# On the Incompleteness of Birkhoff's Theorem: A New Approach to the Central Symmetric Gravitational Field in Vacuum Space

Michel Canac

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Michel Canac. On the Incompleteness of Birkhoff's Theorem: A New Approach to the Central Symmetric Gravitational Field in Vacuum Space. Reports in Advances of Physical Sciences, 2023, Vol.7, 07, 10.1142/S2424942423500020. [https://hal.science/hal-04148209](https://hal.science/hal-04148209)

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# On the Incompleteness of Birkhoff's Theorem:
A New Approach to the Central Symmetric
Gravitational Field in Vacuum Space

Michel Canac

Retired Expert, Framatome, Paris La Défense

michel.canac690@gmail.com

Received 29 October 2022

Accepted 14 February 2023

Published 28 April 2023

Birkhoff's theorem (1923) states that in the framework of General Relativity the only solution to the central symmetric gravitational field in vacuum is the Schwarzschild metric. This result has crucial consequences in the resolution of the dark matter problem. This problem can only be solved through the discovery of a new type of matter particles, or by the introduction of a new theory of gravitation which supplants General Relativity. After reviewing Birkhoff's theorem, it was discovered that by starting the calculation of the metric from an indeterminate metric whose coefficients are locally defined, we obtain a solution containing two arbitrary functions. In general, these functions do not induce any difference between this solution and the Schwarzschild metric. However, it can be seen that if we choose a triangular signal for these functions, the situation changes dramatically: (1) the metric is broken down into four distinct metrics that replace each other cyclically over time, (2) for two of these four metrics, the coordinate differentials *dr* and *dt* switch their spatial/temporal role cyclically, (3) the four metrics are not separable: they form a single logical set that we call a 4-metric and (4) this 4-metric cannot be transformed into the Schwarzschild metric by any coordinate change. According to these findings, there is a second solution in the spherical space, in addition to the Schwarzschild metric, and thus, Birkhoff's theorem is incomplete. In the 4-metric, the orbital velocity of a massive particle does not depend on the radial distance. This 4-metric is thus in agreement with the baryonic Tully-Fisher relation (BTFR), (consequently BTFR is in agreement with a solution of General Relativity without presence of dark matter and without hypothesis on the distribution of stars in galaxies). By combining the 4-metric with the Schwarzschild metric, another 4-metric in agreement with the observed galaxy rotation curve can be obtained. The calculation of the light deflection in this space is also exposed in this paper.

According to these findings: (1) it is not necessary to introduce the notion of dark matter or the notion of distribution of stars in galaxies in order to find the observed galaxy rotation curve in the framework of General Relativity, (2) the modification of the metric with respect to the

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) License lets others remix, tweak, and build upon your work non-commercially, and although their new works must also acknowledge you and be non-commercial, they don't have to license their derivative works on the same terms.
M. Canac

Schwarzschild metric appears to be due to the existence of a lower bound of the space-time
curvature in galaxies (without external field effect), this phenomenon leading to a temporal
oscillation of the space-time curvature, (3) an analysis of the external field effect for the Milky
Way-Andromeda couple allows to model the rotation curve of the two galaxies beyond the
plateau zone. The validation of these findings would be the first step toward challenging the
standard model of cosmology (ΛCDM), as the ΛCDM model cannot be in agreement with the
observed galaxy rotation curve without presence of dark matter. The second step would be the
demonstration that there is no dark matter in intergalactic spaces (not included in this paper).

Keywords: Metric; non-static; galaxy; Schwarzschild; Tully-Fisher; MOND; curvature, bounded; oscillatory; dark matter; Birkhoff.

# 1. Introduction

## 1.1. Fundamentals of our research

For more than 50 years, the problem of dark matter has become one of the most
important scientific enigmas. In 1933, Fritz Zwicky suggested the presence of in-
visible matter in the Coma cluster. In the 1970s, Vera Rubin observed that the
peripheral stars of the Andromeda galaxy moved too fast, with velocities remaining
almost constant when one moved away from the center of the galaxy. This phe-
nomenon was then observed in many galaxies. It was assumed that most galaxies
were surrounded by a halo of invisible matter, called dark matter.

The search for this dark matter has continued since then. Many dark objects have
been discovered such as brown dwarfs, black holes, neutrinos and gas clouds. How-
ever, in 2022, we get a balance of ~5% baryonic matter for ~27% dark matter.
Currently, the nature of this dark matter remains unknown; one possible candidate
could be the sterile neutrino.

According to some scientists, an alternative hypothesis to dark matter could be
the existence of a weakness in the theory of General Relativity. For calculating the
stars velocity, the Schwarzschild metric is used (in fact, the Newton's formulas in this
domain). The astrophysicist Mordehai Milgrom proposed in 1983 the mathematical
model MOdified Newton Dynamics (MOND), which allows calculating correctly the
velocity of stars in many galaxies from the baryonic mass only. This simple model
then gave birth to new relativistic theories proposed to replace General Relativity.
However, these theories pose many unsolved problems of coherence. On the other
hand, General Relativity allows calculating and verifying very fine phenomena like
the propagation of gravitational waves predicted by Einstein.

It seems to us that there is a paradox between the precision with which we can
measure, among other things, the impact of gravitational waves, and the idea that
General Relativity could be at fault in the search for dark matter, where, in this case,
the error would be enormous: ~500%. Given this data, we came to the idea that
General Relativity should not be questioned in the search for dark matter, but that
we should still try to understand why the MOND model gave such good results from
only baryonic mass.
At this point of reflection, it should be remembered that, to date, there is only one solution of General Relativity for the central gravitational field in vacuum: the Schwarzschild metric; Birkhoff's theorem states that this solution is unique. For this reason, the resolution of the dark matter problem within the framework of General Relativity requires the discovery of new particles of matter unknown to date.

The Schwarzschild metric is a static solution of the equations of General Relativity. Birkhoff's theorem states that there is no dynamical solution for the central gravitational field in vacuum. In 1927, Georges Lemaître published a paper showing that a dynamical solution of General Relativity allowed to interpret the galaxies recession velocities as a proof of the expansion of the Universe, and to quantify this phenomenon. This explanation first met with the skepticism of Einstein and Hubble; Einstein seemed to be more than skeptical on the physical validity of the dynamical solutions of General Relativity. The following years gave reason to Georges Lemaître. He was then at the origin of the Big Bang theory, which was confirmed in 1964 with the discovery of the cosmic microwave background radiation (Penzias and Wilson), and he predicted that the Universe could be in accelerated expansion, contrary to most scientists; however, in 1998, this accelerated expansion was proven.

After reading some texts about Georges Lemaître, we came to the conclusion that, in the field of dark matter, there was not only one enigma, but two enigmas: the second enigma was: why is there no dynamical solution to the central symmetric gravitational field in vacuum?

Indeed, we understood from Georges Lemaître's reasoning that the dynamical character of a metric could be inherent to the foundations of General Relativity, and that one could perhaps intuitively suppose that all metrics must be dynamical in the framework of General Relativity. But there was the Birkhoff's theorem!

According to the Birkhoff's theorem, any solution for the central symmetric gravitational field in vacuum, obtained by cancelling the Ricci tensor, is the Schwarzschild metric, or can be transformed into the classical form of the Schwarzschild metric by a convenient coordinate transformation.

We discovered that there was another type of solution, depending on two arbitrary functions. By judiciously choosing these functions, we could obtain a solution composed of a set of four different metrics that follow each other cyclically over time. This set is not separable; each of the four metrics, considered alone, can be transformed into the classical form of the Schwarzschild metric, but the transformation formulas are not identical for the four metrics. We were able to show that it was impossible to transform this non-separable solution into a Schwarzschild metric, and that the geodesics of the resulting metric are different from the Schwarzschild geodesics.

These geodesics are in agreement with the baryonic Tully-Fisher relation (BTFR), which allows to propose the new metric as a candidate to explain BTFR within the framework of General Relativity without any dark matter. Moreover, the new solution can be combined with the Schwarzschild metric to obtain the observed
rotation curve of galaxies without dark matter. In this new solution, the space-time
curvature is oscillatory over time.

Finally, we conclude that we can propose a methodology to solve the dark matter
problem inside a galaxy within the framework of General Relativity without any
dark matter.

## 1.2. Introduction to our calculations

According to Birkhoff's theorem,¹⁻³ a central symmetric gravitational field in vacuum space is automatically static in the framework of General Relativity. This implies that there would be only one solution: the Schwarzschild metric. In spherical coordinates (R, θ, ϕ), the Schwarzschild metric is defined by the space-time interval ds:

$$ds^2 = (1 - R_0/R)dt^2 - dR^2/(1 - R_0/R) - R^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (1.1)$$

where t is the time coordinate in meters (t = c × t_s where t_s is the time coordinate in seconds), c is the light speed, R is the radial space coordinate, θ is the colatitude, ϕ is the longitude, G is the gravitational constant, M is the mass of the central massive body, and R₀ = 2GM/c².

The research findings presented in this paper demonstrate that a variant of the Schwarzschild metric does exist. This variant needs two arbitrary functions. Most of these functions do not induce any difference between this variant and the Schwarzschild metric. However, we will see that there is at least one particular case using a periodic function, which leads to a new type of metric (which we will call a 4-metric) presenting a different behavior with respect to the Schwarzschild metric, and in agreement with the BTFR. According to these findings, there is a second solution in the spherical geometry, in addition to the Schwarzschild metric, and a third solution by combining the Schwarzschild metric and the second solution. This third solution is in agreement with the observed galaxy rotation curve. Thus, it would become possible, in the framework of General Relativity, to explain the observed galaxy rotation curve without the presence of dark matter and without hypothesis on the spatial distribution of stars in galaxies. It would be also possible to evolve the standard model of cosmology (ΛCDM) without the presence of dark matter, provided that we can demonstrate that there is no dark matter in intergalactic spaces (demonstration not included in this paper).

In the mathematical expressions of this paper, the t, r exponents represent the partial derivatives with respect to the t, r coordinates; the u, v exponents represent the full derivatives with respect to the u, v functions.

## 2. Transformation of the Space-Time Interval

Using the transformation

$$dr = dR/|1 - R_0/R|, \quad (2.1)$$
we transform the interval $ds$ (1.1) into the following form, in tortoise coordinates $(r, \theta, \phi)$:

$$ds^2 = [1 - R_0/R(r)](dt^2 - dr^2) - [R(r)]^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (2.2)$$

$$r = R + R_0 \mathrm{Log}|(R/R_0) - 1|, \quad (2.3)$$

The interval $ds$ (2.2) can be rewritten in the equivalent form:

$$ds^2 = e^{2\nu(r)} dt^2 - e^{2\nu(r)} dr^2 - [R(r)]^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (2.4)$$

where $\nu(r)$, $R(r)$ are the functions defined by the relations (2.2) and (2.3). The present hypothesis is that this metric can be extended into the form of a non-static metric, making the hypothesis that the functions $\nu(r)$, $R(r)$ can be extended in the temporal domain with the form $\nu(t, r)$, $R(t, r)$; these functions are defined as the solutions of the tensor equation $R_{ij} = 0$, where $R_{ij}$ is the Ricci tensor. By hypothesis, we define the non-static metric by the following relation:

$$ds^2 = e^{2\nu(t,r)} dt^2 - e^{2\nu(t,r)} dr^2 - [R(t,r)]^2 (d\theta^2 + \sin^2\theta d\phi^2). \quad (2.5)$$

This form has been chosen because it is suspected that when processing the calculation of the Schwarzschild metric by starting from the undefined metric $ds^2 = e^{2\nu(t,r)}dt^2 - e^{2\lambda(t,r)}dr^2 - [R(t,r)]^2(d\theta^2 + \sin^2\theta d\phi^2)$, the classical arbitrary choice $R(t, r) = r$ could be inconvenient (although possible) to obtain a non-static solution for the metric. It is therefore considered more convenient that the undefined metric (2.5) only contains locally defined coefficients and not the radial coordinate itself (which is a global parameter of the system). As a result, $[R(t, r)]^2$ was left as an undefined coefficient. We have set $\lambda = \nu$, this choice following from the Schwarzschild solution written in tortoise coordinates.

### 3. Summary of the Non-Static Metric

The complete demonstration is reported in Appendix A.

Let us set two arbitrary radial coordinates $r_B$ and $r_M$ ($r_B < r_M$), the functions $u = t + r$; $v = t - r$, and two arbitrary two times differentiable functions $HP(u)$ and $HM(v)$. The solution of the tensor equation $R_{ij} = 0$, $R_{ij}$ being the Ricci tensor, is defined by the following relations:

$$r_B \le r \le r_M, \quad (3.1)$$

$$z = R(t, r) + R_0 \mathrm{Log}|(R(t, r)/R_0) - 1|, \quad (3.2)$$

$$u = t + r; \quad v = t - r, \quad (3.3)$$

$$z = \mathrm{HP}(u) - \mathrm{HM}(v) : \frac{\partial^2 z}{\partial t^2} - \frac{\partial^2 z}{\partial r^2} = 0, \quad (3.4)$$

$$e^{2\nu(t,r)} = 4\mathrm{HP}^u\mathrm{HM}^v[1 - R_0/R(t,r)], \quad (3.5)$$

$$ds^2 = 4\mathrm{HP}^u\mathrm{HM}^v[1 - R_0/R(t,r)](dt^2 - dr^2) - [R(t,r)]^2(d\theta^2 + \sin^2\theta d\phi^2). \quad (3.6)$$
The function $R(t, r)$ is implicitly defined by the relations (3.2) and (3.4). The following mathematical transformations show that it seems possible, a priori, in all cases to transform the metric (3.6) into the classical Schwarzschild metric. These transformations are made based upon the assumption that they are all justified.

$$ds^2 = 4\mathrm{HP}^\nu\mathrm{HM}^\nu[1 - R_0/R(t,r)]dudv - [R(t,r)]^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (3.7)$$

$$ds^2 = 4[1 - R_0/R(t,r)]dHPdHM - \{R[\mathrm{HP}(u) - \mathrm{HM}(v)]\}^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (3.8)$$

$$ds^2 = (1 - R_0/R)[(d\mathrm{HP} + d\mathrm{HM})^2 - (d\mathrm{HP} - d\mathrm{HM})^2] - \{R[\mathrm{HP}(u) - \mathrm{HM}(v)]\}^2 \\ \times (d\theta^2 + \sin^2\theta d\phi^2), \quad (3.9)$$

$$z = \mathrm{HP}(u) - \mathrm{HM}(v) = R + R_0\mathrm{Log}|(R/R_0) - 1|. \quad (3.10)$$

By renaming $\mathrm{HP}(u) + \mathrm{HM}(v) = t'$ and $\mathrm{HP}(u) - \mathrm{HM}(v) = r'$, we obtain

$$ds^2 = [1 - R_0/R(r')][(dt')^2 - (dr')^2] - [R(r')]^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (3.11)$$

$$z = r' = R + R_0\mathrm{Log}|(R/R_0) - 1|. \quad (3.12)$$

The relations (3.11) and (3.12) define the classical Schwarzschild metric written in tortoise coordinates. The metric (3.6) can be deduced directly from the static metric by inversely performing the operations (3.7)–(3.12). However, in this case, the functions $\mathrm{HP}(u)$ and $\mathrm{HM}(v)$ are artificially introduced without any justification. In the complete demonstration (cf. Appendix A), these functions are required, and are introduced in the solution of a transport equation (cf. relations (A.21) and (A.22)); moreover, in the inverse calculation process, it has not been proven that the metric (3.6) is the only solution of the tensor equation $R_{ij} = 0$ obtained from the metric (2.5).

Finally, for some kinds of functions $\mathrm{HP}(u)$ and $\mathrm{HM}(v)$, the metric (3.6) is broken down into four distinct metrics that replace each other cyclically over time, and for two of the four metrics it is obtained over time an incessant permutation of the spatial/temporal role between the $dr$ and $dt$ coordinates differentials. In this last case, it becomes impossible to perform the coordinate transformations (3.7)–(3.12) allowing to find the classical Schwarzschild metric.

To obtain a solution, it is imperative to define the functions $\mathrm{HP}(u)$ and $\mathrm{HM}(v)$. Let us suppose that the $R(t, r)$ function has a physical upper bound $R_C$, such as $R(t, r) \le R_C$, where $R_C$ is a physical function depending only on the mass of the central massive body. This research demonstrates in Sec. 5 that it is possible to obtain this result by using the metric (3.6).

## 4. First Integrals of the Geodesics in the General Case of the Non-Static Metric

Let us set (1) $\kappa = 1$ for the massive particles or $\kappa = 0$ for the particles without mass; (2) $\mathrm{Er}$ is the energy of the particle; (3) $h$ is the angular momentum of the particle; (4) $\sigma$ is the proper time of the particle in meters ($\sigma = c\tau$ where $\tau$ is the proper time in
seconds) for a massive particle, or an affine parameter for a particle without mass, and (5) $w = 1/R$.

