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# Inhomogeneous Hubble Flow -Not Dark Matter -Creates the Extra Central Acceleration Affecting Galaxy Rotation Curves

Shlomo Barak

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Shlomo Barak. Inhomogeneous Hubble Flow -Not Dark Matter -Creates the Extra Central Acceleration Affecting Galaxy Rotation Curves. 2025. hal-05112248

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Preprint submitted on 14 Jun 2025

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# Inhomogeneous Hubble Flow - Not Dark Matter - Creates the Extra Central Acceleration Affecting Galaxy Rotation Curves

Shlomo Barak

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

Corresponding author: shlomo@tagapro.com

## Abstract

We contend that the non-Homogenous Space density in and around galaxies, due to the **observed** inhomogeneous Hubble-Flow, creates an additional Central Acceleration. This central acceleration is added to the central acceleration of the galaxy ordinary mass. The combined acceleration affects rotation curves and cause lensing. This reasoning led us to the derivation of the **Milgrom phenomenological** central acceleration $g = -\sqrt{g_0GM}/r$

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 cellular 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.

**Key Words:** Gravitation; Dark Matter; Hubble flow; Light Velocity; General Relativity
# 1 Introduction

The issue of Rotation Curves in galaxies led, in 1933, to the Zwicky hypothesis of Dark Matter (DM) [3]. Alternatively, it led, in 1983, to the suggestions by Milgrom [4] to modify Newtonian Gravity (MOND) and by Bekenstein [5], in 2004, to modify General Relativity (TeVeS).

We show, based on General Relativity, that the gravitational field (central acceleration) around a mass is related to the gradient in its **coordinate light velocity**, which is a real **velocity** [1] [2]. In and around a galaxy the global Hubble flow, affected by the presence of the galaxy mass, is non-homogenous. This non-homogenous space expansion, as we show, creates a non-homogenous space density and an additional gradient in the coordinate light velocity. This gradient is responsible for an additional central acceleration and lensing. In the example, presented here, its value is $g_0 = -1.2 \times 10^{-8} \text{ cm s}^{-2}$, which is responsible for flat rotation curves in our galaxy. We also present an argument explaining why Milgrom's phenomenological MOND equation, which describes rotation curves in galaxies, is approximately correct.

# 2. Light Velocity

We propose that the speed of light is influenced by the density of space, analogous to its dependence on the density of the medium through which it propagates; denser regions of space correspond to a reduced light speed. In previous work [1], we demonstrated that gravitation can be interpreted as a contraction of space, leading to increased spatial density near massive bodies and, consequently, a slower propagation of light. This insight prompted us to investigate the validity of our hypothesis and to affirm that the constancy of light speed for all local observers remains intact [2]. This constancy arises because the measuring instruments of local observers, such as yardsticks, also undergo contraction in regions of higher spatial density [1].
Our investigation is grounded in General Relativity (GR), wherein we replace the traditional Riemannian geometry with the novel framework of Deformed Spaces Geometry [1] [6]. This approach preserves the equations and solutions of GR, along with the concept of curvature, while offering a more tangible and intuitive understanding of gravitational phenomena. It addresses longstanding issues and predicts new phenomena, some of which have been observed in recent experiments conducted at the Hebrew University of Jerusalem (HUJI) [7].

Equation $c(r) = (1-2GM / rc^2) c = ac$ of coordinate light velocity [8] [Section 5.1] suggests that, within the conventional interpretation of GR, space can be attributed a virtual index of refraction:

$$n(r) = c/c(r) = 1/(1-2GM / rc^2) = 1/a$$

Gravitational phenomenon are **traditionally** explained by spacetime curvature and the resulted time dilation. This refractive index with our geometry offers an intuitive mechanism for understanding. By attributing variations in light speed and measuring standards to changes in space density, we provide alternative explanations for several key gravitational effects:

* **Gravitational Lensing**: Interpreted as the refraction of light in regions with a gradient in space density, leading to the bending of light trajectory near massive objects.

* **Shapiro Delay**: Explained by the increased travel time of light signals traversing areas of higher space density, resulting in observable time delays.

* **Gravitational Redshift**: Reconsidered as a frequency shift arising from the energy exchange of photons moving through varying space densities.

