Exotic matter and MOND as special cases of a more general solution in pure general relativity

Several solutions have been proposed to explain the dark matter (DM) component, but two explanations stand out. The most widely accepted is the assumption of the existence of exotic matter, and the second is the assumption of a modification of gravitation called MOND. We show in this article that the solutions of exotic matter and MOND are particular cases of a more general solution of pure general relativity that explains DM with only baryonic matter and without modifying gravity. Within the framework of general relativity linearization (GRL), we first demonstrate that these two explanations (exotic matter and GRL solution) are so close that all explanations within the framework of the exotic matter assumption can be translated within the GRL solution. Conversely, certain effects appear in this alternative that should not be observed in the hypothesis of exotic matter. Different types of alignments (about axes of rotation or planar trajectories) should appear on different types of objects (dwarf satellite galaxies, galaxies, or clusters of galaxies). Observational monitoring of the dynamics of S-2 stars close to our Galactic Center in the future should reveal (due to greater precision) discrepancies from the theory, explainable by the lower mass of SgrA* compensated by a stronger-than-expected GRL field (instead of exotic matter). The implementation of simulations to explain the structuring of the universe on a large scale should be able to reveal filaments or walls and large voids, thanks to a stronger-than-expected GRL field (explaining the DM component instead of exotic matter). The size of these structures would be correlated with the intensity of this GRL field. We show that this solution responds to the points addressed to all alternatives to exotic matter. Conversely, we propose points challenging the exotic matter assumption that are naturally explained in this DM component alternative explanation.

1 Introduction

Recently, we published an article (Le Corre, 2024b) that proposes an alternative to the exotic matter hypothesis to explain the dark matter (DM) component. This solution (Le Corre, 2015) comes from the general relativity (GR) linearization (GRL) and is close to a solution that has recently been called “strong gravitomagnetism” (Astesiano and Ruggiero, 2022b). However, contrary to such “gravitomagnetism,” alternatives where the authors attempt to find a way by which GR could naturally explain the DM component; that is, without ad hoc assumptions, we pretend that ad hoc assumptions are also necessary in our GRL solution. Therefore, it is important to avoid confusion between these gravitomagnetic alternatives and the explanation of Le Corre (2024b).

We demonstrate that our GRL solution is very close to the exotic matter assumption, so close that they are hardly discernible. Two major points ultimately emerge from this study. First, the GRL solution fully includes the exotic matter solution. This means that all articles studying the DM component through the prism of exotic matter can also be translated into our GRL solution without contradicting their results. Second, while the GRL solution implies the exotic matter solution (first point), the opposite is not true; in other words, our solution extends it, in the sense that it allows us to explain observations and relationships that exotic matter cannot explain (alignments, Tully–Fisher relationship, …). Furthermore, we show that the GRL solution for explaining the DM component offers several advantages (unifying solutions of MOND and exotic matter) and also solves the problems associated with exotic matter (insensitive to electromagnetism, negligible on our scale, …). This could also explain the contradictory characteristics of the S-cluster stellar population near the Galactic Center (GC).

This close proximity of these two explanations (exotic matter and our GRL solution) leads us to emphasize the main notable difference between these two explanations, which is the appearance of unexpected alignments on a large scale. This is most likely the only notable difference as long as there is no direct measurement of the hypotheses (i.e., detection of particles of exotic matter or measurement of the Lense–Thirring effect with sufficient precision for the GRL solution). That is why we note that Paudel et al. (2024) observe a very improbable alignment in light of our knowledge. It turns out that this improbable alignment of dwarf galaxies is a prediction made in our article (Le Corre, 2024b). Our solution necessarily implies the appearance of several types of alignment (about axes of rotation or planes of trajectories), more or less observable depending on the studied objects (dwarf satellite galaxies, galaxies, or clusters of galaxies) and their history (interactions with or without their neighborhood).

We also take advantage of this article, on one hand, to verify that our alternative explanation of DM responds to the various required points of Ciotti (2024) addressed to any alternative to the hypothesis of exotic matter and, on the other hand, to conversely address problems that our solution of “gravitic” fields explains but which remain to this day without explanation in the hypothesis of exotic matter.

In summary, compared with Le Corre (2024b), this study newly demonstrates that this GRL solution generates both the exotic matter solution, through Equation 3, and the MOND solution, through Equation 7. Second, our solution “GRL equations + $\overrightarrow{k_0}$” is unfortunately extremely difficult to distinguish experimentally from the exotic matter solution. Third, we provide a detailed comparison of these two solutions. Fourth, we also add new published results—for example, Paudel et al. (2024) for alignments, Le Corre (2024a) for gravitational lens, and MacCrann et al. (2024) for CMB, among others—that corroborate and anchor our GRL solution [dating from 2015 (Le Corre, 2015)] as the most serious candidate to compete with exotic matter. Fifth, we propose an explanation for the S-cluster stellar population near the GC. Sixth, we insist on an important point that also does not appear clearly in Le Corre (2024b) and differentiates our GRL solution from the solutions of “strong gravitomagnetism:” our solution implies an ad hoc hypothesis in the same way as exotic matter: although GR provides the key element of our solution, that is, the gravitic field $\overrightarrow{k_0}$ (non-existent in Newton), GR alone is not sufficient to obtain the high value of this field $k_0$ (hence, our ad hoc hypothesis of gravitic spin fields of galaxy clusters in the same way as the spin fields of atoms explaining the values of the magnetic fields of magnets, too high for Maxwell’s equations alone). To finish, we also propose a potential avenue for explaining astrophysical filaments or walls and large voids that a simulation could reveal, linking the average dimension of voids and filaments or walls to the gravitic field, explaining the DM component. Thus, measuring the dimensions of these structures would indicate the order of magnitude of this field.

To be more explicit about the uniform gravitic field $k_0$ (the keystone of our solution), it must be kept in mind that our results obtained from GRL suggest a possible gravitic effect such as $k_0$, but its persistence and magnitude in full GR remain open questions that would require non-linear treatment.

2 The GRL solution of the DM component with a uniform gravitic field embedding the galaxies

We present the GRL solution in relation to the exotic matter solution to demonstrate, contrary to what one might believe at first glance, that, despite their differences, these two explanations of the DM component are very close, both of the same quality in theoretical terms and difficult to distinguish in terms of observations. In fact, we demonstrate that the GRL solution of Le Corre (2024b) is at least as well founded theoretically as exotic matter and even more solid on certain points. We then demonstrate that all observations explained in the exotic matter assumption can be explained in the gravitic fields assumption (i.e., the GRL solution), and that, to date, there is no decisive observation to reject one or the other solution.

2.1 Two ways to idealize the DM component in GR

In GRL (and therefore in GR), the gravitational field corresponds to the Newtonian term, to which is added a second term defined by a gravitic field similar to the magnetic field of electromagnetism (EM) (Wald, 1984; Hobson et al., 2006; Le Corre, 2015). The equations of motion are then written with $\varphi$, the Newtonian gravitational potential, and $\vec{k}$, the gravitic field:

$\frac{d^2\vec{x}}{d{t}^2}=-\vec{\nabla }\varphi +4\vec{k}\wedge \vec{v}.(1)$

The first term is the Newtonian term (also known as the gravitoelectric field), and the second term (known as the gravitomagnetic field) is a specific term of GR at the origin of the Lense–Thirring effect already observed and measured (Adler, 2015; Everitt, 2015; Ruggiero, 2015). The gravitic field $\vec{k}$ is equivalent to the magnetic field of EM because the GRL field equations (known as Einstein–Maxwell equations) are equivalent to the Maxwell equations of EM (Wald, 1984; Hobson et al., 2006; Le Corre, 2015). Consequently, the gravitic field is defined by the current of mass in the same way as the magnetic field is defined by the current of charge in classical EM. We can then demonstrate that, irrespective of the mass, the second term is always negligible compared to the Newtonian term (Lasenby et al., 2023; Ciotti, 2022; Glampedakis and Jones, 2023).

Within the framework of GRL, the appearance of the DM component, justified and required by observation, can only be obtained in two ways. Using Equation 1, either we act on the first term of GR, or we act on the second term of GR. Acting on the first term can only be carried out via the mass term. This corresponds to the traditional assumption of exotic matter. Acting on the second term can only be carried out via the gravitic field term $\vec{k}$. This corresponds to the GRL assumption of Le Corre (2024b).

