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# Gravity is Contracted -Not Bent -Space Around a Mass

Shlomo Barak

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Shlomo Barak. Gravity is Contracted -Not Bent -Space Around a Mass. 2025. hal-05002534

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# Gravity is Contracted – Not Bent – Space Around a Mass

Shlomo Barak

Taga Innovations 16 Beit Hillel St. Tel Aviv 6701716 Israel 21 Feb 2025

Corresponding author: shlomo@tagapro.com

## Abstract

Demonstrating that gravity results from contracted space around a mass requires replacing Riemannian Geometry—the historical foundation of General Relativity (GR)—with the new geometry of Deformed Spaces. This approach retains the GR equations, their solutions, and the concept of curvature - preserving its definition and meaning - while providing a more tangible and intuitive framework. In Quantum Field Theory (QFT), the electromagnetic (EM) field is embedded in space and follows its topology. A contracted space implies a contracted EM field, leading to a virtual electrical charge at the center of the mass. Our EM theory, 'On the Essence of Electric Charge and Its Field' [1], independently demonstrates that this virtual charge is positive. Recent experiments conducted at the Hebrew University of Jerusalem (HUJI) confirm these assertions [2]. Earth's deformed space induces free fall in masses, exerts an upward force on positive charges and a downward force on negative charges, and globally orients water's polarized molecules, generating an electric potential of approximately 0.5 volts. We interpret these theoretical and experimental results as the essence of **Unification**.

Keywords: Curvature; Space; Geometry; Deformed Space; Riemannian Geometry; General relativity;

## 1 Introduction

### 1.1 General Relativity and the Geometry of Space

Einstein recognized that space is non-homogeneous and non-isotropic. To formulate General Relativity (GR), he required an appropriate mathematical framework. Marcel Grossmann
introduced him to Riemannian geometry, which became the foundation of GR. However, contemporary physics describes the universe as an incomprehensible 3D manifold embedded in a 4D hyperspace. This interpretation poses significant challenges, as it hinders intuitive understanding of GR solutions and obstructs the resolution of persistent issues, such as the universe's observed Euclidean (flat) nature and the enigma of dark matter.

The **current consensus** on space includes the following properties:

**Lattice Structure** – A structured space – a lattice (cellular structure) - is assumed to ensure a finite energy density.

**Elasticity** – The size of space cells can vary, though their structure and composition remain unknown and are considered here irrelevant. The bending of light serves as experimental evidence.

**Vibration** – Space exhibits vibrational characteristics, supported by phenomena such as the Casimir effect and gravitational waves.

**Fluid-like Behavior** – Experimental evidence from frame dragging [3] supports this perspective.

To address the limitations of Riemannian geometry, we propose an alternative isomorphic geometry to reality, that models space as a lattice (a cellular structure). In this framework, differential geometry is employed to attribute curvature to spatial deformations. These deformations correspond to variations in energy within space cells, akin to a mattress of interconnected springs. Consequently, energy and curvature are intrinsically linked, as Einstein expected.

We define **space density** $ρ$ as the number of space cells per cubic centimeter in a deformed 3D region. The quantity $ρ_0$ represents the space density of an undeformed, “standard” space.
Although the fundamental structure and substance of space cells remain unknown, these
properties are not required for our theoretical formulations.

<figure><img src="image_2.png" /><figcaption>Fig. (1) Artist concept of a 3D deformed space-.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. (2) Artist concept of the cells</figcaption></figure>

**1.2 Our Geometry and General Relativity (GR)**

In our framework, we attribute Gaussian curvature to deformed spaces [4]. Positive curvature
around a point arises from **contracted space**, while negative curvature results from **dilated space**, without the need for a saddle-like interpretation.

In a 2D space, curvature at a point has two components along orthogonal directions. In 3D,
considering three orthogonal planes intersecting at a point, there are 3×2=6 curvature
components. These correspond precisely to the six independent components of the symmetric
**Ricci curvature tensor** in three-dimensional space.
Extending this to four-dimensional spacetime, the inclusion of time introduces two additional components, resulting in a total of ten curvature components. These form the Ricci curvature tensor used in General Relativity, leading to Einstein's ten field equations.

