and for a negatively charged particle inwards, as it is in the Maxwell theory of electrostatics (see Fig. 2 in Appendix A).

## 6. Coulomb's Law

Gauss's theorem is: $\int_{\sigma} \nabla \cdot \mathbf{u} \, d\tau = \int_{\sigma} \mathbf{u} \cdot d\boldsymbol{\sigma}$ For a spherical surface with radius $r$ we get:

$\int_{\sigma} \mathbf{u} \cdot d\boldsymbol{\sigma} = u_r \cdot 4\pi r^2$ and since $\int_{\sigma} \nabla \cdot \mathbf{u} \, d\tau = 4\pi H \int_{\sigma} q d\tau = 4\pi HQ$ we get $\mathbf{u} = \frac{Q}{|\mathbf{r}|^3} \mathbf{r}$ or:

$$ \mathbf{E} = \frac{HQ}{|\mathbf{r}|^3} \mathbf{r} \quad \text{Coulomb's Law} \qquad (6) $$

## 7. The Electric Field E and Scalar Potential φ

Every vectorial field can be decomposed into a field that is a gradient of a **scalar potential** (the **polar** part) and a field that is a **vector potential** (the **axial** part), subject to the boundary condition $\mathbf{E} \to 0$ at infinity. Hence:

$$ \mathbf{E} = -\nabla\phi + \nabla \times \mathbf{A} \qquad (7) $$

In the simple static case for the electric field:

$$ \mathbf{E} = -\nabla\phi \qquad (8) $$

and, in case of spherical symmetry, in spherical coordinates:

$$ E_r = -(\nabla\phi)_r = \frac{\partial\phi}{\partial r} \qquad (9) $$

and since: $E_r = \frac{HQ}{r^2}$ we get:
$$ \varphi = \frac{HQ}{r} \qquad [\varphi] = L^2 T^{-2} \qquad (10) $$

From: $\nabla \cdot \mathbf{E} = 4\pi H q$ and $\mathbf{E} = -\nabla \varphi$ we get $\nabla \cdot \nabla \varphi = -4\pi H q$ or:

$$ \Delta\varphi = -4\pi H q \qquad \textbf{Poisson's equation} \qquad (11) $$

In the absence of charge:

$$ \Delta\varphi = 0 \qquad \textbf{Laplace equation} \qquad (12) $$

We can modify equation (11) to incorporate the charge density of the field, which is equivalent to the field energy density. This modification turns (11) into a non-linear equation that resembles the non-linear equation of gravitation but it is beyond the scope of this paper.

## 8. The Electrostatic Force

From (3) $\nabla \cdot \mathbf{u} = 4\pi q$ we get $q = \frac{1}{4\pi} \nabla \cdot \mathbf{u}$, or in tensor notation $q = \frac{1}{4\pi} \frac{\partial u_j}{\partial x_j}$ (Appendix B relates $u_{ij}$ to the metric element $g_{ij}$). Multiplying both sides of (3) by $\mathbf{u}$ gives:

$$ q\mathbf{u} = \frac{\mathbf{u}}{4\pi} \nabla \cdot \mathbf{u} \qquad (13) $$

The left-hand term multiplied by $H$ is denoted by $\mathbf{f}$.

$$ \mathbf{f} = q\mathbf{H}\mathbf{u} \quad \text{or} \quad f_i = qH u_i \qquad (14) $$

At this stage, $\mathbf{f}$ is just a symbol. After a few steps, it is identified as the **electrostatic force density**. Substituting the tensor notation for $\mathbf{u}$, in Equations (13) and (14) gives:

$$ f_i = \frac{H}{4\pi} \frac{\partial u_j}{\partial x_j} u_i = \frac{H}{4\pi} u_i \frac{\partial u_j}{\partial x_j} = \frac{H}{4\pi} \left( \frac{\partial u_i u_j}{\partial x_j} - u_j \frac{\partial u_i}{\partial x_j} \right) \qquad \text{hence:} \\ f_i = \frac{H}{4\pi} \frac{\partial}{\partial x_j} \left( u_i u_j - \frac{1}{2} u^2 \delta_{ij} \right) \qquad (15) $$
where $\delta_{ij}$ is the Kroneker Delta defined by $\delta_{ij} = 1$ for $i = j$, $\delta_{ij} = 0$ for $i \neq j$, and $\mathbf{u}^2 = u_1^2 + u_2^2 + u_3^2$

Hence $f_i$ may be regarded as derived from a tensor:

$P_{ij} = \frac{H}{4\pi} \left( \mathbf{u}_i \mathbf{u}_j - \frac{1}{2} \mathbf{u}^2 \delta_{ij} \right)$. And indeed: $f_i = \frac{H}{4\pi} \frac{\partial P_{ij}}{\partial x_j}$ is identified as the force per unit volume and $P_{ij}$ as the strain tensor.

