arXiv subjects

Mordehai Milgrom

Publications and source records attributed to Mordehai Milgrom.

At least 37 records · Page 2Linked to original sources

arXiv2019

MOND in galaxy groups: a superior sample

Intermediate-richness galaxy groups are an important test ground for MOND. First, they constitute a distinct type of galactic systems, with their own evolution histories and underlying physical processes; secondly, they probe little-chartered regions of parameter space, as they have baryonic masses similar to massive galaxies, and similar velocity dispersions, but much larger sizes -- similar to cluster cores (or even to clusters), but much lower dispersions. Importantly in the context of MOND, they have the lowest internal accelerations reachable inside galactic systems. I analyze a sample of 56 medium-richness groups having a large number ($\ge 15$) of members with measured velocities. The groups obey the deep-MOND, baryonic-mass-velocity-dispersion relation, $M_MGa_0=(81/4)σ^4$, with individual, MOND $M_M/L_K$ ratios of order $1$ solar unit, with $(M_M/L_K)_{median}=0.7$ s.u. compared with the much larger Newtonian $M_d/L_K$ -- several tens s.u., and $(M_d/L_K)_{median}=37$ s.u. The same MOND relation describes dwarf spheroidals -- 2-3 orders smaller in size, and 7-8 orders lower in mass. The groups conformation to the MOND relation is equivalent to their lying on the deep-MOND branch of the `acceleration-discrepancy relation', $g\approx (g_N a_0)^{1/2}$, for $g$ as low as a few percents of $a_0$ ($g_N$ is the baryonic, Newtonian, gravitational acceleration, and $g$ the actual one). This argues against a breakdown of MOND at extremely low accelerations. This conformation also argues against the hypothesis that the remaining MOND conundrum in cluster cores bespeaks a breakdown of MOND on large-distance scales; our groups are as large as cluster cores, but do not show obvious disagreement with MOND. I also discuss the possible presence of the idiosyncratic, MOND external-field effect.

astro-ph.GA↗

arXiv2019

Toomre stability of disk galaxies in quasi-linear MOND

We consider disk stability in the quasi-linear formulation of MOND (QUMOND), the basis for some $N$-body integrators. We generalize the Toomre criterion for the stability of disks to tightly wound, axisymmetric perturbations. We apply this to a family of thin exponential disks with different central surface densities. By numerically calculating their QUMOND rotation curves, we obtain the minimum radial velocity dispersion required for stability against local self-gravitating collapse. MOND correctly predicts much higher rotation speeds in low surface brightness galaxies (LSBs) than does Newtonian dynamics without dark matter. Newtonian models thus require putative very massive halos, whose inert nature implies they would strongly stabilize the disk. MOND also increases the stability of galactic disks, but in contradistinction to Newtonian gravity, this extra stability is limited to a factor of 2. MOND is thus rather more conducive to the formation of bars and spiral arms. Therefore, observation of such features in LSBs could be problematic for Newtonian galaxy models. This could constitute a crucial discriminating test. We quantitatively account for these facts in QUMOND. We also compare numerical QUMOND rotation curves of thin exponential disks to those predicted by two algebraic expressions commonly used to calculate MOND rotation curves. For the choice that best approximates QUMOND, we find the circular velocities agree to within 1.5% beyond $\approx 0.5$ disk scale lengths, regardless of the central surface density. The other expression can underestimate the rotational speed by up to 12.5% at one scale length, though rather less so at larger radii.

astro-ph.GA↗

arXiv2018

MOND in galaxy groups

Galaxy groups, which have hardly been looked at in MOND, afford probing the acceleration discrepancies in regions of system-parameter space that are not accessible in well-studied galactic systems, such as galaxies, galaxy clusters, and dwarf-spheroidal satellites of galaxies. Groups are typically the size of galaxy-cluster cores, but have masses typically only a few times that of a single galaxy. Accelerations in groups get far below those in galaxies, and far below the MOND acceleration. So much so, that many groups might be affected by the external-field effect, which is unique to MOND, due to background accelerations. Here, I analyze the MOND dynamics of 53 galaxy groups, recently catalogued in 3 lists. Their Newtonian, K-band, dynamical $M/L$ ratios are a few tens to several hundreds solar units, with $\langle{M_d/L_K}\rangle= (56, 25, 30)~M_{\odot}/L_{\odot}$, respectively for the 3 lists; thus evincing very large acceleration discrepancies. I find here that MOND requires dynamical $M_M/L_K$ values of order $1~M_{\odot}/L_{\odot}$, with $\langle{M_M/L_K}\rangle=(0.8, 0.56, 1.0) ~M_{\odot}/L_{\odot}$, for the 3 lists, which are in good agreement with population-synthesis stellar values, and with those found in individual galaxies. MOND thus accounts for the observed dynamics in those groups with baryons alone, and no need for dark matter -- an important extension of MOND analysis from galaxies to galactic systems, which, to boot, have characteristic sizes of several hundred kiloparsecs, and accelerations much lower than probed before -- only a few percent of MOND's $a_0$. The acceleration discrepancies evinced by these groups thus conform to the deep-MOND prediction: $g\approx (g_Na_0)^{1/2}$, down to these very low accelerations ($g$ is the measured, and $g_N$ the baryonic, Newtonian acceleration).

