Noncovariance at low accelerations as a route to MOND

Mordehai Milgrom

REVIEW 3 major objections 4 minor 1 cited by

Noncovariance at low accelerations as a route to MOND

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Dropping general covariance at low accelerations yields a pure-metric theory with an AQUAL MOND limit and a GR-plus-cosmological-constant high-acceleration limit.

desk verdict A clear, honestly-scoped existence proof that dropping general covariance can give metric-only MOND in the nonrelativistic limit; the matter sector and Cauchy/ghost checks remain open, exactly as the author says. read the letter →

arxiv 1908.01691 v2 pith:LHVJQ3NS submitted 2019-08-05 gr-qc astro-ph.COastro-ph.GAhep-ph

Mordehai Milgrom This is my paper · ORCID

classification gr-qcastro-ph.COastro-ph.GAhep-ph

keywords MONDmodifiedgravitygeneralcovarianceAQUALbimetricf(Q)preferredinertialframenonrelativisticlimit

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that general covariance—coordinate invariance—may fail below the MOND acceleration $a_0$, and that relaxing it opens a simpler route to relativistic MOND (modified Newtonian dynamics). As a concrete example it constructs a metric-only theory with Lagrangian $\mathcal{L}_M=2\ell_M^{-2}F(\ell_M^2\mathcal{R})$, where $\mathcal{R}$ is the quadratic first-derivative part of the Ricci scalar and $\ell_M=c^2/a_0$. In the static weak-field limit the metric keeps the general-relativity (GR) form, but the potential $\phi$ obeys the AQUAL (aquadratic-Lagrangian) nonlinear Poisson equation, while at high accelerations the same action returns to general relativity with a cosmological constant of order $a_0^2$. If the construction holds, a single metric, with no extra fields and ordinary lensing, can produce MOND phenomenology by giving up one principle rather than adding new degrees of freedom.

What carries the argument

The load-bearing object is $\mathcal{R}$, the quadratic, first-derivative part of the Ricci scalar, built entirely from the Levi-Civita connection of the metric: $\mathcal{R}=\tfrac{1}{2}g^{\mu\nu}(\Gamma^\gamma_{\mu\nu}\Gamma^\lambda_{\lambda\gamma}-\Gamma^\gamma_{\mu\lambda}\Gamma^\lambda_{\nu\gamma})$. Since $\mathcal{R}$ is not a coordinate scalar, the action is not generally covariant, yet it contains only first derivatives and remains Lorentz invariant, matching the idea of an absolute inertial frame supplied by the quantum vacuum. The special index structure of $\mathcal{R}$ is what makes $\bar S^i_{jk}$ independent of $\varphi$ in the weak-field limit, permitting $h_{ij}=0$ so the metric reduces to the single-potential GR form. The interplay between $\mathcal{R}$ and the two asymptotic behaviors of $F$ does the work: $F(z)\to z+\zeta$ restores GR plus a cosmological constant, and $F'(z)\propto z^{1/2}$ produces deep-MOND scale invariance. The constrained-BIMOND and $f(\mathcal{Q})$ formulations are the same mechanism viewed covariantly, with the extra frame-coordinate fields acting as Stückelberg fields.

What would settle it

Measure the relative acceleration of a wide stellar binary in the deep-MOND regime: equation (40) predicts accelerations of order $(G M a_0)^{1/2}/r$ rather than $G M/r^2$; observing the Newtonian inverse-square law in such a binary would falsify the theory's nonrelativistic core.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that noncovariance is not a liability in the MOND regime but a resource: it lets a pure-metric, local action contain first derivatives of the metric and therefore an acceleration scale. The proposed Lagrangian $\mathcal{L}_M=2\ell_M^{-2}F(\ell_M^2\mathcal{R})$ uses the nonscalar $\mathcal{R}=\tfrac{1}{2}g^{\mu\nu}(\Gamma^\gamma_{\mu\nu}\Gamma^\lambda_{\lambda\gamma}-\Gamma^\gamma_{\mu\lambda}\Gamma^\lambda_{\nu\gamma})$, the quadratic first-derivative part of the Ricci scalar. Requiring $F(z)\to z+\zeta$ for $z\gg1$ forces the theory into Einstein–Hilbert form with a cosmological constant at high accelerations; requiring $F'(z)\propto z^{1/2}$ for $z\ll1$ gives the deep-MOND scale-invariant regime. In the nonrelativistic, weak-field limit the metric is $g_{\mu\nu}=\eta_{\mu\nu}-2\phi\,\delta_{\mu\nu}$, with $\phi$ solving $\vec\nabla\cdot[F'(|\vec\nabla\phi|^2/a_0^2)\vec\nabla\phi]=4\pi G\rho$, the AQUAL MOND equation; because the metric has the GR form, lensing uses the same MOND potential. The paper also shows this action is the fixed-gauge form of BIMOND with a flat auxiliary metric and a special case of $f(\mathcal{Q})$ theories, so both provide covariantized, Stückelberg-type versions of the same idea.

