# Free Fall as a Response to a Spatial Gradient in the Speed of Light

Shlomo Barak

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Shlomo Barak. Free Fall as a Response to a Spatial Gradient in the Speed of Light. 2025. hal-05112245

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# Free Fall as a Response to a Spatial Gradient in the Speed of Light

Shlomo Barak

Taga Innovations 18 Yosef Karo St. Tel Aviv 6701422 Israel 09 Jun 2025

Corresponding author: shlomo@tagapro.com

## Abstract

In this paper, we demonstrate that the free-fall acceleration induced by a mass originates from a spatial gradient in the speed of light, resulting from the contracted space surrounding the mass. This gradient is sensed by the elementary particles composing free-falling bodies, compelling them to accelerate toward the mass.

We have previously shown that gravity arises from *contracted* - not curved - space around a mass [1], and that gravitation slows light and contracts measuring rods, thereby ensuring the invariance of the speed of light for all local observers [2]. Building upon our framework on the essence of inertia—where mass is understood as a practical term describing the resistance of a body to acceleration [3] - we conclude that a mass generates a spatial gradient in the speed of light. This gradient, interacting with the internal structure of elementary particles, drives the phenomenon of free fall.

What has traditionally been interpreted as an attractive gravitational force between masses is redefined here as the dynamical response of particles to spatial variations in light speed. From this principle, we derive Newton's law of gravitation and rigorously establish the equivalence of gravitational and inertial mass.

Key Words: Gravitation; Free fall; Elementary particle; Inertia
# 1. Introduction

In this paper, we present a novel explanation of how an elementary particle detects spatial curvature and moves accordingly. Specifically, we:

* Express the free-fall acceleration $g$ as a gradient in the speed of light.

* Construct a simple, yet physically grounded, model of an elementary particle.

* Demonstrate that inertia naturally emerges from this model, determined solely by the particle's internal energy and structure.

* Derive Newton's First and Second Laws from the model's fundamental principles.

* Show how an elementary particle responds to spatial curvature - interpreted here as a gradient in light speed - and how this leads to motion toward mass.

* Derive the universal Newtonian gravitational force and rigorously establish the equivalence of gravitational and inertial mass.

This approach leads to a new understanding of “gravitational attraction” not as a force mediated by a curved geometry, but as the response of particles to variations in light speed caused by contracted space around mass.

# 2. Free-Fall Acceleration as a Gradient in Light Velocity

We propose that the contracted space surrounding a mass gives rise to a spatial gradient in the speed of light. This forms the basis of our central claim:

* Gravity is contracted—not bent—space around a mass [1].

* Gravitation slows light and contracts yardsticks, ensuring that the speed of light remains invariant for all local observers [2].
Within this framework, we argue that an elementary particle - modeled as a circulating
moving wavepacket with a characteristic radius (see Section 3) - is capable of “sensing” these
gradients (see Section 4). This interaction drives the particle to move in the direction of
decreasing light speed, resulting in what we observe as free fall.

## 2.1 The Metric and Light Velocity

In 1916, Karl Schwarzschild was the first to derive a solution to Einstein's field equations - a general spacetime metric - describing the region outside a spherically symmetric mass m with radius R. For r > R according to Schutz [4], the line element $ds^2$ is given by:

$$ds^2 = -(1-2GM/\text{rc}^2)c^2dt^2 + (1-2GM/\text{rc}^2)^{-1}dr^2 + r^2 d\Omega^2 \quad (10.36 \text{ in } [6]) \quad (1)$$

Note that we have included the factor $c^2$ in equation (3), following classical physics conventions. We define a **gravitational scale factor** $a$ as:

$$a = (1-2GM/\text{rc}^2) \quad (2)$$

On the Sun or at the edge of our galaxy, $GM/\text{rc}^2 \sim 10^{-6}$. For

$GM/\text{rc}^2 \ll 1$ and as $r \to \infty$ the scale factor tends to: $a \to 1$.

Rewriting equation (1) using the scale factor $a$ (2) gives:

$$ds^2 = -ac^2dt^2 + a^{-1}dr^2 + r^2 d\Omega^2 \quad (3)$$

The center of the mass $m$ is used as the origin of the coordinate systems.