astro-ph.GA↗

arXiv2018

MOND and the dynamics of NGC 628

Aniyan et al. (2018) have recently published direct measurements of the baryonic mass distribution and the rotation curve of the almost-face-on disc galaxy NGC 628. While its very low inclination makes this galaxy anything but ideal for rotation-curve analysis, these new results, taken at face value, have interesting ramifications for MOND. The methods employed afford a direct determination of the stellar mass in the disc, which, in turn, affords a parameter-free MOND prediction of the rotation curve, which I show. In comparison, the dark-matter fits that Aniyan et al. present have two free parameters. To boot, these results further negate an earlier claim deleterious to MOND. It is that stellar $M/L$ ratios deduced from vertical velocity dispersions in disc galaxies are rather lower than what is required by MOND fits to rotation curves. Specifically, it was claimed that even high-surface-density discs are, by and large, sub maximal; viz., that they show substantial mass discrepancies near their center. This is contrary to the prediction of MOND that in such high-acceleration regions only small discrepancies should appear, if any to speak of. Such claims of low $M/L$ values have been rebutted before, and the fallacy that may have led to them pointed out. The new results strongly buttress these rebuttals.

astro-ph.GA↗

arXiv2017

High-redshift rotation curves and MOND

Genzel et al. have recently published the rotation curves of six high-redshift disc galaxies ($z\sim 0.9-2.4$), which they find to be `baryon dominated' within the studies radii. While not up to the standard afforded by data available for analysis in the nearby Universe, these data are valuable in constraining cosmological evolution of either DM scenarios, or -- as I discuss here -- $z$-dependence of MOND. Indeed, these results, if taken at face value, teach us useful lessons in connection with MOND. a. The dynamical accelerations at the half-light radii, found by Genzel et al., are rather high compared with the MOND acceleration constant, as measured in the nearby Universe: $g(R_{1/2})= (3-11)a_0$. MOND then predicts fractions of `phantom matter' at $R_{1/2}$ of at most a few tens of percents, which, galaxy by galaxy, agree well with what Genzel et al. find. b. The asymptotic rotational speeds predicted by MOND from the baryonic-mass estimates of Genzel et al. are substantially lower ($0.55-0.75$) than the maximal speeds of the RCs. MOND thus predicts a substantial decline of the RCs beyond the maximum. This too is in line with what Genzel et al. find. c. Arguably, the most important lesson is that the findings of Genzel et al. cast very meaningful constraints on possible variation of $a_0$ with cosmic time. For example, they all but exclude a value of the MOND constant of $\sim 4a_0$ at $z\sim 2$, excluding, e.g., $a_0\propto (1+z)^{3/2}$.

astro-ph.GA↗

arXiv2017

Perspective on MOND emergence from Verlinde's "emergent gravity" and its recent test by weak lensing

We highlight phenomenological aspects of Verlinde's recent proposal to account for the mass anomalies in galactic systems without dark matter -- in particular in their relation to MOND. Welcome addition to the MOND lore as it is, this approach have reproduced, so far, only a small fraction of MOND phenomenology, and is still rather tentative, both in its theoretical foundations and in its phenomenology. What Verlinde has extracted from this approach, so far, is a formula -- of rather limited applicability, and with no road to generalization in sight -- for the effective gravitational field of a spherical, isolated, static baryonic system. This formula cannot be used to calculate the gravitational field of disk galaxies, with their rich MOND phenomenology. Notably, it cannot predict their rotation curves, except asymptotically. It does not apply to the few-, or many-body problem; so, it cannot give, e.g., the two-body force between two galaxies, or be used to conduct N-body calculations of galaxy formation, evolution, and interactions. The formula cannot be applied to the internal dynamics of a system embedded in an external field, where MOND predicts important consequences. etc. MOND is backed by full-fledged, Lagrangian theories that can be, and are, routinely applied to all the above phenomena, and more. Verlinde's formula, as it now stands, strongly conflicts with solar-system and possibly earth-surface constraints, and cannot fully account for the mass anomalies in the cores of galaxy clusters (a standing conundrum in MOND). The recent weak-lensing test of the formula is, in fact, testing a cornerstone prediction of MOND, one that the formula does reproduce, and which has been tested before in the very same way.