Concretely, a greater value of mass is assumed in the exotic matter assumption, and because the second term is negligible for irrespective of the mass, Equation 1 is reduced to the Newtonian approximation. The equations of motion with exotic matter are Equation 2

$\frac{d^2\vec{x}}{d{t}^2}=-\vec{\nabla }\varphi -\vec{\nabla }{\varphi }_{DM}.(2)$

In the GRL assumption, a greater value of the gravitic field $\vec{k}$ is assumed, and therefore, the second term of Equation 1 is no longer negligible. The equations of motion then become entire (Equation 1). Thus, in the framework of GR, the DM component can be explained in two different ways: by a purely Newtonian solution, the exotic matter assumption, or by a purely GR solution, the gravitic field assumption.

It is important to note that in both of these explanations, an ad hoc hypothesis is necessary, namely, a new source of matter for exotic matter or a new source of gravitic field $\overrightarrow{k_0}$ for the GRL solution. Some authors try to find a mathematical approach [(Astesiano et al., 2022; Ruggiero, 2024; Astesiano and Ruggiero, 2025; Re and Galoppo, 2025; Galoppo et al., 2025)] specific to GR (via, for example, its nonlinearities) to explain this source of gravitic field. We go further by proposing not only a mathematical but also a physical way. We believe that such a source is generated by galaxy clusters in the same way as electromagnetism generates magnetic fields in ferromagnetic materials that are far too large to be explained by Maxwell’s equations (we explain it later).

2.2 All results explained by exotic matter can be interpreted in the GRL solution

We can now establish the first main result of our article, the equivalence of these two explanations, more precisely. The exotic matter assumption effects are all translatable into the GRL solution. Using Equations 1, 2, we can obtain an equivalence relation between exotic matter and the gravitic field:

$\begin{array}{ll}\hfill & \Vert \vec{\nabla }{\varphi }_{DM}\Vert \approx \frac{G{M}_{DM}}{r^2}=4\Vert \vec{k}\wedge \vec{v}\Vert =4kv|\sin (\widehat{(\vec{k},\vec{v})})|\hfill \\ \hfill & =>M_{DM}\approx \frac{4kv{r}^2}{G}|\sin (\widehat{(\vec{k},\vec{v})})|.\hfill \end{array}(3)$

The translation between these two interpretations of DM can be carried out using Equation 3. Equation 3, which bridges the two explanations of DM, is important because it means that all observations interpreted as the effect of an exotic matter can also be interpreted as the effect of a gravitic field. We can translate all known effects of exotic matter into the gravitic field in a very concrete way. For example, we can use the calculation methods from Le Corre (2024b) to explain the rotation speeds of the Milky Way (MW). We note that the gravitic field $k$ is the sum of the own gravitic field of the studied object $\frac{K_1}{r^2}$ and the uniform gravitic field $k_0$, which embeds the galaxies. The baryonic mass of the MW is $M_{bar\text{\_}MW}\approx 0.8\times 10^{41}$ kg, the gravitic field of the MW to explain DM is $k_{MW}(r)\approx \frac{K_{1\text{\_}MW}}{r^2}{s}^{-1}$ with $K_{1\text{\_}MW}\approx 10^{25.21}{m}^2.s^{-1}$, the uniform gravitic field embedding the MW to also explain DM is $k_0\approx 10^{-16.65}{s}^{-1}$, the typical speed at the end of the MW is $v\approx 2\times 10^{5}m.s^{-1}$, and the typical radius of the MW is $R_{MW}\approx 10^{21}m\approx 33kpc$. With the assumption $\vec{k}\perp \vec{v}$, Equation 3 allows retrieving the value of the mass of exotic matter at the end of the MW:

$\begin{array}{ll}\hfill & M_{DM}\approx \frac{4kv{R_{MW}}^2}{G}\approx \frac{4K_{1\text{\_}MW}v}{G}+\frac{4k_{0}v{R_{MW}}^2}{G}\hfill \\ \hfill & =>M_{DM\text{\_}{K}_{1\text{\_}MW}}\approx \frac{4K_{1\text{\_}MW}v}{G}\approx 2\times 10^{41}kg\hfill \\ \hfill & =>M_{DM\text{\_}{k}_0}\approx \frac{4k_{0}v{R_{MW}}^2}{G}\approx 3\times 10^{41}kg.\hfill \end{array}(4)$

We find the orders of magnitude of the masses (Equation 4) with, in particular, six times more dark matter than baryonic matter. What is admirable is that this GRL solution and its Equation 3 “far from the center of mass” also work with the same value of $k_0$ on the scale of galaxy clusters (therefore, for sizes 30 times greater than the MW). The important point here is to have approximately the same value of $k_0$ because it is a required condition in our GRL solution. $k_0$ is a uniform gravitic field of the environment that is shared by all objects in a given place, regardless of their sizes. For a galaxy cluster (Horner, 2001) of typical radius $R_{Cl}\approx 1Mpc\approx 3\times 10^{22}m$ and typical velocity dispersion $v_{Cl}\approx 10^{6}m.s^{-1}$, we observe a typical total mass (exotic and baryonic matter) of $M_{Cl}\approx 10^{15}{M}_{\odot }$. Approximately at least $60\%$ of this mass is an exotic mass, that is, $M_{DM\text{\_}Cl}\approx 1.2\times 10^{45}$ kg. If we apply (Equation 3) at $r=R_{Cl}$ with always $\vec{k}\perp \vec{v}$, we can translate the expected exotic matter into our expected equivalent gravitic field $k$:

$k\approx \frac{G{M}_{DM\text{\_}Cl}}{4v_{Cl}{R_{Cl}}^2}\approx 10^{-16.7}{s}^{-1}\approx k_0.(5)$

The value of $k$ (Equation 5) matches the expected value of $k_0$ (the gravitic field of the environment). Even with $90\%$ exotic matter among the total mass, we have $k\approx 10^{-16.52}{s}^{-1}$, still within the possible range of expected values of $k_0$ (Le Corre, 2015). We can specify that at this position $r=R_{Cl}$ (“far from the center of mass”), the contribution of the cluster’s own gravitic field to the total DM component is negligible. This can be verified using Le Corre (2024b), in which an approximate calculation yielded $K_{1\text{\_}Cl}\approx 10^{27}{m}^2.s^{-1}$ for Coma Cluster. Such a value yields $M_{DM\text{\_}{K}_{1\text{\_}Cl}}\approx 7\times 10^{43}$ kg. The contribution of the cluster’s own gravitic field in terms of “exotic matter” represents approximately less than $6\%$ of the whole “exotic matter.”

We can also note that (Equation 3) is equivalent to Equation 6 obtained in Le Corre (2024a). Precisely, this relationship allows describing the gravitational lens in the case of the Einstein ring of the JWST-ER1 object with only the baryonic matter.

However, (Equation 3) should not be considered universal. It is essentially valid in weak fields and far from the source (which is approximated as a compact mass). The interest of this relation lies, first, in providing an order of magnitude for the uniform gravitic field $k_0$, and second, potentially for the sum of $k_0$ and the intrinsic gravitic field $k_{\mathit{obj}}$ of the object under consideration.

In our article, this relation is used to support the fact that any exotic matter can be expressed as a gravitic field. It demonstrates this in weak fields and far from the source, but a relation could also be obtained in weak fields (with a mass distribution, i.e., a non-compact object) starting from the GRL metric, Section 6 of Le Corre (2015). For the GR metric, simulations would be necessary to obtain a relation of the NFW/EINASTO type, in the same way that these NFW/EINASTO relations were obtained. In conclusion to this paragraph, we have demonstrated in theoretical terms that any presence of exotic matter can be translated (replaced) in an equivalent way by a gravitic field (potentially the sum of a proper gravitic field $k_{\mathit{obj}}$ and a uniform gravitic field $k_0$). This fact has recently been confirmed experimentally. These gravitic fields $\overrightarrow{k_0}$ and $\overrightarrow{k_{\mathit{obj}}}$ are at the origin of the Lense–Thirring effect (which we discuss later) and correspond to “frame dragging.” Crosta et al. (2020) and Beordo et al. (2024) have recently demonstrated that we can translate the quantity of dark matter into the form of frame dragging in experimental terms (with Gaia DR2 and DR3 data), which corroborates what we have recently demonstrated.

2.3 Explicit assumptions of the two interpretations of DM

We examine the assumptions required in both cases to obtain the DM component. There are two proposals in the exotic matter assumption: a greater value of mass (to explain the great rotation speed at the ends of the galaxies, for example) and a distribution of this mass that covers large areas of the universe (to avoid the gravitational field decrease at the ends of the galaxies, for example). There are also two proposals in GRL assumptions (Le Corre, 2024b): a greater value of the gravitic field (that can also explain the great rotation speed at the ends of the galaxies) and a uniform gravitic field coming from neighboring clusters, which then covers large areas of the universe (to avoid the gravitational field decrease at the ends of the galaxies). In both explanations, two ad hoc assumptions are then made.