The metric presented in Section (3) is derived from the perspective of a faraway observer - one located at a large distance from the center of a mass $m$, which serves as the origin of their coordinate system.

In our framework, space is modeled as a 3D elastic lattice [1]. We define the **space density** $\rho$ as the number of space cells per unit volume. In regions of undeformed space - far from any
mass or charge - this density is denoted $ρ_0$. Crucially, we treat space as a deformable elastic medium, rather than as a bent manifold, in contrast to Riemannian geometry [1].

This conception leads to a key implication: a **standard yardstick** spans the same number of space cells regardless of its location in space. Therefore, any change in measured length reflects a physical contraction or expansion of space itself.

From the viewpoint of the faraway observer, a radial distance interval $dl$, located near the mass $m$, contains fewer of **his** yardstick units $dr$ than it does of the **local** observer's yardstick units $dr$. This discrepancy arises due to the contraction of the local observer's yardstick - an effect of the increased space density near the mass. In essence, local space is contracted, and so is the unit of length defined within it. Therefore:

$$dr_0 = a^{-1/2}dr \qquad a < 1 \qquad (4)$$

where $a(r)$, equation (2), is the **contraction factor** that increases as one approach the mass, representing the ratio of faraway to local yardstick lengths.

From the viewpoint of the faraway observer, a time interval, $dt$, contains a larger number of time units, $dt$, than the number of time units, $dt_0$, for **local** observer. And indeed, from the synchronization of clocks, Rindler [5] p.184 arrives at the conclusion, that:

$$dt_0 = a^{1/2}dt \qquad a < 1 \qquad (5)$$

The 4D spacetime interval between two events; the “emission” of a short pulse of light at point A and its “arrival” at point B is $ds^2 = 0$, [10] p.236. Hence using (3) ignoring $r^2 d\Omega^2$:

$$-ac^2dt^2 + a^{-1}dr^2 = 0$$

$$a^{1/2} cdt = a^{-1/2} dr$$

$$dr/dt = ac \qquad (6)$$

This, $dr/dt = c'$, for a **faraway observer**, is the light velocity close to the mass $M$ and
$dr/dt = c' < c$. $c'$, is a local real and slower light velocity since, according to (10) $a < 1$.

Note that in the literature $dr/dt$ in (6) is called **coordinate speed of light**. This is a misleading name, since $dr/dt$ should be considered a **real speed** [2].

Using (4) and (5) gives:

$dr/dt = a dr_0/dt_0$ Comparing this equation to equation (6), gives:

$$dr_0/dt_0 = c \qquad (7)$$

We thus conclude that **faraway observer** and **local observer** measuring light velocity in their own zones of space arrive at the same result. On the other hand, again:

$$c' = a c \qquad (8)$$

This invariance attributes the title “**a constant of nature**” to light velocity, despite the fact that in different zones of space it can behave differently [2].

## 2.2 The Gravitational Field as a Gradient in Light Velocity

According to (6) and (2) – in which we use $M$ for the gravitating mass and use $m$ for the free-falling mass - for $GM/\text{rc}^2 < 1$:

$$dr/dt = ac = (1 - GM/\text{rc}^2)c = (1 + 2\varphi/c^2)c \qquad (9)$$

For the surface of the sun or the edge of our galaxy: $GM/\text{rc}^2 \sim 10^{-6}$. According to (9), **notating** $dr/dt = c'$, we get:

$$c' = ac \qquad (10)$$

From (9) the gravitational potential is:

$$\varphi = \frac{1}{2} c (c' - c)$$

Note that $c' < c$, and that $\varphi < 0$. The field strength (central acceleration $g$) is thus:
$$E_g = g = - \frac{d\varphi}{dr} = - \frac{1}{2} cdc'/dr \qquad (11)$$

$$E_g = g = - \frac{1}{2} c \nabla c' \qquad (12)$$

Note that $E_g = g$ is the free fall acceleration (central acceleration).

Thus, the gravitational field can be considered a gradient in light velocity $E_g = -1/2c\nabla c'$.

Note that $c'$ is not necessarily a scalar (the case of non-symmetric space density), it is a vector $c'$, and $\nabla c'$ is a gradient of a vector. This gradient involves Christoffel symbols which are involved in the GR alternative to the classical field.