The first integrals can be deduced easily from the first integrals of the Schwarzschild metric; however, their direct calculation has been reported in Appendix B. These first integrals are defined by the following relations, where $R, w, r, t$ are the coordinates of a massive or without mass particle, according to the $\kappa$ value:

$$ \theta = \pi/2, \qquad (4.1) $$

$$ R^2 \frac{d\varphi}{d\sigma} = h, \qquad (4.2) $$

$$ \left(\frac{dR}{d\sigma}\right)^2 + (1 - R_0/R)[(h^2/R^2) + \kappa] = (E_r)^2, \qquad (4.3) $$

$$ \frac{d^2 w}{d\varphi^2} + w - 3w^2 R_0/2 = \kappa R_0/(2h^2), \qquad (4.4) $$

$$ \frac{dt}{d\varphi} = \frac{(R^2 E_r / h)(\text{HM}^v + \text{HP}^u) + (dR/d\varphi)(\text{HM}^v - \text{HP}^u)}{4\text{HP}^u \text{HM}^v (1 - R_0 / R)}, \qquad (4.5) $$

$$ \frac{dr}{d\varphi} = \frac{(R^2 E_r / h)(\text{HM}^v - \text{HP}^u) + (dR/d\varphi)(\text{HM}^v + \text{HP}^u)}{4\text{HP}^u \text{HM}^v (1 - R_0 / R)}. \qquad (4.6) $$

Let us set $V_T$ (in m·s⁻¹) the orbital velocity of a massive particle moving along a circular orbit defined by $R = R_s$; we obtain

$$ (E_r)^2 = \frac{[1 - R_0/R_s]^2}{[1 - (3/2)R_0/R_s]}, \qquad (4.7) $$

$$ \frac{(R_s)^2 E_r / h}{(1 - R_0 / R_s)} = \frac{R_s \sqrt{R_s}}{\sqrt{GM/c^2}}, \qquad (4.8) $$

$$ \frac{dt}{d\varphi} = \frac{(R_s)^2 E_r / h}{(1 - R_0 / R_s)} \frac{(\text{HM}^v + \text{HP}^u)}{4\text{HP}^u \text{HM}^v} = \frac{R_s \sqrt{R_s}}{\sqrt{GM/c^2}} \frac{(\text{HM}^v + \text{HP}^u)}{4\text{HP}^u \text{HM}^v}, \qquad (4.9) $$

$$ (V_T)^2 = \frac{GM/R_s}{(1 - R_0/R_s)} \frac{4\text{HP}^u \text{HM}^v}{(\text{HM}^v + \text{HP}^u)^2}. \qquad (4.10) $$

The detailed calculations of these formulas are reported in Appendix B (Secs. 2 and 3). The relations (4.1)–(4.4) are identical to the corresponding relations of the classical Schwarzschild metric. The angular deflection of the light rays $\delta_R$ is obtained using the following formula:

$$ \delta_R = 4GM/[c^2 R(t, r)]. \qquad (4.11) $$

The Kretschmann scalar $Kr = Ri^{abcd}Ri_{abcd}$, calculated from the Riemann tensor $Ri$ is obtained as

$$ Kr = 48(G^2 M^2)/[c^4 R(t, r)^6]. \qquad (4.12) $$
# 5. On the Incompleteness of Birkhoff's Theorem: The Asymptotic Metric

## 5.1. *Selecting the settings*

Let us define (1) a spherical finite space by $r_B \le r \le r_M$, where $r_B$ and $r_M$ are two arbitrary radial coordinates, (2) two arbitrary positive numbers $\omega$ and $R_C$, and (3) the triangular signal $F_{\text{tr}}^{\omega}(x)$ with pulsation $\omega$ ($\omega$ in $m^{-1}$) and amplitude equal to $\frac{\pi}{2\omega}$; its period (in meters) is equal to $Tp = 2\pi/\omega$.

$$F_{\text{tr}}^{\omega}(x) = -\frac{4}{\pi\omega} \sum_{k=0}^{k=\infty} \frac{\cos(2k + 1)\omega x}{(2k + 1)^2}. \qquad (5.1)$$

Its derivative is the square signal $F_{\text{sq}}^{\omega}(x)$ with amplitude equal to 1

$$\frac{dF_{\text{tr}}^{\omega}(x)}{dx} = F_{\text{sq}}^{\omega}(x) = \frac{4}{\pi} \sum_{k=0}^{k=\infty} \frac{\sin(2k + 1)\omega x}{(2k + 1)}. \qquad (5.2)$$

Let us set:

$$\text{HP}(u) = R_C + F_{\text{tr}}^{\omega}(u)/2; \quad \text{HM}(v) = F_{\text{tr}}^{\omega}(v)/2, \qquad (5.3)$$

$$\text{HP}^u = F_{\text{sq}}^{\omega}(u)/2; \quad \text{HM}^v = F_{\text{sq}}^{\omega}(v)/2, \qquad (5.4)$$

$$z = \text{HP}(u) - \text{HM}(v) = R_C + [F_{\text{tr}}^{\omega}(u) - F_{\text{tr}}^{\omega}(v)]/2, \qquad (5.5)$$

$$dz = du\text{HP}^u - dv\text{HM}^v = duF_{\text{sq}}^{\omega}(u)/2 - dvF_{\text{sq}}^{\omega}(v)/2 = dR/(1 - R_0/R). \qquad (5.6)$$

$R(t, r)$ depends only on cosine functions in $t + r$ and $t - r$, then $R(r, t) = R(t, r)$. The metric (3.6) becomes

$$ds^2 = F_{\text{sq}}^{\omega}(t+r)F_{\text{sq}}^{\omega}(t-r)[1 - R_0/R(t,r)](dt^2 - dr^2) - [R(t,r)]^2(d\theta^2 + \sin^2\theta d\phi^2). \qquad (5.7)$$

Let us set: $\Omega^2 = [R(t, r)]^2(d\theta^2 + \sin^2\theta d\phi^2)$.

## 5.2. *Analysis of the impact of the changes of sign of the derivatives HP$^u$ and HM$^v$ on the metric*

Table 1 summarizes the properties of the four modes obtained from the combinations of the square signals.

By homogenizing the coordinate symbols for obtaining in all modes $dt$ as the temporal differential and $dr$ as the spatial differential, we rewrite the metrics and the relations $dz_i$ in Table 2.

The differentials of the temporal coordinates must be aligned with real time because we have to take into account the first integral $R^2 \frac{d\varphi}{d\sigma} = h \Rightarrow$ variation of the angle $\varphi$ with respect to the proper time is monotonic. Then, we have to change the sign of the temporal coordinate differentials in the third and fourth modes to align them with the first mode unchanged that we set as the reference mode (the sign of $[dt/d\varphi]_3$ and $[dt/d\varphi]_4$ must be positive to be coherent with $[dt/d\varphi]_1$), (cf. also end of
<table><caption>Table 1.</caption><thead><tr><th>Mode</th><th>Square signals</th><th>Metric</th><th>dz</th><th>Geodesic: A</th><th>Geodesic: B</th><th>Symbols</th></tr></thead><tbody><tr><td rowspan="2">1st</td><td>F<sup>ω</sup><sub>sq</sub>(t + r) = 1</td><td>[ds<sup>2</sup>]<sub>1</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>1</sub> = dr</td><td>[dt/dz]<sub>1</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (dR/dz)<sub>1</sub> = (dR/dz)<sub>1</sub> (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>1</sub> = (dR/dz)<sub>1</sub> (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>F<sup>ω</sup><sub>sq</sub>(t − r) = 1</td><td>[ds<sup>2</sup>]<sub>2</sub> = [1 − R<sub>0</sub>/R(t, r)](dr<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>2</sub> = dt</td><td>[dt/dz]<sub>2</sub> = (dR/dz)<sub>2</sub> (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>2</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td rowspan="2">2nd</td><td>F<sup>ω</sup><sub>sq</sub>(t + r) = 1</td><td>[ds<sup>2</sup>]<sub>3</sub> = [1 − R<sub>0</sub>/R(t, r)](dr<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>3</sub> = −dt</td><td>[dt/dz]<sub>3</sub> = (dR/dz)<sub>3</sub> (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>3</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td>F<sup>ω</sup><sub>sq</sub>(t − r) = −1</td><td>[ds<sup>2</sup>]<sub>4</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>4</sub> = −dr</td><td>[dt/dz]<sub>4</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>4</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td rowspan="2">3rd</td><td>F<sup>ω</sup><sub>sq</sub>(t + r) = −1</td><td>[ds<sup>2</sup>]<sub>5</sub> = [1 − R<sub>0</sub>/R(t, r)](dr<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>5</sub> = −dr</td><td>[dt/dz]<sub>5</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>5</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td>F<sup>ω</sup><sub>sq</sub>(t − r) = 1</td><td>[ds<sup>2</sup>]<sub>6</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>6</sub> = −dr</td><td>[dt/dz]<sub>6</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>6</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td>4th</td><td>F<sup>ω</sup><sub>sq</sub>(t + r) = −1</td><td>[ds<sup>2</sup>]<sub>7</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>7</sub> = −dr</td><td>[dt/dz]<sub>7</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>7</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td></td><td>F<sup>ω</sup><sub>sq</sub>(t − r) = −1</td><td>[ds<sup>2</sup>]<sub>8</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>8</sub> = −dr</td><td>[dt/dz]<sub>8</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>8</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr></tbody></table>

<table><caption>Table 2.</caption><thead><tr><th>Mode</th><th>Square signals</th><th>Metric</th><th>dz</th><th>Geodesic: A</th><th>Geodesic: B</th><th>Symbols</th></tr></thead><tbody><tr><td rowspan="2">1st</td><td>F<sup>ω</sup><sub>sq</sub>(t + r) = 1</td><td>[ds<sup>2</sup>]<sub>1</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>1</sub> = dr</td><td>[dt/dz]<sub>1</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (dR/dz)<sub>1</sub> = (dR/dz)<sub>1</sub> (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>1</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>F<sup>ω</sup><sub>sq</sub>(t − r) = 1</td><td>[ds<sup>2</sup>]<sub>2</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>2</sub> = dr</td><td>[dt/dz]<sub>2</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (dR/dz)<sub>2</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>2</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td rowspan="2">2nd</td><td>F<sup>ω</sup><sub>sq</sub>(t + r) = 1</td><td>[ds<sup>2</sup>]<sub>3</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>3</sub> = −dr</td><td>[dt/dz]<sub>3</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (dR/dz)<sub>3</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>3</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>F<sup>ω</sup><sub>sq</sub>(t − r) = −1</td><td>[ds<sup>2</sup>]<sub>4</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>4</sub> = −dr</td><td>[dt/dz]<sub>4</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>4</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td rowspan="2">3rd</td><td>F<sup>ω</sup><sub>sq</sub>(t + r) = −1</td><td>[ds<sup>2</sup>]<sub>5</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>5</sub> = −dr</td><td>[dt/dz]<sub>5</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (dR/dz)<sub>5</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>5</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td>F<sup>ω</sup><sub>sq</sub>(t − r) = 1</td><td>[ds<sup>2</sup>]<sub>6</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>6</sub> = −dr</td><td>[dt/dz]<sub>6</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>6</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td rowspan="2">4th</td><td>F<sup>ω</sup><sub>sq</sub>(t + r) = −1</td><td>[ds<sup>2</sup>]<sub>7</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>7</sub> = −dr</td><td>[dt/dz]<sub>7</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>7</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr><tr><td>F<sup>ω</sup><sub>sq</sub>(t − r) = −1</td><td>[ds<sup>2</sup>]<sub>8</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>8</sub> = −dr</td><td>[dt/dz]<sub>8</sub> = (R<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>[dR/dz]<sub>8</sub> = (dR<sup>2</sup>E<sub>0</sub>/h) (1 − R<sub>0</sub>/R)</td><td>dt:spatial<br>dr:temporal</td></tr></tbody></table>
M. Canac

Sec. 7.3.1). The *dr* differentials are not modified; their changes of sign in the third and fourth modes are necessary to constrain the *R(t, r)* value in a restricted interval (cf. relation (5.6) and also end of Sec. 7.2). Hence the final Table 3 in Sec. 5.3.

**5.3. The asymptotic metric**

Table 3 summarizes the properties of the asymptotic metric: the differentials *dt* of third and fourth modes had been made coherent with first mode.

For the first and the second modes, the metric is the Schwarzschild metric in
tortoise coordinates; for the third and fourth modes, the metric could be identified
with the Schwarzschild metric {$ds^2 = [1 - R_0/R(t,r)][(dt')^2 - (dr')^2] - \Omega^2; dz =
dr'$} by using the coordinates transformations $dt' = dt; dr' = -dr$. Consequently,
when considered alone, the third mode (respectively, the fourth mode) is a
Schwarzschild metric. But in reality, the four modes are not separable; the coordi-
nate differentials $dt$ and $dr$ must be identical in all modes, the four modes consti-
tuting a single logical set, which we will call a 4-metric. Consequently, the third mode
and the fourth mode are not Schwarzschild metrics when we set the first mode
unchanged as the reference mode of the single set. This result is a consequence of the
arbitrary relations (5.3), which maintain the value of the function $z$ in a restricted
interval, whatever the value of the coordinate $r$.

Suppose that there is a function $z^*(t, r)$ derived from the 4 $dz_i$. We would get
$\delta z^* = dr F_{sq}^\omega (t+r)$, which shows that the function $z^*$ does not exist. In fact, if this
function existed, we would obtain the equality $z^* = z$, and the resulting metric would
be the Schwarzschild metric. We must therefore make the hypothesis that there is a
physical metric obtained by averaging over the 4 metrics.

Since it is necessary to determine the frequency of the square signal, we postulate
that this frequency is of the order of magnitude of the Planck frequency $F_p : F_p =
\sqrt{\frac{2\pi c^5}{hG}} \approx 1.855 \times 10^{43}$ Hz ($h$ is the Planck constant), (we would obtain $\omega \sim \frac{2\pi F_p}{c} \Rightarrow$
amplitude of the triangular signal $\sim l_{\text{Planck}}/4$). This assumption is based on the idea that
the frequency must be a constant number with a very high value, involving the
universal constants $G, h, c$ (cf. explanation at the end of Sec. 6). At this frequency
level, it can be considered that the four oscillation modes are no longer individually
discernible. Only the average of these modes, if it is stable, can be physically ob-
served, as we have assumed above.

When the average occurrence of a mode over a time interval *Tp*/4 and a space interval *Tp*/4 is calculated, the occurrence of each mode is found to be equal to 1/4. Consequently, the elements of the average metric can be calculated by performing the sum of the corresponding elements of the 4-metric divided by 4. In the rest of this paper, we will systematically perform this operation to obtain the average metric, considered to be the only observable.
<table><caption>Table 3.</caption><thead><tr><th>Mode</th><th>Square signals</th><th>Metric</th><th>dz</th><th>Geodesic: A</th><th>Geodesic: B</th><th>Symbols</th></tr></thead><tbody><tr><td rowspan="2">1st</td><td>F<sup>sq</sup>(t + r) = 1</td><td>[ds<sup>2</sup>]<sub>1</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>1</sub> = dr</td><td>[d<sub>φ</sub>]<sub>1</sub> = (R<sup>2</sup>E<sub>h</sub>/R<sub>0</sub>)</td><td>[d<sub>φ</sub>]<sub>1</sub> = (dR/dφ)<sub>1</sub> = (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>F<sup>sq</sup>(t − r) = 1</td><td>[ds<sup>2</sup>]<sub>2</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>2</sub> = dr</td><td>[d<sub>φ</sub>]<sub>2</sub> = (R<sup>2</sup>E<sub>h</sub>/R<sub>0</sub>)</td><td>[d<sub>φ</sub>]<sub>2</sub> = (dR/dφ)<sub>2</sub> = (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td rowspan="2">2nd</td><td>F<sup>sq</sup>(t + r) = 1</td><td>[ds<sup>2</sup>]<sub>3</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>3</sub> = −dr</td><td>[d<sub>φ</sub>]<sub>3</sub> = (R<sup>2</sup>E<sub>h</sub>/R<sub>0</sub>)</td><td>[d<sub>φ</sub>]<sub>3</sub> = (dR/dφ)<sub>3</sub> = (1 − R<sub>0</sub>/R)</td><td>dt: temporal<br>dr: spatial</td></tr><tr><td>F<sup>sq</sup>(t − r) = −1</td><td>[ds<sup>2</sup>]<sub>4</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>4</sub> = −dr</td><td>[d<sub>φ</sub>]<sub>4</sub> = (R<sup>2</sup>E<sub>h</sub>/R<sub>0</sub>)</td><td>[d<sub>φ</sub>]<sub>4</sub> = (dR/dφ)<sub>4</sub> = (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td rowspan="2">3rd</td><td>F<sup>sq</sup>(t + r) = −1</td><td>[ds<sup>2</sup>]<sub>5</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>5</sub> = −dr</td><td>[d<sub>φ</sub>]<sub>5</sub> = (R<sup>2</sup>E<sub>h</sub>/R<sub>0</sub>)</td><td>[d<sub>φ</sub>]<sub>5</sub> = (dR/dφ)<sub>5</sub> = (1 − R<sub>0</sub>/R)</td><td>dt: temporal<br>dr: spatial</td></tr><tr><td>F<sup>sq</sup>(t − r) = 1</td><td>[ds<sup>2</sup>]<sub>6</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>6</sub> = −dr</td><td>[d<sub>φ</sub>]<sub>6</sub> = (R<sup>2</sup>E<sub>h</sub>/R<sub>0</sub>)</td><td>[d<sub>φ</sub>]<sub>6</sub> = (dR/dφ)<sub>6</sub> = (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td rowspan="2">4th</td><td>F<sup>sq</sup>(t + r) = −1</td><td>[ds<sup>2</sup>]<sub>7</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>7</sub> = −dr</td><td>[d<sub>φ</sub>]<sub>7</sub> = (R<sup>2</sup>E<sub>h</sub>/R<sub>0</sub>)</td><td>[d<sub>φ</sub>]<sub>7</sub> = (dR/dφ)<sub>7</sub> = (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>F<sup>sq</sup>(t − r) = −1</td><td>[ds<sup>2</sup>]<sub>8</sub> = [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dr<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>8</sub> = −dr</td><td>[d<sub>φ</sub>]<sub>8</sub> = (R<sup>2</sup>E<sub>h</sub>/R<sub>0</sub>)</td><td>[d<sub>φ</sub>]<sub>8</sub> = (dR/dφ)<sub>8</sub> = (1 − R<sub>0</sub>/R)</td><td>dt:temporal<br>dr:spatial</td></tr></tbody></table>
We set, for the time-averaged terms (noted$_{av}$):

$$ [dz]_{\text{av}} = \sum_{i=1}^{i=4} \frac{dz_i}{4}; \quad \left[ \frac{d\varphi}{dt} \right]_{\text{av}} = \sum_{i=1}^{i=4} \frac{1}{4} \left[ \frac{d\varphi}{dt} \right]_i; \quad \left[ \frac{dr}{d\varphi} \right]_{\text{av}} = \sum_{i=1}^{i=4} \frac{1}{4} \left[ \frac{dr}{d\varphi} \right]_i; \tag{5.8} \\ [ds^2]_{\text{av}} = \sum_{i=1}^{i=4} \frac{(ds^2)_i}{4} : [dz]_{\text{av}} = 0 \Rightarrow [dR]_{\text{av}} = 0 \Rightarrow R_{\text{av}} = R_C, $$

$$ \left[ \frac{d\varphi}{dt} \right]_{\text{av}} = \frac{(1 - R_0/R_{\text{av}})}{(R_{\text{av}}^2 E_r / h)}, \quad \left[ \frac{dr}{d\varphi} \right]_{\text{av}} = 0, \tag{5.9} $$

$$ [ds^2]_{\text{av}} = [1 - R_0/R_{\text{av}}](dt^2 - dr^2) - [R_{\text{av}}]^2(d\theta^2 + \sin^2\theta d\phi^2). \tag{5.10} $$

The z-fluctuation obtained from the average is equal to zero, while the z-fluctuation obtained from the formula (5.5) is equal to $\sim \frac{1}{4}$. We assume that the average is justified because both values of the fluctuation can be considered as $\sim 0$ from a physical point of view.

The real time differential interval $dT$ (in seconds) between two events located at $(r, \theta, \varphi)$ is

$$ cdT = \sqrt{1 - R_0/R_{\text{av}}} dt \approx dt \quad (\text{case : } R_0/R_{\text{av}} \ll 1). \tag{5.11} $$

The real radial distance differential interval $dL_r$ is

$$ dL_r = \sqrt{1 - R_0/R_{\text{av}}} dr \approx dr \quad (\text{case : } R_0/R_{\text{av}} \ll 1). \tag{5.12} $$

The average metric (5.10) is similar to the classical Schwarzschild metric written in tortoise coordinates, but in the formula (5.10) $R_{\text{av}} = R_C$ is a constant number.

According to these results, the metric (3.6) cannot be transformed in all cases into the classical Schwarzschild metric. Thus, Birkhoff's theorem can be considered as incomplete. We will analyze in Sec. 6 the physical behavior of this average metric, which we will call the asymptotic metric.

# 6. Geodesics of the Asymptotic Metric

From the relations (4.6)–(4.10) we obtain

$$ \frac{dt}{d\varphi} = \frac{(R_{\text{av}}^2 E_r / h)}{(1 - R_0 / R_{\text{av}})}, \tag{6.1} $$

$$ \frac{dr}{d\varphi} = \frac{(dR_{\text{av}}/d\varphi)}{(1 - R_0/R_{\text{av}})} = 0. \tag{6.2} $$

Let us set the case of a massive particle; its trajectory is defined as $R(\varphi) = R_{\text{av}} = \text{constant} \Rightarrow dR/d\varphi = 0$. Let us set $V_r$ as the radial velocity, $V_T$ as the orbital velocity (in m·s⁻¹). We obtain the following results:

$$ \frac{dr}{d\varphi} = 0, \tag{6.3} $$

$$ (V_r/c) = 0, \tag{6.4} $$
from the relation (4.10):

$$ (V_T)^2 = \frac{(GM/R_{av})}{(1 - R_0/R_{av})}, \qquad (6.5) $$

$$ \frac{(V_T)^2}{R_{av}} = \frac{[GM/(R_{av})]^2}{(1 - R_0/R_{av})} = A_0, \qquad (6.6) $$

$R_{av}$ is a constant number in the observed system ⇒ $R_{av}$ depends only on the mass $M$ of the central massive body. Consequently, it is the same for $A_0 \Rightarrow A_0 = A_0(M)$.