At this step, no explanation can be favored. Both cases propose the same number of assumptions and types kinds of assumptions.

2.4 First problem of exotic matter: insensitive to EM

Assuming a greater value of mass implies the existence of a new type of matter. Its existence being only necessary for gravitation, such an amount of invisible matter must be insensitive to EM. This constitutes the first strange physical behavior. All known matter is sensitive to EM. This new matter then defines a new physical entity. Even if the term “mass” makes this assumption appear natural, we are far from a trivial hypothesis.

Assuming a greater value of the gravitic field does not imply a new entity. The gravitic field is a known entity in GR. The gravitic field, being a component of the gravitational field, is naturally not sensitive to EM, as expected by DM. Using the gravitic field to explain the DM component has the advantage of solving the first problem of exotic matter.

2.5 Second problem of exotic matter: baryonic matter is nearly negligible

The quantity of exotic matter required to obtain the value of the rotation speed is enormous, so large that baryonic matter is nearly negligible. This is the second physical problem of the exotic matter assumption. From a logical point of view, hypothesizing the presence of a non-visible mass is generally done when this missing mass is low compared to the known mass (as, for example, for the discovery of Neptune by Le Verrier) to maintain consistency between theory and observation (otherwise, the theory itself may be called into question). This is clearly not the case for the exotic mass, which is five or six times greater than the known mass; in other words, in the exotic matter assumption, we assume the presence of five other Milky Way galaxies in the MW.

While the large quantity of exotic matter reverses the logical situation that was hitherto coherent between theory and visible mass by a domination of the only known matter for the Newtonian term, the large value of the gravitic field remains sufficiently weak to maintain this coherence between theory and visible mass. The gravitic field is not a mass, so it does not compete with the quantity of baryonic matter. Furthermore, the effect due to the gravitic field, even with these greater values, still remains weak compared to the Newtonian component (Le Corre, 2017). It is only when the Newtonian component becomes very weak that the uniform gravitic field dominates (this situation is similar to the MOND theory, but here, in the weak-field regime, the Newtonian term is supplemented by a GR term and not modified). This avoids the second problem of exotic matter.

2.6 Third problem of exotic matter: negligible and unnecessary at our scale

To obtain the flat rotation speed curve at the ends of the galaxies, exotic matter must cover large areas of the Universe. It must be ubiquitous, and in particular, it should be felt in our solar system. The DM component is, nevertheless, completely negligible and unnecessary in the equations at our scale. It is only required at the scale of the galaxies and beyond, while its presence is everywhere. We note that with exotic matter, the solar system is not surrounded by one Milky Way but by no less than five or even six Milky Ways, and this would strangely have no effect on the solar system. This is the third physical problem.

As demonstrated by Le Corre (2017), these gravitic fields, even greater than expected, which should be detectable by their Lense–Thirring effect, remain below the measurement capacity of our devices, which explains their non-detection at our scale and justifies their being neglected in the solar system. The third problem is once again solved by the GRL solution.

2.7 Fourth problem of exotic matter: the distribution and ubiquity of exotic matter

The ubiquity of exotic matter is quite mysterious, especially because its distribution appears increasingly important the further we look (with a very different ratio of DM to baryon between the galactic scale and the scale of galactic clusters): very present around the galaxies, more weakly present in the galactic disk, and even more strongly present between galaxies. Exotic matter accumulates then differently from visible matter (in a spherical halo around the galaxy, unlike a disk, and in a different proportion depending on the scale), even though it is supposed to undergo the same gravitational interaction. We can also remember the cuspy halo problem (Moore, 1994; Oh et al., 2015). This constitutes the fourth problem.

In the GRL solution, the ubiquity of the DM component is explained by the existence of a uniform gravitic field. Such a uniform field is physically justified by the Einstein–Maxwell equations of GRL. It is the same situation already observed in EM with magnets, where the atomic spin fields extend to the scale of the material, and in particle accelerators, where these magnets can maintain high-speed particles (situation similar to that at the ends of the galaxies). The ubiquity of DM then does not pose a problem in terms of physical feasibility in the GRL solution. Above all, the distribution of DM is clearly explained with the GRL solution because the equations predict a component of DM that is a function of $r$ and $v$ (Le Corre, 2024b) confirmed by Equation 3, that is, the larger the typical size and the greater the rotational speed, the greater the DM component. The GRL solution is therefore consistent with the observed distribution of the DM component. In particular, the cuspy halo problem is naturally solved (Equation 1, due to its dependence on $\vec{v}$ of zero value at its center).

It should also be noted that in our GRL solution, we assumed the existence of a uniform gravitic field $k_0$ in the environment of the galaxies to be able to explain their speed curves. This $k_0$ field must therefore also be experienced by galaxy clusters, which are the preferred environment of galaxies. The GRL solution, therefore, necessarily requires similar $k_0$ values for these two size scales. The fact of obtaining a similar value of $k_0$ (see Section 2.2) calculated independently, on one hand, from galaxies and, on the other hand, from galaxy clusters is an extremely strong point of this GRL solution. If such a field $k_0$ does not exist (i.e., if the GRL solution is false), it is quite miraculous to obtain these similar values of $k_0$ that make it possible to reach the right amount of DM both in galaxies and in clusters of galaxies. In contrast, in the hypothesis of exotic matter, their distributions in these two size scales are different (compared with the proportions of baryonic matter, which nevertheless undergoes the same gravitational interaction). Note also that these values of $k_0$ are sufficiently low to be undetectable locally in our solar system. This $k_0$ value is therefore consistent on three size scales (solar system, galaxy, and galaxy cluster).

In addition, the observation of the CMB shows a universe with zones more or less rich in exotic matter, whose typical size is of the order of a few hundred million light years (MacCrann et al., 2024). This represents approximately 30 Mpc. In the GRL solution, we assume that the uniform gravitic field, which explains the DM component, is generated by the gravitic field of fewer than a dozen neighboring clusters (Le Corre, 2015). We then deduce a typical size of approximately at least three or four clusters, that is to say, approximately at least 20 Ṁpc. The typical size is within the observed order of magnitude.

At this step, the GRL solution appears more suitable than the exotic matter solution.

2.8 The two interpretations of DM lead to new physics

We have observed that exotic matter defines a new physical entity. Consequently, it requires a new physics because we do not know how it interacts with itself, or with the known matter. We do not know its “temperature” or its structuration (baryonic particles form atoms, molecules, stars, black holes, galaxies, … what about exotic particles). It is literally a new physics that is required to integrate this exotic matter into our current physical knowledge.

Nevertheless, assuming a greater value of the gravitic field also implies the existence of a new physical behavior (other than currents of mass). The GRL solution has two advantages. This GRL situation is the same as in classical EM, in which the large values of the magnetic field of magnets cannot be explained by the currents of charge. It is only in the framework of quantum mechanics (QM) that these values could be explained. The fact that the same difficulty has already been encountered and already resolved (which is not the case for exotic matter, unconventional matter with esoteric and heterodox behavior) constitutes the first advantage. If we continue with this analogy, it is very likely that these gravitic field values will also require a quantum mechanics of gravitation, a new physics that is actively being researched. This is the second advantage. While this new physics responds to a natural expectation of theorists (namely, the need for a quantum mechanics of gravitation even if DM did not exist), exotic matter, this new physical entity, and all its new physics are a specific need exclusively dedicated to the DM component. This new physics should explain the previous problems that are naturally solved in the GRL solution. They do not appear in the GRL solution.

3 Some other strong points of the GRL solution

3.1 Rotation speed at the end of the galaxies

First, the uniform gravitic field hypothesis makes it possible to obtain the rotation speeds of the ends of the MW (Le Corre, 2024b) and several other galaxies (Le Corre, 2015) with great precision.

3.2 Gravitational lensing

Some articles claim that the GRL solution cannot explain gravitational lensing (Costa and Natario, 2024). A concrete application on a recent observation of an Einstein ring on the JWST-ER1 object (Le Corre, 2024a) shows, on the contrary, that this gravitational lens effect can be explained very well not only with these gravitic fields but also with the specific values fully explaining the DM component obtained by Le Corre (2024b). This Einstein ring in our solution does not require any exotic matter, only baryonic mass.

3.3 Tully–Fisher relation

This GRL solution with a uniform gravitic field also makes it possible to demonstrate the Tully–Fisher relation (Le Corre, 2023b), and, in particular, the break for super spirals of baryonic mass $M_b>10^{11.5}{M}_{\odot }$ and rotation speeds $v>340km.s^{-1}$, exactly as observed by Ogle (2019). This point is very important because the other interpretations of DM do not explain it. One criticism of the GRL solutions is that they do not yet explain everything that exotic matter explains (which is, in passing, contradicted by Equation 3 in the context of our specific GRL solution). Here is a relationship that the other explanations of DM have not yet explained, while our solution is a young solution that is extremely little studied compared to the decades of study of exotic matter by countless researchers. This demonstrates its effectiveness and relevance (in addition to challenging exotic matter).