### A check of our derivation:

From (10) and (2), we get: $c' = (1 - 2GM/r^2)c$ and $dc'/dr = 2GM/r^2c$. Hence, according to (11)
$$g = - \frac{1}{2} cdc'/dr = - GM/r^2 \quad \text{or:}$$

$$g = - GM/r^3 \cdot r \qquad (13)$$

Note that M is a **gravitational mass**, since it comes from Einstein's field equation.

Note also that equations (12) is a **general equation**, not necessarily limited to the Newtonian gravitational field. This observation is relevant to the **Dark Matter** issue. See sub-section 4.2.

## 3. Inertia

### 3.1 “Rest” and Motion

According to our **GeometroDynamic Model** (GDM) of the Physical Reality [6] every space disturbance, a transversal or longitudinal wavepacket, must propagate at a characteristic velocity: $c_T$ for transverse and $c_L$ for longitudinal disturbances. We distinguish between $c_T$ and $c_L$ when necessary and according to the case. In most cases we use the symbol c instead of $c_T$.
“Rest”

Fig. (1), on the Left, describes “Rest”. In the context of the GDM, “Rest” refers to a condition
in which a wavepacket moves at velocity c along a closed circular path. In such cases, the
geometric center of the circulating wavepacket is defined as its position at rest within space.

Motion

Fig. (1), on the right, illustrates motion. **Motion** arises when the wave packet’s closed circular trajectory transitions into a helical (spiral) path. The GDM assumes that **the length of the wavepacket remains constant** during this transformation. From this single assumption, the results of Special Relativity can be derived (Fig. (1)). This transformation implies a reduction in the spiral radius R compared to the original circular radius R₀. According to Fig. (1):

$$
R = \frac{1}{\gamma} R_0 \tag{14}
$$

The resultant wavepacket (electron) motion, however, is always at the wave velocity c. Thus, a
translatory motion at constant velocity, v, does not involve any exertion of force. It is also obvious
that necessarily v ≤ c.

Newton’s first law and the first postulate of the theory of relativity - which asserts that no
particle or signal can exceed the speed of light c - naturally emerge from the underlying
dynamics described. The helical motion of a charged particle, which manifests as an electric
current formally directed along the negative -z axis, is intrinsically linked to the magnetic field
**B** and the vector potential **A**. The stability of the resulting circular trajectory is maintained by
a precise balance of forces: the outward centrifugal force is exactly countered by a centripetal
force. This centripetal force arises from the Lorentz force, which is generated by the magnetic
field produced by the circulating charge acting back upon itself [7].
<figure><img src="image_1.png" /><figcaption>Fig. (1) Rest and Motion</figcaption></figure>

This understanding led to the derivation and accurate calculation of the masses of the leptons and quarks and their anti-particles, see Appendix A.

## 3.2 The Elementary Particle Inertia

At rest, the energy of the electron is $U_0$, linear momentum is $P_0 = U_0/c$ and the angular momentum is $L_0 = P_0 R_0$.

In motion, energy, linear momentum and angular momentum are $U$, $P = U/c$ and $L = PR$ respectively.

The conservation of angular momentum $L = L_0$ (which is $1/2\hbar$ for the electron) implies that:

$PR = P_0R_0$ hence $UR = U_0R_0$ but:

$$R = 1/\gamma \cdot R_0 \qquad (14)$$

Thus:

$$U = \gamma U_0 \qquad (15)$$

Considering the energy as purely electromagnetic, where the elementary charge $Q$ is a sphere of radius $r_0$ [7], the relations $U_0= Q^2/2r_0$, $U= Q^2/2r$ and (2) give:
$$r = 1/\gamma \cdot r_0 \qquad (16)$$

For $v \ll c$ (3) is:

$$r = (1 - v^2/2c^2) r_0 \qquad (17)$$

Acceleration is accompanied by an increase in energy U and a corresponding reduction in the radii R and r. The external force applied to accelerate a particle performs work by further curving the surrounding space [7], thereby reducing r and R. This force increases internal stress and induces greater strain in the deformed space lattice, effectively enlarging its curvature. In this framework, Newton's Second Law emerges naturally: the need for force arises from the resistance of space to further deformation, giving rise to inertia.