A key consequence of our approach is that the size of space cells is directly linked to their energy of contraction or dilation, which in turn determines the curvature. Thus, the right-hand side of Einstein's field equations expresses not only energy (as Einstein famously questioned) but also curvature, aligning it conceptually with the left-hand side.

This insight shows that the Einstein field equations and the Schwarzschild metric remain intact and valid within both geometries. However, as the following sections will reveal, there is a fundamental difference in how reality is perceived in these two frameworks.

## 1.3 Observers Perception of the Reality Based on the Two Geometries

**Local Observer:** An observer measuring physical phenomena within their immediate vicinity.

**Far-Away Observer:** An observer who is distant from the system or region of interest, where the local observer resides, and who indirectly measures or relates to the same physical phenomena.

### Agreement and Disagreement Between Observers

Two local observers, residing at the same spatial location at the same time, but each equipped with a different geometric framework, will agree on all physical measurements. For example, both will conclude that their radial distance is elongated (stretched) due to the gravitational field, as described by the Schwarzschild metric in Riemannian geometry. However, far-away observers in the same scenario will reach different conclusions depending on the geometry they employ: A far-away observer using Riemannian geometry will **agree** with their
corresponding local observer's measurements. A far-away observer using Deformed Space Geometry will **disagree** with their local observer. This observer will argue that the Schwarzschild metric is misleading and that the radial distance is actually contracted. In this framework, it is understood that the radial distance is contracted and not elongated and that gravitation is space contraction.

## 1.4 Gravitation

Gravitation is the deformation of space around a mass. This deformation is a contraction rather than a bending, as suggested by Riemannian geometry. This contraction is expressed as positive curvature in both Riemannian geometry and our proposed geometry. We relate space contraction to frame dragging, showing that frame dragging by the spins of material particles is the source of contraction, namely gravitation [5]. Hence, singularities are merely mathematical constructs, and gravitation is inherently quantized. The gravitational field, like the electromagnetic field, is quantized by the presence of gravitons.

<figure><img src="image_4.png" /><figcaption>Fig. (3) The True Deformation of Space by a Mass</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. (4) The True Deformation of Space by a Mass at Point P</figcaption></figure>
And Not

<figure><img src="image_6.png" /><figcaption>Fig. (5) The Curving of Space by a Stellar Mass</figcaption></figure>

**1.5 Geometry of Deformed Space and GR**

The New Meaning of Gravity in General Relativity

Fig. (5) is a typical illustration presented in textbooks, showing how a mass curves space.

However, we cannot imagine a curved three-dimensional space manifold bending within a four-dimensional hyperspace. Fig. (3) depicts the actual deformation of space caused by a mass, which is, in reality, simply the contraction of space. Fig. (4) illustrates this true deformation of space around point P- a contraction of space at P that reduces the size of the space cells, resulting in a positive curvature. This observation led us to conclude that gravity is not the bending of space, but rather the contraction of space.

**1.6 Local versus Global Curvature**

In a curved manifold, several points close to each other can exhibit the same curvature. For instance, on the surface of a sphere, all points have the same curvature. This is known as **global curvature**. In contrast, in a deformed space, if there is contraction or dilation around the deformation is necessarily different at adjacent points, which means the curvature point, will vary. This phenomenon is called **local deformation (curvature)**.
# 2 The Geometry of Deformed Lattices - Definitions and Axioms [4]

## 2.1 Definitions

**Space Density ρ at a point within a singular space cell** is the inverse of the volume of this particular cell, $[ρ] = L^{-3}$.

**Space Density ρ at a point on the boundary between cells** is the inverse of the average volume of the cells that share this boundary, $[ρ] = L^{-3}$.

For physicists who consider the linear dimension of a space cell to be the **Planck Length**

1.6·10⁻³³ cm, **Space Density**, is approximately the **number of space cells per unit volume**.

**Line Density t at a point within a singular space cell** is the inverse of the linear dimension of this particular cell, $[t] = L^{-1}$

**Line Density t at a point on the boundary between cells** is the inverse of the average linear dimensions of the cells that share this boundary, $[t] = L^{-1}$.