Load-bearing premise

The construction stands on the assumption that a well-defined preferred inertial frame exists—likely supplied by the quantum vacuum—in which the noncovariant connection terms in the action are evaluated; if no such frame exists, or if the matter action also needs noncovariant modification, the explicit AQUAL limit and the lensing conclusion do not follow.

Editorial extensions

If this is right

In the static weak-field limit, a nonrelativistic system obeys a single nonlinear Poisson equation $\vec\nabla\cdot[F'(|\vec\nabla\phi|^2/a_0^2)\vec\nabla\phi]=4\pi G\rho$, so Newtonian behavior emerges only where $|\vec\nabla\phi|\gg a_0$.
The metric $g_{\mu\nu}=\eta_{\mu\nu}-2\phi\,\delta_{\mu\nu}$ has the same form as in GR, so light deflection and gravitational lensing are computed with the MOND potential, not with an extra field.
High accelerations automatically return to general relativity plus a cosmological constant of order $a_0^2$, linking the dark-energy scale to the MOND acceleration.
The action admits covariant completions: constrained BIMOND with a flat auxiliary metric, and $f(\mathcal{Q})$ theories; in both, the lost coordinate freedom reappears as Stückelberg fields.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not claimed in the paper, but if covariance is only approximate and restored above $a_0$, then the decisive empirical arena shifts to low-acceleration systems; Solar-System tests at high accelerations cannot constrain the theory, while wide binaries and dwarf satellites can.
Not claimed in the paper, but a natural next step would be to derive $a_0$ from vacuum physics: if the quantum vacuum supplies the preferred frame, the near equality $a_0\sim c^2/\ell_\Lambda$ becomes a calculational target rather than an input.
Not claimed in the paper, but if matter actions must also become noncovariant in the same regime, the exact AQUAL equation and the strength of equivalence-principle violations would change; measuring such violations in low-acceleration systems could distinguish this route from covariant MOND theories.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper is an existence proof with the limitations printed in the text. What is actually new is the specific noncovariant action L_M = 2ℓ_M^{-2}F(ℓ_M^2 R), where R is the quadratic connection part of the Ricci scalar, and the demonstration that its static weak-field limit is AQUAL: for g = η − 2φδ, the potential solves ∇·[F'(|∇φ|²/a0²)∇φ] = 4πGρ, while high accelerations return GR with a cosmological constant. The connection to constrained BIMOND with a flat auxiliary metric, and to f(Q) gravity, is worked out explicitly and is the part I found most useful; it places the construction in a familiar landscape.

The weak-field and nonrelativistic derivations (Secs. III D and III E) are coherent. The steps that set q=0 and h_ij=0 follow from the stated boundary conditions and the specific quadratic argument, not from hidden assumptions. The author also gives fair credit to earlier work and lists the obvious gaps: matter action noncovariance, Cauchy problem, stability/ghosts, and gravitational-wave constraints.

The soft spots are real but not concealed. The stress-test note is on target: equation (40) is derived from the gravitational action while the matter action is kept covariant, and once diffeomorphism invariance is broken the standard conservation law is not automatic. The paper openly says this at the start of Sec. III and again in Sec. III F. So this is not a hidden flaw; it is an acknowledged limitation of a deliberately simplified demonstration. The second soft spot is that the deep-MOND behavior F'(z) ∝ z^{1/2} is imposed so that the NR limit reproduces AQUAL; the MOND content is an input encoded in F, not derived. That is fine for an existence proof, but it means the theory does not predict the low-acceleration regime, it encodes it.