If the x-axis is chosen parallel to a line of force at any point, then $u_y = u_z = 0$ and $u_x = u$, and:

$$P_{xx} = -P_{yy} = -P_{zz} = \frac{H}{8\pi} \mathbf{u}^2.$$

Thus, the pressure perpendicular to the surface is equal to the energy density.

From (4), $\mathbf{E} = -\mathbf{H}\mathbf{u}$, we get the expression for the energy density in the field:

$$\epsilon = \frac{1}{8\pi H} \mathbf{E}^2 \qquad (16)$$

Since $\mathbf{f}$ is identified as the force per unit volume, we can return to the expression $\mathbf{f} = q\mathbf{H}\mathbf{u}$, and recognize the **electrostatic force density**:

$$\mathbf{f} = q\mathbf{E} \qquad [f] = LT^{-2} \quad \text{and the electrostatic force:} \quad \mathbf{F} = QE \qquad [F] = L^4 T^{-2}$$

## 9. Coulomb's Force Law

Equation (16) expresses the field energy density of a system of charges. Hence:

$$U = \frac{1}{8\pi H} \int_{\tau} E^2 \cdot d\tau \quad \text{where } \mathbf{E} \text{ is the field produced by these charges, and the integral}$$

goes over all space. Substituting $E = -\nabla\phi$, $U$ can be expressed as follows:

$$U = \frac{1}{8\pi H} \int_{\tau} E^2 d\tau = \frac{1}{8\pi H} \int_{\tau} \nabla \cdot \phi \mathbf{E} d\tau + \frac{1}{8\pi H} \int_{\tau} \phi \nabla \cdot \mathbf{E} d\tau$$
According to Gauss's theorem, the first integral is equal to the integral of $\nabla \cdot \varphi \mathbf{E}$ over the surface bounding the volume of integration, but since the integral is taken over all space and since the field is zero at infinity, this integral vanishes. Substituting (5), $\nabla \cdot \mathbf{E} = 4\pi H\mathbf{q}$, in the second integral, gives the expression for the energy of a system of charges: $U = \frac{1}{2} \int_{\tau}^{\tau} q\varphi \cdot d\tau$. For a system of point charges, $Q_i$ we can replace the integral with a sum over the charges $U = \frac{1}{2} \sum_i Q_i \varphi_i$ where $\varphi_i$ is the potential of the field produced by all the charges, at the point where the charge $Q_i$ is located. From Coulomb's law:

$$ \varphi_i = \sum_{i \neq j} \frac{HQ_j}{r_{ij}} \quad \text{where } r_{ij} \text{ is the distance between the charges } Q_i, Q_j \text{ we get:} $$

$$ U = \frac{1}{2} \sum_{i \neq j} \frac{HQ_i Q_j}{r_{ij}} \quad \text{In particular, the energy of interaction of two charges is:} $$

$$ U = \frac{HQ_i Q_j}{r_{ij}} \quad \text{and the force is:} \quad F = \frac{\partial U}{\partial r} = \frac{HQ_i Q_j}{r_{ij}^2} \quad \text{or:} $$

$$ F = H \frac{Q_1 Q_2}{|r_{12}|^3} r_{12} \quad \text{Coulomb's force law} \quad (17) $$

In Appendix C we present some preliminary experimental evidence that our model of the elementary charge and its field is isomorphic to reality.

## 10. Summery

Our definition of electric charge density - rooted in the density variations of an elastic space lattice - allows us to conceive of an elementary charge not as a point-like entity or a string, which are foreign to space itself, but as a region of contracted or dilated space. Consequently,
elementary particles must also possess finite size and internal structure. This insight enables us
to:

* Derive the Maxwell equations of electrostatics directly from space geometrodynamics, without any phenomenology.

* Extend Einstein's General Relativity field equations to naturally incorporate electromagnetism.

* Derive the entire Maxwell theory as a manifestation of space geometrodynamics.

* Predict and calculate the attributes of elementary charges and particles from first principles.

**Acknowledgments**

We would like to thank the late Professor J. Bekenstein and Professor A. Zigler of the Hebrew
University of Jerusalem, and the late Professor Y. Silberberg of the Weizmann Institute of
Science, for their careful reading, of early drafts and versions, and helpful comments. Thanks
to Dr Moshe Klein of the Hebrew University of Jerusalem for his careful reading and helpful
comments.