Crucially, we have replaced Newton's axiomatic laws with a postulated physical model of the elementary particle as a moving space deformation. Among the significant merits of this replacement are:

* It leads to a **proof of the equivalence of gravitational and inertial mass**.

* It demonstrates that **gravitational mass is inertial mass** - not merely equivalent, but identical in origin.

* It reveals that **mass is not a fundamental attribute of matter**, but a practical construct derived from energy and space deformation.

* It proves that **inertia is an intrinsic property of matter**, arising from the geometry of space itself, thus removing the need for an additional field (such as the Higgs field) to account for it.

Note that $L= 1/2 \hbar$, $R=1/2 \hbar c/U$, $U_0= Q^2/2r_0$ and $r = Q^2/2U$ give:

$$\alpha = r/R = Q^2/\hbar c \qquad \text{Fine Structure Constant.} \qquad (18)$$
Note also that at “rest” the angular momentum can point in any direction, whereas in motion it can point, generally (see Section 3.1), in the direction of motion or opposite to it. This attribute is related to **Space Quantization**.

## 3.2 Mass m of an Elementary Particle

Our “relativistic” relation (15) gives:

$$U = \gamma U_0 = U_0 \left( 1 - \frac{v^2}{c^2} \right)^{-\frac{1}{2}} \approx U_0 + \frac{\frac{1}{2} U_0 v^2}{c^2} \quad \text{The second term:} \quad \frac{\frac{1}{2} U_0 v^2}{c^2} = \frac{1}{2} (U_0/c^2) v^2,$$

is identified as the **kinetic energy** where:

$$m_0 = U_0 / c^2 \qquad (19)$$

Thus, **inertial mass m** becomes merely a **practical term**. We also get:

$$m = \gamma m_0 \qquad (20)$$

Conventional physics distinguishes between **kinetic energy** - $1/2mv^2$ - which a particle possesses due to its motion, and its **internal energy** - $1/2m_0c^2$ - associated with intrinsic structure or rest mass. In contrast, within our framework, kinetic energy is also interpreted as **internal deformation energy**: it arises from the reduction in the characteristic radii r and R of the space deformation that constitutes the particle. As such, this energy cannot be meaningfully separated from the particle's rest energy. Both arise from the same geometric mechanism - changes in the curvature of space - which implies a **unified origin for motion and mass**.

It is important to note that the **electron model** presented here is a simplified representation of a more comprehensive model detailed in [7], which successfully derives the **masses of the leptons and quarks** from the geometry and dynamics of deformed space, see Appendix A.
# 4. Free Fall Acceleration (Central Acceleration)

## 4.1 Free fall

Free fall occurs because elementary particles are **moving circulating wave packets**, and as such, they are inherently sensitive to **spatial variations in the speed of light**. When a portion of a particle's circular path enters a region where the local speed of light is lower, the quantity $U/c$ (linear momentum per unit light speed) increases in that region. To conserve angular momentum $L=(U/c) \cdot R$, the radius $R$ of the circulation must decrease. This geometric adjustment leads to a net increase in the particle's **translational motion** - that is, it accelerates toward the region of slower light speed (see Section 3).

This process occurs **continuously**, giving rise to a **constant free-fall acceleration**, since Earth (like all masses) contracts the surrounding space and thus reduces the local speed of light [2]. Importantly, as discussed in Section 3.1, the **circulation plane of the wave packet is not perpendicular** to the direction of fall. This non-orthogonal orientation ensures that the particle persistently "feels" the gradient in light speed and continues to accelerate accordingly.

Because the orientations of circulating planes are **random across particles**, free fall emerges as a **statistical phenomenon** at the macroscopic scale - yet with a highly predictable and consistent result.

## 4.2 Dark Matter

Equation (12), which expresses **central acceleration as a consequence of an Inhomogeneous Deformed Space**, combined with a fundamental observation about the **Hubble flow**, has led us to a novel and straightforward resolution of the long-standing dark matter problem. The key observation is simple yet profound: within galaxies, the Hubble flow is nearly absent, while outside them, the flow continues unabated. This discontinuity generates a gradient in space density across the galactic outskirts - their 'skirt' - which gives rise to an additional central
acceleration. “Building on this insight, we demonstrate - theoretically and for the first time, and
**in contrast to MOND’s empirical approach** - that this spatial gradient results in an **additional**
**central acceleration**. In short:

It is the Inhomogeneous Hubble Flow - not dark matter - that generates the extra central
acceleration observed in galaxy rotation curves.