astro-ph.GA↗

arXiv2016

The $Λ$CDM simulations of Keller and Wadsley do not account for the MOND mass-discrepancy-acceleration relation

Keller and Wadsley (2016) have smugly suggested, recently, that the end of MOND may be in view. This is based on their claim that their highly-restricted sample of $Λ$CDM-simulated galaxies are "consistent" with the observed MOND mass-discrepancy-acceleration relation (MDAR), in particular, with its recent update by McGaugh et al. (2016), based on the SPARC sample. From this they extrapolate to "$Λ$CDM is fully consistent" with the MDAR. I explain why these simulated galaxies do not show that $Λ$CDM accounts for the MDAR. a. Their sample of simulated galaxies contains only 18 high-mass galaxies, within a narrow range of one order of magnitude in baryonic mass, at the very high end of the observed, SPARC sample, which spans 4.5 orders of magnitude in mass. More importantly, the simulated sample has none of the low-mass, low-acceleration galaxies -- abundant in SPARC -- which encapsulate the crux and the nontrivial aspects of the predicted and observed MDAR. The low-acceleration part of the simulated MDAR is achieved, rather trivially, from the flattish-velocity-curve regions of the simulated high-mass galaxies. b. Half of the simulated galaxies have "wrong" rotation curves that differ greatly from any observed ones. This, does not prevent these wrong galaxies from lying on the observed MDAR (for trivial reasons, again). They, in fact, define the high-acceleration branch of the simulated MDAR. c. To boot, even if $Λ$CDM were made "consistent" with the MDAR through the elaborate adjustments that the simulations allow, this would not obviate MOND, which predicts much more than the MDAR.

astro-ph.GA↗

arXiv2016

MOND impact on and of the recently updated mass-discrepancy-acceleration relation

McGaugh et al. (2016) have used their extensive SPARC sample to update the well-known mass-discrepancy-acceleration relation (MDAR), which is one of the major predicted "MOND laws". This is not a newly discovered relation. Rather, it improves on the many previous studies of it, with more and better data. Like its precedents, it bears crucial ramifications for the observed dynamical anomalies in disc galaxies, and, in particular, on their resolution by the MOND paradigm. Their result, indeed, constitute a triumph for MOND. However, unlike previous analyses of the MDAR, McGaugh et al. have chosen to obfuscate the MOND roots of their analysis, and its connection with, and implications for, this paradigm. For example, the fitting formula they use, seemingly as a result of some unexplained inspiration, follows in its salient properties from the basic tenets of MOND, and has already been used in the past in several MOND analyses. No other possible origin for such a function is known. Given that this formula had already been shown to reproduce correctly the observed rotation curves from the baryon distribution (as a MOND effect), it must have been clear, a priory, that it should describe correctly the MDAR, which is but a summary of rotation curves. The present paper corrects these oversights -- bringing to light the deep connections with MOND, suppressed by McGaugh et al. It also gives due credit to previous works, and discusses some new, important, but less known, aspects of this MOND relation. (Substantially abridged.)

astro-ph.GA↗

arXiv2016

Universal MOND relation between the baryonic and `dynamical' central surface densities of disc galaxies

I derive a new MOND relation for pure-disc galaxies: The `dynamical' central surface density, $Σ^0_D$, deduced from the measured velocities, is a universal function of only the true, `baryonic' central surface density, $Σ^0_B$: $Σ^0_D=Σ_M \mathcal{S}(Σ^0_B/Σ_M)$, where $Σ_M\equiv a_0/2πG$ is the MOND surface density constant. This surprising result is shown to hold in both existing, nonrelativistic MOND theories. $\mathcal{S}(y)$ is derived: $\mathcal{S}(y)=\int_0^yν(y')dy'$, with $ν(y)$ the interpolating function of the theory. The relation aymptotes to $Σ^0_D=Σ^0_B$ for $Σ^0_B\ggΣ_M$, and to $Σ^0_D=(4Σ_MΣ^0_B)^{1/2}$ for $Σ^0_B\llΣ_M$. This study was prompted by the recent finding of a correlation between related attributes of disc galaxies by Lelli et al. (2016). The MOND central-surface-densities relation agrees very well with these results.