3.4 MOND and exotic matter unified in the GRL solution

Our specific DM solution (GRL equations with an additional uniform gravitic field $\left.k_0\approx 10^{-16.5}{s}^{-1}\right)$ also makes it possible to explain the MOND theory as an approximation of GRL (Le Corre, 2023a) because, in GRL, the equations of motion are Equation 6:

$\frac{v{(r)}^2}{r}=\frac{G{M}_{\mathit{Gal}}}{r^2}\left(1+\frac{4k(r)v(r)r^2}{G{M}_{\mathit{Gal}}}\right).(6)$

It is interesting to note that this GRL solution allows justifying “the proper functioning” of the hypotheses of both exotic matter because, due to Equation 3, the uniform gravitic field can be approximated by an equivalent mass, and MOND, because the value of the uniform gravitic field obtained by Le Corre (2024b) makes it possible to obtain the characteristic value of MOND $a_0\approx 10^{-10}m.s^{-2}$ (Le Corre, 2023a):

$a_0\approx \frac{G{M}_{\mathit{Gal}}}{r^2}{\left(1+\frac{4k_{0}v(r)r^2}{G{M}_{\mathit{Gal}}}\right)}^2.(7)$

This GRL solution (GRL equations with an additional uniform gravitic field $\left.k_0\approx 10^{-16.5}{s}^{-1}\right)$ is therefore unifying, which is interesting because it justifies posteriori the other explanations of DM as an approximation of GRL (even the exotic matter). In other words, this GRL solution has as its limiting case the two most-studied interpretations of DM and thus serves as a unifying solution that is more general than exotic matter and MOND. This theoretical unification should be enough to make this specific GRL solution a serious contender against both exotic matter and MOND theory. In general, a theory’s ability to unify multiple viewpoints reveals its relevance:

GRL equations + $k_0\approx 10^{-16.5}{s}^{-1}\Rightarrow$ MOND parameter. $a_0\approx \frac{G{M}_{\mathit{Gal}}}{r^2}{\left(1+\frac{4k_{0}v(r)r^2}{G{M}_{\mathit{Gal}}}\right)}^2$.

GRL equations + $k_0\approx 10^{-16.5}{s}^{-1}\Rightarrow$ exotic matter. $M_{DM}\approx \frac{4kv{r}^2}{G}|\sin (\widehat{(\vec{k},\vec{v})})|$.

We now explore three situations that could help distinguish the GRL solution from other DM solutions.

3.5 Observation of improbable alignments of galaxies and their axis of rotation

To summarize, due to Equation 3, we have demonstrated that these two explanations (exotic matter and gravitic fields) are indistinguishable or, rather, that it is very difficult to find situations where these two solutions could be differentiated. This first main result of our article is obtained because we omit the vector aspect of the gravitic field and look only at its intensity and not its direction. This is the reason why the GRL solution entirely contains the exotic matter solution; that is, all exotic matter can be replaced by gravitic fields, but the reverse is not true. The vector aspect of the gravitic field leads to a remarkable consequence, which is the major difference between these two solutions. The GRL solution implies the existence of numerous kinds of alignment that the exotic matter hypothesis predicts with more difficulty.

As indicated by Le Corre (2024b), one of the predictions of our GRL solution is that the neighboring galaxies that have not interacted with other galaxies must present a tendency toward the alignment of their axis of rotation. The observation of Paudel et al. (2024) corresponds exactly to this situation. As indicated by Paudel et al. (2024), the studied galaxies are isolated (except for two), which corresponds to our condition of “not having undergone interaction with other galaxies” and “not too far from each other,” which corresponds to our condition of “neighboring galaxies” as defined by Le Corre (2024b). Under these conditions, they clearly observe an improbable alignment of their axis of rotation, which corresponds to the consequence of our solution, namely, “a tendency toward the alignment of their axis of rotation.”

It is also important to remember what they wrote (Paudel et al., 2024): “[…] the satellites around our MW galaxy are distributed in an extremely thin plane with an axis ratio of 0.29 (Pawlowski et al., 2012; Pawlowski et al., 2014), and there are a dozen similar structures that have been identified in various galaxy groups (Libeskind et al., 2019). Extending this phenomenon, we may have observed similar structures on a much smaller scale in contrast to previously discovered planar satellite systems around the massive host with a stellar mass of $\approx 10^{11}{M}_{\odot }$. In addition to their planar distribution, we also find that the member galaxies in this group have similar rotational directions.”

These planar distributions of dwarf satellites around a host are also a crucial prediction of our solution caused by this weak uniform gravitic field of the environment (Le Corre, 2024b; Le Corre, 2015). We write “crucial” because all these improbable alignments are certainly the main way to distinguish the gravitic fields assumption from the exotic matter assumption to explain the DM component because, as shown in Section 2, these two explanations of the DM component are similar and then very difficult to distinguish with current observations, cf. (Equation 3).

Several other observations have already shown the existence of such planar movements or unexpected alignments: quasar polarization vectors not randomly oriented as naturally expected, but concentrated around one preferential direction (Hutsemekers, 1998; Hutsemekers et al., 2005); planes of co-rotating satellites, similar to those seen around the Andromeda galaxy, are ubiquitous, and their coherent motion suggests that they represent a substantial repository of angular momentum on scales of approximately 100 kpc (Ibata et al., 2014); all but 2 of the 29 galaxies in the nearby Centaurus A Group are aligned in either of two thin planes roughly parallel with the supergalactic equator (Tully et al., 2015); prominent alignment of jet position angles of radio galaxies over an area of 1 square degree along a “filament” of about 1 (Taylor and Jagannathan, 2016); four active galactic nuclei roughly oriented along a line (Hennawi et al., 2015); observations of 65 distant galaxy clusters showing alignments at earlier epochs when the Universe was only one-third of its current age (West et al., 2019)…

In a deep field survey in relative proximity to the Galactic pole, Shamir (2025) observed that the number of galaxies that rotate in the opposite direction relative to the Milky Way galaxy is $50\%$ higher than the number of galaxies that rotate in the same direction as the Milky Way. This unexpected behavior is also a prediction of our GRL solution (Le Corre, 2024b; Le Corre, 2015). As indicated by Le Corre (2024b), “statistically, 3 neighboring clusters which follow each other should have their spins included in the same half-space […]. On the scale of approximately a diameter of 3 or 4 clusters, the Universe would then be slightly anisotropic.” In addition, in Le Corre (2015), it was stated that “in our solution, $\overrightarrow{k_0}$ (from cluster) and $\overrightarrow{\frac{K_1}{r^2}}$ (from galaxy) must be in the same half-space, for nearly all the galaxies inside a cluster. Said differently, a galaxy’s spin vector and the spin vector of the cluster (that contains these galaxies) must be in the same half-space.”

3.6 Potential solution to the three problems of stars near the Galactic Center: “Paradox of youth,” conundrum of old age, and top-heavy initial mass function

Current models suggest that stars cannot form near the Galactic Center (GC) (Sabha et al., 2012). The existence of the S-cluster stars (near the GC) then poses three problems: the first is that the stars appear as bright as young stars but the spectroscopic features reveal more evolved stars (Ghez et al., 2005; Ghez et al., 2003), a problem known as the “paradox of youth.” The second problem is that we observe a paucity of old and evolved stars (Buchholz et al., 2009; Merritt, 2010). The third problem is that the initial mass function (IMF) of stars near to the GC appears to be biased toward high-mass stars (Lu et al., 2013; Bartko et al., 2009). With the GRL solution, the existence of a higher-than-expected uniform gravitic field could solve all three problems. Indeed, with a higher value of $\overrightarrow{k_{GC}}$, the rotational speed of matter near the GC should be higher than expected (as with $\overrightarrow{k_0}$ at the end of the galaxies). The consequence could then be that the matter that clumps together to form stars makes them spin faster. The mass needed to overcome nuclear pressure would then have to be greater than without this external gravitic field to compensate for the centrifugal force. In this environment with a high gravitic field, a higher mass for a star does not mean a shorter lifespan but an apparent change in the gravitational interaction. Globally, the main sequence must be moved to higher masses. This would explain why stars appear younger than expected due to their rotation speed, brighter due to their greater mass, and why the IMC favors high-mass stars. John et al. (2025) suggest that exotic matter could explain these problems: “the continuous accretion of dark matter fuel allows these stars to maintain equilibrium eternally.” Once again, we find that exotic matter and the GRL solution are very difficult to discern by their effects. Nevertheless, the high variation of the speed of the “S-stars” near GC could be a way to detect this higher gravitic field (Le Corre, 2018) because its effects directly depend on the rotational speed ($4\vec{k}\wedge \vec{v}$). In the future, with the improvement in the precision of measurements of the speed and mass of SgrA*, we should see a discrepancy with the predictions on the orbit of these “S-stars,” which should be able to be explained by a lower mass of SgrA* but compensated by a larger gravitic field of GC ($\overrightarrow{k_{GC}}$). A mathematical solution could also be proposed to “save” the exotic matter assumption, albeit perhaps with a complicated distribution of exotic matter.