If $R_0/R_{av} \ll 1$:

$$ (V_T)^2 \approx (GM/R_{av}), \qquad (6.7) $$

$$ \begin{gathered} (V_T)^2/R_{av} \approx [GM/(R_{av})]^2 \approx A_0(M) \Rightarrow R_{av} \approx \sqrt{GM/A_0(M)}, \qquad (6.8) \\ (V_T)^4 \approx GMA_0(M), \end{gathered} $$

$A_0(M)$ is the radial acceleration. The orbital velocity (6.5) is similar to the classical Schwarzschild formula, provided that we replace in the formula the distance coordinate by the local value of the $R$ function. The $R_{av}$ function value being constant in this space, the radial acceleration and the orbital velocity of the stars located in this zone are constant. By identifying the mass $M$ and the baryonic mass $M_b(M \equiv M_b)$, we obtain

$$ R_{av} \approx \sqrt{GM_b/A_0(M_b)}, \qquad (6.9) $$

$$ (V_T)^4 \approx GM_bA_0(M_b). \qquad (6.10) $$

The relation (6.10) has the form of the BTFR.⁴⁻⁶ By using the formula (6.9), we obtain

$$ \text{Deflection of the light rays: } \delta_R = 4\sqrt{GM_bA_0(M_b)}/c^2 = 4(V_T)^2/c^2, \qquad (6.11) $$

$$ \text{Kretschmann scalar: } Kr = 48[A_0(M_b)]^3/(c^4GM_b). \qquad (6.12) $$

From the formula of the Kretschmann scalar, we can observe that there is a particular case if $(A_0)^3 \propto M_b$. In this case, the Kretschmann scalar does not depend on $M_b$, and the formula of the asymptotic velocity becomes

$$ (V_T)^4 \approx GM_bA_0(M_b) \propto (M_b)^{4/3} \Rightarrow M_b \propto (V_T)^3. \qquad (6.13) $$

If, for very large values of the central baryonic mass, the function $A_0(M_b)$ were obtained by the asymptotic formula: $(A_0)^3 = k_\infty M_b$ ($k_\infty$ = constant), we would obtain for the asymptotic Kretschmann scalar: $Kr_\infty = 48 k_\infty/(c^4 G)$, which would mean that the space-time curvature has an absolute lower bound in a spherical space with a central massive body (without external field effect).

In Table 1 of the paper by Lelli et al.⁵ different results are reported ⇒ $M_b \propto (V_T)^s$, $s$ varying from 3.06 to 3.85. In his MOND theory,⁷ Milgrom used $s = 4$ and $A_0(M_b) = a_0 = 1.2 \times 10^{-10} \text{ m} \cdot \text{s}^{-2}$. To fit the galaxy radial acceleration relation on
the SPARC database, Li et al.¹⁰ used the same parameters as Milgrom and obtained a tight fit.

According to this result, BTFR is in agreement with the asymptotic metric defined within the framework of General Relativity, without any dark matter. We conclude that the asymptotic metric is a candidate to explicate BTFR without any developments such as new massive particles (dark matter) or a new gravitational theory or a particular distribution of stars in galaxies. This finding challenges the standard model of cosmology (ΛCDM), because this standard model needs the presence of dark matter to explain BTFR. We have now to analyze the transition part of the galaxy rotation curve, located between the Schwarzschild zone and the BTFR zone.

It can be observed that the Kretschmann scalar Kr = 48[A₀(M_b)]³/(c⁴GM_b) is constant in average in this space, which signifies that the space-time curvature is constant in average. If we accept the idea that this 4-metric is the asymptotic metric in a galaxy (without external field effect), we deduce that the space-time curvature has a lower bound in a galaxy. The existence of this lower bound led us to the idea that the oscillation frequency of the system could be of the order of magnitude of the Planck frequency, because this lower bound could have a quantum origin. The Planck frequency, involving G, h, c would seem to us well adapted to occur in a metric of General Relativity involving probably a phenomenon of quantum origin.

# 7. The Hybrid Metric

## 7.1. Selecting the settings

This research defines a hybrid metric, which is a combination of the classical Schwarzschild metric and of the preceding 4-metric. It is being used to model the transition zone of the galaxy rotation curve. Let us define (1) a finite spherical space by $r_A \le r \le r_B$, where $r_A$ and $r_B$ are two arbitrary radial coordinates, (2) two arbitrary positive numbers $\alpha$ and $\omega$, and (3) two functions HP($u$) and HM($v$):

$$ \mathrm{HP}(u) = r_A + \alpha u/2 + F_{\mathrm{tr}}^{\omega}(u)/2; \quad \mathrm{HM}(v) = \alpha v/2 + F_{\mathrm{tr}}^{\omega}(v)/2, \qquad (7.1) $$

$$ \mathrm{HP}^u = [\alpha + F_{\mathrm{sq}}^{\omega}(u)]/2; \quad \mathrm{HM}^v = [\alpha + F_{\mathrm{sq}}^{\omega}(v)]/2, \qquad (7.2) $$

$$ z = \mathrm{HP}(u) - \mathrm{HM}(v) = r_A + \alpha(u - v)/2 + [F_{\mathrm{tr}}^{\omega}(u) - F_{\mathrm{tr}}^{\omega}(v)]/2, \qquad (7.3) $$

$$ \begin{aligned} dz &= du\mathrm{HP}^u - dv\mathrm{HM}^v = dR/(1 - R_0/R) \Rightarrow dz \\ &= du[\alpha + F_{\mathrm{sq}}^{\omega}(u)]/2 - dv[\alpha + F_{\mathrm{sq}}^{\omega}(v)]/2 = dR/(1 - R_0/R). \end{aligned} \qquad (7.4) $$

The metric (3.6) becomes

$$ ds^2 = [\alpha + F_{\mathrm{sq}}^{\omega}(u)][\alpha + F_{\mathrm{sq}}^{\omega}(v)](1 - R_0/R)(dt^2 - dr^2) - [R(t, r)]^2(d\theta^2 + \sin^2\theta d\varphi^2). \qquad (7.5) $$

Let us set: $\Omega^2 = [R(t, r)]^2(d\theta^2 + \sin^2\theta d\varphi^2)$.
$$ \text{Geodesic relations (4.5), (4.6):} \quad \frac{dt}{d\varphi} = \frac{(R^2 E_r / h)(\text{HM}^v + \text{HP}^u) + (dR/d\varphi)(\text{HM}^v - \text{HP}^u)}{4\text{HP}^u \text{HM}^v (1 - R_0 / R)}; \quad \frac{dr}{d\varphi} = \frac{(R^2 E_r / h)(\text{HM}^v - \text{HP}^u) + (dR/d\varphi)(\text{HM}^v + \text{HP}^u)}{4\text{HP}^u \text{HM}^v (1 - R_0 / R)} $$

In the following tables, the indices on the parameters $\frac{dt}{d\varphi}$ and $\frac{dr}{d\varphi}$ have been omitted. Table 4 summarizes the properties of the four modes, according to the combination of the square signals.

By performing a convenient coordinates transformation on $dt$ and $dr$ we can verify that each mode, considered alone, can be deduced from the Schwarzschild metric $\{ds^2 = [1 - R_0/R(t,r)][(dr')^2 - (dr')^2] - \Omega^2; dz = dr'\}$.

We obtain two kinds of solution, depending on whether $\alpha > 1$ or $\alpha < 1$.

## 7.2. First case $\alpha > 1$

For the circular geodesics of a massive particle, we set $dr = 0$; hence Table 5, deduced from Table 4.

The variable $w$ is defined by the equation $\frac{d^2w}{d\varphi^2} + w - 3w^2 R_0 / 2 = \kappa R_0 / (2h^2)$; this equation is verified approximately by the relations $\frac{dw}{d\varphi} = \pm \frac{E_r}{\alpha h}$ considered on a very short time, by taking into account the relation $(\omega \gg \pi / 2)$; on average, we obtain $dw/d\varphi = 0$.

We set, for the time-averaged terms: $[dz]_{\text{av}} = \sum_{i=1}^{i=4} \frac{dz_i}{4}$; $[d\varphi]_{\text{av}} = \sum_{i=1}^{i=4} \frac{d\varphi}{4}$; $[ds^2]_{\text{av}} = \sum_{i=1}^{i=4} \frac{(ds^2)_i}{4}$:

$$ \begin{aligned} [dz]_{\text{av}} &= \frac{(\alpha dr + dr) + (\alpha dr + dt) + (\alpha dr - dt) + (\alpha dr - dr)}{4} \\ &= \alpha dr = \frac{dR_{\text{av}}}{(1 - R_0 / R_{\text{av}})}, \end{aligned} \qquad (7.6) $$

$$ \begin{aligned} \left[ \frac{d\varphi}{dt} \right]_{\text{av}} &= \frac{(\alpha + 1) + 2\alpha + (\alpha - 1)}{4} \frac{(1 - R_0 / R_{\text{av}})}{(R_{\text{av}})^2 E_r / h} = \alpha \frac{(1 - R_0 / R_{\text{av}})}{(R_{\text{av}})^2 E_r / h} \Rightarrow \left[ \frac{d\varphi}{\alpha dt} \right]_{\text{av}} \\ &= \frac{(1 - R_0 / R_{\text{av}})}{(R_{\text{av}})^2 E_r / h}, \end{aligned} \qquad (7.7) $$

$$ \begin{aligned} [ds^2]_{\text{av}} &= \frac{(\alpha + 1)^2 + 2(\alpha^2 - 1) + (\alpha - 1)^2}{4} [1 - R_0/R_{\text{av}}(r)](dt^2 - dr^2) - \Omega^2 \Rightarrow [ds^2]_{\text{av}} \\ &= [1 - R_0/R_{\text{av}}(r)][(\alpha dr)^2 - (\alpha dr')^2] - [R_{\text{av}}(r)]^2(d\theta^2 + \sin^2\theta d\varphi^2), \end{aligned} \qquad (7.8) $$

$$ [dz]_{\text{av}} = \alpha dr = \frac{dR_{\text{av}}}{(1 - R_0 / R_{\text{av}})}, \qquad (7.9) $$

The average metric defined by the relations (7.8) and (7.9) is the Schwarzschild metric, written in tortoise coordinates. Consequently, if $\alpha > 1$, the metric (7.5) is a Schwarzschild metric. We can see that the sign – in the fluctuating term of $dz_4$ is essential to find exactly the Schwarzschild metric.
<table><caption>Table 4.</caption><thead><tr><th>Mode</th><th>Square signals</th><th>Metric</th><th>dz</th><th>Geodesic: A</th><th>Geodesic: B</th></tr></thead><tbody><tr><td>1st</td><td>F<sup>2</sup><sub>sq</sub>(t + r) = 1<br>F<sup>2</sup><sub>sq</sub>(t - r) = 1</td><td>ds<sup>2</sup> = (α + 1)<sup>2</sup>[1 - R<sub>0</sub>/R(t, r)](dt<sup>2</sup> - dr<sup>2</sup>) - Ω<sup>2</sup></td><td>dz<sub>1</sub> = αdr + dr</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α+1)(1-R<sub>0</sub>/R)</td><td>du/du = (dR/dc)<br>(α+1)(1-R<sub>0</sub>/R)</td></tr><tr><td>2nd</td><td>F<sup>2</sup><sub>sq</sub>(t + r) = 1<br>F<sup>2</sup><sub>sq</sub>(t - r) = -1</td><td>ds<sup>2</sup> = (α<sup>2</sup> - 1)[1 - R<sub>0</sub>/R(t, r)](dt<sup>2</sup> - dr<sup>2</sup>) - Ω<sup>2</sup></td><td>dz<sub>2</sub> = αdr + dt</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α<sup>2</sup>-1)(1-R<sub>0</sub>/R)</td><td>du/du = -(R<sup>2</sup>E<sub>h</sub>/h)<br>(α<sup>2</sup>-1)(1-R<sub>0</sub>/R)</td></tr><tr><td>3rd</td><td>F<sup>2</sup><sub>sq</sub>(t + r) = -1<br>F<sup>2</sup><sub>sq</sub>(t - r) = 1</td><td>ds<sup>2</sup> = (α<sup>2</sup> - 1)[1 - R<sub>0</sub>/R(t, r)](dt<sup>2</sup> - dr<sup>2</sup>) - Ω<sup>2</sup></td><td>dz<sub>3</sub> = αdr - dt</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α<sup>2</sup>-1)(1-R<sub>0</sub>/R)</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α<sup>2</sup>-1)(1-R<sub>0</sub>/R)</td></tr><tr><td>4th</td><td>F<sup>2</sup><sub>sq</sub>(t + r) = -1<br>F<sup>2</sup><sub>sq</sub>(t - r) = -1</td><td>ds<sup>2</sup> = (α - 1)<sup>2</sup>[1 - R<sub>0</sub>/R(t, r)](dt<sup>2</sup> - dr<sup>2</sup>) - Ω<sup>2</sup></td><td>dz<sub>4</sub> = αdr - dr</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α-1)(1-R<sub>0</sub>/R)</td><td>du/du = (dR/dc)<br>(α-1)(1-R<sub>0</sub>/R)</td></tr></tbody></table>

<table><caption>Table 5.</caption><thead><tr><th>Mode</th><th>Metric</th><th>dz</th><th>Geodesic: dr/dφ = 0</th><th>Geodesic: dr/dφ = 0</th><th>w = 1/R</th></tr></thead><tbody><tr><td>1st</td><td>ds<sup>2</sup> = (α + 1)<sup>2</sup>[1 - R<sub>0</sub>/R(t, r)](dt<sup>2</sup> - dr<sup>2</sup>) - Ω<sup>2</sup></td><td>dz<sub>1</sub> = αdr + dr</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α+1)(1-R<sub>0</sub>/R)</td><td>dR/dφ = 0</td><td>du/du = 0</td></tr><tr><td>2nd</td><td>ds<sup>2</sup> = (α<sup>2</sup> - 1)[1 - R<sub>0</sub>/R(t, r)](dt<sup>2</sup> - dr<sup>2</sup>) - Ω<sup>2</sup></td><td>dz<sub>2</sub> = αdr + dt</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α-1-R<sub>0</sub>/R)</td><td>(R<sup>2</sup>E<sub>h</sub>/h) - (dR/dφ)α = 0</td><td>du/du = -E<sub>h</sub>/α<sub>h</sub></td></tr><tr><td>3rd</td><td>ds<sup>2</sup> = (α<sup>2</sup> - 1)[1 - R<sub>0</sub>/R(t, r)](dt<sup>2</sup> - dr<sup>2</sup>) - Ω<sup>2</sup></td><td>dz<sub>3</sub> = αdr - dt</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α-1-R<sub>0</sub>/R)</td><td>(R<sup>2</sup>E<sub>h</sub>/h) + (dR/dφ)α = 0</td><td>du/du = E<sub>h</sub>/α<sub>h</sub></td></tr><tr><td>4th</td><td>ds<sup>2</sup> = (α - 1)<sup>2</sup>[1 - R<sub>0</sub>/R(t, r)](dt<sup>2</sup> - dr<sup>2</sup>) - Ω<sup>2</sup></td><td>dz<sub>4</sub> = αdr - dr</td><td>du/du = (R<sup>2</sup>E<sub>h</sub>/h)<br>(α-1)(1-R<sub>0</sub>/R)</td><td>dR/dφ = 0</td><td>du/du = 0</td></tr></tbody></table>
### 7.3. Second case $0 \le \alpha < 1$

#### 7.3.1. Analysis of the impact of the changes of sign of the derivatives HP$^u$ and HM$^v$ on the metric

We rewrite Table 4 in Table 6, with a few scheduling changes.

By homogenizing the coordinate symbols to obtain in all modes $dt$ as the temporal differential and $dr$ as the spatial differential, we obtain Table 7.

The differentials of the temporal coordinates must be aligned with real time because we have to take into account the first integral $R^2 \frac{d\varphi}{d\sigma} = h \Rightarrow$ variation of the angle $\varphi$ with respect to the proper time is monotonic. Then, we have to change the sign of the temporal coordinate differentials in the third and fourth modes to align them with the first mode unchanged that we set as the reference mode. Hence Table 8.

We can see that the change in sign of $dt$ for the third metric not only regularized the sign of $dt/d\varphi$, but also stabilized the average of $dz_i$. Without this change of sign, the average would have diverged over time.

#### 7.3.2. The hybrid metric ($0 \le \alpha < 1$)

For the circular geodesics of a massive particle, let us set $dr = 0$. Table 9 summarizes the properties of the hybrid metric.

The variable $w$ is defined by the equation: $\frac{d^2w}{d\varphi^2} + w - 3w^2R_0/2 = \kappa R_0/(2h^2)$; this equation is verified approximately by the relations $\frac{dw}{d\varphi} = \mp \frac{\alpha E_r}{h}$ considered on a very short time, by taking into account the relation ($\omega \gg \pi/2$); on average over time: $dw/d\varphi = 0$.