astro-ph.GA↗

arXiv2016

Scale Invariance at low accelerations (aka MOND) and the dynamical anomalies in the Universe

Galactic systems, and the Universe at large, exhibit large dynamical anomalies: The observed matter in them falls very short of providing enough gravity to account for their dynamics. The mainstream response to this conundrum is to invoke large quantities of `dark matter' -- which purportedly supplies the needed extra gravity -- and also of `dark energy', to account for further anomalies in cosmology, such as the observed, accelerated expansion. The MOND paradigm offers a different solution: a breakdown of standard dynamics (gravity and/or inertia) in the limit of low accelerations -- below some acceleration $a_0$. In this limit, dynamics become space-time scale invariant, and is controlled by a gravitational constant $\mathcal{A}_0\equiv Ga_0$, which replaces Newton's $G$. With the new dynamics, the various detailed manifestations of the anomalies in galaxies are predicted with no need for dark matter. The cosmological anomalies could, but need not have to do with small accelerations. For example, the need for dark matter in accounting for the expansion history of the Universe is eliminated if the relevant gravitational constant is $\approx 2πG$. Such a `renormalization' of $G$ could be a dimensionless parameter of a MOND theory. The constant $a_0$ turns out to carry cosmological connotations, in that $2πa_0\approx cH_0\approx c^2(Λ/3)^{1/2}$, where $H_0$ is the present expansion rate of the Universe, and $Λ$ the measured `cosmological constant'. There are MOND theories in which this `coincidence' is natural. I draw on enlightening historical and conceptual analogies from quantum theory to limelight aspects of MOND. I also explain how MOND may have strong connections with effects of the quantum vacuum on local dynamics.

astro-ph.GA↗

arXiv2016

The response of linear inhomogeneous systems to coupled fields: Bounds and perturbation expansions

We consider the response of a multicomponent body to $n$ fields, such as electric fields, magnetic fields, temperature gradients, concentration gradients, etc., where each component, which is possibly anisotropic, may cross couple the various fields with different fluxes, such as electrical currents, electrical displacement currents, magnetic induction fields, energy fluxes, particle fluxes, etc. We obtain the form of the perturbation expansions of the fields and response tensor in powers of matrices which measure the difference between each component tensor and a homogeneous reference tensor ${\bf L}_0$. For the case of a statistically homogeneous or periodic composite the expansion coefficients can be expressed in terms of positive semidefinite normalization matrices alternating with positive semidefinite weight matrices, which at each given level sum to the identity matrix. In an appropriate basis the projection operators onto the relevant subspaces can be expressed in block tridiagonal form, where the blocks are functions of these weight and normalization matrices. This leads to continued fraction expansions for the effective tensor, and by truncating the continued fraction at successive levels one obtains a nested sequence of bounds on the effective tensor incorporating successively more weight and normalization matrices. The weight matrices and normalization matrices can be calculated from the series expansions of the fields which solve the conductivity problem alone, without any couplings to other fields, and then they can be used to obtain the solution for the fields and effective tensor in coupled field problems in composites.

cond-mat.mtrl-sci↗

arXiv2016

Critical take on "Mass models of disk galaxies from the DiskMass Survey in MOND"