**The Space density** and **Line Density of an un-deformed (uniform) space** are denoted $ρ_0$ and $t_0$ respectively. Note that, $t^3 \sim ρ$.

In a uniform space, all of its elementary cells are of the same size, and Euclidean geometry is valid.

Note that in our figures, we **represent** a space cell (of unknown geometry and structure) by a circle in 2D and a sphere in 3D.
## 2.2 The Axioms

1. Space is a Lattice - its structure is cellular

2. Space is elastic – its cells sizes can change

3. Space is without boundaries – infinite

4. The linear dimension of a space cell serves as the intrinsic yardstick unit of length

## 2.3 Local and Far-Away Observers

**Local Observer:** An observer measuring physical phenomena within their immediate vicinity.

(An observer that resides in a specific zone of space is necessarily subjected- he himself and his yardstick- to space deformations in this zone. When the density is not uniform, i.e., cells are of different sizes, the local observer discovers that Euclidean geometry **is not** valid).

**Far-Away Observer:** An observer who is distant from the system or region of interest, where the local observer resides, and who measures (relates to) the same physical phenomena indirectly.

**Measurements** taken, by a **Local Observer** or **Far-Away Observer**, are with standard tools and units of length and time like the centimeter and the second - tools that once at the same time and place where identically produced. At new different places and times, however, these tools are subject to modifications due space density. This understanding is in the center of our theory.

Axiom 4 means that taking a length measurement, by an internal observer, can be the counting of the number of cells between two points. In this case, **the dimension of length is a pure**
**number.** The internal observer can also use his elastic 1 cm yardstick and count the number of times it goes into the above distance. In this case **the dimension of length is** given in cm.

## 2.4 Parameter value at a point

Parameter value at a point is the average value in the close vicinity of the point. This is especially relevant to the term deformation or curvature that is not an intrinsic parameter of a point but a parameter that relates to the topology of its surroundings.

$$ \textbf{Excess Radius} \delta r \equiv r_{\text{measured}} - c_{\text{measured}} / 2\pi. $$

This term is used in differential geometry.

When an internal observer measures circles, their circumferences c and radii r, they find an **Excess Radius** δr that differs from zero if Euclidian geometry is not valid.

This happens since the observer and their yardstick are both deformed exactly as the space in which they are immersed.

Note that, the reality is granular. Even space is made of cells. Hence continuous equations and parameters are only approximations, appropriate for phenomena with scales much larger than the granularity scale.

Note that, in our geometry the radius r can simply be the number of space cells it includes regardless of their sizes. The length of the circumferences, however, depends on space density there. Hence the meaning of **Excess Radius**.

## 2.5 The Gaussian Curvature K versus the Gaussian Deformation K

For a 2D bent surface (2D manifold) the Gaussian curvature is [6]:

$$ K = \lim_{r \to 0} \frac{6\delta r}{r^3} \qquad (1) $$

Here, a measured distance is notated by r.
For the case of symmetry, K for small r is:

$$K \approx 6\delta r/r^3 \qquad (2)$$

When space is deformed, every point in it is displaced. A point P with a position vector **r** before the deformation, is displaced to a point P' with a position vector **r'** due to the deformation.

$$\mathbf{u} = \mathbf{r}' - \mathbf{r}$$

is the **displacement vector**.

In Riemannian geometry the yardstick unit of length is retained. In our geometry, however, the yardstick units are also deformed, like space itself, and hence the displacement vector is **u** = 0.

$$\mathbf{r}' = \mathbf{r} + \mathbf{u}$$

is the **intrinsic radius**, and in our geometry **r'** = **r**.

To calculate **deformation**, we use the definition (1):

Note, see (1), that in differential geometry the dimension of curvature is [K] = L⁻².

Note that a space cell resembles a three-dimensional spring. A **dilation or contraction** is **accompanied with energy. This energy affects space density and hence the deformation**

# 3 Deformation (Curvature) in the Geometry of Deformed Lattices

In a deformed space a yardstick does not retain its length but is contracted or dilated (stretched) like its local space.