We obtain, for the time-averaged terms $[dz]_{av}$, $[d\varphi/dt]_{av}$, $[ds^2]_{av}$:

$$ 
\begin{aligned}
[dz]_{av} &= \frac{(\alpha dr + dr) + (\alpha dt + dr) + (-\alpha dt - dr) + (\alpha dr - dr)}{4} \\
&= \frac{\alpha}{2} dr = \frac{dR_{av}}{(1 - R_0/R_{av})},
\end{aligned}
\quad (7.10) $$

$$ \left[ \frac{d\varphi}{dt} \right]_{\text{av}} = \frac{(1 + \alpha) + 2 + (1 - \alpha)}{4} \frac{(1 - R_0/R_{\text{av}})}{(R_{\text{av}})^2 E_r / h} \Rightarrow \left[ \frac{d\varphi}{dt} \right]_{\text{av}} = \frac{(1 - R_0/R_{\text{av}})}{(R_{\text{av}})^2 E_r / h}, \quad (7.11) $$

$$ 
\begin{aligned}
[ds^2]_{\text{av}} &= \frac{(1 + \alpha)^2 + 2(1 - \alpha^2) + (1 - \alpha)^2}{4} [1 - R_0/R_{\text{av}}(r)](dt^2 - dr^2) - \Omega^2 \\
&\Rightarrow [ds^2]_{\text{av}} = [1 - R_0/R_{\text{av}}(r)](dt^2 - dr^2) - [R_{\text{av}}(r)]^2(d\theta^2 + \sin^2\theta d\varphi^2),
\end{aligned}
\quad (7.12) $$

$$ [dz]_{\text{av}} = \frac{\alpha}{2} dr = \frac{dR_{\text{av}}}{(1 - R_0/R_{\text{av}})} \quad (0 \le \alpha < 1). \quad (7.13) $$

The average metric defined by the relations (7.12) and (7.13) is not the Schwarzschild metric. We will use this metric to model the transition zone of the galaxy rotation curve, between the Schwarzschild zone and the BTFR zone.
<table><caption>Table 6.</caption><thead><tr><th>Mode</th><th>Metric</th><th>dz</th><th>Geodesic: A</th><th>Geodesic: B</th><th>Symbols</th></tr></thead><tbody><tr><td>1st</td><td>ds<sup>2</sup> = (1 + α)<sup>2</sup>[1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>1</sub> = αdr + dr</td><td><sup>dt</sup>&frasl;<sub>dc</sub> = <sup>(R<sup>2</sup>E<sub>h</sub>/(h))</sup>&frasl;<sub>(1+α)(1−R<sub>0</sub>/R)</sub></td><td><sup>dc</sup>&frasl;<sub>dc</sub> = <sup>(dR/(dc)<sup>2</sup>)</sup>&frasl;<sub>(1+α)(1−R<sub>0</sub>/R)</sub></td><td>dt: temporal<br>dr:spatial</td></tr><tr><td>2nd</td><td>ds<sup>2</sup> = (1 − α<sup>2</sup>)[1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>2</sub> = αdr + dt</td><td><sup>dt</sup>&frasl;<sub>dc</sub> = − <sup>(R<sup>2</sup>E<sub>h</sub>/(h)α−(dR/(dc)<sup>2</sup>)</sup>&frasl;<sub>(1−α<sup>2</sup>)(1−R<sub>0</sub>/R)</sub></td><td><sup>dc</sup>&frasl;<sub>dc</sub> = <sup>(R<sup>2</sup>E<sub>h</sub>/(h)−(dR/(dc)<sup>2</sup>)α</sup>&frasl;<sub>(1−α<sup>2</sup>)(1−R<sub>0</sub>/R)</sub></td><td>dt: temporal<br>dr:spatial</td></tr><tr><td>3rd</td><td>ds<sup>2</sup> = (1 − α<sup>2</sup>)[1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>3</sub> = αdr − dt</td><td><sup>dt</sup>&frasl;<sub>dc</sub> = − <sup>(R<sup>2</sup>E<sub>h</sub>/(h)α+(dR/(dc)<sup>2</sup>)</sup>&frasl;<sub>(1−α<sup>2</sup>)(1−R<sub>0</sub>/R)</sub></td><td><sup>dc</sup>&frasl;<sub>dc</sub> = − <sup>(R<sup>2</sup>E<sub>h</sub>/(h)+dR/(dc)<sup>2</sup>)α</sup>&frasl;<sub>(1−α<sup>2</sup>)(1−R<sub>0</sub>/R)</sub></td><td>dt:spatial<br>dr:temporal</td></tr><tr><td>4th</td><td>ds<sup>2</sup> = (1 − α)<sup>2</sup>[1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>4</sub> = αdr − dr</td><td><sup>dt</sup>&frasl;<sub>dc</sub> = − <sup>(R<sup>2</sup>E<sub>h</sub>/(h))</sup>&frasl;<sub>(1−α)(1−R<sub>0</sub>/R)</sub></td><td><sup>dc</sup>&frasl;<sub>dc</sub> = − <sup>(dR/(dc)<sup>2</sup>)</sup>&frasl;<sub>(1−α)(1−R<sub>0</sub>/R)</sub></td><td>dt:temporal<br>dr:spatial</td></tr></tbody></table>

<table><caption>Table 7.</caption><thead><tr><th>Mode</th><th>Metric</th><th>dz</th><th>Geodesic: A</th><th>Geodesic: B</th><th>Symbols</th></tr></thead><tbody><tr><td>1st</td><td>ds<sup>2</sup> = (1 + α)<sup>2</sup>[1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>1</sub> = αdr + dr</td><td><sup>dt</sup>&frasl;<sub>dc</sub> = <sup>(R<sup>2</sup>E<sub>h</sub>/(h))</sup>&frasl;<sub>(1+α)(1−R<sub>0</sub>/R)</sub></td><td><sup>dc</sup>&frasl;<sub>dc</sub> = <sup>(dR/(dc)<sup>2</sup>)</sup>&frasl;<sub>(1+α)(1−R<sub>0</sub>/R)</sub></td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>2nd</td><td>ds<sup>2</sup> = (1 − α<sup>2</sup>) [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>2</sub> = αdt + dr</td><td><sup>dt</sup>&frasl;<sub>dc</sub> = <sup>(R<sup>2</sup>E<sub>h</sub>/(h)−(dR/(dc)<sup>2</sup>)α</sup>&frasl;<sub>(1−α<sup>2</sup>)(1−R<sub>0</sub>/R)</sub></td><td><sup>dc</sup>&frasl;<sub>dc</sub> = − <sup>(R<sup>2</sup>E<sub>h</sub>/(h)α−(dR/(dc)<sup>2</sup>)α</sup>&frasl;<sub>(1−α<sup>2</sup>)(1−R<sub>0</sub>/R)</sub></td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>3rd</td><td>ds<sup>2</sup> = (1 − α<sup>2</sup>) [1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>3</sub> = αdt − dr</td><td><sup>dt</sup>&frasl;<sub>dc</sub> = <sup>(R<sup>2</sup>E<sub>h</sub>/(h)+dR/(dc)<sup>2</sup>)α</sup>&frasl;<sub>(1−α<sup>2</sup>)(1−R<sub>0</sub>/R)</sub></td><td><sup>dc</sup>&frasl;<sub>dc</sub> = − <sup>(R<sup>2</sup>E<sub>h</sub>/(h)α+(dR/(dc)<sup>2</sup>)α</sup>&frasl;<sub>(1−α<sup>2</sup>)(1−R<sub>0</sub>/R)</sub></td><td>dt: temporal<br>dr: spatial</td></tr><tr><td>4th</td><td>ds<sup>2</sup> = (1 − α)<sup>2</sup>[1 − R<sub>0</sub>/R(t, r)](dt<sup>2</sup> − dt<sup>2</sup>) − Ω<sup>2</sup></td><td>dz<sub>4</sub> = αdr − dr</td><td><sup>dt</sup>&frasl;<sub>dc</sub> = − <sup>(R<sup>2</sup>E<sub>h</sub>/(h))</sup>&frasl;<sub>(1−α)(1−R<sub>0</sub>/R)</sub></td><td><sup>dc</sup>&frasl;<sub>dc</sub> = − <sup>(dR/(dc)<sup>2</sup>)</sup>&frasl;<sub>(1−α)(1−R<sub>0</sub>/R)</sub></td><td>dt:temporal<br>dr:spatial</td></tr></tbody></table>
<table><caption>Table 8.</caption><thead><tr><th>Mode</th><th>Metric</th><th>dz</th><th>Geodesic: A</th><th>Geodesic: B</th><th>Symbols</th></tr></thead><tbody><tr><td>1st</td><td>$ds_1^2 = (1 + \alpha)^2[1 - R_0/R(t, r)](dt^2 - dr^2) - \Omega^2$</td><td>$dz_1 = \alpha dr + dr$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h)}{(1+\alpha)(1-R_0/R)}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(dR/d\varphi)}{(1+\alpha)(1-R_0/R)}$</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>2nd</td><td>$ds_2^2 = (1 - \alpha^2)[1 - R_0/R(t, r)](dt^2 - dr^2) - \Omega^2$</td><td>$dz_2 = \alpha dt + dr$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h) - (dR/d\varphi)}{(1-\alpha)(1-R_0/R)}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = -\frac{(R^2 E_e / h)\alpha - (dR/d\varphi)}{(1-\alpha)(1-R_0/R)}$</td><td>dt:temporal<br>dr:spatial</td></tr><tr><td>3rd</td><td>$ds_3^2 = (1 - \alpha^2)[1 - R_0/R(t, r)](dt^2 - dr^2) - \Omega^2$</td><td>$dz_3 = -\alpha dt - dr$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h) + (dR/d\varphi)}{(1-\alpha)(1-R_0/R)}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = -\frac{(R^2 E_e / h)\alpha + (dR/d\varphi)}{(1-\alpha)(1-R_0/R)}$</td><td>dt: temporal<br>dr: spatial</td></tr><tr><td>4th</td><td>$ds_4^2 = (1 - \alpha)^2[1 - R_0/R(t, r)](dt^2 - dr^2) - \Omega^2$</td><td>$dz_4 = \alpha dr - dr$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h)}{(1-\alpha)(1-R_0/R)}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = -\frac{(dR/d\varphi)}{(1-\alpha)(1-R_0/R)}$</td><td>dt:temporal<br>dr:spatial</td></tr></tbody></table>

<table><caption>Table 9.</caption><thead><tr><th>Mode</th><th>Metric</th><th>dz</th><th>Circular geodesic: A</th><th>Circular geodesic: B</th><th>$w = 1/R$</th></tr></thead><tbody><tr><td>1st</td><td>$ds_1^2 = (1 + \alpha)^2[1 - R_0/R(t, r)](dt^2 - dr^2) - \Omega^2$</td><td>$dz_1 = \alpha dr + dr$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h)}{(1-\alpha)(1-R_0/R)}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = 0$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = 0$</td></tr><tr><td>2nd</td><td>$ds_2^2 = (1 - \alpha^2)[1 - R_0/R(t, r)](dt^2 - dr^2) - \Omega^2$</td><td>$dz_2 = \alpha dt + dr$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h)}{(1-R_0/R)}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = -\frac{(R^2 E_e / h)\alpha - (dR/d\varphi)}{\alpha}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = -\frac{\alpha E_e}{\alpha}$</td></tr><tr><td>3rd</td><td>$ds_3^2 = (1 - \alpha^2)[1 - R_0/R(t, r)](dt^2 - dr^2) - \Omega^2$</td><td>$dz_3 = -\alpha dt - dr$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h)}{(1-R_0/R)}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h)\alpha + (dR/d\varphi)}{\alpha}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{\alpha E_e}{\alpha}$</td></tr><tr><td>4th</td><td>$ds_4^2 = (1 - \alpha)^2[1 - R_0/R(t, r)](dt^2 - dr^2) - \Omega^2$</td><td>$dz_4 = \alpha dr - dr$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = \frac{(R^2 E_e / h)}{(1-\alpha)(1-R_0/R)}$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = 0$</td><td>$\frac{d\bar{\varphi}}{d\varphi} = 0$</td></tr></tbody></table>
The real time differential interval $dT$ between two events located at $(r, \theta, \varphi)$ is

$$dT = \sqrt{1 - R_0/R_{av}(r)} dt \approx dt \quad (\text{case: } R_0/R_{av}(r) \ll 1). \quad (7.14)$$

The real radial distance differential interval $dL_r$ is

$$dL_r = \sqrt{1 - R_0/R_{av}(r)} dr \approx dr \quad (\text{case: } R_0/R_{av}(r) \ll 1). \quad (7.15)$$

From the relation (7.13), we obtain $dr = (2/\alpha)dR_{av}/(1 - R_0/R_{av})$. From the relation (7.15), we obtain

$$dL_r \approx \left(\frac{2}{\alpha}\right) dR_{av} \quad (\text{case: } R_0/R_{av}(r) \ll 1). \quad (7.16)$$

From the relations (4.8) and (7.11), we obtain on average $[\frac{d\varphi}{dt}]_{av} = \frac{\sqrt{GM/c^2}}{R_s \sqrt{R_s}}$.

The orbital velocity $V_T$ of a massive particle located on a circular orbit defined by $R_{av}(r_s) = R_s$ is defined by the following relation:

$$(V_T)^2 = \frac{GM/R_s}{(1 - R_0/R_s)}. \quad (7.17)$$

The radial acceleration $a_r$ is defined by the approximate relation

$$a_r \approx [GM/(R_s)^2]. \quad (7.18)$$

The relation (7.17) is identical to the relation (6.5). Consequently, we conclude that these formulas are valid in all the zones of space under the simplified form $(R_0/R_{av}(r) \ll 1)$

$$(V_T)^2 \approx \frac{GM}{R_{av}(r_s)}, \quad (7.19)$$

$$a_r \approx \frac{(V_T)^2}{R_{av}(r_s)} \approx \frac{GM}{[R_{av}(r_s)]^2}. \quad (7.20)$$

Table 10 summarizes the definitions of the function $z$, according to the three zones of the galaxy. In the transition zone, we will have to redefine the coefficient $\alpha$, in order to get a physical result for the galaxy rotation curve.

<table><caption>Table 10.</caption><thead><tr><th>Zone</th><th>$dz/dr \approx dR_{av}/dr$</th><th>$z \approx R_{av}$</th></tr></thead><tbody><tr><td>Schwarzschild</td><td>1</td><td>r</td></tr><tr><td>Transition ($0 \le \alpha < 1$)</td><td>$\alpha/2$</td><td>cf. Sec. 9</td></tr><tr><td>BTFR</td><td>0</td><td>$R_C(M_b)$</td></tr></tbody></table>
## 8. Discussion on the Hybrid Metric

We can look for a more general solution in the following form, where $\mu(x)$ is an arbitrary function:

$$HP(u) = r_A + \mu(u)/2 + F_{\text{tr}}^{\omega}(u)/2; \quad HM(v) = \mu(v)/2 + F_{\text{tr}}^{\omega}(v)/2, \quad (8.1)$$

$$HP^u = [\mu^u + F_{\text{sq}}^{\omega}(u)]/2; \quad HM^v = [\mu^v + F_{\text{sq}}^{\omega}(v)]/2; \quad 0 \le \mu^u < 1; 0 \le \mu^v < 1. \quad (8.2)$$

After all calculations have been completed, we find $[dz]_{\text{av}} = \frac{\mu^u}{2}dr = \frac{dR_{\text{av}}}{(1-R_0/R_{\text{av}})}$. It follows that $[dz]_{\text{av}}$ is a full differential only if $\mu^u$ is a constant number, which is the case we have studied. The maximum value of $z$: $z_{\text{max}}$, is obtained in the second mode as: $z_{\text{max}} = z(t + \frac{T_p}{4}, r) = z_{\text{av}}(r) + \frac{\alpha}{2} \frac{T_p}{4}$. We can obtain the inequality: $z_{\text{max}} > R_C$ if $z_{\text{av}}(r) + \alpha \frac{T_p}{8} > R_C$. If we assume that $z_{\text{av}}$ tends asymptotically to $R_C$, there will be always a zone where $z_{\text{max}} > R_C$, if $\alpha > 0$. This issue will be dealt with in Sec. 9.

## 9. The Galaxy Rotation Curve According to our General Relativity Solution

The first part of the galaxy rotation curve is defined by the Schwarzschild metric. In this zone, the following relation: $R_{\text{av}}(r) \approx r$ applies ($\alpha > 1$); the function $z$ oscillates according to the following formalism (Table 11).

In the transient zone ($0 < \alpha < 1$), the function $z$ oscillates according to the following formalism (Table 12).

Consequently, locally, the maximum value over time of the function $z$ is reached in the second mode, in both cases.

In the asymptotic zone, where BTFR applies, we have obtained $R(t, r) = R_C \approx \sqrt{GM_b/A_0(M_b)}$; then, if we make the assumption that this $R_C$ value

<table><caption>Table 11.</caption><thead><tr><th>Mode</th><th>dz</th></tr></thead><tbody><tr><td>1st</td><td>dz<sub>1</sub> = dr + dr</td></tr><tr><td>2nd</td><td>dz<sub>2</sub> = dr + dt</td></tr><tr><td>3rd</td><td>dz<sub>3</sub> = dr - dt</td></tr><tr><td>4th</td><td>dz<sub>4</sub> = dr - dr</td></tr></tbody></table>

<table><caption>Table 12.</caption><thead><tr><th>Mode</th><th>dz</th></tr></thead><tbody><tr><td>1st</td><td>dz<sub>1</sub> = αdr + dr</td></tr><tr><td>2nd</td><td>dz<sub>2</sub> = αdt + dr</td></tr><tr><td>3rd</td><td>dz<sub>3</sub> = -αdt - dr</td></tr><tr><td>4th</td><td>dz<sub>4</sub> = αdr - dr</td></tr></tbody></table>
cannot be exceeded in the whole galaxy (without External Field Effect), it is necessary to have a damping mechanism to control the oscillations of the function z over time. We postulate this mechanism by setting the damping function α:

$$ \alpha = 4[1 - z/R_C(M_b)]^2. \qquad (9.1) $$

The asymptotic metric describes a quasi-frozen space in which the space-time curvature does not vary in average. We can perhaps understand this state as being in perfect thermodynamic equilibrium; in this hypothesis, the thermodynamic disequilibrium would be caused by the oscillation of the space-time curvature; the damping function (9.1) would describe the mechanism of return to the thermodynamic equilibrium.

We postulated the formula (9.1) in order to obtain a galaxy rotation curve in agreement with the radial acceleration relation (RAR).10,11 The function $α(z/R_C)$ plays two roles: on the one hand, it is a damping function of the temporal oscillations of the function z; on the other hand, this function defines the spatial gradient of the function z.

The Schwarzschild zone ends at the boundary with the transition zone at $α = 1$; in the Schwarzschild zone ($α > 1$) the relation $\frac{dR_{av}}{dr} = 1$ applies, whatever the value of $α > 1$; in the transient zone ($0 ≤ α < 1$), the relation $\frac{dR_{av}}{dr} = \frac{α}{2}$ applies. Therefore, the derivative $dR_{av}/dr$ is discontinuous at the end of the Schwarzschild zone, since $\frac{dR_{av}}{dr} = 1$ to the left of the boundary ($α = 1^+$), and $\frac{dR_{av}}{dr} = \frac{1}{2}$ to the right of the boundary ($α = 1^−$).

The function z being very close to the R function ($R_0 ≪ R$), we replace z by R (the function $R_{av}$ is the average over time of the function R).