Angus et al. (2015) have recently faulted MOND as follows: Studying thirty disc galaxies from the DiskMass survey, they derive the profiles of velocity dispersion perpendicular to the discs as predicted by MOND, $σ_M(r)$, and compare them with $σ(r)$, measured as part of the DiskMass project. This is a new test of MOND, different from rotation-curve analysis. A nontrivial accomplishment of MOND -- not discussed by Angus et al. -- is that the shapes of $σ_M$ and $σ$ agree very well, i.e, $η(r)\equivσ_M(r)/σ(r)$ is well consistent with being $r$-independent (while $σ$ and $σ_M$ are strongly $r$ dependent). The fault found with MOND was that $η$ is systematically above 1 (with an average of about 1.3). I have suggested to Angus et al. that the fault may lie with the DiskMass dispersions, which may well be too low for the purpose at hand: Being based on population-integrated line profiles, they may be overweighed by younger populations, known to have much smaller dispersions, and scale heights, than the older populations, which weigh more heavily on the light distributions. I discuss independent evidence that supports this view, and show, besides, that if the DiskMas dispersions are underestimates by only 25 %, on average, the MOND predictions are in full agreement with the data, in shape and magnitude. Now, Aniyan et al. (2015) have questioned the DiskMass $σ$ on the same basis. They show for the solar column in the Milky Way that: " Combining the (single) measured velocity dispersion of the total young + old disc population... with the scale height estimated for the older population would underestimate the disc surface density by a factor of $\sim 2.$" If this mismatch found for the Milky Way is typical, correcting for it would bring the measured DiskMass $σ(r)$ to a remarkable agreement with the predicted MOND $σ_M(r)$.

astro-ph.GA↗

arXiv2015

Ultra-diffuse cluster galaxies as key to the MOND cluster conundrum

MOND reduces greatly the mass discrepancy in clusters of galaxies, but does leave a consistent global discrepancy of about a factor of two. It has been proposed, within the minimalist and purist MOND, that clusters harbor some indigenous, yet-undetected, cluster baryonic (dark) matter (CBDM). Its total amount has to be comparable with that of the observed hot gas. Following an initial discovery by van Dokkum & al. (2015a), Koda & al. (2015) have recently identified more than a thousand ultra-diffuse galaxy-like objects (UDGs) in the Coma cluster. Robustness of the UDGs to tidal disruption seems to require, within Newtonian dynamics, that they are much more massive than their observed stellar component. Here, I propound that a considerable fraction of the CBDM is internal to UDGs, which endows them with robustness. The rest of the CBDM objects formed in now-disrupted kin of the UDGs, and is dispersed in the intracluster medium. While the discovery of cluster UDGs is not in itself a resolution of the MOND cluster conundrum, it lends greater qualitative plausibility to CBDM as its resolution, for reasons I discuss. Alternatively, if the UDGs are only now falling into Coma, their large size and very low surface brightness could result from the adiabatic inflation due to the MOND external-field effect, as described in Brada & Milgrom (2000). I also consider briefly solutions to the conundrum that invoke more elaborate extensions of purist MOND, e.g., that in clusters, the MOND constant takes up larger-than-canonical values of the MOND constant.

astro-ph.GA↗

arXiv2015

The road to MOND--a novel perspective

Accepting that galactic mass discrepancies are due to modified dynamics, I show why it is specifically the MOND paradigm that is pointed to cogently. MOND is thus discussed here as a special case of a larger class of modified dynamics theories whereby galactic systems with large mass discrepancies are described by scale-invariant dynamics. This is a novel presentation that uses more recent, after-the-fact insights and data (largely predicted beforehand by MOND). Starting from a purist set of tenets, I follow the path that leads specifically to the MOND basic tenets. The main signposts are: (i) Space-time scale invariance underlies the dynamics of systems with large mass discrepancies. (ii) In these dynamics, $G$ must be replaced by a single "scale-invariant" gravitational constant, Q0 (in MOND, Q0=A0=Ga0, where a0 is MOND's acceleration constant). (iii) Universality of free fall points to the constant q0=Q0/G as the boundary between the G-controlled, standard dynamics, and the Q0-controlled, scale-invariant dynamics (in MOND, q0=a0). (iv) Data clinches the case for q0 being an acceleration (MOND).

astro-ph.GA↗

arXiv2015

Cosmological variation of the MOND constant: secular effects on galactic systems

The proximity of the MOND acceleration constant with cosmological accelerations -- for example, a0 cH0/2pi -- points to its possibly decreasing with cosmic time. I begin to consider the secular changes induced in galactic systems by such presumed variations, assumed adiabatic. It is important to understand these effects, in isolation from other evolutionary influences, in order to identify or constrain a0 variations by detection of induced effects, or lack thereof. I find that as long as the system is fully in the deep-MOND regime -- as applies to many galactic systems -- the adiabatic response of the system obeys simple scaling laws. For example, in a system that would be stationary for fixed a0, the system expands homologously as a0^{-1/4}, while internal velocities decrease uniformly as a0^{1/4}. If a0 is proportional to cH at all relevant times, this change amounts to a factor of ~ 2.5 since redshift 10. For a system stationary in a rotating frame, the angular frequency decreases as a0^{1/2}. The accelerations increase relative to a0 as a0^{-1/4}, pushing the system towards the Newtonian regime. All this follows from the appearance of a0 in MOND and the scale invariance of the deep-MOND limit -- two basic tenets of MOND. More complicated evolution ensues when parts of the system become Newtonian, or are so from inception. For example, these parts may become unstable, not being protected by MOND's stabilizing effects. The existence of such regions also modifies the MONDian regime, since they affect the potential everywhere, and since constituents might migrate between the Newtonian and MONDian regimes. Studying these last effects would require detailed numerical calculations.

