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# Gravitation Slows Light and Contracts Yardsticks, Ensuring Invariant Light Speed for All Local Observers

Shlomo Barak

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Shlomo Barak. Gravitation Slows Light and Contracts Yardsticks, Ensuring Invariant Light Speed for All Local Observers. 2025. hal-05053730

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# Gravitation Slows Light and Contracts Yardsticks, Ensuring Invariant Light Speed for All Local Observers

Shlomo Barak

Taga Innovations 16 Beit Hillel St. Tel Aviv 6701716 Israel 26 Apr 2025

Corresponding author: shlomo@tagapro.com

## Abstract

The title, of this paper, encapsulates a central conclusion emerging from the reformulation of General Relativity using Deformed Space Geometry in place of the standard Riemannian Geometry framework. This reformulation enables the explanations of some gravitational effects by local variations in the speed of light and yardstick contraction, while **preserving the invariance of light speed for all local observers**. In General Relativity (GR), local observers consistently measure the speed of light as the invariant constant c, regardless of their position within a gravitational field. However, observers at a radial distance r from a gravitating mass m, perceive the coordinate speed of light as $c(r) = (1-2Gm / rc^2) c = ac$.

Traditionally, this coordinate-dependent speed is considered a mathematical artifact without physical significance. It is the change in time with the radial distance that is considered a real effect that explains astrophysical phenomena. We however, consider time as merely a practical term, whereas distance and velocity are fundamental. Thus, we are proposing that $c(r)$ is real while the locally measured invariant $c$ results from gravitational effects on measuring instruments. This perspective is developed within the framework of Deformed Spaces Geometry, an alternative to Riemannian geometry, which retains the formalism and solutions of GR but offers a more intuitive understanding of gravitational phenomena. Our approach addresses longstanding issues in GR and predicts novel phenomena, some of which have been observed in recent experiments conducted at the Hebrew University of Jerusalem.

Keywords: Space; Geometry; Deformed Space; Riemannian Geometry; General Relativity
# 1. Introduction

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. 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 [2] [1]. 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) [3].

Equation $c(r) = (1-2Gm / rc^2) c = ac$ of coordinate light velocity [4] [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 phenomena 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.

# 2. Arguments for the Space Density Dependence of Light Velocity in a Deformed 3D Space Lattice [4]

## 2.1 Einstein's Thought Experiment

Fig. (1) illustrates a miniature elevator acting as a "clock". A beam of light enters the elevator from the left at point A1. In the absence of a gravitational field, the beam travels straight
parallel to the floor (dashed line) and strikes the opposite wall. The two events: The entry at A1 and the hitting of the right wall - define a measured time interval $d\tau$. Now consider the case where there is still no gravitational field, but the elevator is uniformly accelerated upwards. In this case the light beam, reaches point A2 due to the elevator's upward displacement during the beam's horizontal motion. If this acceleration matches the local free-fall gravitational acceleration $g$, then, according to the **equivalence principle**, it is impossible to distinguish between two situations: an elevator accelerating through empty space or remaining stationary within a gravitational field, with the same free fall acceleration $g$. This is the foundation of Einstein's thought experiment, which led to the conclusion that gravitation causes the deflection of light.

<figure><img src="image_2.png" /><figcaption>Fig. (1) The Elevator “Clock”</figcaption></figure>

This deflection, entering A1 and reaching A2, is the reason why the time $d\tau'$ between the events of entering A1 and reaching A2 is a longer time interval than $d\tau$, this is the known **time dilation** $d\tau' > d\tau$. On the basis of this discussion, the above result is accurately given by: $d\tau' = a^{-1/2} d\tau$

where $a = (1-2Gm/c^2)$, $a < 1$ see section 4.

## 2.2 Our Thought Experiment

Fig. (2a Left) shows a miniature elevator acting as a "clock." A beam of light enters the elevator from below at point A1. In the absence of a gravitational field, the beam travels straight upward, perpendicular to the floor, and strikes the ceiling at point A2. The two events - entering at A1 and hitting the ceiling at A2 - define a measured time interval, $d\tau$. Now,
consider the case where there is still no gravitational field, but the elevator is uniformly
accelerating upward. In this situation, the light beam reaches A2 later than it would without
acceleration, due to the elevator's upward motion during the beam's travel. This acceleration
causes the time interval $d\tau'$ between the two events to be longer than $d\tau$. This well-known
phenomenon is called time dilation, with $d\tau' > d\tau$. The precise relationship is given by:
$$d\tau' = a^{-1/2}d\tau.$$ Pound and Rebka (1959) experimentally confirmed this effect.