**Funding declaration**

The author declares no funding.

**Conflict of interest**

The author declares no conflicts of interest.

**Data available with the paper**

The authors declare that the data supporting the findings of this study are available within the
paper.
References

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## Appendix A: Contraction and Dilation of Space, and the Strain Tensor $u_{ij}$

The aim of this appendix is to prove that: $\nabla \cdot \mathbf{u} = -\frac{\rho - \rho_0}{\rho}$

Fig. (2) shows the position vector $\mathbf{r}$ of a spot, $\mathbf{p}$, in space, with no strain. When stress is applied on space and a deformation occurs, the location of $\mathbf{p}$ becomes $\mathbf{p}'$ with a position vector $\mathbf{r}'$. The vector $\mathbf{u}$ is the Elastic Displacement Vector (theory of Elasticity). The origin in Fig. (2) is arbitrary and does not play any role in our discussion.

<figure><img src="image_3.png" /><figcaption>Fig. (2) The Displacement Vector $\mathbf{u}$ in the Elastic Space</figcaption></figure>

In this Appendix, we show that the divergence of the elastic displacement vector $\nabla \cdot \mathbf{u}$ in an elastic medium equals the relative change in the volume $\frac{dV' - dV}{dV}$ of a strained medium. The following discussion is based on a derivation made by Landau and Lifshitz [22].

$\mathbf{u} = \mathbf{r}' - \mathbf{r}$ can be denoted by its components:

$u_i = x_i' - x_i$ Let $dl'$ be the deformed distance between adjacent points, since:

$$dx_i' = dx_i + du_i \qquad dl^2 = dx_i^2 \qquad dl'^2 = dx_i'^2 = (dx_i + du_i)^2$$
by the substitution of $du_i = \left(\frac{\partial u_i}{\partial x_k}\right) dx_k$ above we get:

$$dl^2 = dl^2 + 2 \frac{\partial u_i}{\partial x_i} dx_i dx_k + \frac{\partial u_i}{\partial x_k} \frac{\partial u_i}{\partial x_l} dx_k dx_l \quad \text{Since the summation is taken over both suffixes i and k in the second term on the right, we get:}$$

$$\left( \frac{\partial u_i}{\partial x_k} + \frac{\partial u_k}{\partial x_i} \right) dx_i dx_k$$

In the third term, we interchange the suffixes i and l. Then $dl^2$ takes the final form:

$$dl^2 = dl^2 + 2u_{ik}dx_i dx_k \quad \text{where the strain tensor } u_{ik} \text{ is defined as:}$$

$$u_{ik} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_k} + \frac{\partial u_k}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \frac{\partial u_j}{\partial x_k} \right) \quad (A1)$$

If $u_i$ and their derivatives are small, we can neglect the last term as being of the second order of smallness. Thus, for small deformations, the strain tensor is given by:

$$u_{ik} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_k} + \frac{\partial u_k}{\partial x_i} \right) \quad \text{We see that it is symmetrical: } u_{ik} = u_{ki}$$

$u_{ik}$, can be diagonalized, like any symmetrical tensor, at any given point. Thus, at any given point, we can choose coordinate axes, the principal axes of the tensor, in such a way that only the diagonal components $u_{11}$, $u_{22}$, $u_{33}$ of the 3D tensor $u_{ik}$ are different from zero. These components, the principal values of the strain tensor, are denoted by $u^{(1)}$, $u^{(2)}$, $u^{(3)}$. We should remember that, if the tensor $u_{ik}$ is diagonalized at a specific point in the body, it is not, in general, diagonal at any other point.

If this strain tensor is diagonalized at a given point, the element of length near it becomes:

$$\begin{aligned} dl^2 &= (\delta_{ik} + 2u_{ik})dx_i dx_k \\ &= (1 - 2u^{(1)})dx_1^2 + (1 - 2u^{(2)})dx_2^2 + (1 - 2u^{(3)})dx_3^2 \end{aligned}$$
We see that the expression is the sum of three independent terms. This means that the strain in any volume element may be regarded as composed of independent strains in three mutually perpendicular directions, namely those of the principal axes of the strain tensor. Each of these strains is a simple dilation, or contraction, in the corresponding direction: the length $dx_1$ along the first principal axis becomes $dx'_1 = \sqrt{(1+2u^{(1)})}dx_1$, and similarly for the other two axes. The quantity $\sqrt{(1+2u^{(1)})}-1$ is consequently equal to the relative extension $(dx'_i - dx_i)/dx_i$ along the i^th^ principal axis. The relative extension of the elements of length along the principal axes of the strain tensor, at a given point, is, to within higher-order quantities $\sqrt{(1+2u^{(1)})}-1 \approx u^{(i)}$, i.e., they are the principal values of the tensor $u_{ik}$.