In the transient zone, for the temporal role, we obtain, in the second mode $\frac{dR}{dt} = \frac{α}{2} = 2[1 - \frac{R}{R_C(M_b)}]^2$. The maximum value of R: $R_{max}$, is $R_{max} = R(t + \frac{T_p}{4}, r) ⇒ (1 - \frac{R_{max}}{R_C(M_b)}) = \frac{[1 - R_{av}(r)/R_C(M_b)]}{1 + [1 - R_{av}(r)/R_C(M_b)][T_p/(2R_C(M_b))]} > 0 ⇒ R_{max} < R_C(M_b)$

For the spatial role:

$$ \frac{dR_{av}}{dr} = \frac{α}{2} = 2 \left[ 1 - \frac{R_{av}}{R_C(M_b)} \right]^2 \qquad (9.2) $$

In formula (9.2), we have approximated the function R by its temporal average $R_{av}$. In a real galaxy, we have to take into account the mass variation over space; we generalize this last formula in the following form, where $M_{b<r}$ is equal to the baryonic mass internal to the sphere defined by the coordinate r:

$$ \beta = \frac{R_{av}}{R_C(M_{b<r})}; \quad \rho = \frac{r}{R_C(M_{b<r})}; \quad \beta_0 = \frac{r_0}{R_C(M_{b<r})} \Rightarrow \frac{d\beta}{d\rho} = \frac{\alpha}{2} = 2(1 - \beta)^2, \qquad (9.3) $$

where $r_0$ is the radial coordinate value at which the transition zone begins; at this coordinate: $α = 1 + ε \sim 1(ε ≪ 1)$.
In the Schwarzschild zone : $\rho \le \beta_0 : \frac{d\beta}{d\rho} = 1 \Rightarrow \beta = \rho$

The transition zone begins at $\alpha = 1 + \varepsilon$; $\beta = \beta_0 = \frac{1}{2}$; $\frac{d\beta}{d\rho} = \frac{1}{2}$

In the transition zone : $\rho > \beta_0 = \frac{1}{2} \Rightarrow \frac{d\beta}{d\rho} = 2(1 - \beta)^2 \Rightarrow \beta = \frac{2\rho}{1 + 2\rho}$

Summary of the formulas:

$$R_C \approx \sqrt{GM_{b<r}/A_0(M_{b<r})}, \qquad (9.4)$$

$$\rho = \frac{r}{R_C(M_{b<r})}, \qquad (9.5)$$

$$\beta = \frac{R_{av}}{R_C(M_{b<r})}, \qquad (9.6)$$

$$\text{if } \rho \le \frac{1}{2} \Rightarrow \beta = \rho; \quad \text{if } \rho > \frac{1}{2} \Rightarrow \beta = \frac{2\rho}{1 + 2\rho}, \qquad (9.7)$$

radial acceleration:

$$a_r \approx \frac{GM_{b<r}}{(R_{av})^2}, \qquad (9.8)$$

orbital velocity of a massive particle:

$$(V_T)^2 \approx \frac{GM_{b<r}}{R_{av}}, \qquad (9.9)$$

angular deviation of a particle of zero mass:

$$\delta_R = \frac{4GM_{b<r}}{c^2 R_{av}} = 4 \frac{(V_T)^2}{c^2}, \qquad (9.10)$$

Kretschmann scalar:

$$\mathrm{Kr} = 48 \frac{G^2 (M_{b<r})^2}{c^4 (R_{av})^6}. \qquad (9.11)$$

According to the hybrid metric coupled to the damping function (9.1), the asymptotic metric is reached only at infinity; without any external field effect, we obtain in the asymptotic regime: an upper bound $R_C \approx \sqrt{GM_b/A_0(M_b)}$ for the $R(t, r)$ value, a lower bound $a_r = A_0(M_b)$ for the radial acceleration, and a lower bound for the Kretschmann scalar $= \frac{48[A_0(M_b)]^3}{c^4GM_b}$. Consequently, the space-time curvature has a lower bound in the spherical space without EFE.
## 10. The Galaxy Rotation Curve According to the MOND, McGaugh and GR Models

In his MOND theory,⁷⁻⁹ in 1983, Milgrom postulated a formulation of the radial acceleration in galaxies with the formalism: $g_{\text{bar}} = \frac{GM_{\text{bar}}}{r^2}$; $g_{\text{obs}} = \frac{V^2}{r}$; $g_{\text{bar}} = g_{\text{obs}}\mu(g_{\text{obs}}/a_0)$; $a_0 \approx 1.2 \times 10^{-10} \text{ m} \cdot \text{s}^{-2}$; the function $\mu(g_{\text{obs}}/a_0)$ is an empirical interpolation function, equal to 1 if $g_{\text{obs}} \gg a_0$, and equal to: $\mu(g_{\text{obs}}/a_0)$ if $g_{\text{obs}} \ll a_0$. Two current functions $\mu$ used by Milgrom are $\mu_1 = \sqrt{\frac{(g_{\text{obs}}/a_0)^2}{1+(g_{\text{obs}}/a_0)^2}}$; $\mu_2 = \frac{(g_{\text{obs}}/a_0)}{1+(g_{\text{obs}}/a_0)}$; $g_{\text{bar}}$ is the baryonic acceleration; $g_{\text{obs}}$ is the observed acceleration. We will use $\mu_2$ in this paper.

McGaugh¹⁰,¹¹ postulated the formulation $g_{\text{obs}} = g_{\text{bar}}/[1 - e^{-\sqrt{g_{\text{bar}}/a_0}}]$, where $a_0 = 1.2 \times 10^{-10} \text{ m} \cdot \text{s}^{-2}$. In,¹⁰ Li et al. showed that this last correlation gives a tight fit for the Radial Acceleration Relation (RAR) to the SPARC database. For this reason, the GR damping function (9.1) has been defined to fit the McGaugh correlation. We can obtain, for our GR formulation, the formalism used by Milgrom and McGaugh by deducing $g_{\text{bar}} = \frac{a_0}{\rho^2}$; $g_{\text{obs}} = \frac{a_0}{\beta\rho}$. The three functions $g_{\text{obs}}(g_{\text{bar}})$ are reported in Table 13, where $A_0(M_\text{b}) = a_0$ for the GR model.

The functions $\beta(\rho)$ are reported in Table 14.

The three formulas give similar results, as we can see in Figs. 1 and 2. In Fig. 2, $g_{\text{obs}}$ and $g_{\text{bar}}$ are in $\log_{10}$ coordinates.

The formulas by Milgrom ($\mu_2$) and McGaugh give almost indistinguishable results. The GR damping function $\alpha = 4(1 - \beta)^2$ has been determined as the simplest formula to be in agreement with the McGaugh and Milgrom ($\mu_2$) formulas. Consequently, the galaxy rotation curve is similar for the three calculations. There is a difference in the formulation of the radial acceleration between the MOND model and the GR model. In the MOND theory, the radial acceleration $g_{\text{obs}}$ is postulated with the following formalism $g_{\text{obs}} = \frac{V^2}{r}$; $g_{\text{bar}} = g_{\text{obs}}\mu(g_{\text{obs}}/a_0)$. In our reasoning, within the framework of General Relativity, the radial acceleration $a_r$ is a result of the theory. We have obtained the formulas $a_r \approx V^2/R_{\text{av}}(r) \approx GM_\text{b}/[R_{\text{av}}(r)]^2$.

<table><caption>Table 13.</caption><tr><td>Milgrom ($\mu_2$)</td><td>$g_{\text{obs}} = g_{\text{bar}}/2 + \sqrt{g_{\text{bar}}a_0 + (g_{\text{bar}}/2)^2}$</td></tr><tr><td>McGaugh</td><td>$g_{\text{obs}} = g_{\text{bar}}/[1 - e^{-\sqrt{g_{\text{bar}}/a_0}}]$</td></tr><tr><td>GR</td><td>$g_{\text{bar}} \ge 4a_0 \Rightarrow g_{\text{obs}} = g_{\text{bar}}; g_{\text{bar}} < 4a_0 \Rightarrow g_{\text{obs}} = g_{\text{bar}}/2 + \sqrt{g_{\text{bar}}a_0}$</td></tr></table>

<table><caption>Table 14.</caption><tr><td>Milgrom ($\mu_2$)</td><td>$\beta = \frac{2\rho}{1+\sqrt{1+4\rho^2}}$</td></tr><tr><td>McGaugh</td><td>$\beta = \rho(1 - e^{-1/\rho})$</td></tr><tr><td>GR</td><td>$\rho \le \frac{1}{2} \Rightarrow \beta = \rho; \rho > \frac{1}{2} \Rightarrow \beta = \frac{2\rho}{1+2\rho}$</td></tr></table>
<figure><img src="image_4.png" /><figcaption>Fig. 1. Plan($g_{\text{bar}}$, $g_{\text{obs}}$).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Plan($\rho$, $\beta$).</figcaption></figure>

Consequently: $g_{\text{obs}} \neq a_r : g_{\text{obs}} \approx \sqrt{g_{\text{bar}}} a_0 \propto \frac{1}{r}$ in the MOND asymptotic regime, while $a_r$ tends to $A_0(M_b)$ in the GR asymptotic regime; $a_r$ cannot be lower than $A_0(M_b)$ (without External Field Effect). However, the galaxy rotation curve is quite similar in both cases.

# 11. The External Field Effect (EFE)

The asymptotic metric does not describe the real asymptotic behavior of a galaxy rotation curve, because of the external field effect induced by the nearby galaxies. We have performed a preliminary study, reported in Appendix D, to evaluate the impact
<figure><img src="image_6.png" /><figcaption>Fig. 3. MW and M31 rotation curves ($\alpha_{m\_MW} = 0.40$; $\alpha_{m\_M31} = 0.15$).</figcaption></figure>

of the external field effect, in the case of the couple MW-M31. The results obtained
are as follows:

(1) The $\alpha$ formula: $\alpha = 4(1 - \beta)^2$ is modified: $\Rightarrow \alpha = \max[\alpha_m, 4(1 - \beta)^2]$, $\alpha_m$ being a constant number for a galaxy, depending on the nearby galaxies. Consequently, we have evaluated two values: $\alpha_{m\_MW}$ and $\alpha_{m\_M31}$.

(2) Our preliminary assessment is $\alpha_{m\_MW} = 0.40$; $\alpha_{m\_M31} = 0.15$.

(3) We observe that the gravitational field of MW is significantly more distorted than that of M31 ($\alpha_{m\_MW} \approx 3 \times \alpha_{m\_M31}$) (cf. Fig. 3),

These results should be viewed with caution. This is a first analysis of the defor-
mation of the gravitational field of galaxies caused by their nearby environment,
based only on a particular case: the Milky Way — Andromeda couple.

12. Gravitational Lensing

In this paper, we have not analyzed detailed measurements of gravitational lensing.
However, we can affirm that, if in a given configuration, the velocities of the stars
obtained with our model are similar to the velocities obtained with a dark matter
model, the angular deflection of the light rays will be in agreement in both models,
because of the formula $\delta_R = 4(V_T)^2/c^2$, valid in both models.

Consequently, we obtain a new result for the MOND theory: although MOND is
not a relativistic theory, if we apply our results obtained in the framework of General
Relativity, the deflection of light rays $\delta_R$ must be calculated by the formula
$\delta_R = 4(V_T)^2/c^2$; it is the same for the RAR.

13. Conclusion

In the framework of General Relativity, in symmetric spherical vacuum space, we
have obtained a new kind of metric: the so-called 4-metric, composed of four distinct
metrics which, locally, follow each other cyclically over time. The time-averaged metric obtained from a 4-metric can be different from the Schwarzschild metric, depending on its four component metrics. A particular case: our asymptotic 4-metric, is in agreement with the BTFR: the orbital velocity of a massive particle is constant in this metric. Consequently, BTFR can be considered as resulting from General Relativity, without any dark matter, and without any assumption on the spatial distribution of the stars. Physically, this regime is the asymptotic regime obtained in the spherical space (without external field effect). According to our findings, in this metric, the radial acceleration is equal to a universal function of the baryonic mass of the central massive body: $A_0(M_b)$. Consequently, in later work, to deal with the problem of dark matter in intergalactic space, it will be possible, if necessary, to use an extended function $A_0(M_b)$, to fit the relative velocity measurements between galaxies. These findings suggest that Birkhoff's theorem is incomplete.

Until the end of its flat behavior, the galaxy rotation curve is explained by our research as resulting from the existence of a lower bound of the radial acceleration (respectively, the Riemann space-time curvature), leading to an oscillation of the radial acceleration (respectively, the Riemann space-time curvature) in the whole galaxy. These oscillations make the Schwarzschild metric unstable from a certain radial distance. From this distance, the metric converges asymptotically to the regime of the BTFR, until the external field effect acts on the intrinsic gravitational field of the galaxy. We have outlined a method to calculate all these phenomena. By combining the Schwarzschild metric with the asymptotic 4-metric, we have obtained the hybrid 4-metric, in which the parameter $R(t,r)$ is oscillatory over time. By coupling the hybrid 4-metric with a damping function including a preliminary version of the external field effect, we have obtained a metric which is in agreement with the observed galaxy rotation curve.

As these findings are obtained in the framework of General Relativity without presence of dark matter and without any assumption about the distribution of stars in galaxies, their validation would challenge the standard model of cosmology (ΛCDM).

We have performed the calculation of the light deflection, but we have not presented comparisons with the deflection measurements in this paper; however, we have shown that the calculation of the light deflection will agree with a dark matter model if the velocity of the stars agrees in both models.

There is a point of disagreement between our work and the MOND theory: the radial acceleration resulting from our calculations is different from the acceleration postulated in the MOND theory (but our galaxy rotation curve is very close to that of MOND).

The aim of this paper being essentially to present our calculations concerning the incompleteness of Birkhoff's theorem, and the first consequences resulting from it (BTFR and the galaxy rotation curve), the interaction of galaxies in clusters resulting from our findings has not been presented in this paper.
# Appendix A. Calculating the Non-Static Metric

In the mathematical expressions of the appendices, the exponents $t, r$ define the partial derivatives with respect to the $t, r$ coordinates. The exponents $u, v, z, R$ define the full derivatives with respect to functions $u, v, z, R$. $R_0$ is defined by the relation: $R_0 = 2GM/c^2$.

## A.1. The Ricci tensor

For the space-time interval $ds^2 = e^{2\nu(t,r)}dt^2 - e^{2\lambda(t,r)}dr^2 - [R(t,r)]^2(d\theta^2 + \sin^2\theta d\phi^2)$, the non-zero Ricci tensor components are defined by the following expressions (cf. calculations in Appendix C: relations (C.1)–(C.5)):

$$R_{00} = \nu^t(\lambda^t + 2R^t/R) - (\lambda^t\lambda^t + \lambda^{tt} + 2R^{tt}/R) + e^{2\nu-2\lambda}[\nu^{rr} + \nu^r(\nu^r - \lambda^r) + 2\nu^r R^r/R] = 0, \quad (A.1)$$

$$R_{11} = \lambda^r(\nu^r + 2R^r/R) - (\nu^r\nu^r + \nu^{rr} + 2R^{rr}/R) + e^{2\lambda-2\nu}[\lambda^{tt} - \lambda^t(\nu^t - \lambda^t) + 2\lambda^t R^t/R] = 0, \quad (A.2)$$

$$R_{22} = 1 + e^{-2\nu}[R^{tt}R + R^t R^t - R^t R(\nu^t - \lambda^t)] - e^{-2\lambda}[R^{rr}R + R^r R^r + R^r R(\nu^r - \lambda^r)] = 0, \quad (A.3)$$

$$R_{33} = \sin^2\theta\{1 + e^{-2\nu}[R^{tt}R + R^t R^t - R^t R(\nu^t - \lambda^t)] - e^{-2\lambda}[R^{rr}R + R^r R^r + R^r R(\nu^r - \lambda^r)]\} = 0, \quad (A.4)$$

$$R_{01} = R_{10} = (2/R)(-R^{\mathrm{tr}} + \nu^r R^t + \lambda^t R^r) = 0. \quad (A.5)$$

By setting $R(t, r) = r$, we can verify that we obtain the Ricci tensor leading to the definition of the classical Schwarzschild metric. Now, instead of setting $R(t, r) = r$, we set $\lambda(t, r) = \nu(t, r)$. We make this choice to obtain a Schwarzschild metric in tortoise coordinates, extended in the temporal domain. We neglect the $R_{33}$ component, which is proportional to $R_{22}$, and we multiply each tensor component by a convenient term, to obtain the following new set of four equations:

$$ds^2 = e^{2\nu(t,r)}dt^2 - e^{2\nu(t,r)}dr^2 - [R(t,r)]^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (A.6)$$

$$RR_{00} = 2\nu^t R^t - R\nu^{tt} - 2R^{tt} + R\nu^{rr} + 2\nu^r R^r = 0, \quad (A.7)$$

$$RR_{11} = 2\nu^r R^r - R\nu^{rr} - 2R^{rr} + R\nu^{tt} + 2\nu^t R^t = 0, \quad (A.8)$$

$$e^{2\nu}R_{22} = e^{2\nu} + (R^t R^t + RR^{tt}) - (R^{rr}R + R^r R^r) = 0, \quad (A.9)$$

$$(R/2)R_{01} = -R^{\mathrm{tr}} + \nu^r R^t + \lambda^t R^r = 0. \quad (A.10)$$

We perform two combinations of Eqs. (A.7) and (A.8):

The opposite of half-sum: Eq. (A.7)/2 + Eq. (A.8)/2 and its consequences

$$-2\nu^t R^t + R^{tt} - 2\nu^r R^r + R^{rr} = 0 \Rightarrow e^{-2\nu}(-2\nu^t R^t + R^{tt}) + e^{-2\nu}(-2\nu^r R^r + 2R^{rr}) = 0 \Rightarrow (R^t e^{-2\nu})^t + (R^r e^{-2\nu})^r = 0.$$
The half-difference: Eq. (A.7)/2–Eq. (A.8)/2

$$-R\nu^{tt} - R^{tt} + R\nu^{rr} + R^{rr} = 0$$

$$\text{From Eq. (A.9)} \Rightarrow e^{2\nu} + (R^t R^t + RR^{tt}) - (R^{rr} R + R^r R^r) = 0$$

$$\Rightarrow (RR^r)^r - (RR^t)^t = e^{2\nu}.$$

Let us set an arbitrary two times differentiable function $F(t, r)$; we obtain the final set of four equations:

$$(R^t e^{-2\nu} + F^r)^t + (R^r e^{-2\nu} - F^t)^r = 0, \qquad (A.11)$$

$$(RR^r)^r - (RR^t)^t - e^{2\nu} = 0, \qquad (A.12)$$

$$R^{tt} - R^{rr} + R\nu^{tt} - R\nu^{rr} = 0, \qquad (A.13)$$

$$-R^{tr} + \nu^r R^t + \nu^t R^r = 0. \qquad (A.14)$$

## A.2. Solving Eqs. (A.11) and (A.14)

With the $F$ function being arbitrary, we can split Eq. (A.11) by setting both Eqs. (A.15):

$$(R^t e^{-2\nu} + F^r) = 0; \quad (R^r e^{-2\nu} - F^t) = 0, \qquad (A.15)$$

$$R^t = -F^r e^{2\nu}; \quad R^r = F^t e^{2\nu}, \qquad (A.16)$$

$$R^t F^t + R^r F^r = 0. \qquad (A.17)$$

Let us calculate the derivatives of relations (A.16) and replace in Eq. (A.14) $R^t$ and $R^r$ by their values.

$$\begin{aligned} R^{tr} &= -F^{rr} e^{2\nu} - 2\nu^r F^r e^{2\nu} = R^{rt} = F^{tt} e^{2\nu} + 2\nu^t F^t e^{2\nu} \\ &= -\nu^r F^r e^{2\nu} + \nu^t F^t e^{2\nu} \Rightarrow F^{tt} = F^{rr} = -\nu^r F^r - \nu^t F^t \end{aligned} \qquad (A.18)$$

$$F^{tt} - F^{rr} = 0.$$

Let us set $u = t + r$; $v = t - r \Rightarrow \text{FP}(u)$ and $\text{FM}(v)$ being two unknown two times differentiable functions, we obtain

$$F = \text{FP}(u) + \text{FM}(v) \Rightarrow F^t = \text{FP}^u + \text{FM}^v; \qquad (A.19)$$

$$F^r = \text{FP}^u - \text{FM}^v \Rightarrow \rho = (F^t)^2 - (F^r)^2 = 4\text{FP}^u\text{FM}^v.$$

Rewriting Eq. (A.17) in coordinates $(u, v)$ instead of $(t, r)$

$$\begin{aligned} R^t F^t + R^r F^r &= 0 \\ \Rightarrow \left[ \left( \frac{\partial R}{\partial u} \right)_v + \left( \frac{\partial R}{\partial v} \right)_u \right] (\text{FP}^u + \text{FM}^v) + \left[ \left( \frac{\partial R}{\partial u} \right)_v - \left( \frac{\partial R}{\partial v} \right)_u \right] (\text{FP}^u - \text{FM}^v) &= 0 \\ \Rightarrow \left( \frac{\partial R}{\partial u} \right)_v \text{FP}^u + \left( \frac{\partial R}{\partial v} \right)_u \text{FM}^v &= 0. \end{aligned}$$
Let us set:

$$ \mathrm{HP}^u = 1/(4\mathrm{FP}^u); \quad \mathrm{HM}^v = 1/(4\mathrm{FM}^v) \Rightarrow \left(\frac{\partial R}{\partial u}\right)_v + \left(\frac{\partial R}{\partial v}\right)_u = 0, \qquad (\mathrm{A.20}) $$

$$ 4\mathrm{HP}^u\mathrm{HM}^v = 1/(4\mathrm{FP}^u\mathrm{FM}^v) = 1/\rho. $$

Rewriting Eq. (A.20) in coordinates (HP, HM) instead of (u, v):

$$ \left(\frac{\partial R}{\partial u}\right)_v = \left(\frac{\partial R}{\partial \mathrm{HP}}\right)_{\mathrm{HM}} \mathrm{HP}^u; \\ \left(\frac{\partial R}{\partial v}\right)_u = \left(\frac{\partial R}{\partial \mathrm{HM}}\right)_{\mathrm{HP}} \mathrm{HM}^v \Rightarrow \left(\frac{\partial R}{\partial \mathrm{HP}}\right)_{\mathrm{HM}} + \left(\frac{\partial R}{\partial \mathrm{HM}}\right)_{\mathrm{HP}} = 0. \qquad (\mathrm{A.21}) $$