* **Casimir Effect Variations**: Predicted changes in the speed of light between closely spaced plates due to vacuum energy density levels affecting space density (assuming
nonlinearity of the elasticity of space). In this case a local observer can observe directly
the change in light velocity.

This fourth predicted phenomenon can be explored locally, where no gravitational gradient is involved. The first three phenomena are interpreted by faraway observers far from the places in which they occur. The Deformed Space Geometry framework not only preserves the mathematical structure of General Relativity but also provides a more tangible interpretation of gravitational interactions. By focusing on the material-like properties of space, this approach offers new insights into the nature of gravity and its effects on light and measurement standards.

Recent experimental observations, such as those conducted at the Hebrew University of Jerusalem, reveal phenomena consistent with the predictions of Deformed Space Geometry.

**3. Central 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.
## 3.1 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 $r > r_c$ according to Schutz [9], the line element $ds^2$ is given by:

$$ds^2 = -(1-2GM/rc^2)c^2dt^2 + (1-2GM/rc^2)^{-1}dr^2 + r^2 d\Omega^2 \quad (10.36 \text{ in } [9]) \quad (1)$$

Note that we have included the factor $c^2$ in equation (1), following classical physics conventions. We define a **gravitational scale factor** $a$ as:

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

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

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

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

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

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 $\rho_0$. 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 [10] 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 $r^2 d\Omega^2$:

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

$$a^{1/2} c dt = a^{-1/2} dr$$

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

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:

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

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].

## 3.2 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/ $rc^2 < 1$:

$$dr/dt = ac = (1 - GM/rc^2)c = (1 + 2\phi/c^2)c \qquad (9)$$

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

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

From (9) the gravitational potential is:

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

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

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

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)
$g = -\frac{1}{2}cdc'/dr = -GM/r^2$ 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.

## 4. The Overlooked Central Acceleration Due to the Inhomogeneous Hubble Flow in and Around Galaxies

The **cosmological scale factor** (CSF), $\alpha$, in the epoch of galaxies formation 500–700 Myr

$(z = 8–11)$ after the Big-Bang [11], is notated $\alpha_b$. Taking $z = 9$ gives:

$\alpha_b = 1/(z+1) = 0.1$, whereas the present CSF in the intergalactic space is $\alpha_0 = 1$.

$$\alpha_b = 0.1 \qquad \alpha_0 = 1 \qquad (14)$$
Note that the CSF, $\alpha$, in this section is not the gravitational scale factor, $\alpha$, of Section 2.

**Space in the universe expands, but space within galaxies does not** [12] [13]. We, however, assume that at some region in the galaxy or on its skirt space starts to expand gradually to reach asymptotically $\alpha_0=1$.

A simple **toy function** for a variable CSF, in and around galaxies, is:

$$ \alpha = \alpha_b + (\alpha_0 - \alpha_b) [1 - \exp(-r/(R/4))] \quad (15) $$

R is the **Hubble sphere radius**. For $r=0$, $\alpha = \alpha_b$ and for $r \to R$, $\alpha = 0.98$, which is close to $\alpha_0 = 1$.

Note that the radius of the universe is many times larger than the Hubble sphere radius R. Taking R/4 is **arbitrary**, but based on the size of “dark matter halos” it is **reasonable**; it should, however, be supported by observations.

Substituting the values $\alpha_b = 0.1$ and $\alpha_0 = 1$, of equation (14), in equation (15) gives:

$$ \alpha = 0.1 + 0.9[1 - \exp(-4r/R)] \quad (16) $$

For $r \ll R$ equation (16) becomes:

$$ \alpha = 0.1 + 3.6r/R \quad (17) $$

According to equations (11) and (17) and using the Hubble parameter $H = c/R$,

defined as $H = \alpha'/\alpha$, gives for $r \ll R$ the following value for $dc'/dr$:

$$ dc'/dr = c d \alpha^2/dr = c2\alpha d\alpha/dr = 2 \cdot \alpha \cdot 3.6 \cdot c/R $$