3.7 Potential solution to explain the appearance of cosmic filaments or walls and large voids

Our large-scale gravitic field $\overrightarrow{k_0}$ solution implies that the universe is decomposed into three characteristic scales with respect to its isotropy. This depends on the expression of the gravitational forces of GR (and more explicitly by GRL). A first “locally” isotropic scale where the first term of GR dominates, the Newtonian term is an isotropic and central force that cannot extend “indefinitely.” It roughly extends from our scale to that of galaxies or even perhaps galaxy clusters. Then, a second, much larger scale, where the second term of GR takes the advantage, the gravitic term is an anisotropic and non-central force that can extend “indefinitely” in the manner of the spin fields of the EM. The universe then becomes anisotropic and more precisely isotropic in $2D$, where the gravitic forces dominate (the mathematical expression of the gravitic force, $\overrightarrow{f_k}=\vec{k}\wedge \vec{v}$, implies a planar arrangement), while in the third dimension (that in the direction of the gravitic field, that is, perpendicular to the planar movement), the gravitational force, the Newtonian and gravitic terms, is drastically reduced. However, the weakness of this gravitic term (sufficient nevertheless to explain the DM component) implies that this anisotropy (generating planar arrangements) propagates without maintaining its direction. In other words, at an even larger third scale, the universe becomes isotropic again because this “local” anisotropy is found in all directions.

This is precisely what is observed because inside the thickness of the galactic disk (and even perhaps a cluster), matter is distributed in a roughly isotropic manner (first scale). Beyond the clusters of galaxies, the universe becomes anisotropic with the appearance of filaments or walls and large voids (second scale). Finally, at the scale of the entire universe, where this structure of filaments or walls and large voids is found in all dimensions, the universe then becomes isotropic again in its entirety (third scale).

Our solution implies, on the second scale, the appearance of filaments or walls, which must be long $2D$ ribbons that turn and twist on themselves like garlands of paper rather than tubes or like very flattened tubes. The counterpart of this “$2D$” anisotropy is that these ribbons of matter (the filaments or walls) necessarily border a void because perpendicular to the filaments or walls along the third dimension, the gravitational force is drastically reduced. That would correspond to the large voids. The appearance of this non-isotropic intermediate scale would be exclusively due to this gravitic field $\vec{k}$. This gravitic field exists naturally in GR (even outside our solution). It could then be argued that this goes against the difficulty the current simulations have with demonstrating the appearance of such structures by this gravitic field.

Our solution implies that this gravitic field is of the order of $10^6$ times more intense than that expected “classically” by GR, at least for the large structures of the universe (galaxies and clusters of galaxies). We can even venture into a “dimensional” calculation. If we consider that this gravitic force evolves roughly in $r^{-2}$ (far from its source), we can expect that along one spatial dimension, that is, along “$r$,” we will have an effect $10^3$ times greater than what we would naturally obtain by GR. This anisotropic attractive force will then, on the one hand, reduce the scale of the structures in its $2D$ plane (where there is matter) and, on the other hand, less impact the bubbles (without matter) perpendicular to $\vec{k}$. We can, therefore, expect the ratio of the sizes of the bubbles (roughly the second scale) to the lower structures (roughly the first scale) to be in the same proportion of $10^3$. The radius of the large voids is of the order of $10^{8}pc$. Our proportion leads to a radius of the order of $10^{5}pc\approx 100kpc$, which corresponds to sizes barely larger than the size of galaxies, as expected for the first scale. This therefore explains why simulations with the “natural” gravitic field of GR do not show this anisotropic intermediate scale of $2D$ ribbons and large $3D$ voids.

It would be interesting to demonstrate by numerical simulations what structures are put in place with a gravitic field much higher than that of the GR base. We can also expect that the characteristic size of the intermediate structures (ribbons and large voids) is directly linked to the intensity of this gravitic field. It would also be interesting to determine which intensity of this gravitic field would be necessary to obtain large voids of the dimension of those that we observe and determine whether it corresponds to the order of magnitude of $10^6$ times more intense than that expected “classically” by GR and necessary to also explain the DM component. It could be added that the potential tendency toward alignment in this context of clusters could also perhaps explain the appearance of cluster walls.

4 Problems to be addressed by GR-based alternatives

Some criticisms note that GRL solutions do not explain everything that exotic matter can explain. These criticisms, even if they are fair (because ultimately, these solutions must explain all that exotic matter can explain), are nevertheless not a reason to reject this solution. Indeed, the hypothesis of exotic matter has the advantage of several decades of study. The most recent hypothesis of a uniform gravitic field surrounding the galaxies is much younger and, unfortunately, cannot claim to respond immediately to all phenomena. This does not mean a priori that it is less good, but simply that it is much less studied. Our solution of a uniform gravitic field surrounding galaxies and originating from galaxy clusters has, in fact, never been studied. Let us even add that this longevity of absence of direct proof of the exotic matter would rather work to its disadvantage.

Despite the youth of our solution, due to Equation 3, we understand that most of the points noted by Ciotti (2024) and that any alternative to exotic matter must address are actually explained by our GRL solution. The following points take up one by one all the points of Ciotti (2024). Ciotti’s points are marked in quotation marks:

• “Stellar velocities in massive galaxies are $v\approx 300km.s^{-1}$ and proportionally less in lower mass stellar systems (some of them requiring in Newtonian gravity even larger values of DM-to-baryon ratios than disk galaxies!), with expected corrections of ${(\frac{v}{c})}^2\approx 10^{-6}$. The needed GR corrections of the Newtonian predictions instead should be of the same order as the observed velocities: a physically clear explanation of this huge and unexpected effect is needed, hopefully presented with a simple but robust order-of-magnitude estimate, involving the observed galactic properties.” As demonstrated in Equation 20 of Le Corre (2024b), the stellar velocities can be $\approx 270km.s^{-1}$ and even greater at $r\approx 100kpc$ if we take the value of the uniform gravitic field $k_0$ that embeds the MW. Moreover, our solution allows obtaining the Tully–Fisher relation and, in particular, the break for super spirals of baryonic mass $M_b>10^{11.5}{M}_{\odot }$ and rotation speeds $v>340km.s^{-1}$, exactly as observed by Ogle (2019). Our solution allows obtaining the velocities of both the MW and massive galaxies. The DM-to-baryons ratios depend not only on the mass of the disk galaxies in our solution but also on $k_0$, which is independent of the galaxies. This explains the statement that “some of them requiring in Newtonian gravity even larger values of DM-to-baryon ratios than disk galaxies!” This explanation is physically clear (similar to the uniform magnetic field of a magnet that generates a huge and unexpected effect in EM without the support of the QM) and has a robust order of magnitude, $10^{-16.62}{s}^{-1}<k_0<10^{-16.3}{s}^{-1}$ (Le Corre, 2015), which could be measured by the Lense–Thirring effect if the sensitivity of the measuring tools were sufficient, which is not yet the case (Le Corre, 2017). We note that the very unexpected distribution of exotic matter deduced from observations is physically unexplainable because no physics to date can explain the behavior of exotic matter, although it is physically justified within the framework of our GRL solution (cf. Section 2).

• “Several GR-based conceptual frameworks have been proposed, from gravitomagnetism, to geometric dragging, to GR effects of boundary conditions/vacuum solutions, to retarded potential effects due to unsteady accretion of gas on galaxies, to gravitomagnetic dipole effects produced by pairs of rotating BHs, and so on. To the astronomer, there appear to be too many GR proposed solutions: are they different manifestations of the same phenomenon? Are they physically different? Are they compatible?” The GRL solution is clearly defined in the same way that the uniform magnetic field of magnets in electromagnetism (EM) can explain very high-speed trajectories in particle accelerators. Our solution is physically equivalent to this EM situation. Instead of the uniform magnetic field, we have the uniform gravitic field $k_0$ due to neighboring clusters (and instead of the electric field, we have the Newtonian gravity field) to explain very high-speed trajectories in galaxies. Our solution is then clearly physically known and mathematically verified. Our physical entity (the gravitic field) is, in fact, more justified than a new entity of matter insensitive to EM, which is unprecedented in physics, outside the current known theoretical and observational framework. In contrast, the gravitic field has already been observed by its Lense–Thirring effect, although not yet with sufficient precision to detect the effect of $\overrightarrow{k_0}$ (Le Corre, 2017). In our GRL solution, we agree with the exotic matter that ad hoc hypotheses are necessary. These assumptions are ultimately of the same nature in the two explanations (cf. Section 2). What is certain is that the GRL solution is compatible with exotic matter, as revealed by Equation 3. We could also add that several concepts have been proposed for exotic matter, cold dark matter, hot dark matter, warm dark matter, WIMPS, axions, sterile neutrinos … This diversity (for exotic matter and for GR solutions) above all reveals our lack of knowledge and the certainty that no solution to date is truly satisfactory (even for exotic matter).