The solution of Eq. (A.21) is an arbitrary function of the z parameter defined hereafter:

$$ z = \mathrm{HP}(u) - \mathrm{HM}(v), \qquad (\mathrm{A.22}) $$

$$ R = R(z), \qquad (\mathrm{A.23}) $$

$$ z^t = \mathrm{HP}^u - \mathrm{HM}^v; \quad z^r = \mathrm{HP}^u + \mathrm{HM}^v \Rightarrow z^{tt} - z^{rr} = 0; \qquad (\mathrm{A.24}) $$

$$ (z^r)^2 - (z^t)^2 = 4\mathrm{HP}^u\mathrm{HM}^v = 1/\rho, $$

$$ R = R(z) \Rightarrow R^t = R^z(\mathrm{HP}^u - \mathrm{HM}^v); \quad R^r = R^z(\mathrm{HP}^u + \mathrm{HM}^v), \qquad (\mathrm{A.25}) $$

from relations (A.16): $e^{2\nu} = -R^t/F^r = -R^z \frac{\mathrm{HP}^u - \mathrm{HM}^v}{\mathrm{FP}^u - \mathrm{FM}^v} = -R^z \frac{\mathrm{HP}^u - \mathrm{HM}^v}{\left(\frac{1}{\mathrm{HP}^u} - \frac{1}{\mathrm{HM}^v}\right)}$,

$$ e^{2\nu} = 4\mathrm{HP}^u\mathrm{HM}^v R^z = R^z/\rho. \qquad (\mathrm{A.26}) $$

### A.3. Solving Eq. (A.12)

$$ \begin{aligned} & (RR^r)^r - (RR^t)^t - e^{2\nu} = 0 \\ & \Rightarrow (RR^z z^r)^r - (RR^z z^t)^t - R^z/\rho = 0 \\ & \Rightarrow (RR^z)^z z^r z^r + RR^z z^{rr} - (RR^z)^z z^t z^t - RR^z z^{tt} - R^z/\rho = 0 \\ & \Rightarrow (RR^z)^z (z^r z^r - z^t z^t) + RR^z (z^{rr} - z^{tt}) - R^z/\rho = 0 \\ & \Rightarrow (RR^z)^z/\rho - R^z/\rho = 0 \Rightarrow (RR^z)^z - R^z = 0 \\ & \Rightarrow RR^z - R = -R_0 \Rightarrow R^z = (1 - R_0/R), \end{aligned} \qquad (\mathrm{A.27}) $$

$$ e^{2\nu} = 4\mathrm{HP}^u\mathrm{HM}^v R^z = 4\mathrm{HP}^u\mathrm{HM}^v (1 - R_0/R). \qquad (\mathrm{A.28}) $$

At the limit of the static metric, we obtain $\mathrm{HP}^u = \mathrm{HM}^v = 1/2$.

$$ R^z = dR/dz = (1 - R_0/R) \Rightarrow dz = dR/(1 - R_0/R) \Rightarrow \qquad (\mathrm{A.29}) $$

$$ z = \mathrm{HP}(u) - \mathrm{HM}(v) = R + R_0 \mathrm{Log}|R/R_0 - 1|. \qquad (\mathrm{A.30}) $$

The relation (A.30) allows determining implicitly the $R(t, r)$ function as a function of the two differentiable functions $\mathrm{HP}(u)$ and $\mathrm{HM}(v)$.
In the case of the static metric: $HP(u) = (t+r)/2$, $HM(v) = (t-r)/2 \Rightarrow z = r$. The relation (A.28) defines $e^{2\nu(t,r)}$; the definition of the non-static metric is completed, but we have to verify Eq. (A.13).

### A.4. Verification of Eq. (A.13): $R^{tt} - R^{rr} + R\nu^{tt} - R\nu^{rr} = 0$

$$ \text{From relation (A.23) } \Rightarrow R^t = R^z z^t; \quad R^r = R^z z^r \Rightarrow R^{tt} = R^{zz} z^t z^t + R^z z^{tt}; \qquad (A.31) \\ R^{rr} = R^{zz} z^r z^r + R^z z^{rr}, $$

$$ z^{rr} - z^{tt} = 0; \quad (z^r)^2 - (z^t)^2 = 1/\rho; \qquad (A.32) \\ R^{tt} - R^{rr} = R^{zz}(z^t z^t - z^r z^r) + R^z(z^{tt} - z^{rr}) \Rightarrow R^{tt} - R^{rr} = -R^{zz}/\rho, $$

$$ \text{from relations (A2.13) } \Rightarrow e^{2\nu} = 4\text{HP}^u\text{HM}^v R^z \Rightarrow 2\nu \\ = \text{Log}(R^z) + \text{Log}(2\text{HP}^u) + \text{Log}(2\text{HM}^v), \qquad (A.33) $$

$$ 2\nu^t = (R^{zz}/R^z)z^t + (\text{HP}^{uu}/\text{HP}^u) + (\text{HM}^{vv}/\text{HM}^v) \Rightarrow 2\nu^{tt} \\ = (R^{zz}/R^z)z^t z^t + (R^{zz}/R^z)z^{tt} + (\text{HP}^{uu}/\text{HP}^u)^u + (\text{HM}^{vv}/\text{HM}^v)^v, \qquad (A.34) $$

$$ 2\nu^r = (R^{zz}/R^z)z^r + (\text{HP}^{uu}/\text{HP}^u) - (\text{HM}^{vv}/\text{HM}^v) \Rightarrow 2\nu^{rr} \\ = (R^{zz}/R^z)z^r z^r + (R^{zz}/R^z)z^{rr} + (\text{HP}^{uu}/\text{HP}^u)^u + (\text{HM}^{vv}/\text{HM}^v)^v, \qquad (A.35) $$

$$ R\nu^{tt} - R\nu^{rr} = (R^{zz}/R^z)(z^t z^t - z^r z^r)R/2 + (R^{zz}/R^z)(z^{tt} - z^{rr})R/2 \\ = -(R/2\rho)(R^{zz}/R^z)^z, \qquad (A.36) $$

$$ \begin{aligned} R^{tt} - R^{rr} + R\nu^{tt} - R\nu^{rr} &= -R^{zz}/\rho - (R/2\rho)(R^{zz}/R^z)^z \\ &= -(1/2\rho)[2R^{zz} + (RR^{zz}/R^z)^z - R^z R^{zz}/R^z] \\ &= -(1/2\rho)[R^z + RR^{zz}/R^z]^z = -(1/2\rho)[(RR^z)^z/R^z]^z \\ &= 0\{(RR^z)^z = R^z \text{ from relations (A.27)}\} \\ & \qquad \times R^{tt} - R^{rr} + R\nu^{tt} - R\nu^{rr} = 0. \end{aligned} $$

### A.5. Summary of the non-static metric

Let us set $u = t + r$, $v = t - r$, $HP(u)$, $HP(v)$, $R_0 = 2GM/c^2$.

$HP(u)$ and $HM(v)$ are arbitrary two times differentiable functions.

$$ ds^2 = e^{2\nu(t,r)}dt^2 - e^{2\nu(t,r)}dr^2 - [R(t,r)]^2(d\theta^2 + \sin^2\theta d\phi^2) \qquad (A.37) \\ e^{2\nu} = 4\text{HP}^u\text{HM}^v(1 - R_0/R) $$

$$ z = \text{HP}(u) - \text{HM}(v) = R + R_0\text{Log}|R/R_0 - 1|. \qquad (A.38) $$

The $R(t,r)$ function is implicitly defined by the relation (A.38). A priori, we have to deduce the functions $HP(u)$ and $HM(v)$ from the boundary conditions (one upstream and one downstream); but we have another possibility: a physical law forcing the $R(t,r)$ value. The spatial domain of the non-static metric is finite. Due to the wave equation: $z^{tt} - z^{rr} = 0$, internal self-maintained waves are possible in the finite space.
The Kretschmann scalar Kr: Kr = $R^{abcd}R_{abcd}$ obtained from the Riemann tensor $R$, is equal to

$$ \mathrm{Kr} = 48G^2 M^2 / \{c^4 [R(t, r)]^6\}. \qquad (A.39) $$

## Appendix B. Geodesics First Integrals

Let us set $κ = 1$ for the massive particles, $κ = 0$ for the particles without mass, $w = 1/R$; $R_0 = 2GM/c^2$

Let us set $E_r$: the particle energy, $h$: the particle angular momentum.

### B.1. Geodesics of the non-static metric

The Christoffel numbers different from zero are

$$ \begin{aligned} \Gamma^0_{00} &= \Gamma^0_{11} = \nu^t; & \Gamma^0_{01} &= \Gamma^0_{10} = \nu^r; & \Gamma^0_{22} &= e^{-2\nu}RR^t; & \Gamma^0_{33} &= e^{-2\nu}RR^t\sin^2\theta, \\ \Gamma^1_{00} &= \Gamma^1_{11} = \nu^r; & \Gamma^1_{01} &= \Gamma^1_{10} = \nu^t; & \Gamma^1_{22} &= -e^{-2\nu}RR^r; & \Gamma^1_{33} &= -e^{-2\nu}RR^r\sin^2\theta, \\ \Gamma^2_{02} &= \Gamma^2_{20} = R^t/R; & \Gamma^2_{12} &= \Gamma^2_{21} = R^r/R; & \Gamma^2_{33} &= -\sin\theta\cos\theta, \\ \Gamma^3_{03} &= \Gamma^3_{30} = R^t/R; & \Gamma^3_{13} &= \Gamma^3_{31} = R^r/R; & \Gamma^3_{23} &= \Gamma^3_{32} = \cot g\theta. \end{aligned} $$

The geodesics are defined by the following equation: $\frac{d^2x^\lambda}{d\sigma^2} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{d\sigma} \frac{dx^\nu}{d\sigma} = 0$, $\sigma$ is the proper time in meters ($\sigma = c\tau$) where $\tau$ is the proper time in seconds. For a massive particle, and an affine parameter for a particle without mass. In this appendix, the $\sigma, \varphi$ exponents define the full derivatives with respect to the $\sigma, \varphi$ parameters. Using the Christoffel numbers, we obtain the four geodesics equations:

$$ t^{\sigma\sigma} + \nu^t(t^\sigma)^2 + 2\nu^r t^\sigma r^\sigma + \nu^t(r^\sigma)^2 + e^{-2\nu}RR^t(\theta^\sigma)^2 + e^{-2\nu}RR^t\sin^2\theta(\varphi^\sigma)^2 = 0, \quad (B.1) $$

$$ r^{\sigma\sigma} + \nu^t(t^\sigma)^2 + 2\nu^r t^\sigma r^\sigma + \nu^r(r^\sigma)^2 - e^{-2\nu}RR^r(\theta^\sigma)^2 - e^{-2\nu}RR^r\sin^2\theta(\varphi^\sigma)^2 = 0, \quad (B.2) $$

$$ \theta^{\sigma\sigma} + 2(R^t/R)t^\sigma\theta^\sigma + 2(R^r/R)r^\sigma\theta^\sigma - \sin\theta\cos\theta(\varphi^\sigma)^2 = 0, \quad (B.3) $$

$$ \varphi^{\sigma\sigma} + 2(R^t/R)t^\sigma\varphi^\sigma + 2(R^r/R)r^\sigma\varphi^\sigma + 2\cot g\theta\theta^\sigma\varphi^\sigma = 0, \quad (B.4) $$

Simple transformations lead to the following relations:

$$ (R^2\theta^\sigma)^\sigma - R^2 \sin\theta \cos\theta (\varphi^\sigma)^2 = 0, \quad (B.5) $$

$$ (R^2 \sin^2 \theta \varphi^\sigma)^\sigma = 0 \Rightarrow R^2 \sin^2 \theta \varphi^\sigma = h. \quad (B.6) $$

Due to the symmetry of the problem, the trajectory is contained in a plane; we set $\theta = \pi/2$ (the usual choice). We obtain two relations:

$$ \theta = \pi/2, \quad (B.7) $$

$$ R^2 \varphi^\sigma = h. \quad (B.8) $$

From the space-time interval: $ds^2 = e^{2\nu}dt^2 - e^{2\nu}dr^2 - R^2(d\theta^2 + \sin^2\theta d\varphi^2)$, we obtain the following first integral:

$$ e^{2\nu}(t^\sigma)^2 - e^{2\nu}(r^\sigma)^2 - (h^2/R^2) = \kappa \Rightarrow (t^\sigma)^2 - (r^\sigma)^2 = e^{-2\nu}[(h^2/R^2) + \kappa]. \quad (B.9) $$
For obtaining the last first integrals, we make the following calculations:
Equations (B.1) and (B.2) are rewritten

$$t^{\sigma\sigma} + 2\nu^t(t^\sigma)^2 + 2\nu^r r^\sigma t^\sigma + \nu^t[(r^\sigma)^2 - (t^\sigma)^2] + e^{-2\nu}RR^t(h^2/R^4) = 0, \quad (B.10)$$

$$r^{\sigma\sigma} + 2\nu^t t^\sigma r^\sigma + 2\nu^r (r^\sigma)^2 - \nu^r [(r^\sigma)^2 - (t^\sigma)^2] - e^{-2\nu}RR^r(h^2/R^4) = 0, \quad (B.11)$$

$$e^{2\nu}t^{\sigma\sigma} + 2e^{2\nu}\nu^{\sigma}t^{\sigma} + e^{-2\nu}\{-\nu^t e^{2\nu}[(h^2/R^2) + \boldsymbol{\kappa}] + e^{2\nu}RR^t(h^2/R^4)\} = 0 \quad (B.12)$$
$$\Rightarrow (e^{2\nu}t^{\sigma})^{\sigma} - (e^{-2\nu}/2)\{e^{2\nu}[(h^2/R^2) + \boldsymbol{\kappa}]\}^t = 0,$$

$$e^{2\nu}r^{\sigma\sigma} + 2e^{2\nu}\nu^{\sigma}r^{\sigma} + e^{-2\nu}\{\nu^r e^{2\nu}[(h^2/R^2) + \boldsymbol{\kappa}] - e^{2\nu}RR^r(h^2/R^4)\} = 0 \quad (B.13)$$
$$\Rightarrow (e^{2\nu}r^{\sigma})^{\sigma} + (e^{-2\nu}/2)\{e^{2\nu}[(h^2/R^2) + \boldsymbol{\kappa}]\}^r = 0,$$

$$e^{2\nu} = (1 - R_0/R)/\rho;$$

$$\text{Let us set: } \Omega = (1 - R_0/R)[(h^2/R^2) + \boldsymbol{\kappa}] = \rho e^{2\nu}[(h^2/R^2) + \boldsymbol{\kappa}] \Rightarrow \quad (B.14)$$

$$\Omega/\rho = e^{2\nu}[(h^2/R^2) + \boldsymbol{\kappa}],$$
$$2(e^{2\nu}t^{\sigma})^{\sigma} - e^{-2\nu}(\Omega/\rho)^t = 0, \quad (B.15a)$$

$$2(e^{2\nu}r^{\sigma})^{\sigma} + e^{-2\nu}(\Omega/\rho)^r = 0. \quad (B.15b)$$

Multiplying Eq. (B.15a) by $-F^r$, Eq. (B.15b) by $F^t$ and adding, we obtain

$$-2F^r(e^{2\nu}t^{\sigma})^{\sigma} + e^{-2\nu}F^r(\Omega/\rho)^t + 2F^t(e^{2\nu}r^{\sigma})^{\sigma} + e^{-2\nu}F^t(\Omega/\rho)^r = 0 \Rightarrow \quad (B.16)$$

$$(-2F^r e^{2\nu} t^{\sigma} + 2F^t e^{2\nu} r^{\sigma})^{\sigma} + 2e^{2\nu} (F^r)^{\sigma} t^{\sigma} - 2e^{2\nu} (F^t)^{\sigma} r^{\sigma} + e^{-2\nu} F^r (\Omega/\rho)^t + e^{-2\nu} F^t (\Omega/\rho)^r = 0, \quad (B.17)$$

$R^t = -F^r e^{2\nu}; R^r = F^t e^{2\nu}; dR = -F^r e^{2\nu} dt + F^t e^{2\nu} dr$; Eq. (B.17) becomes

$$2R^{\sigma\sigma} + 2e^{2\nu}(F^r)^{\sigma}t^{\sigma} - 2e^{2\nu}(F^t)^{\sigma}r^{\sigma} + e^{-2\nu}F^r[(\Omega^R/\rho)R^t - (\Omega/\rho^2)\rho^t] + e^{-2\nu}F^t[(\Omega^R/\rho)R^r - (\Omega/\rho^2)\rho^r] = 0, \quad (B.18)$$

$$2e^{2\nu}(F^r)^{\sigma}t^{\sigma} - 2e^{2\nu}(F^t)^{\sigma}r^{\sigma} = 2e^{2\nu}(F^{rt}t^{\sigma} + F^{rr}r^{\sigma})t^{\sigma} - 2e^{2\nu}(F^{tt}t^{\sigma} + F^{tr}r^{\sigma})t^{\sigma} = 2F^{rt}[(h^2/R^2) + \boldsymbol{\kappa}].$$

From relations (A.19): $\rho = (F^t)^2 - (F^r)^2 \Rightarrow \rho^t = 2F^t F^{tt} - 2F^r F^{tr}; \rho^r = 2F^t F^{tr} - 2F^r F^{rr}$

$$e^{-2\nu}(\Omega/\rho^2)(\rho^t F^r + \rho^r F^t) = 2e^{-2\nu}e^{2\nu}\rho[(h^2/R^2) + \boldsymbol{\kappa}][(F^t F^{tt} - F^r F^{tr})F^r + (F^t F^{tr} - F^r F^{rr})F^t]/\rho^2 = 2F^{tr}[(h^2/R^2) + \boldsymbol{\kappa}].$$

Equation (B.17) becomes

$$2R^{\sigma\sigma} + 2F^{rt}[(h^2/R^2) + \boldsymbol{\kappa}] + \Omega^R - 2F^{tr}[(h^2/R^2) + \boldsymbol{\kappa}] = 0 \quad (B.19)$$
$$\Rightarrow 2R^{\sigma\sigma} + \Omega^R = 0 \Rightarrow 2R^{\sigma}R^{\sigma\sigma} + \Omega^R R^{\sigma} = 0 \Rightarrow$$

$$[(R^{\sigma})^{2}]^{\sigma} + \Omega^{\sigma} = 0 \quad (B.20)$$
$$\Rightarrow (R^{\sigma})^{2} + \Omega = (E_{r})^{2} \quad \text{where } E_{r}$$

is a constant number : the energy of the particle,
$$(R^\sigma)^2 + (1 - R_0/R)[(h^2/R^2) + \kappa] = (E_r)^2. \qquad (B.21)$$