We propose that the **non-homogeneous space density** in and around galaxies, induced by the
observed **inhomogeneity in the Hubble flow**, gives rise to an **extra central acceleration**. This
additional acceleration is superimposed on the Newtonian acceleration produced by the
galaxy’s visible mass. Together, they determine the observed **flat rotation curves** and also
account for **gravitational lensing** effects typically attributed to dark matter.

This reasoning has led us to a **theoretical derivation of Milgrom's phenomenological
acceleration law**, firmly grounded in the geometry of deformed space and without the need for
exotic matter or modified dynamics:

$$
g = -(g_0GM)^{1/2} / r
$$

with $g_0 = -1.2 \pm 0.2 \times 10^{-8} \text{ cm s}^{-2}$. This result emerges from our understanding of space
as a deformable medium, as elaborated in:

*Gravity is Contracted - Not Bent - Space Around a Mass* [1]

*Gravitation Slows Light and Contracts Yardsticks, Ensuring Invariant Light Speed for All*
*Local Observers* [2].

Thus, there is no need for Dark Matter, or MOND. We do not have to modify known and
accepted physics to explain rotation curves and lensing.

On this subject in a yet to come dedicated paper.
## 5. Equivalence of Gravitational Mass M and Inertial Mass M'

Taking (13) and $F = gM'$, where M' is an inertial mass, gives: $F = -GMM'/r^2$ or

$$F = -GMM'/r^3 \quad (21)$$

In (21), M' is the inertial mass of the falling particle whereas M is the gravitational mass of the curving mass that creates the gradient in light velocity. Each and every mass falls free in the gradient of light velocity created by the presence of all other gravitational masses.

Let $M_1$ and $M_2$ be the gravitational masses of two particles and $M_1'$, $M_2'$ their respective inertial masses. According to (21):

$F_{12} = -GM_1M_2'/r^2$ and $F_{21} = -GM_2M_1'/r^2$. If $F_{12} = F_{21}$, which is Newton's third law (applicable only for the weak field case in which space translational symmetry holds) then:

$M_1M_2' = M_2M_1'$ or:

$$M_1'/M_1 = M_2'/M_2 = \text{constant.} \quad (22)$$

This is a proof of the **equivalence of gravitational mass and inertial mass**.

## 6. The Repulsive Force between Masses

Positive electric charges are known to **repel** each other. In the GeometroDynamic Model (GDM), this repulsion is attributed to the **positive curvature** assigned to positive charges.

Similarly, in General Relativity (GR), **positive curvature** is associated with **mass**. For GDM to be consistent with GR, it follows that **masses must also repel each other** under certain conditions.

Indeed, the **cosmological constant** $\Lambda$ in GR introduces a **repulsive component** to gravity, evident when GR is reduced to its weak-field limit - the **Poisson equation**. The resulting expression for the total field strength or gravitational acceleration is:
$$g = -\nabla \varphi = -GM/r^3 \cdot \mathbf{r} + 1/3 \cdot \Lambda c^2 \cdot \mathbf{r} \qquad (23)$$

This equation shows that at **large distances** - on the order of **hundreds of kiloparsecs** - the **repulsive acceleration** due to $\Lambda$ can dominate over the **attractive Newtonian term**, implying that **galaxies may repel one another**.

This leads to a compelling hypothesis:

**If galaxies repel each other, and space within galaxies does not expand, then this large-scale repulsion could be the driver behind the expansion of space between galaxies.**

Supporting this interpretation, our **recent experiments at the Hebrew University of Jerusalem (HUJI)** [10] have demonstrated that **positive ions ascend in Earth's gravitational field**, suggesting a repulsive interaction. This experimental evidence supports the idea that **both positive charges and masses induce positive curvature**, contracting space locally, and therefore must **mutually repel**.