<figure><img src="image_3.png" /><figcaption>Fig. (2) The Elevator “Clock”</figcaption></figure>

If this acceleration matches the local free-fall gravitational acceleration g, then, according to the **equivalence principle**, it is impossible to distinguish between two situations: an elevator accelerating through empty space or remaining stationary within a gravitational field experiencing the same free-fall acceleration g, see Fig. (2b Right). This thought experiment led us to consider that gravitation might cause a contraction of space, resulting in a gradient in its index of refraction.

In Fig. (2b Right), the index of refraction n, illustrated by the grayscale shading, is higher at the bottom and lower at the top (n1 > n2). We argue that this gradient in the index of refraction - caused by a gradient in space density - modifies the speed of light, thereby leading to time dilation. This alternative viewpoint inspired the development of the Geometry of
Deformed Spaces, proposed as a new foundation for General Relativity, replacing the traditional Riemannian geometry of curved manifolds.

Note that if, in Fig. (2b), the light beam enters from the left side, as shown in Fig. (1), it must bend. In this case, however, we attribute the bending to the gradient in the index of refraction (space density) across the beam, see Fig. (2b). Some may correctly point out that the classical derivation of the bending angle based on this reasoning produces only half of the result predicted by General Relativity (GR) - Einstein (1907). However, in our approach, we derived and calculated the bending angle according to GR principles, with the crucial distinction that space is contracted around a mass, not curved (1). We assure the reader that our careful derivation fully reproduces the conventional GR result.

Note that both the upward acceleration $g$ in Fig. (2a) and the gravitational field $g$ in Fig.(2b) cause the same contraction in the elevator's height. Therefore, the equivalence principle holds.

## 2.3 The Consequence

In both thought experiments, if the elevator “clock” were realized as a physical clock operating under gravity, it would exhibit a longer time interval - meaning it would tick more slowly, so $dt' < dt$. The deflection of the light beam shown in Fig. (1) or the slowing of the beam in Fig.(2b) illustrates the curvature/deformation, of the temporal dimension, as described by the first term $-ac^2 dt^2$ in the Schwarzschild metric (3) in section4. Thus:
$$ic(r)dt' = ia^{1/2} cdt \quad (i=\sqrt{-1}) \quad \text{and} \quad dt' = a^{1/2}dt.$$

## 2.4 Arguments for the Space Density Dependence of Light Velocity

Gravitational waves are fundamentally waves of space itself, propagating at the same speed as electromagnetic waves - that is, the speed of light. According to Quantum Field Theory (QFT), all 17 known fields are embedded within space and conform to its topology. It is therefore reasonable to consider light as a space wave as well. This view is supported by our
models of the photon and graviton, which conceptualize these particles as manifestations of localized space vibrations. When ensembles of such particles are in phase, they exhibit classical wave behavior. In our theory, space is treated as a physical entity possessing a cellular internal structure characterized by local properties such as density and degree of deformation.

Furthermore, classical electromagnetism relates the speed of light to the permittivity $\epsilon$ and permeability $\mu$ of space via the well-known relation:

$$c(r) = (\epsilon\epsilon_0 \cdot \mu\mu_0)^{-1/2}$$

Here, $\epsilon$ and $\mu$ are the relative values associated with matter, while $\epsilon_0$ and $\mu_0$ pertain to the vacuum properties of space. Since $\epsilon$ and $\mu$ depend on matter density, we posit that $\epsilon_0$ and $\mu_0$ are, analogously, functions of the local space density $\rho(r)$ [1]. This assumption aligns with our foundational view that matter is not foreign to space but is instead a localized configuration of the vibrating space itself - space is the only substance that exists. In our paper titled "Electric Charge and Its Field as Deformed Space" [5] we define electric charge density in terms of space density. Remarkably, this definition alone - without relying on phenomenological evidence - leads directly to the theory of Electrostatics. Moreover, when combined with the Lorentz Transformation, Electrostatics, without any phenomenology, yields the complete Maxwell Electromagnetic Theory, as we have demonstrated. From this, we infer that the true (local) speed of light is inversely proportional to the local space density:

$$c(r) \propto 1/\rho(r)$$

In section 5.1, we present the well-known expression for coordinate light velocity in the presence of gravitation, which corresponds to space contraction:

$$c(r) = (1 - 2Gm/rc^2) c = ac$$

Our interpretation of this equation proving the invariance of light velocity for all local observers in Section 5 - leads to the logical conclusion that this expression describes a physically real
variation of light speed due to space deformation. This conclusion stands in contrast to the
conventional interpretation, which attributes the variation to time of observers' clocks.