## 3.1 Positive Symmetric Deformation (Curvature) at a Point P in a 2D Deformed Space

Consider Fig.6, in which the circles represent space cells, or that a circle's diameter represents the length of a yardstick. The yardstick at position P is the smallest and from P outward the
yardstick increases in size symmetrically. This situation is analogous to a metallic plate, where the temperature increases from the center of the surface outwards, and thus the density decreases, i.e., the cell size increases. The 2D, local observer, in Figure (6), finds that the ratio of the circumference C of the circle to the radius r, as measured by the intrinsic yardstick, is: $C/r < 2\pi$.

The Excess Radius $\delta r$, in this case, is the same as for a curved manifold around a point P:

$$ \delta r = r_{\text{measured}} - c_{\text{measured}} / 2\pi. \qquad (3) $$

For a positive curved 2D space, as Figure (3) shows, $\delta r = r_{\text{meas}} - r_{\text{cal}} > 0$.

The 2D observer, therefore, concludes that there are two possibilities: they live in a two-dimensional space with a variable density, or on a curved two-dimensional surface “manifold” bent in a three-dimensional hyper-space.

The 2D observer cannot imagine a three-dimensional space, but may be able to accept the necessary abstraction. In any case, they are not able to decide between the two possibilities just by examining their locality.

$$ r_{\text{measured}} = 8 $$

$$ c_{\text{measured}} = 36 $$

$$ c/r = 36/8 = 4.5 < 2\pi $$

$$ r_{\text{measured}} > r_{\text{calculated}} $$

The Excess Radius

$$ \delta r = r_{\text{meas}} - r_{\text{cal}} > 0 $$

<figure><img src="image_7.png" /><figcaption>Fig. (6) Positive Deformation</figcaption></figure>
## 3.2 The Gaussian Deformation (Curvature)

We define the Gaussian curvature in the same way as in Differential Geometry

$$ K = \lim_{r \to 0} \frac{6\delta r}{r^3} \qquad (1) $$

$$ K = 1/R_c^2 \qquad (4) $$

$R_c$ is defined as the “radius of curvature” at the point P.

For the case of symmetry, (1) for small r gives:

$$ K \approx 6\delta r/r^3 \qquad (2) $$

## 3.3 Asymmetric Positive Deformation (Curvature) at a Point P

For no radial symmetry, K is taken as the geometric mean (average) of the largest and smallest curvatures, $k_1$, $k_2$; as if we have two surfaces with their corresponding excess radii $\delta r_1$ and $\delta r_2$.

A more accurate K should be taken based on the space density around P.

# 4. Negative Deformation in a 2D Deformed Space

## 4.1 Symmetric Deformation at a Point P

Fig. (7) shows a cell at point P, which is the largest, and from P outwards the cells decrease in size. In this case, $\frac{C}{r} > 2\pi$ and $\delta r = r_{\text{meas}} - r_{\text{cal}} < 0$. In this case, the curvature is negative

$K < 0$. Fig. (7) is analogous to a metal plate, where the temperature decreases from the point P outwards, as if the point P is heated with a blow-torch.

## 4.2 Asymmetric Deformation at a Point P

For no radial symmetry, K is taken, with a minus sign, as the geometric mean (average) of the excess radii $\delta r_1$ and $\delta r_2$. An accurate K should be taken based on the space density around P.
$$r_{\text{meas}} = 8$$

$$C_{\text{meas}} = 64$$

$$\frac{C}{r} = 10.2 > 2\pi$$

$$r_{\text{meas}} < r_{\text{cal}}$$

The Excess Radius

$$\delta r = r_{\text{meas}} - r_{\text{cal}} < 0$$

<figure><img src="image_8.png" /><figcaption>Fig. (7) Negative Deformation</figcaption></figure>

## 5. A Saddle

At point P in the center, see Fig. (8), in 1/4 π direction, the outward dilation of space means that the curvature is positive, with a radius of curvature R+. In the 3/4 π direction the outward contraction of space means that the curvature is negative, with a radius of curvature R-.