In this way, **GDM aligns with GR** both theoretically and experimentally: the **positive curvature** associated with mass and charge leads to a consistent framework.

## 7. Summary

In this paper, we demonstrated how a mass responds to the **curvature of space** and moves accordingly. This process of **sensing and responding to spatial curvature** manifests as **free fall**, which is often - though incorrectly - referred to as “gravitational attraction.”

This understanding arises from our **model of elementary particles** as localized moving deformations in space, and from our understanding of **inertia**, where **mass is treated not as a practical construct**.
# Acknowledgements

We would like to thank the late **Professor Y. Silberberg** of the Weizmann Institute of
Science, as well as **Professor A. Zigler** and **Dr. Moshe Klein** of the Hebrew University of
Jerusalem, for their careful reading of the manuscript and their insightful comments.

# Declarations

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

[2] S. Barak: Gravitation Slows Light and Contracts Yardsticks, Ensuring Invariant Light Speed for All Local Observers

[3] S. Barak: The Essence of Inertia - Mass is Merely a Practical Term

[4] B. F. Schutz: General Relativity, Cambridge (2003)

[5] W. Rindler: Relativity, Oxford (2004)

[6] S. Barak: GeometroDynamic Model (GDM) of the Physical Reality August 2022

ISBN: ISBN 978-965-90727-1-2 https://www.researchgate.net/profile/Shlomo-Barak-2/research

A Spacetime GeometroDynamic Model (GDM) of Reality (a Theory Of Everything - TOE) A Realization of Einstein's Vision September 2021

DOI: 10.13140/RG.2.2.19796.00644

[7] S. Barak: On the Essence of Electric Charge Part 2: How Charge Curves Space. hal-01402667 (2016) https://hal.archives-ouvertes.fr/hal-01402667

[8] S. Barak: On the Essence of Electric Charge, Part 1: Charge as Deformed Space.
hal-01401332 (2016) https://hal.archives-ouvertes.fr/hal-01401332

[9] C.M. Ho and S.D.H. Hsu: Astronomical Constraints on Dark Energy. arXiv:1501.05952

[10] S. Barak Y. Neve-Oz A. Zigler: Experimental Evidence that Positive Ions Ascend and Negative Ions Descend Towards the Center of Earth August 2024 DOI: 10.13140/RG.2.2.28392.89609

## Appendix A The Electron/Positron Mass

Our geometry of deformed space lattices (Barak 2019), as opposed to the Riemannian geometry of curved manifolds, allows us to assign positive curvature to contracted regions of space - corresponding to positive electric charges - and negative curvature to dilated regions, corresponding to negative charges. This framework enables us to apply General Relativity (GR) in our derivations and demonstrate that a positive elementary charge can be modeled as a type of black hole, while a negative elementary charge resembles a white hole.

Using this approach, we construct a physical model of the electron (and other elementary particles) and analytically derive their fundamental properties, including inertial mass and spin. These results are beyond the predictive scope of both the Standard Model and String Theory. The equations for these particle masses involve only the fundamental constants G, c, $\hbar$, and $\alpha$ (the fine-structure constant). Our calculated values are consistent with the CODATA 2014 recommended data.

### Our result for the Inertial Mass of the Electron

$$M_e = \frac{s^2 \sqrt{2}}{\pi(1+\pi\alpha)} \sqrt{G^{-1}\alpha\hbar c^{-3}} \quad s = 1, [s] = LT^{-1}$$

$$M_e (\text{calculated}) = 0.910,36 \ 10^{-27} \text{g}$$

$$M_e (\text{measured}) = 0.910,938,356(11) \ 10^{-27} \text{g}$$

A dimensionality check: $[G^{-1}] = ML^{-3}T^2$, $[\alpha] = 1$, $[\hbar] = ML^2T^{-1}$, $[c^{-3}] = L^{-3}T^3$. Thus:

$$M = [s^2 \sqrt{G^{-1} \alpha \hbar c^{-3}}] = L^2 T^{-2} (ML^{-3} T^2 \cdot ML^2 T^{-1} \cdot L^{-3} T^3)^{1/2} = L^2 T^{-2} (M^2 L^{-4} T^4)^{-1/2} = L^2 T^{-2} ML^{-2} T^2 = M$$