We consider time and mass as merely practical terms whereas distance velocity and energy are
fundamental. This subject, however, lies beyond the scope of this paper.

3. The Meaning of Coordinate Quantities in Deformed Space Geometry

In conventional geometries like Riemannian geometry, the coordinate system is a purely
mathematical construct, selected based on convenience or symmetry. Infinitesimal quantities
such as dr and dt have no inherent physical meaning; they are simply coordinate differentials,
defined arbitrarily and applied uniformly throughout the manifold. Within this framework, a
physical "yardstick"- a fixed reference for length - might be represented by dr, but its value
remains invariant for all observers, regardless of their location in the manifold or the local
physical conditions. This universality underlies the view that the **coordinate velocity**, defined
as dr/dt, is often not considered a physically real quantity but rather a formal abstraction. It
lacks physical substance because both dr and dt are coordinate-dependent constructs,
disconnected from any underlying local structure of space.

In contrast, within the geometry of Deformed Spaces, dr is directly and physically linked to
the local space density. The core assumption is that space consists of discrete, deformable
cells, and that no measurable length can be smaller than the linear size of a space cell at a
given location. Consequently, dr is bounded by the **minimum local spatial scale** and cannot
be reduced arbitrarily. Within this framework, a standard yardstick (refer to section 4.3.2), is
defined as either a single cell or a sequence of *n* cells represented by **dr**. It is standard in the
sense that the value of *n* remains the same for all observers. However, the physical length
measured by a distant observer reflects the local deformation of space.
It is important to note that observers located in regions with different space densities are themselves deformed, consistent with the deformation of the space they inhabit.

This fundamental dependence of dr on space density reshapes the meaning of the coordinate velocity of light. Both dr and dt are now rooted in physically constrained units - dr by the local space density and dt by the ratio of a fixed yardstick to the local speed of light. Thus, the ratio dr/dt acquires a real physical significance. It becomes a **local observable** that varies with the intrinsic geometry of space itself. Hence, the coordinate velocity of light is no longer an abstract mathematical artifact, but a real, measurable quantity that depends on the local space density.

# 4. The GR Schwarzschild Metric

## 4.1 Schwarzschild Metric

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 [6], the line element $ds^2$ is given by:

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

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

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

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:

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

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

## 4.2 Observers and Coordinates

A **Local Observer** is an observer situated at a specific point within an n-dimensional
manifold or an *n*-dimensional space. This observer is embedded within their immediate
surroundings, shares the same dimensional structure, and is affected by any deformations
occurring nearby.

A Faraway Observer is also located at a point within an *n*-dimensional manifold or space and shares the same dimensional framework. However, this observer is not subject to any local deformations, as none occur in their vicinity. In other words, their local space is Euclidean, or very close to it.

**The variables** $t$, $r$, $\theta$, and $\phi$ ($\theta$, and $\phi$ included in $\Omega$) appearing in the metric (1) are the coordinates used by the faraway observer, who is positioned at a large radial distance $r$ from the origin at the center of the gravitating mass $m$ (see [4], p.629). These are the coordinates associated with the Schwarzschild metric (1), a solution to the General Relativity field equations.

**4.3 Coordinates and Prime Coordinates**

The variables $t$, $r$, $\theta$ and $\phi$, which appear in equation (1), when marked with a prime notation
$t'$, $r'$, $\theta'$ and $\phi'$ represent measurements made by the local observer. For example, the second
term in (3) connects $r'$ to $r$ through the relation $dr'^2 = a^{-1} dr^2$, or $dr' = a^{-1/2} dr$.