Thus, we can define the overall curvature K as:

$$K = 1/(R_{+} R_{-}) \qquad (5)$$

If $R_{+} - R_{-} > 0$ then approximately $K > 0$, and if $R_{+} - R_{-} < 0$ then $K < 0$.

Note that an accurate definition is related to space density in all the area.

<figure><img src="image_9.png" /><figcaption>Fig. (8) Saddle-like Elastic 2-D Space</figcaption></figure>
## 6. Deformation at a Point P in a 3D Deformed Space

Fig. (9) shows a small cell located at point P, and cells of increasing size radiating from P with spherical symmetry. If we measure the circumference of a great circle whose center is P, in any direction, a measurement of the Excess Radius yields:

$$ \delta r = r_{\text{meas}} - r_{\text{cal}} > 0 $$

<figure><img src="image_10.png" /><figcaption>Fig. (9) Three-dimensional Positive Deformation</figcaption></figure>

Measurements of circumferences and radii can be taken around any point inside the sphere. But for any point, except point P in Fig. 9, there is no symmetry.

For a deformed 3D space with spherical symmetry around P the Gaussian curvature at P is also expressed by equation (2).

For the case of no symmetry, we must determine the degree of deformation of circles around point P, in three orthogonal planes through P. For each of the orthogonal planes we must determine the largest and smallest curvatures, $k_1$, $k_2$, called the principal curvatures. Thus, to specify a deformed three-dimensional space around a point P we need 3 x 2 numbers. The average of these six principal curvatures is the average Gaussian curvature of the deformed space at the locality of P. According to the above, a complete definition of curvature in close proximity to a point in three-dimensional space requires six “curvature numbers”. These represent three pairs of curvature numbers for each of the three intersecting planes perpendicular to each other. These curvature numbers are components of a symmetric tensor of
2<sup>nd</sup> rank called the contracted Riemannian tensor of curvature, or the Ricci tensor.

Geodesics

Fig. (10) shows the shortest distance between two points, A and B, in a deformed two-
dimensional space. This figure shows that the shortest distance between points A and B for a
local observer, with their changing yardstick, is the solid line path that passes through the
centers of seven cells, and not the “straight” dashed line through A and B, that passes through
nine cells. The shortest distance between points A and B, solid line, is the **geodesic line**. There
is no difficulty in imagining this in three dimensions.

<figure><img src="image_11.png" /><figcaption>Fig. (10) A Geodesic on a Spherical Surface (Right) and on a
Deformed 2D Space (Left)</figcaption></figure>

Theorems of this geometry appear in [4].

7. Riemannian Geometry

**7.1 1D and 2D manifolds** Fig. (11) presents 2D manifolds curved in a hyper 3D

space.

<figure><img src="image_12.png" /><figcaption>Fig. (11) Manifolds: Plane (Euclidean) & Bent (Non-Euclidean)</figcaption></figure>
On a manifold a yardstick retains its length, but is bent like the manifold.

**A curved single-dimensional line in a two - dimensional plane**

<figure><img src="image_13.png" /><figcaption>Fig. (12) a Curved Line on a Plane</figcaption></figure>

At a point, where the line is curved, a circle is tightly pressed to the curve – an **Osculatory circle**. The circle’s radius at this point is called the **radius of curvature**. The smaller the radius the larger the curvature. The curvature of a line is termed positive if the osculatory circle is below the line and negative if above.

## 7.2 2D Curved Manifolds in a 3D Euclidean Space

To obtain the curvature of a two-dimensional surface (manifold) at a point, P, that is intrinsic to the surface, consider the length C of a closed path that is the locus of all points that are at the same geodesic distance, r from the point.

For a sphere of radius $r_0$, the above circumference C for small $r/r_0$, as Fig. (13) shows, is:

$$C = 2\pi r_0 \sin\left(\frac{r}{r_0}\right) \approx 2\pi \left(r - \frac{r^3}{6r_0^2}\right) = 2\pi \left(r - \frac{Kr^3}{6}\right)$$

Here $K = \frac{1}{r_0^2}$ is a natural **definition** for the curvature of a sphere of radius $r_0$. By extracting K from the above equation, we obtain the **curvature** K and re-define it for a more general case:

$$K = \frac{3}{\pi} \lim_{r \to 0} \frac{2\pi r - C}{r^3} \qquad (6)$$

This K depends only on the intrinsic properties of the metric of the two-dimensional surface
with no reference to the embedding space. The sign of the curvature is positive if $C < 2\pi r$, negative if $C > 2\pi r$, and zero if $C = 2\pi$.