This relation is identical to the corresponding relation of the static metric. We obtain a Binet equation

$$(R^2 \varphi^\sigma w^\sigma \sigma^\varphi)^2 + (1 - R_0 w)(h^2 w^2 + \kappa) = (E_r)^2, \qquad (B.22)$$

$$\text{from } R^2 \varphi^\sigma = h \Rightarrow h^2 (w^\varphi)^2 + (1 - R_0 w)(h^2 w^2 + \kappa) = (E_r)^2, \qquad (B.23)$$

$$2h^2 w \varphi \varphi w^\varphi + w^\varphi (2h^2 w - 3w^2 h^2 R_0 - \kappa R_0) = 0, \qquad (B.24)$$

$$w \varphi \varphi + w - 3w^2 R_0 / 2 = \kappa [R_0 / (2h^2)]. \qquad (B.25)$$

We are calculating now the last first integral:

$$\text{From Eqs (B.15): } 2F^t (e^{2\nu} t^\sigma)^\sigma - e^{-2\nu} F^t (\Omega/\rho)^t = 0; \qquad (B.26) \\ 2F^r (e^{2\nu} r^\sigma)^\sigma + e^{-2\nu} F^r (\Omega/\rho)^r = 0 \Rightarrow$$

$$(2F^t e^{2\nu} t^\sigma)^\sigma - 2e^{2\nu} (F^t)^\sigma t^\sigma - e^{-2\nu} F^t (\Omega/\rho)^t - (2F^r e^{2\nu} r^\sigma)^\sigma \\ + 2e^{2\nu} (F^r)^\sigma r^\sigma - e^{-2\nu} F^r (\Omega/\rho)^r = 0 \Rightarrow \qquad (B.27)$$

$$[2e^{2\nu}(F^t t^\sigma - F^r r^\sigma)]^\sigma - 2e^{2\nu}[(F^t)^\sigma t^\sigma - (F^r)^\sigma r^\sigma] - e^{-2\nu}F^t(\Omega/\rho)^t - e^{-2\nu}F^r(\Omega/\rho)^r = 0, \qquad (B.28)$$

$$(F^t)^\sigma t^\sigma - (F^r)^\sigma r^\sigma = t^\sigma (F^{tt} t^\sigma + F^{tr} r^\sigma) - r^\sigma (F^{rt} t^\sigma + F^{rr} r^\sigma) \\ = F^{tt} (t^\sigma t^\sigma - r^\sigma r^\sigma) = F^{tt} e^{-2\nu} [(h^2/R^2) + \kappa], \qquad (B.29)$$

$$e^{-2\nu} F^r (\Omega/\rho)^r + e^{-2\nu} F^t (\Omega/\rho)^t \\ = e^{-2\nu} F^r [(\Omega^R/\rho) R^r - (\Omega/\rho^2) \rho^r] + e^{-2\nu} F^t [(\Omega^R/\rho) R^t - (\Omega/\rho^2) \rho^t], \\ = e^{-2\nu} F^r [(\Omega^R/\rho) F^t e^{2\nu} - (\Omega/\rho^2) \rho^r] + e^{-2\nu} F^t [-(\Omega^R/\rho) F^r e^{2\nu} - (\Omega/\rho^2) \rho^t],$$

$$e^{-2\nu} F^r (\Omega/\rho^2)^r + e^{-2\nu} F^t (\Omega/\rho^2)^t \\ = -e^{-2\nu} (\Omega/\rho^2) (\rho^r F^r + \rho^t F^t) = -e^{-2\nu} e^{2\nu} [(h^2/R^2) + \kappa] (\rho^r F^r + \rho^t F^t) / \rho, \qquad (B.30)$$

$$\text{from relations (A.19): } \rho = (F^t)^2 - (F^r)^2 \Rightarrow \rho^t = 2F^t F^{tt} - 2F^r F^{rt}; \quad \rho^r = 2F^t F^{tr} - 2F^r F^{rr}$$

$$(\rho^r F^r + \rho^t F^t) = 2F^r (F^t F^{tr} - F^r F^{rr}) + 2F^t (F^t F^{tt} - F^r F^{rt}) = 2F^{tt} \rho, \qquad (B.31)$$

$$e^{-2\nu} F^r (\Omega/\rho^2)^r + e^{-2\nu} F^t (\Omega/\rho^2)^t = -2F^{tt} [(h^2/R^2) + \kappa], \qquad (B.32)$$

From relations: (B.28), (B.29), (B.32): $[2e^{2\nu}(F^t t^\sigma - F^r r^\sigma)]^\sigma - 2F^{tt}[(h^2/R^2) + \kappa] + 2F^{tt}[(h^2/R^2) + \kappa] = 0$; $[2e^{2\nu}(F^t t^\sigma - F^r r^\sigma)]^\sigma = 0 \Rightarrow e^{2\nu}(F^t t^\sigma - F^r r^\sigma) = E_r$; $E_r$ is the particle energy; $4HP^u HM^v (1 - R_0/R)(F^t t^\sigma - F^r r^\sigma) = R^2 E_r/h$; $z = HP(u) - HM(v)$; $\Rightarrow z^\varphi = HP^u u^\varphi - HM^v v^\varphi = R^\varphi/(1 - R_0/R)$; $\{\text{from relation (A.29)}\}$; $t^\varphi(HP^u + HM^v) + (HP^u - HM^v)r^\varphi = (R^2 E_r/h)(1 - R_0/R)$; $t^\varphi(HP^u - HM^v) + r^\varphi(HP^u + HM^v) = R^\varphi/(1 - R_0/R)$; $2HP^u u^\varphi = [(R^2 E_r/h) + R^\varphi]/(1 - R_0/R)$; $2HM^v v^\varphi = [(R^2 E_r/h) - R^\varphi]/h - R^\varphi/(1 - R_0/R)$.
## B.2. Summary of the first integrals

$$ \theta = \pi/2, \qquad (B.33) $$

$$ R^2 d\varphi/d\sigma = h, \qquad (B.34) $$

$$ (R^\sigma)^2 + (1 - R_0/R)[(h^2/R^2) + \kappa] = (E_r)^2, \qquad (B.35) $$

$$ w^{\varphi\varphi} + w - 3w^2GM/c^2 = \kappa GM/h^2c^2, \qquad (B.36) $$

$$ e^{2\nu}(F^t t^\sigma - F^r r^\sigma) = E_r, \qquad (B.37) $$

$$ 2\mathrm{HP}^u u^\varphi = [(R^2 E_r / h) + R^\varphi] / (1 - R_0 / R), \qquad (B.38) $$

$$ 2\mathrm{HM}^v v^\varphi = [(R^2 E_r / h) - R^\varphi] / (1 - R_0 / R), \qquad (B.39) $$

$$ \frac{dt}{d\varphi} = \frac{(R^2 E_r / h)(\mathrm{HM}^v + \mathrm{HP}^u) + (dR/d\varphi)(\mathrm{HM}^v - \mathrm{HP}^u)}{4\mathrm{HP}^u \mathrm{HM}^v (1 - R_0 / R)}, \qquad (B.40) $$

$$ \frac{dr}{d\varphi} = \frac{(R^2 E_r / h)(\mathrm{HM}^v - \mathrm{HP}^u) + (dR/d\varphi)(\mathrm{HM}^v + \mathrm{HP}^u)}{4\mathrm{HP}^u \mathrm{HM}^v (1 - R_0 / R)}, \qquad (B.41) $$

Relations (B.33), (B.34), (B.35), (B.36) are identical to the corresponding relations of the static metric. Consequently, the function $R(\varphi) = 1/w(\varphi)$ is strictly identical to the corresponding function of the static metric. Differences appear only in the calculation of the t, r coordinates, and in the calculation of the real time and distance intervals.

## B.3. Ellipsoidal geodesics

Let us consider the case of a massive particle whose the $R(\varphi)$ function is approximately an ellipse which rotates around its vertical axis. Let us calculate the $h$ and $E_r$ values. We can set, from Eq. (B.35), for the $R$ minimal value $R_{\min}$ and for the $R$ maximal value $R_{\max}$:

$$ (dR/d\sigma)_{\min} = 0 \Rightarrow (1 - R_0 u_{\min})[h^2(u_{\min})^2 + 1] = (E_r)^2 \quad (u_{\min} = 1/R_{\min}), $$

$$ (dR/d\sigma)_{\max} = 0 \Rightarrow (1 - R_0 u_{\max})[h^2(u_{\max})^2 + 1] = (E_r)^2 \quad (u_{\max} = 1/R_{\max}), $$

$$ h^2(u_{\min})^2 - R_0 h^2(u_{\min})^3 - R_0 u_{\min} = h^2(u_{\max})^2 - R_0 h^2(u_{\max})^3 - R_0 u_{\max}, $$

$$ h^2(u_{\max} + u_{\min}) - R_0 h^2[(u_{\max})^2 + (u_{\min})^2 + u_{\max} u_{\min}] - R_0 = 0. $$

Let us consider now the case of a circle of radius $R: (u_{\max} = u_{\min} = 1/R = \text{constant number})$; we obtain

$$ 2h^2/R - 3R_0h^2/R^2 - R_0 = 0 \Rightarrow h^2[1 - 3R_0/(2R)] = RR_0/2 \Rightarrow h^2/R^2 = [R_0/(2R)]/[1 - 3R_0/(2R)]. $$

From Eq. (B.35), as the $R$ function is constant $(1 - R_0/R)(h^2/R^2 + 1) = (E_r)^2$

$$ (E_r)^2 = (1 - R_0/R) \left[ \frac{[R_0/(2R)]}{[1 - 3R_0/(2R)]} + 1 \right] = (1 - R_0/R)^2 [1 - 3R_0/(2R)] $$
$$ \frac{h^2/R^2}{(E_r)^2} = \frac{[R_0/(2R)]/[1-3R_0/(2R)]}{(1-R_0/R)^2/[1-3R_0/(2R)]} = \frac{[R_0/(2R)]}{(1-R_0/R)^2} \Rightarrow \frac{RE_r/h}{(1-R_0/R)} = \sqrt{\frac{2R}{R_0}} = \sqrt{\frac{R}{GM/c^2}} \Rightarrow \frac{R^2 E_r/h}{(1-R_0/R)} = \frac{R\sqrt{R}}{\sqrt{GM/c^2}} $$

Let us set: $V_r$ the radial velocity, $V_T$ the orbital velocity.

$$ \frac{dL_r}{cdT} = (V_r/c) = \frac{(R^2 E_r/h)(\text{HM}^v - \text{HP}^u) + (dR/d\varphi)(\text{HM}^v + \text{HP}^u)}{(R^2 E_r/h)(\text{HM}^v + \text{HP}^u) + (dR/d\varphi)(\text{HM}^v - \text{HP}^u)} $$

$$ (V_T)^2 = (R)^2 \left(\frac{d\varphi}{dT}\right)^2 = (R)^2 \left(\frac{d\varphi}{dt}\right)^2 \left(\frac{dt}{dT}\right)^2, $$

$$ (cdT)^2 = 4\text{HM}^v\text{HP}^u(1 - R_0/R)dt^2; \quad \frac{dt}{d\varphi} = \frac{(R^2 E_r/h)}{(1 - R_0/R)} \frac{(\text{HM}^v + \text{HP}^u)}{4\text{HP}^u\text{HM}^v} \Rightarrow (V_T)^2 = R^2 \frac{GM/c^2}{R^3} \frac{(4\text{HP}^u\text{HM}^v)^2}{(\text{HM}^v + \text{HP}^u)^2} \frac{c^2}{4\text{HM}^v\text{HP}^u(1 - R_0/R)}, $$

$$ (V_T)^2 = \frac{GM/R}{(1 - R_0/R)} \frac{4\text{HP}^u\text{HM}^v}{(\text{HM}^v + \text{HP}^u)^2}, \qquad (B.42) $$

$$ (V_r/c) = \frac{\text{HM}^v - \text{HP}^u}{\text{HM}^v + \text{HP}^u}. \qquad (B.43) $$

## Appendix C. The Ricci Tensor

To calculate the Ricci tensor, we use the Harrison formulas (1960) [Théorie du champ: Landau & Lifschitz; p. 339]

The metric: $ds^2 = g_{\alpha\beta}dx^\alpha dx^\beta = e^{2\nu}c^2dt^2 - e^{2\lambda}dr^2 - R^2(d\theta^2 + \sin^2\theta d\phi^2)$.

Non-zero components with the Harrison form: $g_{ii} = e_i e^{2F_i}$ with $e_0 = 1$, $e_\alpha = -1(\alpha = 1, 2, 3)$:

$$ g_{00} = e^{2\nu}; \quad g_{11} = -e^{2\lambda}; \quad g_{22} = -R^2; \quad g_{33} = -R^2\sin^2\theta; $$

$$ e^{2F_0} = e^{2\nu}; \quad e^{2F_1} = e^{2\lambda}; \quad e^{2F_2} = R^2; \quad e^{2F_3} = R^2\sin^2\theta $$

$$ F_0 = \nu; \quad F_1 = \lambda; \quad F_2 = \text{Log}R; \quad F_3 = \text{Log}R + \text{Log} \sin\theta $$

The Harrison formulas for the Ricci tensor: $R_{ij}$

$$ R_{ik} = \sum_{l \neq i, k} (F_{l,k} F_{k,i} + F_{i,k} F_{l,i} - F_{l,i} F_{k,i} - F_{i,k} F_{l,i}), \quad i \neq k $$

$$ R_{ii} = \sum_{l \neq i} \left[ F_{i,i} F_{l,i} - (F_{l,i})^2 - F_{l,i,i} + e_i e_l e^{2(F_i - F_l)} (F_{l,l} F_{i,l} - (F_{i,l})^2 - F_{i,l,l} - F_{i,l} \sum_{m \neq i, l} F_{m,l}) \right] $$
Calculating $R_{00}$

$$R_{00} = \sum_{l \neq 0} \left[ F_{0,0}F_{l,0} - (F_{l,0})^2 - F_{l,0,0} \right. \\ \left. \qquad + e_0 e_l e^{2(F_0 - F_l)} (F_{l,l}F_{0,l} - (F_{0,l})^2 - F_{0,l,l} - F_{0,l} \sum_{m \neq 0,l} F_{m,l}) \right] \\ \Rightarrow R_{00} = \sum_{l \neq 0} R_{00l},$$

$$l = 1 : R001 = F_{0,0}F_{1,0} - (F_{1,0})^2 - F_{1,0,0} \\ + e_1 e^{2(F_0 - F_1)} \left( F_{1,1}F_{0,1} - (F_{0,1})^2 - F_{0,1,1} - F_{0,1} \sum_{m \neq 0,1} F_{m,1} \right),$$

$$l = 1 : R001 = \nu^t \lambda^t - (\lambda^t)^2 - \lambda^{tt} - e^{2(\nu - \lambda)} (\nu^r \lambda^r - (\nu^r)^2 - \nu^{rr} - \nu^r(F_{2,1} + F_{3,1})),$$

$$l = 1 : R001 = \nu^t \lambda^t - (\lambda^t)^2 - \lambda^{tt} - e^{2(\nu - \lambda)} [\nu^r(\lambda^r - \nu^r) - \nu^{rr} - 2\nu^r R^r / R],$$

$$l = 2 : R002 = F_{0,0}F_{2,0} - (F_{2,0})^2 - F_{2,0,0} \\ - e^{2(F_0 - F_2)} \left( F_{2,2}F_{0,2} - (F_{0,2})^2 - F_{0,2,2} - F_{0,2} \sum_{m \neq 0} F_{m,2} \right),$$

$$l = 2 : R002 = \nu^t \frac{R^t}{R} - \left(\frac{R^t}{R}\right)^2 - \left(\frac{R^{tt}}{R} - \frac{R^t}{R} \frac{R^t}{R}\right) = \nu^t \frac{R^t}{R} - \frac{R^{tt}}{R},$$

$$l = 3 : R003 = F_{0,0}F_{3,0} - (F_{3,0})^2 - F_{3,0,0} \\ - e^{2(F_0 - F_3)} \left( F_{3,3}F_{0,3} - (F_{0,3})^2 - F_{0,3,3} - F_{0,3} \sum_{m \neq 0} F_{m,3} \right),$$

$$l = 3 : R003 = \nu^t \frac{R^t}{R} - \left(\frac{R^t}{R}\right)^2 - \left(\frac{R^{tt}}{R} - \frac{R^t}{R} \frac{R^t}{R}\right) = \nu^t \frac{R^t}{R} - \frac{R^{tt}}{R},$$

$$R_{00} = \lambda^t(\nu^t - \lambda^t) - \lambda^{tt} + 2\nu^t \frac{R^t}{R} - 2\frac{R^{tt}}{R} \\ + e^{2(\nu - \lambda)} [\nu^r(\nu^r - \lambda^r) + \nu^{rr} + 2\nu^r R^r / R].$$

Calculating $R_{11}$

$$R_{11} = \sum_{l \neq 1} \left[ F_{1,1}F_{l,1} - (F_{l,1})^2 - F_{l,1,1} \right. \\ \left. \qquad - e_l e^{2(F_1 - F_l)} \left( F_{l,l}F_{1,l} - (F_{1,l})^2 - F_{1,l,l} - F_{1,l} \sum_{m \neq 1,l} F_{m,l} \right) \right] \Rightarrow R_{11} \\ = \sum_{l \neq 1} R_{11l},$$

$$l = 0 : R110 = F_{1,1}F_{0,1} - (F_{0,1})^2 - F_{0,1,1} \\ - e^{2(F_1 - F_0)} (F_{0,0}F_{1,0} - (F_{1,0})^2 - F_{1,0,0} - F_{1,0}(F_{2,0} + F_{3,0})),$$
$$l = 0 : R110 = \nu^r(\lambda^r - \nu^r) - \nu^{rr} + e^{2(\lambda-\nu)} \left( \lambda^t(\lambda^t - \nu^t) + \lambda^{tt} + 2\lambda^t \frac{R^t}{R} \right),$$

$$l = 2 : R112 = F_{1,1}F_{2,1} - (F_{2,1})^2 - F_{2,1,1} + e^{2(F_1-F_2)}(F_{2,2}F_{1,2} - (F_{1,2})^2 - F_{1,2,2} - F_{1,2}(F_{0,2} + F_{3,2})),$$

$$l = 2 : R112 = \lambda^r \frac{R^r}{R} - \frac{R^{rr}}{R},$$

$$l = 3 : R113 = F_{1,1}F_{3,1} - (F_{3,1})^2 - F_{3,1,1}$$

$$+ e^{2(F_1-F_3)} \left( F_{3,3}F_{1,3} - (F_{1,3})^2 - F_{1,3,3} - F_{1,3} \sum_{m \neq 1,l} F_{m,l} \right),$$

$$l = 3 : R113 = \lambda^r \frac{R^r}{R} - \frac{R^{rr}}{R},$$

$$R_{11} = \nu^r(\lambda^r - \nu^r) - \nu^{rr} + 2\lambda^r \frac{R^r}{R} - 2 \frac{R^{rr}}{R} + e^{2(\lambda-\nu)} \left( \lambda^t(\lambda^t - \nu^t) + \lambda^{tt} + 2\lambda^t \frac{R^t}{R} \right).$$

