Deur's work on gravity (which he considers to be within general relativity but which deviates from how it is conventionally applied in significant ways) is one of the most promising approaches to understanding dark matter and dark energy phenomena. In his latest paper, below, he predicts and confirms as association between disk thickness and the magnitude of dark matter phenomena observed.

# Relativistic corrections to the rotation curves of disk galaxies

(Submitted on 10 Apr 2020)

We present a method to investigate the effect of relativistic corrections arising from large masses to the rotation curves of disk galaxies. The method employs a mean-field approximation and gravitational lensing. Applying it to a basic model of disk galaxy, we find that these corrections become important and magnified at large distances. The magnitude of the effect is sufficient to explain the galactic missing mass problem without requiring a significant amount of dark matter.A prediction of the model is that there should be a strong correlation between the inferred galactic dark mass and the galactic disk thickness. We use two independent sets of data to verify this.

## Bibliographic data

## Submission history

From: Alexandre Deur [view email]**[v1]**Fri, 10 Apr 2020 11:23:04 UTC (1,188 KB)

The introduction to the latest paper explains that:

The total mass of a nearby disk galaxy is typically obtained from measuring its rotation curve and deducing from it the mass using Newton’s dynamics. The rationale for this non-relativistic treatment is the small velocity of stars: v/c << 1 sufficiently far from the central galactic black hole. However, the assumption that relativistic corrections are negligible may be questioned on several grounds. Inspecting the post-Newtonian [1] Lagrangian, e.g. for two masses M1 and M2 separated by r, shows non-Newtonian potential terms of the type G^2M1M2(M1+M2)/2r^2 (G is the gravitational constant) that are independent of v, thus not suppressed at small v, and can be non-negligible for large enough M1 and M2. These terms express the non-linear nature of General Relativity (GR), which arises from its field self-interaction: the gravitational field has an energy and hence gravitates too.

Field self-interactions are well-known in particle physics: Quantum Chromodynamics (QCD), the gauge theory of the strong force between quarks, features color-charged fields that self-interact. In fact, GR and QCD have similar Lagrangians, including self-interacting terms, as can be seen when the Einstein-Hilbert Lagrangian of GR is expanded in a polynomial form [2, 3]. Field self-interaction in QCD, which causes quark confinement, exists even for static sources, as shown by the existence of numerous heavy quark bound states (in which v ≈ 0 for quarks) [4] and by classic numerical lattice calculations for v = 0 quarks [5]. This, as well as the correspondence between the respective terms of the GR and QCD Lagrangians, shows that for bodies massive enough a relativistic treatment is required regardless of their velocity. Finally, the measured speeds at the rotation curve plateaus are of several hundreds of km/s, e.g. 300 km/s (or v/c = 0.1%) for NGC 2841. They are similar to that of stars orbiting the central black hole of our galaxy and clearly display the relativistic dynamics expected in the strong regime of GR [6].

These arguments suggest that one should investigate the importance of relativistic dynamics in galaxies and how it affects the missing mass problem. From experience with QCD, a non-perturbative approach is required to fully account for field self-interaction, making post-Newtonian formalism inadequate. In Refs. [2, 3], a nonperturbative numerical lattice method was used. Here, we propose to approach the problem with a mean-field technique combined with gravitational lensing. There are several advantages of the approach compared to the lattice method used in [2, 3]: (1) it is an entirely independent method, thereby providing a thorough check of the lattice result; (2) it is not restricted to the static limit of the lattice method and can be applied to systems with complex geometries; (3) it is significantly less CPU-intensive than a lattice calculation, and hence much faster; (4) it clarifies that the effect calculated in Refs. [2, 3] is classical.The lattice approach – an inherently quantum field theory (QFT) technique – used in Refs. [2, 3] may misleadingly suggest that a quantum phenomenon is involved. In fact,the classical nature of the effect is consistent with these lattice calculations being performed in the high-temperature limit in which quantum effects disappear, as discussed in Ref. [3]; (5) the lensing formalism is more familiar to astrophysicists and cosmologists, in contrast to lattice techniques with its QFT underpinning and terminology.