Note that for a two-dimensional surface (manifold) **Gaussian curvature** can be defined as:

$$K = \frac{1}{(r_0)_1 (r_0)_2} \qquad (7)$$

<figure><img src="image_14.png" /><figcaption>Fig. (13) Curvature of a Sphere</figcaption></figure>

where $(r_0)_1$ and $(r_0)_2$ are the radii of the largest and smallest osculating circles of sections formed by the corresponding planes perpendicular to a third plane, tangential to the surface at P. If the two osculating circles are on the same side of this third plane, we define the curvature as **positive**, and if on opposite sides - as **negative**.

For a saddle surface, as shown in Fig. (15), the two osculating circles at point P in the middle of the saddle are on opposite sides of the third tangential plane to the surface at P, and thus the saddle at P has a negative curvature.

## 7.3 The Excess Radius $\delta r$ of Curved Manifolds

The **Excess Radius** $\delta r$ of curved manifolds around a point P, see Fig. (14). is:

$$\delta r \equiv r_{\text{measured}} - c_{\text{measured}} / 2\pi. \qquad (8)$$
<figure><img src="image_15.png" /><figcaption>Fig. (14) Excess Radius</figcaption></figure>

Measuring distances with an intrinsic local yardstick gives the radius, r(measured) from point P to point P', and the circumference, C(measured), of a circle centered at P passing through P'.

The calculated radius is:

$$r_{\text{calculated}} = c_{\text{measured}} / 2\pi. \qquad (9)$$

## 7.4 The Gaussian Curvature

According to [6] equation (10.7), for a 2D bent surface (manifold) the Gaussian curvature is:

$$K = \lim_{r \to 0} \frac{6\delta r}{r^3} \qquad (1)$$

$R_c$ is the radius of curvature at the point P on the surface, and:

$$K = 1/R_c^2$$

According to [6], for the case of spherical symmetry, for small r:

$$K \approx 6\delta r/r^3 \qquad (2)$$

For no radial symmetry, K is taken as the geometric mean (average) of the largest and smallest curvatures, k1, k2, one for each of the osculating circles, as if we have two separated surfaces.
## 7.5 Negative Curvature at a Point P on a 2D Manifold - A Saddle

<figure><img src="image_16.png" /><figcaption>Fig. (15) A Saddle (2D Manifolds in 3D Space) Around Point P</figcaption></figure>

In Fig. (15) an osculating circle, of radius r, on the saddle around the point P has a longer circumference than that of a circle on a flat surface, because the radius wiggles up and down [6]. The Excess Radius δr of our curved manifold is negative. Gaussian curvature is:

$$K = \lim_{r \to 0} \frac{6\delta r}{r^3} \qquad (1)$$

and hence the saddle is curved negatively K < 0.

Let us draw a plane tangential to the saddle at P. Perpendicular to this plane; we draw two more planes with osculating circles through P from the two sides of the plane tangential to the saddle at P. Let $r_{c1}$ be the radius of an up circle with the smallest radius, and $r_{c2}$ be the radius of a down circle with, also, the smallest radius. In this case:

$$K = -1/(r_{c1} r_{c2})$$

If the radii are equal and denoted $r_c$ then:

$$K = -1/r_c^2$$

This is our conclusion as Far-Away Observers. The Local Observer, located on the manifold should again make measurements around P. They would see that the calculated R', the result of dividing the circumference of the circle around P by $2\pi$, is greater than the measured R. Thus,
$$δr = R - R' < 0$$

and again, the local observer residing on the manifold and the far-away observer agree on the nature of the curvature of the manifold. Chapter 7. shows the **close resemblance** between the geometry of deformed spaces and Riemannian geometry. The next chapters will show **the difference**.