Calculating $R_{22}$:

$$R_{22} = \sum_{l \neq 2} \left[ F_{2,2}F_{l,2} - (F_{l,2})^2 - F_{l,2,2} + e_2e_l e^{2(F_2-F_l)} \left( F_{l,l}F_{2,l} - (F_{2,l})^2 - F_{2,l,l} - F_{2,l} \sum_{m \neq 2,l} F_{m,l} \right) \right] \Rightarrow R_{22} = \sum_{l \neq 2} R_{22l},$$

$$l = 0 : R220 = F_{2,2}F_{0,2} - (F_{0,2})^2 - F_{0,2,2}$$

$$- e^{2(F_2-F_0)} \left( F_{0,0}F_{2,0} - (F_{2,0})^2 - F_{2,0,0} - F_{2,0} \sum_{m \neq 2,0} F_{m,0} \right),$$

$$l = 0 : R220 = -e^{-2\nu}R^2 \left( \nu^t \frac{R^t}{R} - \frac{R^t}{R} \frac{R^t}{R} - \left( \frac{R^{tt}}{R} - \frac{R^t}{R} \frac{R^t}{R} \right) - \frac{R^t}{R} (F_{1,0} + F_{3,0}) \right),$$

$$l = 0 : R220 = -e^{-2\nu}R^2 \left[ \nu^t \frac{R^t}{R} - \frac{R^{tt}}{R} - \frac{R^t}{R} \left( \lambda^t + \frac{R^t}{R} \right) \right]$$

$$= -e^{-2\nu}R^2 \left[ (\nu^t - \lambda^t) \frac{R^t}{R} - \frac{R^{tt}}{R} - \frac{R^t}{R} \frac{R^t}{R} \right],$$

$$l = 0 : R220 = -e^{-2\nu}[(\nu^t - \lambda^t)RR^t - RR^{tt} - R^t R^t],$$

$$l = 1 : R221 = F_{2,2}F_{1,2} - (F_{1,2})^2 - F_{1,2,2}$$

$$+ e^{2(F_2-F_1)} \left( F_{1,1}F_{2,1} - (F_{2,1})^2 - F_{2,1,1} - F_{2,1} \sum_{m \neq 2,1} F_{m,1} \right),$$
$$l = 1 : R221 = e^{-2\lambda} R^2 \left( \lambda^r \frac{R^r}{R} - \frac{R^r}{R} \frac{R^r}{R} - \left( \frac{R^{rr}}{R} - \frac{R^r}{R} \frac{R^r}{R} \right) - \frac{R^r}{R} (F_{0,1} + F_{3,1}) \right),$$

$$l = 1 : R221 = e^{-2\lambda} R^2 \left[ \lambda^r \frac{R^r}{R} - \frac{R^{rr}}{R} - \frac{R^r}{R} \left( \nu^r + \frac{R^r}{R} \right) \right] \\ = e^{-2\lambda} R^2 \left[ \left( \lambda^r - \nu^r \right) \frac{R^r}{R} - \frac{R^{rr}}{R} - \frac{R^r}{R} \frac{R^r}{R} \right],$$

$$l = 1 : R221 = e^{-2\lambda} [(\lambda^r - \nu^r)RR^r - RR^{rr} - R^r R^r],$$

$$l = 3 : R223 = F_{2,2}F_{3,2} - (F_{3,2})^2 - F_{3,2,2} \\ + e_2e_3e^{2(F_2-F_3)} \left( F_{3,3}F_{2,3} - (F_{2,3})^2 - F_{2,3,3} - F_{2,3} \sum_{m \neq 2,3} F_{m,3} \right),$$

$$l = 3 : R223 = - \left( \frac{\cos \theta}{\sin \theta} \right)^2 - \left( - \frac{\sin \theta}{\sin \theta} - \frac{\cos \theta}{\sin \theta} \frac{\cos \theta}{\sin \theta} \right) = 1,$$

$$R_{22} = -e^{-2\nu}[(\nu^t - \lambda^t)RR^t - RR^{tt} - R^t R^t] \\ + e^{-2\lambda}[(\lambda^r - \nu^r)RR^r - RR^{rr} - R^r R^r] + 1,$$

$$R_{22} = e^{-2\nu}[-(\nu^t - \lambda^t)RR^t + RR^{tt} + R^t R^t] \\ - e^{-2\lambda}[-(\lambda^r - \nu^r)RR^r + RR^{rr} + R^r R^r] + 1.$$

Calculating $R_{33}$:

$$R_{33} = \sum_{l \neq 3} \left[ F_{3,3}F_{l,3} - (F_{l,3})^2 - F_{l,3,3} \right. \\ \left. + e_3e_l e^{2(F_3-F_l)} (F_{l,l}F_{3,l} - (F_{3,l})^2 - F_{3,l,l} - F_{3,l} \sum_{m \neq 3,l} F_{m,l}) \right] \Rightarrow R_{33} \\ = \sum_{l \neq 3} R_{33l},$$

$$l = 0 : R330 = F_{3,3}F_{0,3} - (F_{0,3})^2 - F_{0,3,3} \\ + e_3e_0e^{2(F_3-F_0)} \left( F_{0,0}F_{3,0} - (F_{3,0})^2 - F_{3,0,0} - F_{3,0} \sum_{m \neq 3,0} F_{m,0} \right),$$

$$l = 0 : R330 = -e^{-2\nu} R^2 \sin^2\theta \left( \nu^t \frac{R^t}{R} - \frac{R^t}{R} \frac{R^t}{R} - \left( \frac{R^{tt}}{R} - \frac{R^t}{R} \frac{R^t}{R} \right) \right. \\ \left. - \frac{R^t}{R} (F_{1,0} + F_{2,0}) \right),$$

$$l = 0 : R330 = -e^{-2\nu} R^2 \sin^2\theta \left[ \nu^t \frac{R^t}{R} - \frac{R^{tt}}{R} - \frac{R^t}{R} \left( \lambda^t + \frac{R^t}{R} \right) \right] \\ = -e^{-2\nu} R^2 \sin^2\theta \left[ (\nu^t - \lambda^t) \frac{R^t}{R} - \frac{R^{tt}}{R} - \frac{R^t}{R} \frac{R^t}{R} \right],$$
$$l = 0: R330 = e^{-2\nu}\sin^2\theta[-(\nu^t - \lambda^t)RR^t + RR^{tt} + R^t R^t],$$

$$l = 1: R331 = F_{3,3}F_{1,3} - (F_{1,3})^2 - F_{1,3,3} \\ + e_3e_1e^{2(F_3-F_1)} \left[ F_{1,1}F_{3,1} - (F_{3,1})^2 - F_{3,1,1} - F_{3,1} \sum_{m \neq 3,1} F_{m,1} \right],$$

$$l = 1: R331 = e^{-2\lambda}R^2\sin^2\theta \left[ \lambda^r \frac{R^r}{R} - \frac{R^r}{R} \frac{R^r}{R} - \left( \frac{R^{rr}}{R} - \frac{R^r}{R} \frac{R^r}{R} \right) \right. \\ \left. - \frac{R^r}{R} (F_{0,1} + F_{2,1}) \right],$$

$$l = 1: R331 = e^{-2\lambda}R^2\sin^2\theta \left[ \lambda^r \frac{R^r}{R} - \frac{R^{rr}}{R} - \frac{R^r}{R} \left( \nu^r + \frac{R^r}{R} \right) \right] \\ = e^{-2\lambda}R^2\sin^2\theta \left[ (\lambda^r - \nu^r) \frac{R^r}{R} - \frac{R^{rr}}{R} - \frac{R^r}{R} \frac{R^r}{R} \right],$$

$$l = 1: R331 = -e^{-2\lambda}\sin^2\theta[-(\lambda^r - \nu^r)RR^r + RR^{rr} + R^r R^r],$$

$$l = 2: R332 = F_{3,3}F_{2,3} - (F_{2,3})^2 - F_{2,3,3} \\ + e_3e_2e^{2(F_3-F_2)} \left( F_{2,2}F_{3,2} - (F_{3,2})^2 - F_{3,2,2} - F_{3,2} \sum_{m \neq 3,2} F_{m,2} \right),$$

$$l = 2: R332 = \sin^2\theta \left[ - \left( \frac{\cos\theta}{\sin\theta} \right)^2 - \left( - \frac{\sin\theta}{\sin\theta} - \frac{\cos\theta}{\sin\theta} \frac{\cos\theta}{\sin\theta} \right) \right] = \sin^2\theta,$$

$$R_{33} = e^{-2\nu}\sin^2\theta[-(\nu^t - \lambda^t)RR^t + RR^{tt} + R^t R^t] \\ - e^{-2\lambda}\sin^2\theta[-(\lambda^r - \nu^r)RR^r + RR^{rr} + R^r R^r] + \sin^2\theta,$$

$$R_{33} = \sin^2\theta\{e^{-2\nu}[-(\nu^t - \lambda^t)RR^t + RR^{tt} + R^t R^t] \\ - e^{-2\lambda}[-(\lambda^r - \nu^r)RR^r + RR^{rr} + R^r R^r] + 1\}.$$

Calculating the cross terms:

$$R_{ik} = \sum_{l \neq i, k} (F_{l,k}F_{k,i} + F_{i,k}F_{l,i} - F_{l,i}F_{k,i} - F_{i,k}F_{l,i}), \quad i \neq k.$$

Calculating $R_{01}$:

$$R_{01} = \sum_{l \neq 0,1} (F_{l,1}F_{1,0} + F_{0,1}F_{l,0} - F_{l,0}F_{l,1} - F_{l,0,1}),$$

$$R_{01} = (F_{2,1}F_{1,0} + F_{0,1}F_{2,0} - F_{2,0}F_{2,1} - F_{2,0,1}) \\ + (F_{3,1}F_{1,0} + F_{0,1}F_{3,0} - F_{3,0}F_{3,1} - F_{3,0,1}),$$

$$R_{01} = \frac{R^r}{R}\lambda^t + \nu^r \frac{R^t}{R} - \frac{R^t}{R} \frac{R^r}{R} - \left( \frac{R^{tr}}{R} - \frac{R^t}{R} \frac{R^r}{R} \right) \\ + \frac{R^r}{R} \lambda^t + \nu^r \frac{R^t}{R} - \left( \frac{R^{tr}}{R} - \frac{R^t}{R} \frac{R^r}{R} \right).$$
$$R_{01} = 2 \frac{R^r}{R} \lambda^t + 2 \nu^r \frac{R^t}{R} - 2 \frac{R^{tr}}{R} = \frac{2}{R} (R^r \lambda^t + R^t \nu^r - R^{tr}),$$

$$R_{01} = \frac{2}{R} (R^r \lambda^t + R^t \nu^r - R^{tr}).$$

Calculating $R_{02}$:

$$R_{02} = \sum_{l \neq 0,2} (F_{l,2}F_{2,0} + F_{0,2}F_{l,0} - F_{l,0}F_{l,2} - F_{l,0,2}),$$

$$R_{02} = (F_{1,2}F_{2,0} + F_{0,2}F_{1,0} - F_{1,0}F_{1,2} - F_{1,0,2}) + (F_{3,2}F_{2,0} + F_{0,2}F_{3,0} - F_{3,0}F_{3,2} - F_{3,0,2}),$$

$$R_{02} = 0.$$

Calculating $R_{03}$:

$$R_{03} = \sum_{l \neq 0,3} (F_{l,3}F_{3,0} + F_{0,3}F_{l,0} - F_{l,0}F_{l,3} - F_{l,0,3}) = 0.$$

Calculating $R_{12}$:

$$R_{12} = \sum_{l \neq 1,2} (F_{l,2}F_{2,1} + F_{1,2}F_{l,1} - F_{l,1}F_{l,2} - F_{l,1,2}),$$

$$R_{12} = (F_{0,2}F_{2,1} + F_{1,2}F_{0,1} - F_{0,1}F_{0,2} - F_{0,1,2}) + (F_{3,2}F_{2,1} + F_{1,2}F_{3,1} - F_{3,1}F_{3,2} - F_{3,1,2}),$$

$$R_{12} = \left( \left( \frac{\cos \theta}{\sin \theta} \right) \frac{R^r}{R} - \frac{R^r}{R} \left( \frac{\cos \theta}{\sin \theta} \right) \right) = 0.$$

the non-zero components of the Ricci tensor:

$$R_{00} = \nu^t(\lambda^t + 2R^t/R) - (\lambda^t\lambda^t + \lambda^{tt} + 2R^{tt}/R) + e^{2\nu-2\lambda}[\nu^{rr} + \nu^r(\nu^r - \lambda^r) + 2\nu^rR^r/R], \quad (C.1)$$

$$R_{11} = \lambda^r(\nu^r + 2R^r/R) - (\nu^r\nu^r + \nu^{rr} + 2R^{rr}/R) + e^{2\lambda-2\nu}[\lambda^{tt} - \lambda^t(\nu^t - \lambda^t) + 2\lambda^tR^t/R], \quad (C.2)$$

$$R_{22} = 1 + e^{-2\nu}[R^{tt}R + R^t R^t - R^t R(\nu^t - \lambda^t)] - e^{-2\lambda}[R^{rr}R + R^r R^r + R^r R(\nu^r - \lambda^r)], \quad (C.3)$$

$$R_{33} = \sin^2\theta\{1 + e^{-2\nu}[R^{tt}R + R^t R^t - R^t R(\nu^t - \lambda^t)] - e^{-2\lambda}[R^{rr}R + R^r R^r + R^r R(\nu^r - \lambda^r)]\}, \quad (C.4)$$

$$R_{01} = R_{10} = (2/R)(-R^{tr} + \nu^r R^t + \lambda^t R^r). \quad (C.5)$$

## Appendix D. Extension of the Metric for Large Distances: The External Field Effect

At large distances ($r \sim 25$ kpc), the galaxy rotation curve is no more flat in the MW and M31 galaxies (cf. Dai et al.,¹² Sofue¹³⁻¹⁵). In our GR model, the so-called
“asymptotic metric” can only evolve under the action of an external gravitational field. In the following, we propose an extension of the metric based on a heuristic hypothesis.

In the transition zone: $\rho > \beta_0 = \frac{1}{2} \Rightarrow \frac{d\beta}{d\rho} = \frac{\alpha}{2} = 2(1 - \beta)^2 \Rightarrow \beta = \frac{2\rho}{1+2\rho} \Rightarrow \rho = \frac{1}{2} \frac{\beta}{1-\beta}$. When $\rho$ increases, $\alpha \to 0$ and $\beta \to 1$; when $\beta$ is close to 1, if a small perturbation results in $\beta = \beta_{mB} > 1$, the solution of the differential equation is modified: it becomes $\frac{1}{\beta-1} = \frac{1}{\beta_{mB}-1} - 2(\rho - \rho_{mB})$. We make the hypothesis that such a perturbation occurs due to the external field effect (EFE), and makes unstable the solution: $\beta = \frac{2\rho}{1+2\rho}$ at large distances.

Hence our heuristic assumption when $\alpha \to 0$, the solution $\beta = \frac{2\rho}{1+2\rho}$ becomes unstable due to the EFE; consequently, we assume that $\alpha$ has a lower bound $\alpha_m$, due to EFE.

$$ \alpha = \max[\alpha_m, 4(1 - \beta)^2], \qquad (D.1) $$

$$ \alpha_m = 4(1 - \beta_m)^2, \qquad (D.2) $$

$$ \beta_m = 1 - \frac{\sqrt{\alpha_m}}{2}; \quad \rho_m = \frac{1}{2} \frac{\beta_m}{1 - \beta_m} = \frac{1}{\sqrt{\alpha_m}} - 1/2 $$

$$ \text{For } \rho \ge \rho_m : \frac{d\beta}{d\rho} = \frac{\alpha_m}{2} \Rightarrow \beta = \beta_m + \frac{\alpha_m}{2}(\rho - \rho_m) $$

$$ \text{for the third zone of the rotation curve } \rho \ge \rho_m : \beta = \left(1 - \frac{\sqrt{\alpha_m}}{2}\right)^2 + \frac{\alpha_m}{2}\rho. \qquad (D.3) $$

To analyze the relevance of this model, we use data from Dai et al.,¹² and Sofue.¹⁵ The data (Bulge, Disk, Halo contributions, ΛCDM calculation, and measure) are captured from Figs. D.1 (MW) and D.2 (M31).¹²

<figure><img src="image_7.png" /><figcaption>Fig. D.1. MW grand rotation curve ($\alpha_{m\_MW} = 0.40$).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. D.2. M31 grand rotation curve ($\alpha_{m\_M31} = 0.15$).</figcaption></figure>

<table><caption>Table D.1. Summary of the $\beta$ variation: $\alpha$ in the three zones.</caption><thead><tr><th>Zone</th><th>ρ interval</th><th>β formula</th><th>β interval</th></tr></thead><tbody><tr><td>1st zone</td><td>ρ ≤ 1/2</td><td>β = ρ</td><td>β ≤ 1/2</td></tr><tr><td>2nd zone</td><td>1/2 ≤ ρ ≤ 1/√α<sub>m</sub> - 1/2</td><td>β = 2ρ/(1+2ρ)</td><td>1/2 ≤ β ≤ 1 - √α<sub>m</sub>/2</td></tr><tr><td>3rd zone</td><td>ρ ≥ 1/√α<sub>m</sub> - 1/2</td><td>β = (1 - √α<sub>m</sub>/2)<sup>2</sup> + α<sub>m</sub>/2 ρ</td><td>β ≥ 1 - √α<sub>m</sub>/2</td></tr></tbody></table>

<table><caption>Table D.2. The baryonic masses.<sup>15</sup></caption><thead><tr><th>Galaxy</th><th>M31</th><th>MW</th></tr></thead><tbody><tr><td>Baryonic mass (10<sup>11</sup> M<sub>☉</sub>)</td><td>1.61 ± 0.084</td><td>1.37 ± 0.42</td></tr></tbody></table>

$\alpha_{m\_MW}$ is the coefficient quantifying the deformation of the Milky Way gravitational field caused by the Andromeda galaxy. Similarly, $\alpha_{m\_M31}$ is the coefficient quantifying the deformation of Andromeda galaxy gravitational field caused by the Milky Way. It seems clear to us that these two coefficients can only be equal if the two galaxies have very close baryonic masses. In the present case, masses from Table 2<sup>15</sup> by Sofue are reported in Table D.2.

With the mass of M31 being higher than that of MW, we should expect to obtain the inequality $\alpha_{m\_MW} > \alpha_{m\_M31}$: the deformation of the MW gravitational field is expected greater than that of M31. In a preliminary study, we obtained $\alpha_{m\_MW} \approx 0.40$ and $\alpha_{m\_M31} \approx 0.15$.

An empiric formulation for $\alpha_{m\_M31}$ and $\alpha_{m\_MW}$ can be obtained as reported in Table D.3.

The deformation of the MW curve is significantly greater than that of the M31 curve ($\alpha_{m\_MW} \approx 3 \times \alpha_{m\_M31}$). This result suggests that the mass difference between M31 and MW could be significantly larger than indicated in Table D.2. Further studies are needed to justify this simple model.
<table><caption>Table D.3. $\alpha_m$ formulas.</caption><tr><td>M31</td><td>$\alpha_{mM31} = K(r_{mW\_M31})(M_{b\_M31}/M_{b\_MW})^3$</td></tr><tr><td>MW</td><td>$\alpha_{m\_MW} = K(r_{mW\_M31})(M_{b\_MW}/M_{b\_M31})^3$</td></tr><tr><td></td><td>$r_{mW\_M31}$ = distance between MW and M31</td></tr><tr><td></td><td>$K(r_{mW\_M31}) \approx 0.245$</td></tr></table>

<figure><img src="image_9.png" /><figcaption>Fig. D.3. MW and M31 grand rotation curves ($\alpha_{m\_MW} = 0.40$, $\alpha_{m\_M31} = 0.15$).</figcaption></figure>

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