• “Exponential disks in Newtonian gravity produce reasonably flat rotation curves in the radial range from 1 to 3 scale-lengths, even in the absence of a DM halo. DM is only required by HI rotation curves in the external regions, well beyond the edge of the bright optical part of the galaxy. A GR model predicting a flat rotation curve inside the stellar disk is in accordance with Newtonian gravity.” Our approach is focused on the end of the galaxies where the DM component is required. It effectively explains the flat rotation curve and its values in the external regions of the galaxies. Because of our approximation, Equation 20 of Le Corre (2024b) is verified for $r>15kpc$ far from all matter, and this is precisely why (Equation 20) of Le Corre (2024b) challenges the assumption of exotic matter, which does not follow such a point approximation.

• “If DM is mimicked by GR effects due to rotation (such as in gravitomagnetism), why are massive DM halos required also in clusters of galaxies and elliptical galaxies, systems that often show a very low rotational support? It should be recalled that rotation curves of disk galaxies are not the only (and not the most important) phenomenological indication of the possible presence of DM halos.” In our solution, DM is not essentially explained by rotation (by mass currents). It is mainly explained by the uniform gravitic field $\overrightarrow{k_0}$. This field also embeds elliptical galaxies with the same intensity as spiral galaxies. This $k_0$ is assumed to be generated by the clusters of galaxies. Therefore, they must have a DM component (and certainly in greater quantity). We recall that our GRL solution also explains gravitational lensing (Le Corre, 2024a), the Tully–Fisher relation (Le Corre, 2023b), MOND theory (Le Corre, 2023a), alignments of dwarf satellite galaxies (Le Corre, 2024b), and also exotic matter (cf. Section 2).

• “For galaxies/clusters with gravitational lensing (a GR weak-field prediction), DM halos are inferred, with properties in remarkable agreement with the predictions based on stellar dynamics and/or hydrostatic equilibrium of hot, X-ray emitting gaseous coronae. Should we conclude that also gravitational lensing is affected by (unexpected) GR corrections?” The GRL solution with the gravitic field works very well with gravitational lensing (Le Corre, 2024a), as demonstrated with the Einstein ring of the JWST-ER1 object obtained with only the baryonic matter and the gravitic field. More in-depth articles would be necessary to fully confirm these agreements. Nevertheless, note that mathematically, Equation 3 allows translating the explanation by exotic matter into an explanation by gravitic field. It seems unlikely that the gravitic field will fail to replace exotic mass. These two explanations are very close because for exotic matter, we act on the first component of GR (the Newtonian field) thanks to the term of mass, and we act on the second component of GR for our GRL solution thanks to the term of gravitic field. It is like a principle of communicating vessels where the fluid is the DM component. Due to Equation 3, if $M_{DM}$ generates gravitational lensing, $\frac{4kv{r}^2}{G}|\sin (\widehat{(\vec{k},\vec{v})})|$ will generate the same gravitational lensing.

• “Dwarf spheroidal galaxies and ultra-faint galaxies are low-mass stellar systems, with very small values of v/c but with inferred very high DM-to-baryon ratios, even larger than in disk galaxies. Why are GR effects in these very weak-field systems proportionally more important than in more massive galaxies?” In our solution, DM is essentially explained by the gravitic field $k_0$ of the environment that is independent of the mass of the galaxy. Two neighboring galaxies should then be embedded into a gravitic field $k_0$ of the same order of magnitude. If their masses are very different, the DM-to-baryon ratios will necessarily be larger for ultra-faint galaxies than for massive galaxies.

• “In Globular Clusters, some of them in the same shape/velocity dispersion/mass range of galaxies in the previous point, DM is not required: why should GR behave so differently in stellar systems of similar mass?” As in the previous point, DM depends less on the mass of the system than on the gravitic field of the environment $k_0$. GR does not behave differently in stellar systems of similar mass, but $k_0$ (in which the stellar systems are embedded) is different and independent of stellar systems. In the GRL solution, as we can observe in Equation 3, the DM component is proportional to $\vec{k}\wedge \vec{v}$. This means that the DM component can be more or less important depending on the mutual direction of $\vec{k}$ and $\vec{v}$ and also on $r^2$. As explained by Le Corre (2024b), there are several ways to obtain stellar systems without DM (i.e., $k_0\approx 0$) in our solution.

• “Cosmological simulations with DM and initial conditions derived from the observed CMB appear to be highly successful in reproducing the growth of the large-scale structure of the Universe. What about the cosmological predictions of GR in the absence of DM?” A first study by Le Corre (2015) indicates the viability of our hypothesis of a gravitic field in the context of the CMB (albeit certainly oversimplifying), but more in-depth studies and simulations are necessary. We must also emphasize once again that our solution does not mean the absence of DM; it means that DM is not explained by exotic matter but by the gravitic fields. Thanks to Equation 3, these gravitic fields can be seen as an equivalent mass (the “exotic matter”). As a first approximation, our gravitic field should probably behave like exotic matter. More in-depth studies are necessary to confirm the cosmological simulations. Perhaps, our solution could even improve these simulations by potentially explaining structures such as the BOSS Great Wall or other “local” anisotropies because our gravitic field embedding a large zone of our Universe implies “local” alignments. As said previously (cf. Section 2), our GRL solution implies a typical size of “exotic matter” of the order of magnitude observed in CMB (the size of a few neighboring clusters).

• “Disk galaxies in the absence of DM halos are unstable in Newtonian gravity (Ostriker and Peebles, 1973): can pure GR stabilize disk galaxies?” More in-depth studies are necessary to confirm the stability of disk galaxies. In our solution, the absence of exotic matter does not mean the absence of a DM component. Therefore, there is no reason that disk galaxies are unstable because the DM component is still present in the galaxies (in the form of a gravitic field rather than exotic matter). Again, Equation 3 indicates that if $M_{DM}$ allows stabilizing disk galaxies, $\frac{4kv{r}^2}{G}|\sin (\widehat{(\vec{k},\vec{v})})|$ will stabilize disk galaxies in the same way.

• “Finally, it would be of the utmost importance to have clear and well-defined mathematical procedures to be applied to the observed baryonic components of galaxies to unambiguously predict the expected GR effects in the proposed scenario, to be compared with the observed kinematical fields” Our solution offers a clear and well-defined mathematical solution and even more importantly, a clear and well-defined physical solution that is not the case for exotic matter, which is a mathematical solution but had not been a physical solution until now (cf. Section 2, which compares this unconventional matter to the physical gravitic field already observed). Our explanation is physically equivalent to what happens in particle accelerators but within the framework of the Einstein–Maxwell equations.

5 Problems to be addressed by exotic matter

It is not surprising that all these previous points are also verified by the GRL solution because we can repeat what we indicated in the first sections: the gravitic fields can be interpreted as a component of matter, according to Equation 3. It is in the detail of this expression that we understand their difference. The gravitic fields $k$ are so weak that they need the support of high speeds and large radii to become dominant (cf. Equation 3), and we cannot easily differentiate these two explanations by local detection. However, gravitic fields should cause slight anisotropies, while the hypothesis of exotic matter is isotropic. This is why observations of different types of alignments, such as Paudel et al. (2024) on different types of objects (dwarf satellite galaxies, galaxies, or galaxy clusters), are important: they can make it possible to dissociate these two explanations of dark matter. Here are some points from our GRL solution that challenge the assumption of exotic matter, which exotic matter should address to compete with this GRL solution:

• The GRL solution explains the Tully–Fisher relation (Le Corre, 2023b). How can exotic matter explain it?

• The GRL solution explains why DM is insensitive to EM (because the gravitic field is a component of gravitation). How can exotic matter explain that it is insensitive to EM (other than calling these particles “exotic,” when to date, no known matter is insensitive to EM)?