## 8. On the Metric, Ricci Tensor and General Relativity (GR)

The metric tensor $g_{ij}$ and Ricci tensor $R_{ij}$ appear in the left-hand side of the Einstein equation of GR:

$$R_{ij} - \frac{1}{2}Rg_{ij} = 8\pi G/c^4 \cdot T_{ij} \qquad (10)$$

A metric tensor represents local deformations, around a given point, of both a continuum and a lattice. For a lattice, this representation is legitimate only if the sizes of its cells are orders of magnitude smaller than the scale of the deformation in this locality since we are using continuous differential equations. An example of such a metric is the Schwarzschild metric [7].

The Ricci curvature tensor is the corresponding matrix of traces of the Riemannian curvature tensor. An element $R_{ij}$ of the Ricci tensor is an average of sectional curvatures, around a given point, of the intersection line of a manifold with the $x_i x_j$ plane. The scalar curvature $R$ is defined as the trace of the Ricci curvature.

## 9. On the 4D Deformed Spacetime – 3D Space and 1D Time

To a far-away observer, the universe behaves, for all intents and purposes, like an elastic body, and the theory of elasticity serves as the tool for performing measurements and calculations. For a local observer within the universe, rulers, clocks, and even themselves change according to the deformation of space at a given point. However, this change is
imperceptible to the local observer. The only way a local observer can detect deformation-i.e.,
a change in space density-is by measuring geometric properties such as triangles and circles.
Any deviation from the expected sum of 180 degrees for a triangle's angles in undeformed
space, or from the expected ratio of a circle's circumference to its radius (which should be 2π
in undeformed space), would indicate deformation. The local observer interprets such
deviations as a failure of Euclidean geometry, whereas the far-away observer simply sees
them as expressions of variations in space density.

In this scenario, where rulers and clocks change but do not appear to change, the local
observer may adopt either the geometry of Deformed Spaces or Riemannian geometry.

However, as seen from the far-away observer's perspective, adopting the Deformed Space
geometry allows for a simple and tangible picture of the universe. This picture describes a
deformed three-dimensional space in which the speed of light-and consequently, the rate of
clocks at any given point-is determined by space density.

To solve any dynamic problem involving the motion of a light beam or a particle, we must
work within an abstract, four-dimensional mathematical spacetime. Here, tangibility is
unnecessary, as this spacetime is not a “physical space” but a mathematical construct
designed for calculation.

10. The Metric in the Riemannian Geometry

Schwarzschild, in 1916, was the first to find a solution to Einstein's field equation - a general spacetime metric - for the exterior of a spherically-symmetric star of mass M and radius R. For r > R according to Schutz [7] the line element $ds^2$ is:

$$ds^2 = -(1-2M/rc^2)c^2dt^2 + (1-2M/rc^2)^{-1}dr^2 + r^2 d\Omega^2 \quad (10.36 \text{ in } [7]) \quad (11)$$

Note our added term $c^2$ in (11), according to **classical physics**.
We denote a **gravitational scale factor**, a:

$$a = (1-2M/rc^2) \qquad (12)$$

For the surface of the sun or the edge of our galaxy:

$$GM/rc^2 \sim 10^{-6} \text{ and thus } GM/rc^2 \ll 1. \qquad (13)$$

For $GM/rc^2 \ll 1$ equation (2), for $r \to \infty$, is approximated as:

$$a \to 1 \qquad (14)$$

We rewrite equation (11) to become:

$$ds^2 = - a c^2 dt^2 + a^{-1} dr^2 + r^2 d\Omega^2 \qquad (15)$$

The metric in equation (15) is the same for both the Local Observer and the Far-Away Observer.

The center of a mass, M, serves as the origin of their co-ordinates.