I don't recall if I blogged his 2019 paper on gravity (which was updated in January of this year) or not, so rather than checking, I am cutting and pasting that here as well. Notably, he has two collaborators on the 2019 paper. I have updated my permanent page on Deur's gravitational work to reflect both of these papers.

# Significance of Gravitational Nonlinearities on the Dynamics of Disk Galaxies

(Submitted on 31 Aug 2019 (v1), last revised 11 Jan 2020 (this version, v2))

The discrepancy between the visible mass in galaxies or galaxy clusters, and that inferred from their dynamics is well known. The prevailing solution to this problem is dark matter. Here we show that a different approach, one that conforms to both the current Standard Model of Particle Physics and General Relativity, explains the recently observed tight correlation between the galactic baryonic mass and its observed acceleration.Using direct calculations based on General Relativity's Lagrangian, and parameter-free galactic models, we show that the nonlinear effects of General Relativity make baryonic matter alone sufficient to explain this observation.

## Bibliographic data

## Submission history

From: Balša Terzić [view email]

**[v1]**Sat, 31 Aug 2019 00:02:04 UTC (278 KB)

**[v2]**Sat, 11 Jan 2020 04:32:06 UTC (1,168 KB)

From Part II:

Field self-interaction makes GR non-linear. The phenomenon is neglected when Newton’s law of gravity is used, as typically done in dynamical studies of galaxies or galaxy clusters. However, such a phenomenon becomes significant once the masses involved are large enough. Furthermore, it is not suppressed by low velocity—unlike some of the more familiar relativistic effects—as revealed by e. g. the inspection of the post-Newtonian equations (Einstein et al. 1938). In fact, the same phenomenon exists for the strong nuclear interaction and is especially prominent for slow-moving quark systems (heavy hadrons), in which case it produces the well-known quark confining linear potential.

The connection between self-interaction and non-linearities is seen e.g. by using the polynomial form of the Einstein-Hilbert field Lagrangian (see e.g. Salam 1974; Zee 2013)

L = sqrt(det(gµν)) gµνR µν 16πG = sigma(∞ to n=0) (16πGM) n/2 [ϕ n ∂ϕ∂ϕ] (1)

where gµν is the metric, Rµν the Ricci tensor, M the system mass and G is the gravitational constant. In the natural units (~ = c = 1) used throughout this article, [G] = energy−2 . The polynomial is obtained by expanding gµν around a constant metric ηµν of choice, with ϕµν ≡ gµν − ηµν the gravitational field. The brackets are shorthands for sums over Lorentz-invariant terms (Deur 2017). Field self-interaction originates from the n > 0 terms in Eq. (1), distinguishing GR from Newton’s theory, for which the Lagrangian is given by the n = 0 term. One consequence of the n > 0 terms is that they effectively increase gravity’s strength. It is thus reasonable to investigate whether they may help to solve the missing mass problem. In fact, it was shown that they allow us to quantitatively reproduce the rotation curves of galaxies and the dynamics of clusters without need for dark matter, also providing a natural explanation for the flatness of the rotation curves (Deur 2009).

The phenomenon underlying these studies is ubiquitous in Quantum Chromodynamics (QCD, the gauge theory of the strong interaction). The GR and QCD Lagrangians are similar and both contain field self-interaction terms. In QCD, their effect is well-known as they are magnified by the large QCD coupling, typically αs = 0.1 at the transition between perturbative and strong regimes (Deur et al. 2016). In GR, self-interaction becomes important for non-negligible values of the coupling sqrt(GM/L) (L is the system’s characteristic scale, used here to form the dimensionless coupling needed to heuristically assess self-interaction’s importance), typically for sqrt(GM/L) > 10^−3 (Deur 2017). In QCD, a critical effect of self-interaction is a stronger binding of quarks, resulting in their confinement. In GR, self-interaction likewise increases gravity’s strength, which can explain the missing mass problem (Deur 2009).

One may question the relevance of field self-interaction at large galactic radii r, where on the one hand the matter density is small, so field self-interaction is negligible, and where the matter density is small, so field self-interaction is negligible, but where the missing mass problem grows worst. However, once the gravity field lines are distorted at small r due to large matter density there, they evidently remain so even if the matter density becomes negligible (no more field self- interaction, i.e. no further distortion of the field lines), preserving a form of potential different to that of Newton. Thus, even if the gravity field becomes weak, the deviation from Newton’s gravity remains1.