astro-ph.GA↗

arXiv2014

MOND theory

A general account of MOND theory is given. I start with the basic tenets of MOND, which posit departure from standard dynamics in the limit of low acceleration -- below an acceleration constant a0 -- where dynamics become scale invariant. I list some of the salient predictions of these tenets. The special role of a0 and its significance are then discussed. In particular, I stress its coincidence with cosmologically relevant accelerations. The deep-MOND limit and the consequences of its scale invariance are considered in some detail. General aspects of MOND theories are then described, after which I list briefly presently known theories, both nonrelativistic and relativistic. Most full-fledged theories modify the gravitational action, hinge on a0, introduce an interpolating function between the low and high accelerations, and obey MOND requirements in the opposite limits. These theories have much heuristic value as proofs of various concepts (e.g., that covariant MOND theories can be written with correct gravitational lensing). But, probably, they are, at best, effective theories of limited applicability. I then outline several other promising approaches to constructing MOND theories that strive to obtain MOND as an effective theory from deeper concepts, for example, by modifying inertia and/or gravity as a result of interactions with some omnipresent agent. Some theories do enjoy a natural appearance of a cosmological-constant-like contribution that, furthermore, exhibits the observed connection with a0. However, none were shown to address fully the mass discrepancies in cosmology and structure formation that are otherwise explained by cosmological dark matter. We have no clues as to whether and how MOND aspects enter non-gravitational phenomena, but I discuss briefly some possibilities.

astro-ph.CO↗

arXiv2014

General virial theorem for modified-gravity MOND

An important and useful relation is known to hold in two specific MOND theories. It pertains to low-acceleration, isolated systems of pointlike masses, m_p, at positions r_p, subject to gravitational forces F_p. It reads sum_p r_p.F_p=-(2/3)(Ga0)^{1/2}[(\sum_p m_p)^{3/2}-\sum_p m_p^{3/2}]; a0 is the MOND acceleration constant. Here I show that this relation holds in the nonrelativistic limit of any modified-gravity MOND theory. It follows from only the basic tenets of MOND (as applied to such theories): departure from standard dynamics at accelerations below a0, and space-time scale invariance in the nonrelativistic, low-acceleration limit. This implies space-dilatation invariance of the static, gravitational-field equations, which, in turn, leads to the above point-mass virial relation. Thus, the various MOND predictions and tests based on this relation hold in any modified-gravity MOND theory. Since we do not know that any of the existing MOND theories point in the right direction, it is important to identify such predictions that hold in a much larger class of theories. Among these predictions are the MOND two-body force for arbitrary masses, and a general mass-velocity-dispersion relation of the form sigma^2=(2/3)(MGa0)^{1/2}[1-\sum_p (m_p/M)^{3/2}], where M is the total mass.

gr-qc↗

arXiv2014

Gravitational waves in bimetric MOND

I consider the weak field limit (WFL) of bimetric MOND (BIMOND): the lowest order in the small departures Hmn and H'mn from double Minkowski space-time. In particular, I look at propagating solutions, for a favorite subclass of BIMOND. The WFL splits into two sectors for two linear combinations, Hmn+ and Hmn-, of Hmn and H'mn. The Hmn+ sector is equivalent to the WFL of general relativity (GR), with its gauge freedom, and has the same vacuum gravitational waves. The Hmn- sector is fully nonlinear even for the weakest Hmn-, and inherits none of the coordinate gauge freedom. The equations of motion are scale invariant in the deep-MOND limit of purely gravitational systems. In these last two regards, the BIMOND WFL is greatly different from that of other bimetric theories studied to date. Despite the strong nonlinearity, an arbitrary pair of harmonic GR wave packets of Hmn and H'mn moving in the same direction, is a solution of the (vacuum) BIMOND WFL.

gr-qc↗