Taking for $\alpha$ its average value $(\alpha_b + \alpha_0)/2 \sim 0.5$ gives:

$$ dc'/dr = 0.36H \quad (18) $$
The H value as of today - the Hubble constant $H_0$, [14], is:

$$H_0 = 2.26 \pm 0.25 \times 10^{-18} \text{ sec}^{-1}.$$

Substituting this value (without the error range, since we are using an artificial toy function) in equation (18) gives:

$$\frac{dc'}{dr} = 0.36 H_0 = 0.81 \times 10^{-18} \text{sec}^{-1} \qquad (19)$$

The value for the **central acceleration**, due to the inhomogeneous Hubble Flow, as of today, is calculated using equations (11) and (19):

$$g = - \frac{1}{2} \frac{dc'}{dr} = -1.22 \times 10^{-8} \text{ cm s}^{-2}. \qquad (20)$$

This acceleration, notated $g_0$, is:

$$g_0 = -1.22 \times 10^{-8} \text{ cm s}^{-2} \qquad (21)$$

Note the fit of our value for $g_0$, in equation (21), to the observed [2] MOND $g_0$, which is:

$$g_0 = -1.2 \pm 0.2 \times 10^{-8} \text{ cm s}^{-2}. \qquad (22)$$

MOND theory uses the notation $a_0$ rather than $g_0$. Note that this kind of central acceleration (21) is responsible for **flat rotation curves**. The real situation, though, is much more complicated since space is kind of a fluid and the inhomogeneous Hubble Flow creates complex space density patterns in and around galaxies. How the mass of a galaxy and its motion affect the Hubble flow should be further explored.

Gravitation is the contraction of space, whereas space expansion is the dilation of space. $g_N$ is the result of gravitational space contraction (curving) whereas $g_0$ is the result of space dilation (curving). Let $r_0$ denote the distance from the center of a galaxy at which space contraction was balanced by space dilation, in the epoch of the galaxy's creation. This balance at $r_0$, with the larger $g_0$ **of that time**, is expressed by the equality $g_N = g_0$, or:
$$GM/r_0^2 = g_0 \qquad (23)$$

Thus:

$$r_0 = (GM/g_0)^{1/2} \qquad (24)$$

Note the following: With time H becomes smaller and so does the gradient in light velocity, see equation (18). Thus, the zone of balance, at $r_0$, see (24), moves forward, away from the center of the galaxy, as if “Dark Matter Halos” grow with time.

Our central acceleration, equation (12), is based on a gradient in the coordinate light velocity; hence we can explain **lensing**, including the lensing of empty zones of space.

Note that the **geometric average (mean)** of the Newtonian **central acceleration** $g_N = -GM/r^2$, and $g_0$ gives:

$$g = - \frac{\sqrt{g_0 GM}}{r} \qquad (25)$$

We take the geometric mean, as central accelerations are inherently related to the radii of space curvatures. A detailed discussion of this relationship, however, lies beyond the scope of this paper.

Equation (25) bears resemblance to the phenomenological MOND equation proposed by Milgrom [4]. In MOND, when central accelerations fall below the characteristic scale $g_0$, Newtonian dynamics are modified accordingly. In our model, **by contrast**, $g_0$ represents a genuine additional central acceleration rather than a threshold for modifying Newtonian gravity. It is important to emphasize that our approach remains fully consistent with General Relativity and its weak-field Newtonian limit, and does not require any modification to these established framework
# 5. Summary

In this paper, we present a novel derivation of a MOND-like acceleration law grounded in the geometry of deformed 3D space, as proposed by the GeometroDynamic Model (GDM). Without modifying Newtonian dynamics or General Relativity, we show that the additional central acceleration observed in galaxy rotation curves can naturally emerge from spatial inhomogeneities in the Hubble flow. Unlike Milgrom's MOND, where a characteristic acceleration $g_0$ signals the breakdown of Newtonian dynamics, our $g_0$ represents a real, physically derived central acceleration due to the curvature properties of space. We demonstrate that the geometric mean is the appropriate averaging method for such accelerations, reflecting their deep connection to the radii of spatial curvature. This approach offers a unified, geometry-based explanation of the phenomena commonly attributed to dark matter, while remaining fully compatible with the existing framework of General Relativity in its weak-field limit.

# Acknowledgements

We would like to thank Dr. Moshe Klein of the Hebrew University of Jerusalem, for his 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
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