• In our GRL solution, (Equation 10) of Le Corre (2024b) can explain DM exclusively without matter far from the center of the galaxy and with great accuracy for the measurements in the MW. How can exotic matter obtain such a relation? Indeed, if exotic matter is a correct solution, mathematically, (Equation 10) of Le Corre (2024b) should nevertheless be an approximation of the exotic matter idealization (because of its mathematical agreement with observation). Note that in this solution, (Equation 10) is obtained at the ends of the MW (which is a relation of general relativity) because we are far from the vast majority of the mass, contrary to the exotic matter distribution.

• How can the exotic matter assumption explain that, in our solar system, exotic matter does not need to exist, but, at large scale, baryonic matter is nearly negligible (representing only $25\%$ of the matter)? Our GRL solution explains why DM is undetectable in our scale [value of $k$ too weak for the precision of our current capacity of measurement (Le Corre, 2024b)] and dominating at scale of the galaxy at its ends (large values of $r$ and $v$ that compensate the extreme low value of $k$) according to Equation 3.

• We demonstrate in Le Corre (2023a) that MOND idealization is a particular solution of the GRL solution: it means that the gravitic field explains why and how MOND and exotic matter (regarded as an equivalent mass according to Equation 3) can “work” to be a viable approximate solution of the DM component. How can exotic matter explain that MOND is also an approximated possible solution of the DM component? Even if MOND is not the correct physical explanation, mathematically, it can be regarded as an approximated solution of the DM component. Our GRL solution plays a unifying role—perhaps, an indication that the gravitic field approach is more fundamental.

• Which observations, explained in the context of exotic matter, cannot be explained by the gravitic fields? That is, how can (Equation 3) be verified by its left-hand side, “$M_{DM}$,” without being verified by the right-hand side “$\frac{4kv{r}^2}{G}|\sin (\widehat{(\vec{k},\vec{v})})|$?” Equation 3 is perhaps the most destabilizing point for the exotic matter assumption. This makes exotic matter a subset of the gravitic field solution. Conversely, alignments that are outside the similarity with exotic matter (outside of Equation 3 due to the vectorial aspect of such observations) constitute observations explainable by gravitic fields but not foreseen by exotic matter.

• How can we explain that exotic matter agglutinates around galaxies while it agglutinates less in the galactic disk? We can reiterate this question on a larger scale: how can we explain that exotic matter agglutinates in greater quantities around clusters of galaxies while it agglutinates in smaller quantities around galaxies (in proportion to the visible mass, that is, with very different DM-to-baryon ratios)? The uniform gravitic field $k_0$ avoids this strangeness because the DM component is no longer associated with the visible mass of the object studied (except for galaxy clusters, which would be at the origin of this field $k_0$). It is, therefore, logically disturbing that the quantities of exotic matter are deduced from gravitation (experienced by visible matter), but that this exotic matter does not experience gravitation in the same way.

6 Discussion

The ad hoc hypothesis of a gravitic field greater than that predicted by mass currents may seem artificial, but it is no less so than that of the exotic mass. It is even more justified by physics because it is strictly the same physical situation as in EM, with the existence of magnetic fields such as those generated by magnets (to explain the greater values) and the same physical situation as in EM, with the particle accelerators (to explain how a uniform gravitic field can maintain high-speed orbits). In comparison, the existence of exotic matter is an extremely esoteric and heterodox physical hypothesis, with no equivalent in current physics (i.e., a mass insensitive to EM). It is as if we explained the magnetic field of magnets, in the theoretical framework of EM in the absence of QM, by the existence of a dark electric charge that would be insensitive to gravity and not by a magnetic field larger than that due to charge currents (justifiable only in the framework of QM).

To support this point, we note the observations of the Lense–Thirring effect (due to the gravitic field) recently measured in the disk of M87 (Cui et al., 2023). The observed period of its jet $T_{\mathit{jet}}\approx 11.25years$ is in excellent agreement with the Kerr solution of GR. The results are consistent with spin-down of a Kerr black hole (BH) (van Putten et al., 2024). Concretely, the value of ${\mathrm{\Omega }}_{LT}$ (yielding $T_{\mathit{jet}}$) is $10^{-8}rad.s^{-1}$ (Van Putten et al., 2024). In our solution, on the one hand, the main part of the DM component is the uniform gravitic field $k_0$ (generated by the galaxy clusters), which yields ${\mathrm{\Omega }}_{LT}=2k_0\approx 10^{-16}rad.s^{-1}$ (Le Corre, 2017). The Lense–Thirring effect of our $k_0$ is then $10^{-8}$ times smaller than what is measured. Its effect is completely indistinguishable from the Lense–Thirring effect of the BH of M87. Such a measure is not yet sufficiently precise to reject our solution. On the other hand, the fact that the Lense–Thirring effect for a BH, which is an extreme object for GR, does not reveal any excess tends to demonstrate that GR, even under extreme conditions, does not show any sign of a larger gravitic field. It then justifies the need for an ad hoc hypothesis (as postulated in the GRL solution). This result could confirm that GR is not susceptible to producing native effects (non-linear terms or others) that would explain this DM component, as expected for some alternative explanation (Astesiano and Ruggiero, 2022b; a). In our solution, we consider GR in the same way as the classical EM (without QM). In classical EM, the Biot–Savart law is always true and provides the magnetic fields of a large number of materials. However, for ferromagnetic materials, the Biot–Savart law is unable to account for the magnetization because it is necessary to postulate the existence of spin (which is only justified in the theoretical frame of QM). Our solution leads to postulating that GR and the Lense–Thirring effect are verified for a large number of objects, but that some objects (galaxies and galaxy clusters) are in some way “ferrogravitic” objects for which the Lense–Thirring effect will not be sufficient to explain their gravitic field.

To summarize, the results of Van Putten et al. (2024) seem to demonstrate that a GR capable of natively resolving the DM component without an ad hoc hypothesis is unlikely. They do not contradict our solution or that of exotic matter. Finally, we can add that in the objective of a quantum gravitation, our approach can potentially provide an idea of the maximum gravitic field generated by a potential gravitic moment “$\overrightarrow{{\mu }_k}$” that elementary particles could carry, if we assume that globally all these elementary gravitic moments are aligned inside very large structures (galaxies, clusters of galaxies, and perhaps black holes that present, in our solution of DM, gravitic fields greater than their single “classical” gravitic moment expected by GR alone). Our studies, in particular, that by Le Corre (2024b), provide for our galaxy a mass of $M_{\mathit{Gal}}\approx 8\times 10^{40}kg$ for a gravitic field $K_{1\text{\_}\mathit{Gal}}\approx 10^{25.21}{m}^2{s}^{-1}$ and for the Coma Cluster, a mass of $M_{Cl}\approx 6\times 10^{43}kg$ for a gravitic field $K_{1\text{\_}Cl}\approx 10^{27.24}{m}^2{s}^{-1}$. Very roughly, if we count the number ($\frac{M}{m_{p\text{\_}n}}$) of protons and neutrons ($m_{p\text{\_}n}\approx 10^{-27}kg$) to which these masses correspond, and we divide the global gravitic field “$\frac{K1}{r^2}$” on each of these elementary particles, we find a same potential elementary value of $K_{1\text{\_}p\text{\_}n}\approx 10^{-43}{m}^2{s}^{-1}$, for both the Milky Way and the Coma cluster. In other words, a potential elementary gravitic moment $\overrightarrow{{\mu }_{k\text{\_}p\text{\_}n}}$ would produce a gravitic field around it (far from the source of this gravitic field) of $\frac{10^{-43}}{r^2}{s}^{-1}$. At the proton or neutron radius, this yield give $k_{{\mu }_{k\text{\_}p\text{\_}n}}\approx 10^{-13}{s}^{-1}$. If we used an equivalent of the Biot–Savart law for GRL, we would have $k_{{\mu }_{k\text{\_}p\text{\_}n}\text{\_}GR}\approx \frac{G}{c^2}\frac{m_p\wedge v}{r^2}$. With $v=c$ and $r$ as the proton or neutron radius, this would yield $k_{{\mu }_{k\text{\_}p\text{\_}n}\text{\_}GR}\approx 10^{-19}{s}^{-1}$. We find again that our expected gravitic field (even at the elementary scale) has a discrepancy of a factor $10^6$ larger than the value potentially expected by GRL.

This does not demonstrate that the exotic matter assumption is not correct, but it does suggest that the gravitic field’s assumption is more reasonable because it does not encounter the same problems as exotic matter. At the same time, this is no guarantee that it is correct either. All articles should present their results both in the assumption of exotic matter and in the assumption of gravitic fields.