**4.3.1 Riemannian curved manifolds**

Let **dr** represent the length of a yardstick according to a faraway observer. In the geometry of Riemannian curved manifolds, the local observer's yardstick maintains the same length. In the equation above, **dr'** denotes the radial distance, as measured by the local observer using the same standard yardstick **dr**. On a positively curved manifold, it necessarily holds that **dr'** > **dr**.

The first term in equation (3), $ac^2dt^2$ (ignoring the negative sign), is derived through a more
involved process as follows.
**Time Length Measurements:**

The measured length of a time interval $t'$ between two events depends on the duration of the unit time $d\tau'$ on the local clock. A longer time unit on the clock results in a smaller numerical value for the time interval.

Time dilation, in both General Relativity (GR) and Special Relativity (SR), arises from the stretching of the faraway observer's **time unit** $d\tau$ to become, for the local observer, the **local clock's time unit** $d\tau'$. This time dilation is due to either gravitation (GR) or motion (SR). In this context, $d\tau' > d\tau$, and more precisely, $d\tau' = a^{-1/2} d\tau$.

Gravitation stretches the unit time $d\tau$ into $d\tau'$, causing the clock to run more slowly.

Consequently, the measured interval $t'$ between two events, from the perspective of the faraway observer, is shorter: specifically, $dt' < dt$, with the precise relation $dt' = a^{1/2} dt$.

**4.3.2 Deformed Spaces Geometry**

In Barak's Deformed Space Geometry, a yardstick represented by $dr$ is not identical for different observers - it reflects the local deformation of space. Space consists of discrete, deformable cells, and no measurable length can be smaller than the linear size of a cell at a given location.

Therefore, $dr$ is constrained by the minimum local spatial scale and cannot be reduced arbitrarily. A standard unit of length or a yardstick, can be defined as $n$ contiguous space cells aligned in a straight row, where $n$ is the same for all observers and is chosen arbitrarily.

Within this framework, a yardstick is standard in the sense that the value of $n$ remains the same for all observers. However, the physical length measured by a distant observer reflects the local deformation of space. It is important to note that observers located in regions with different space densities are themselves deformed, consistent with the deformation of the space they inhabit.
The radial distance dr' represents the actual measured distance within the contracted space. It appears larger than the corresponding calculated radius, indicating a space with positive curvature (see Fig. 6 in [1]). However, the number of length units (or "cells") contained in this distance depends on the observer's location. In regions of higher space density, such as near a gravitational source, the unit of length (cell size or n times it) is smaller, and more cells are counted over the same distance. In contrast, farther from the gravitational source where space density is lower, the unit of length is larger, so fewer cells are counted over the same distance.

This relation is captured by: $dr' = a^{1/2} dr$ ($a < 1, dr' < dr$)

This fundamental dependence of dr on space density redefines the meaning of the coordinate velocity of light. Since both dr and dt are now physically grounded - dr by space density and dt by the ratio of a given yardstick to the local speed of light - the ratio dr/dt acquires a real physical interpretation. It becomes a locally measurable quantity that varies with the intrinsic geometry of space. Thus, in Deformed Space Geometry, the coordinate velocity of light is no longer a mere coordinate artifact but a genuine, density - dependent observable. The variation of dr across space is not a coordinate ambiguity but a direct manifestation of the local physical reality of space.

Similarly, the duration of a time interval t' between two events depends on the local unit of time dτ' of the clock. A longer time unit yields a smaller numerical value for the measured time interval. In this framework, this effect arises from the gradient of space density - and consequently the gradient of the space refractive index - around a gravitating mass (see Section 2.2). Time dilation in both General Relativity (GR) and Special Relativity (SR) is related to the elongation of the faraway observer's time unit dτ into the local observer's dτ' due to gravity or motion. Here, dτ' > dτ, more precisely: dτ' = a⁻¹/² dτ. Gravitation dilates (stretches) the unit of time from dτ to dτ' causing **the local clock to run slower**. Thus, the time interval t' measured
locally by the local observer between two events is shorter than the faraway observer's
corresponding measurement t, following dt' < dt and accurately dt' = a¹/₂dt.