The yardsticks of both observers are rigid and have the same unit of length. Distances measured by them are denoted dr. The radial distance is the **measured distance dr'** on the curved manifold (it is larger than the calculated radius - positive curvature) and, hence, it contains more units of length. **As if** the gravitational field **elongates or stretches** the distance:

$$dr' = a^{-1/2}dr \quad (a < 1, dr' > dr) \qquad (16)$$

## 11. The Metric in the Geometry of Deformed Space

For the Local Observer that adopts our geometry of deformed space (16) is **valid**, but for the Far-Away Observer, that adopts our geometry, **it is not**. For the Far-Away Observer, in the Deformed Space Geometry it is:

$$dr' = a^{1/2} dr \quad (a < 1, dr' < dr) \qquad (17)$$

This observation is the main reason for adopting our Deformed space Geometry.
The radial distance is the **measured distance dr**` in the contracted space. It is larger than the calculated radius - positive curvature, see Fig (6). However, it contains more units of length (cells) for the Local Observer and Less for the Far-Away Observer, since the unit of length (cell size) is affected by the density of space. It is smaller for the Local Observer and the larger for the Far-Away Observer, whereas in Riemannian Geometry the yardstick of the Local Observer and that of the Far-Away Observer are the same. Hence, **gravitation is the contracted space around a mass**, with all the implications mentioned in the abstract.

Equation (17) is our logical inference.

## 12. On Our Experiments [2]

**Experimental Evidence that on Earth Negative Ions Descend, Positive Ions Ascend and Pollard Water is Globally Oriented.**

According to General Relativity (GR), the Schwarzschild metric, in our geometry, shows that space-and consequently, our measuring instruments-contracts around every mass, including the Earth. According to Quantum Field Theory (QFT), its 17 fields are embedded in space and follow its topology, including the electromagnetic field. Therefore, in the contracted space around the Earth, the electric vector field is also contracted, as if the Earth carries a positive or negative charge. However, QFT cannot determine which charge it is, as this is a matter of convention.

We propose that contracted space embeds the field of a positive charge, as confirmed by our experiments. Specifically, our findings show that positively charged ions move upward, while negatively charged ions move downward toward the center of the Earth. These ions accelerate up and down at rates several times greater than the free-fall acceleration. Additionally, polar water molecules in a horizontally oriented rectangular block of aerogel generate a voltage of approximately 0.5 volts between its upper and lower sides. This phenomenon confirms the
global orientation of water molecules in the Earth’s contracted space, behaving as if space itself
embeds an electric field.

Beyond these experimental results, we constructed models of the elementary particles, rather
than treating them merely as excitations of QFT fields. We derived and calculated their
attributes, such as their masses (for those with inertia), instead of considering them as constants
of nature – as QFT does. This work led to the development of our Theory of Everything (TOE),
in which all fields are simply manifestations of space itself—ultimately demonstrating that
**Space Is All There Is.**

**13. Summary**

The understanding that mass contracts space around it, rather than bending it, opens the path
for unifying gravitation and electromagnetism, as demonstrated by our experiments conducted
at HUJI. This insight also enables us to explain how mass contracts space due to Frame
Dragging, a phenomenon generated at the microscopic level by the spins of particles. These
findings are made possible by adopting a new geometry of Deformed Spaces, as opposed to
the traditional Riemannian geometry of bent manifolds in higher-dimensional hyperspaces.

**Acknowledgments**

We would like to thank 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

[1] S. Barak: Electric Charge and Its Field as Deformed Space
Applied Physics Research; Vol. 11, No. 4; 2019 http://dx.doi.org/10.5539/apr.v11n4p29

[2] S. Barak, Y. Neve-Oz, A. Zigler: Experimental Evidence that Positive Ions Ascend and Negative Ions Descend Towards the Center of Earth https://hal.science/hal-04664992

[3] S. Barak, A. Zigler, J. Papeer: Experimental Evidence in Support of the Possibility That the Local Space of Earth Is Confined Applied Physics Research; Vol. 11, No. 4; 2019 http://dx.doi.org/10.5539/apr.v11n4p7

[4] S. Barak: The Universe Geometry of a Deformed 3D Space Lattice
science/hal-04670068

[5] S. Barak: On the Essence of Gravitation and Inertia, Part 2: The Curving of Space by an Elementary Particle (Nov 2016) https://hal.archives-ouvertes.fr/hal-01405460

[6] A.M. Steane: Relativity made relatively easy. Oxford (2013)

[7] S. B. F. Schutz: General Relativity, Chapter 10, Cambridge (2003)