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# Geometric construction of gravitation

Stefan Catheline

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Stefan Catheline. Geometric construction of gravitation. 2025. inserm-04924918

HAL Id: inserm-04924918

https://inserm.hal.science/inserm-04924918v1

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# Geometric construction of gravitation

Stefan Catheline

LabTAU, INSERM, Centre Léon Bérard, Université Lyon 1, Univ Lyon, F-69003, LYON,
France

The equations of classical mechanics derive from Kepler's laws as interpreted by Isaac Newton. Relativistic solid rotation makes it possible to rehabilitate René Descartes's vision of gravitation and to give it a geometric construction in an Euclidian space. The profoundly relativistic nature of Newton's gravitation law is recovered.

ORCID: 0000-0002-5524-4120 stefan.catheline@inserm.fr

Plunged into the gravitational field of a mass of classical gravitational radiusª r<sub>G</sub>, any inertial mass at distance r obeys Kepler's third law: ω²r³ = c²r<sub>G</sub>. By removing the rotation ω = v/r from this expression, Isaac Newton revealed a relationship between accelerations, v²/r = c²r<sub>G</sub>/r², with inertial acceleration as the term on the left and a form of the universal law of gravitation as the term on the right. Let us rehabilitate rotation. Newton's equation can be simplified slightly, v²r = c²r<sub>G</sub> to exhibit the velocity ratio v/c = √(r<sub>G</sub>/r). This latter expression is fundamental because it allows us to introduce an angle of aberration θ through cosθ = v/c whose sine is none other than the contraction coefficient α = √(1 - v²/c²) from special relativity [1]. Gravitation can therefore be written in trigonometric form as follows:

$$ \cos\theta^2 = \frac{r_G}{r} \qquad (1). $$

<footnote>ª The classical gravitational radius of a mass m, r<sub>G</sub> = Gm/c², where G and c stand for the gravitational constant and the speed of light, should not be confused with the Schwarzchild radius (black hole), which is twice as large: r<sub>S</sub> = 2Gm/c².</footnote>
Gravitational rotation, deduced from Kepler's third law, $\omega = \frac{c}{r}\sqrt{\frac{r_G}{r}}$, can now take another form, $\omega = \frac{c}{r}\cos\theta = \frac{v}{r}$, and thus expresses classical rotation again. Knowing that in the theory of relativistic solid rotation [2] $\cos\theta = \frac{r}{R}$, it also follows that $\omega = \frac{c}{R}$. The radius R designates both the distance from the horizon of the solid rotation undergone by the satellite mass and the curvature of its space. These results eventually led to the geometric construction of gravitation of Figure 1. From this ground breaking point of view, the law of gravitation set out in equation 1 can be geometrically translated by the application of right-angled triangles of angle at the vertex $\theta$. This work thus confirms once again the profoundly relativistic nature of Newton's gravitation law.

<figure><img src="image_3.png" /><figcaption>FIG. 1: Geometric construction of gravitation from the rotation of the gravitational mass $r_G$ and the relativistic solid rotation R.</figcaption></figure>

Let us now look for the geometric origin of gravitation within the interaction between the two rotations introduced so far: $r_G$ and R. There comes a curvature potential $V_R$ associated with the interaction of the rotation $\omega$ with the unit angular momentum of the gravitational mass $\Gamma = cr_G$ so that $V_R = \omega cr_G = \frac{c^2 r_G}{R}$.
Recalling the relation $cos\theta = \frac{r}{R}$, we obtain the relation between curvature potential and gravitational potential $V_G$:

$$V_{\Gamma} = \frac{c^2 r_G}{r} \cos\theta = V_G \cos\theta \qquad (2)$$

This relationship has important implications. Firstly, it explains that the gradient according to $R$ of the curvature potential $V_{\Gamma} = \frac{c^2 r_G}{R}$ gives the law of universal gravitation, as does the gradient according to $r$ of the gravitation potential $V_G = \frac{c^2 r_G}{r}$. In addition, the field of classical mechanics is characterised by a curvature potential in $\frac{1}{R}$ that is small compared to that of gravitation in $\frac{1}{r}$, $V_{\Gamma} \ll V_G$. This explains how a photon emitted from a satellite follows a quasi-rectilinear trajectory, independent of its orbit [3]. Finally, and perhaps most importantly, equation 2 introduces a $cos\theta$ factor between classical and relativistic curvature mechanics. This crucial parameter also plays a central role in quantum mechanics [4].

To sum up, gravitation, radiated outside a mass and postulated in a Newtonian vision, originates from the interaction of unit angular momentum with rotation in Descartes' vortex vision. Gravitation then results from a geometric construction involving four circles of the Euclidian space.

<footnote n="1">¹ The meaning of relativity, Albert Einstein, Ed, Barnes and Nobles Book, New York, 5th edition, p. 32, 1984.</footnote>

<footnote n="2">² Catheline, S. A mechanistic interpretation of relativistic rigid body rotation. Sci Rep 13, 9047, 2023.</footnote>

<footnote n="3">³ R.C. Jennison, Ray paths in a rotating system, Nature 199, 739, 1963.</footnote>

<footnote n="4">⁴ Unified theory of relativistic rotation, S. Catheline, Relativistic rigid body rotation and quanta, Chapter V, Editions Universitaires Européennes, 2024.</footnote>