**5. The Invariant Light Speed for All Local Observers**

**5.1 Coordinate Light Velocity [4]**

The 4D **spacetime interval** ds² between two events; the “emission” of a short pulse of light at point A and the “arrival” of this pulse at point B, is the basis of the measurement of the speed of light. In this case:

$$
ds^2 = 0.
$$

Using equation (3), (where r² dΩ² is ignored [4]) we get:

$$
0 = -a \c^2 dt^2 + a^{-1} dr^2 \quad \text{thus} \quad a^{1/2} c \cdot dt = a^{-1/2} dr \quad \text{that gives:}
$$

$$
\frac{dr}{dt} = ac = c(r) \tag{4}
$$

This, c(r)= dr/dt = ac, for a **faraway observer**, is the light velocity close to a mass m, where
c' < c. This, dr/dt = c(r), is a slower light velocity since, according to equation (2), a < 1.

In the literature dr/dt in equation (10) (1) is called **coordinate speed of light**, [2].

**5.2 The Local Observer speed of Light Based on Riemannian Manifolds Geometry**

The relationship between local observers' radial distant r' and the coordinate distant r that
reflects gravitational length elongation (due to the curving), is given by:

$$
dr' = (1 - 2Gm / rc^2)^{-1/2} dr = a^{-1/2} dr \quad dr' > dr \tag{5}
$$

The relationship between local observers' time t' and the **coordinate time** t, that reflects
**gravitational time dilation**, is given by:

$$
dt' = (1 - 2Gm / rc^2)^{1/2} dt = a^{1/2} dt \quad dt' > dt \tag{6}
$$

The argument here is that the bending of the trajectory of light, due to the curved manifold,
result in an extended duration of travel. Accordingly, the speed of light as perceived by local observers becomes:

$$dr'/dt' = a^{-1/2} dr / a^{1/2} dt = a^{-1} dr/dt \quad \text{and substituting (4) gives:}$$

$$dr'/dt' = c \qquad (7)$$

This is the **invariance of the speed of light for all local observers.**

## 5.3 The Local Observer speed of Light Based on Barak Deformed Spaces Geometry

The relationship between local observers' radial distant r' and the coordinate distant r is understood, similarly to the effect of time, and reflects the **gravitational length contraction** (due to space contraction). This is given by:

$$dr' = (1-2Gm / rc^2)^{-1/2}dr = a^{-1/2} dr \quad dr' > dr \qquad (5)$$

The local observer unit of length or yardstick is contracted and hence a distant measurement dr' will take more measuring steps and dr' > dr.

The relationship between local observers' time t' and the **coordinate time** t, that reflects **gravitational time dilation**, is given, as before in the Riemannian geometry, by:

$$dt' = (1-2Gm / rc^2)^{1/2} dt = a^{1/2} dt \quad dt' > dt \quad (6)$$
The argument presented is that the trajectory of light through a region of denser, contracted space near a mass result in an extended duration of travel. Accordingly, the speed of light as perceived by local observers is given by:

$$dr'/dt' = a^{-1/2} dr / a^{1/2} dt = a^{-1} dr/dt \quad \text{and substituting (4) gives:}$$

$$dr'/dt' = c \qquad (7)$$

This is the **invariance of the speed of light for all local observers.**

## 5.4 In Conclusion

In both geometries, metric (1) applies, and for all local observers - including a faraway observer, who, being situated within his own local region, is likewise treated as a local observer - the invariance of c is preserved.

$$dr'/dt' = c \qquad (7)$$
And in words:

**The invariance of the speed of light for all local observers holds under both Riemannian manifold geometry and Barak's deformed space geometry.**

**Q.E.D.**

**A remainder:** This conclusion does not, in any way, preclude the potential validity of a coordinate – real light velocity.

# 6. Poetic Insight and Summary

Pythagoras envisioned reality as fundamentally numerical. In our view, this poetic notion finds a concrete form: space possesses a cellular, lattice-like structure, and the geometry of Deformed Spaces is rooted in counting these fundamental units. According to our Theory of Everything (TOE), **space is all there is**—thus, reality itself is constructed from numbers.

Building on this framework, we propose a shift from Riemannian Geometry - the traditional backbone of General Relativity - to the new geometry of Deformed Spaces. Within this updated framework, we reach a pivotal conclusion: gravitation slows down light and contracts yardsticks, preserving the invariance of light speed for all local observers. Consequently, the speed of light should no longer be regarded as an absolute constant like $\pi$, but rather as a locally maintained quantity shaped by the structure of space itself.

# Acknowledgments

We would like to thank Dr Moshe Klein of the Hebrew University of Jerusalem for his careful reading and helpful 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

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