Let us consider an infinitesimal volume element dV, and find its volume dV' after a deformation. To do so, we take the principal axes of the strain tensor, at the point considered, as the coordinate's axes. Then the elements of length $dx_1$, $dx_2$, $dx_3$ along these axes become, after the deformation, $dx'_1 = (1+u^{(1)})dx_1$, etc. The volume dV is the product $dx_1dx_2dx_3$, while dV' is $dx'_1dx'_2dx'_3$. Thus $dV' = dV(1+u^{(1)})(1+u^{(2)}) \times (1+u^{(3)})$.

Neglecting higher-order terms, we therefore have $dV' = dV(1+u^{(1)}+u^{(2)}+u^{(3)})$. The sum $u^{(1)}+u^{(2)}+u^{(3)}$ of the principal values of a tensor is well known to be invariant, and is equal to the sum of the diagonal components $u_{ii} = u_{11}+u_{22}+u_{33}$ in any coordinate system. Thus:

$dV' = dV(1+u_{ii})$ or: $\frac{dV' - dV}{dV} = u_{ii}$ or: $\frac{dV' - dV}{dV} = \nabla \cdot u$ and according to (2) we get:

$$ \nabla \cdot u = - \frac{\rho - \rho_0}{\rho} \qquad (A2) $$
# Appendix B: The Small Deformation Strain Tensor and the Fundamental Metric Tensor

In the peer reviewed paper [23], the authors demonstrate that the small deformation strain tensor, see (A1), could be used as a fundamental metric tensor, instead of the usual fundamental metric tensor. We quote their **Conclusion**:

> "Through the present paper, it was possible to demonstrate that **the small deformation strain tensor could be used as a fundamental metric tensor, instead of the usual fundamental metric tensor.** Also, it was possible to prove that from that tensor, not only other mathematical structures could be constructed, but also another fundamental tensor was obtained; that was to say, we had constructed two of them, $u_{\mu\nu}$, and $B_{\mu\nu\sigma\rho}$. It is through these tensors that the gap between pure geometry and physics is bridged. In particular, $u_{\mu\nu}$ relates the observed interval $ds$ to the mathematical coordinate specification $dx_{\mu}$. Also, the $u_{\mu\nu}$ appear as the potentials of the inertial field [6]. Therefore, it is reasonable to assume that, in the presence of a gravitational field, the $u_{\mu\nu}$ is again the potential which determines the accelerations of free bodies; in other words, the $u_{\mu\nu}$ is the potential of the gravitational field. Thus, a stage has been reached at which the results obtained can be applied to the theory of gravitation [4]. However, that task that would not be repeated here was established by *Albert Einstein*, and finally formulated by him in 1916, as probably the most beautiful of the physical theories."

Note however that space deformation is a local feature whereas curvature can also be a global feature.
**Appendix c: Experimental Evidence that Earth's Contracted Space Causes Free Fall of Masses and Exerts Upward Force on Positive Charges and Downward Force on Negative Charges**

According to General Relativity (GR) the Schwarzschild Metric shows that space, ourselves and our yardstick are contracted around each and every mass, including the Earth [24]. According to Quantum Field Theory (QFT) its 17 fields are embedded in space and follow its topology – the electromagnetic field included. Consequently, the electric vector field in the contracted space around Earth, is also contracted, as if the Earth were a positive or negative charge. However, QFT cannot determine which is which, as this is a matter of convention. We propose that the contracted space embeds the field of a positive charge, a hypothesis confirmed by our experiments [25]. The arguments, supporting this convention, are beyond the scope of this paper. Our experiments demonstrate that positively charged ions ascend while negatively charged ions descend towards the Earth's center. These ions are accelerated upward and downward at rates several times greater than free fall acceleration. Additionally, polar water molecules, in a horizontally oriented rectangular bloc of aerogel, generate a voltage of about 0.5 volt between the bloc's upper and lower sides. This phenomenon confirms the global orientation of water in the Earth's contracted space, as if space embeds a positive electrical field.

Beyond these experimental results we have also undertaken the following: We developed models of Elementary Particles rather than merely treating them as excitations of the QFT fields and we derived and calculated their attributes- such as mass (for those with inertia)-rather than treating them as fundamental constants of nature in QFT. These insights have led to the emergence of our Evolutionary Theory of Everything (TOE), in which all the fields are simply manifestations of space itself – suggesting that Space Is All There Is.