Equation 3 may appear trivial, but it should be able to convince the most skeptical of an alternative explanation for exotic matter, not by rejecting the DM component but by showing the extreme proximity of these two explanations. It is obvious that the exotic matter hypothesis benefits from many theoretical successes in explaining many observations. Any alternative should be able to obtain a relationship that connects its own interpretation to the interpretation of exotic matter as a limiting case. Equation 3 achieves this, demonstrating that the DM component can be interpreted either as exotic matter or as a gravitic field. As detailed in Sections 2.2 and 2.7, this GRL explanation demonstrates a formidable consistency because of obtaining the same value for the shared field $k_0$ on three size scales that present very different characteristics in the context of exotic matter: in our solar system (negligible and undetectable), in galaxies (where it begins to dominate compared to baryonic matter), and in galaxy clusters (where it dominates in even greater proportions than in the galaxies).

We note that during the merger of two galaxies or clusters of galaxies, due to Equation 3, because their matter will have different velocity vectors (in direction and intensity) and then also different own gravitic fields (but with the same uniform gravitic field $k_0$), their matter should undergo slightly different effects ($\vec{k}\wedge \vec{v}$). This could explain that in the “Bullet Cluster” 1E 0657-558 (Markevitch et al., 2002), we find different distinct centers of inertia and as in MACS J0018.5 + 1626 (Silich et al., 2024) with a displacement $\vec{k}\wedge \vec{v}$ not in the direction of $\vec{v}$. Furthermore, because the main component of DM is due to $k_0$ shared by the two mergers but imposed by the environment (i.e., independent of them), this main part of the DM component should be less impacted by the interaction than the baryonic mass, which could explain the apparent offset between the primary mass peaks of the baryonic mass and of the “exotic matter $k_0$.” In passing, we can note that this environmental component $k_0$ of the gravitic field corresponds in terms of exotic matter to a DM component that is collisionless because it is independent of the interacting objects (unlike their own component $\frac{K_1}{r^2}$).

Equation 3 shows three situations without DM. Two situations consist of having $k_0\simeq 0$. Either we have an isolated object that is far from neighboring clusters (source of $k_0$), or we have a non-isolated object in a zone where the gravitic fields of neighboring clusters do not align and whose sum cancels out, that is, probably with neighboring clusters with quite different axes of rotation. Finally, a third situation consists of having a product $\vec{k}\wedge \vec{v}$, which is zero when the rotation speed of the object is aligned with $k_0$.

Other consequences are proposed by Le Corre (2024b) and Le Corre (2015) that are not a priori predicted by the hypotheses of exotic matter or MOND. For example, the warping of the galaxies could be explained by the non-perpendicularity of $k_0$ with their rotation speed vectors (i.e., with their own gravitic field).

7 Conclusion

To avoid any misunderstanding, we insist that our solution (Le Corre, 2024b) does not imply that the DM component does not exist. On the contrary, our solution consists of obtaining this DM component with an ad hoc assumption. All situations where the DM component appears face the same two gravitational problems. The first problem with gravitation is that observation requires a much larger gravitational field than expected. The second problem is that the distribution of this gravitational field extends well beyond visible matter and even tends to increase with the distance from it.

According to the equations of motion (Equation 1) due to gravitation, these problems can be solved either by acting on the mass for the first Newtonian term (the method followed by the exotic matter assumption) or the gravitic field for the second term (the method followed by our GRL solution). We can, for example, focus on the rotation speed of the ends of the galaxies for which the value of the rotation speed is extremely high (the first problem) and which presents a flat speed curve (the second problem). In the exotic matter assumption, the galaxies would be composed of a greater quantity of mass than expected (to solve the first problem). Their distribution, to compensate for the natural decrease of the gravitational field, would increase with distance (to solve the second problem). In the GRL solution, we assume a greater value of the gravitic field than expected (to solve the first problem) and a uniform gravitic field in which the galaxies would be embedded (to solve the second problem).

The solution of the exotic matter leads to a new physical entity, a new mass, for two reasons: because it is insensitive to EM and because of its unconventional distribution (differently distributed compared to known matter, even though it is deduced from the same gravitational interaction). This situation has no equivalent in physics. In contrast, in the GRL solution, the gravitic field is not a new entity. It is a known physical entity at the origin of the Lense–Thirring effect, which was confirmed several years ago. The existence of a uniform gravitic field is also a natural solution of the Einstein–Maxwell equations. This uniform gravitic field would be generated by neighboring clusters. This situation is physically well defined because it is equivalent to the physical situation of EM, where the magnetic field of magnets is generated on a large scale by neighboring atomic spins (with large values unexplainable by the charge currents of classical EM). Furthermore, the physical situation at the ends of the galaxies, in our GRL solution, is also physically well defined because it is equivalent to the physical situation of EM encountered within particle accelerators, where very high-speed trajectories are maintained in orbits.

More concretely, in the exotic matter assumption, the Milky Way would be composed of five to six other MWs, and yet this excess weight would strangely have no influence on the solar system. In the GRL solution, the larger value of the gravitic field still remains below our detection capabilities. The GRL solution is then naturally in agreement with observation, undetectable on our scale, and dominant on the scale of the galaxies and beyond. The assumption of such a quantity of exotic matter makes the only known matter (the baryonic matter) almost negligible. These quantities deduced from the equations of gravitation, equations themselves based on this only known a priori minority matter, raise questions about the logical process of this deduction (without calling into question the equations themselves). In contrast, according to Equation 3, the ubiquity of the DM component on a very large scale is justified and can be explained by the parameters $r$ and $v$, which make this component large where $r$ and $v$ are large.

Equation 3 is also important because it demonstrates that any explanation in the form of exotic matter can be translated into the form of a gravitic field. We thus demonstrate that the exotic matter solution is entirely contained in the GRL solution. This result is important because it demonstrates the equivalence of these two explanations. The explanation of the DM component by gravitic fields explains the same observations as exotic matter. For example, the gravitational lensing can be explained thanks to this relation, as demonstrated by Le Corre (2024a).

Without direct measurements of gravitic fields or direct detections of exotic particles, it will be very difficult to discern and favor one or the other of these two explanations of the DM component. This then means that the results obtained over the years in the context of exotic matter should also be valid in the context of gravitic fields. Nevertheless, observations strongly constrain large-scale gravitic fields, such as $k_0$ (CMB anisotropy and Bianchi anisotropy limits, along with the statistical isotropy of large-scale galaxy velocity fields). It would be necessary to confirm whether the value of $k_0$ assumed in our solution is consistent with these constraints. We note that the GRL solution, in terms of direct measurements of the gravitic fields, implies an effect of Lense–Thirring between $0.3mas.y^{-1}$ and $1.5mas.y^{-1}$ (Le Corre, 2024b), in addition to the expected $6606mas.y^{-1}$ geodetic precession and the expected $39mas.y^{-1}$ frame-dragging precession.

Consequently, it is the vector aspect of (Equation 1) that should open the way to discriminative observations. This is why all observations of all types of alignment (axes of rotation, orbital planes, etc.) of the large structures of the universe (dwarf satellite galaxies, clusters, etc.) should constitute very important tests of these two explanations. Many types of alignments (unexpected in the context of exotic matter) have already been observed. This vector aspect should also explain the appearance of cosmic filaments or walls and large voids, a zone of anisotropy at an intermediate scale between the scale of galaxy clusters and that of the universe as a whole. Another situation that should distinguish the GRL solution from other solutions is that of a system subject to very large speed variations, such as the S2 stars around SgrA*. With sufficient precision in the measurements, deviations in the orbits should appear and be explained by a lower mass of SgrA* compensated by a larger gravitic field of SgrA* than expected.

Our GRL solution also challenges exotic matter because, in addition to containing the exotic matter solution, it explains the MOND theory. The GRL solution appears to be a more general solution for which the two main explanations of the DM component to date (exotic matter and MOND) are limiting cases of GRL (and consequently of GR). The GRL solution unifies these two explanations and justifies their validity. In addition, the gravitic field assumption explains the Tully–Fisher relationship, which remains unexplained in the exotic matter assumption.

It is difficult not to conclude that the GRL solution (gravitic field assumptions) is a better explanation than the exotic matter assumption, not because the exotic matter assumption would be false, but because the exotic matter solution is entirely contained and translatable in terms of the gravitic field. The GRL solution resolves the logical problems of exotic matter. The only new physics required by exotic matter is also required by the GRL solution. As we have observed in this article, even on this subject, the GRL solution presents itself favorably because it very probably responds to the expected and sought-after theoretical need for a quantum mechanics of gravitation, unlike a new physics dedicated specifically to exotic matter, which would not a priori meet this need.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

SL: Conceptualization, Investigation, Methodology, Writing – original draft, Writing – review and editing.

Funding

The author(s) declared that financial support was not received for this work and/or its publication.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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