A key feature for this article is the suppression of self-interaction effects in isotropic and homogeneous systems (Deur 2009):

• In a two-point system, large p GM/L or αs values lead to a constant force between the two points (and a vanishing force elsewhere), i.e. the string-like flux-tube, well-known in QCD.

• For a homogeneous disk, because of the symmetry, the flux collapses only outside the disk plane. The resulting force between the disk center and a point in the disk at distance r decreases as 1/r.

• For a homogeneous sphere, the force recovers its usual 1/r^2 behavior since the flux has no particular direction or plane of collapse.

This symmetry dependence has led to the discovery of a correlation between the missing mass of elliptical galaxies and their ellipticity (Deur 2014). This also illustrates the point of the previous paragraph: even if the matter density in the disk decreases quickly with r, the missing mass problem—which in our approach comes from the difference between the GR and Newtonian treatments—grows worst since the difference between the 1/r GR force in the 2D disk and the 1/r^2 Newtonian force grows with r. This offers a simple explanation for the relation reported in MLS2016: although densities, and thus accelerations, are largest at small r, the 1/r − 1/r^2 difference between the GR and Newtonian treatments remains moderate. However, it becomes important at large r although accelerations are small. Furthermore, at small r, the 1/r^2 force is restored for GR due to finite disk thickness hz , since isotropy is restored for r < hz . Moreover, since disk galaxies often contain a central high-density bulge that is usually nearly spherical (M'endez-Abreu et al. 2008), self-interaction effects are suppressed there by the near-spherical symmetry, and the departure from the 1/r^2 behavior occurs after the bulge-disk transition.

1 An analogous phenomenon exists for QCD: the parton distribution functions (PDFs) that characterize the structure of the proton are nonperturbative objects even if they are defined and measured in the limit of the asymptotic freedom of quarks where αs tends to zero. Thus,From the conclusion:PDFs are entirely determined by the self-interaction/non-linearities of QCD, although those are negligible at the large energy-momentum scale where PDFs are relevant.

Our findings support the possibility that GR’s self-interaction effects increase the gravitational force in large, nonisotropic mass distributions. When applied to disk galaxies, the increased force on the observed matter transposes to the missing mass needed in the traditional Newtonian analyses. We have thus proposed a plausible explanation for the correlation between the luminous mass in galaxies and their observed gravitational acceleration shown in MLS2016. That this correlation is encapsulated in our basic, parameter-free, models indicates its fundamental origin.

The explanation proposed here is natural in the sense that it is a consequence of the fundamental equations of GR and of the characteristic magnitudes of the galactic gravitational fields, and in the sense that no fine tuning is necessary. This contrasts with the dark matter approach that necessitates both yet unknown particles and a fine tuning in galaxy evolution and baryon-dark matter feedbacks (see e.g. Ludlow et al. 2017). We used several approaches that are quite different, thus leading to a robust conclusion.

The work presented here adds to a set of studies that provide straightforward and natural explanations for the dynamical observations suggestive of dark matter and dark energy, but without requiring them nor modifying the known laws of Nature. This includes flat rotation curves of galaxies (Deur 2009), the Bullet cluster (Clowe et al. 2006; Deur 2009), galaxy cluster dynamics (Deur 2009), and the evolution of the universe (Deur 2019). The TullyFisher relation (Tully & Fisher 1977) also finds an immediate explanation (Deur 2009). There are compelling parallels between those observations and QCD phenomenology, e.g. the equivalence between galaxies’ Tully-Fisher relation, and hadrons’ Regge trajectories (Deur 2009, 2017), plausibly due to the similarity between GR’s and QCD’s underlying fundamental equations. The fact that these phenomena are well-known for other areas of Nature that possess a similar basic formalism; the current absence of natural and compelling theory for the origin of dark matter (supersymmetry being now essentially ruled out); and the yet unsuccessful direct detection of a dark matter candidate or its production in accelerators despite coverage of the phase-space expected for its characteristics; all support the approach we present here as a credible solution to the missing mass problem.

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