Narrow Results

We review and emphasize the importance of the Satellite Test of the Equivalence Principle (STEP) in probing one of the most successful and popular alternative theories to dark matter known as Modified Newtonian Dynamics (MOND), on Earth. This would be achieved with no modification of STEP’s current design and sensitivity and if MOND exists STEP could in principle easily detect it.

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 been 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 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 ($\mathrm{\Lambda }$CDM), as the $\mathrm{\Lambda }$CDM 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).

The GAIA DR3 measurement campaign has produced a MW rotation curve with significantly improved accuracy compared with previous campaigns. In 2023, several authors presented accurate rotation curves calculated from these measurements, within the framework of the dark matter hypothesis. They established new estimates of the Milky Way’s dynamical mass of around $2\times {10}^{11}{M}_{\odot }$. Some of these authors showed that, from a radial distance of around 18kpc, the Milky Way presents a significant Keplerian decay in the rotation curve, rather than a plateau zone. In this paper, we use a set of data tables from four different authors in a single database. Without making any assumptions about the existence or absence of dark matter, we analyze the observed radial acceleration as a function of the baryonic acceleration. We show that the observed acceleration is an accurate linear function of the baryonic acceleration, making the dark matter hypothesis problematic. Furthermore, we prove that the MW rotation curve can be calculated within the framework of General Relativity without dark matter, using the dynamic metric we published earlier in 2023: “On the incompleteness’ of Birkhoff’s theorem: A new approach to the central symmetric Gravitational Field in Vacuum Space.” In particular, this metric allows us to predict and model the Keplerian decay zone of the rotation curve. Our dynamical mass evaluation does not differ significantly from the value $2\times 10^{11}{M}_{\odot }$.