Who is this for? People thinking about relativistic MOND and about whether covariance is emergent, and people working on modified gravity with preferred frames. Not for phenomenology. It deserves a serious referee because the construction is explicit, the mapping to BIMOND/f(Q) is useful, and the author has not oversold it. I would send it to review, with the expectation that the referee pushes on the matter sector and well-posedness rather than on the existence-proof core.

Referee Report

3 major / 4 minor

Summary. The paper proposes that general covariance, like other principles of relativistic dynamics, may break down at accelerations below MOND's a0, and that this breakdown is a useful route for constructing MOND theories. The central construction is a pure-metric, local, but noncovariant gravitational Lagrangian L_M = 2 ℓ_M^{-2} F(ℓ_M^2 R), where R is the quadratic, first-derivative part of the Ricci scalar built from the Levi-Civita connection. With F(z)→z+ζ for z≫1 the theory reduces to GR with a cosmological constant, and with F'(z)∝z^{1/2} for z≪1 the nonrelativistic weak-field limit yields the AQUAL equation ∇·[F'(|∇φ|²/a0²)∇φ]=4πGρ for a static source. The author shows that the resulting metric has the same form as in GR, g_{μν}≈η_{μν}−2φδ_{μν}, implying standard gravitational lensing with the MOND potential. The theory is then identified as a fixed-gauge version of BIMOND with the auxiliary metric constrained to be flat, and also as a special case of f(Q) theories, providing a Stückelberg-style covariantization.

Significance. If the construction is accepted as an existence proof, the paper makes a useful conceptual point: relaxing general covariance removes one of the obstacles that forces relativistic MOND theories to introduce extra gravitational degrees of freedom or nonlocality. The explicit nonrelativistic derivation leading to Eq. (40), the clean identification of the metric form (41), and the equivalence mappings to constrained BIMOND and f(Q) are valuable and clearly presented. The paper is also honest about its limitations: the deep-MOND behavior is encoded in F rather than derived, and the author lists open issues including the matter action, Cauchy problem, ghosts, and gravitational-wave propagation. Because the low-acceleration content is an input and the coupled matter-gravity system is not closed, the paper should be read as a heuristic existence argument rather than a complete theory; with that framing, its significance is moderate but genuine.

major comments (3)

[Sec. III (discussion before Eq. 7) and Sec. III F] The derivation of Eq. (40) uses the energy-momentum tensor defined by the standard covariant matter action (3), while the gravitational action (8) is not diffeomorphism invariant. Once general covariance is broken, the full action no longer has the symmetry that guarantees ∇_ν T^{μν}=0, and the field equations (24) need not be mutually consistent for the matter sources used in Sec. III E. The paper itself states that ignoring noncovariance of matter actions 'might lead to inconsistencies' and lists 'questions related to modifying the matter actions' as open in Sec. III F. This is a load-bearing gap: Eq. (40) is a gravitational-sector equation for a prescribed static density, not a closed coupled system that can be said to reproduce MOND. The claims about the theory and its lensing behavior should be explicitly restricted to the formal gravitational-sector result, or the matter sector must be specified.
[Sec. III E, Eqs. (11) and (40)] The deep-MOND asymptotic F'(z) ∝ z^{1/2} and the normalization α=2/3 in Eq. (11) are imposed precisely so that the nonrelativistic equation (40) reduces to the standard AQUAL equation. The low-acceleration content of the theory is therefore an input encoded in F, not a prediction that follows from relaxing general covariance. This is legitimate for an existence proof, but the abstract's phrasing, which emphasizes that the metric 'produces gravitational lensing as in GR only with the MOND potential,' should be accompanied by a clear statement that this is a consistency check of the construction rather than an independent derivation of MOND phenomenology.
[Sec. III E, Eqs. (39)–(41)] The conclusion that the nonrelativistic metric has the GR form (41) with h_ij=0 rests on showing that h_ij=0 is a solution of Eq. (39) under the stated boundary conditions. The paper does not address uniqueness or stability of this branch. If other static solutions with h_ij≠0 exist for the same source, the actual metric need not have the form (41), and the lensing conclusion 'light and massive, slow test bodies see the same potential' would not follow. The branch selection should be justified, or the claim should be softened to say that (41) is one consistent solution branch.

minor comments (4)

[Throughout] There are several typographical slips in the running text, such as 'ge neral', 't he', and 'hard to palate'; these should be corrected in a final version.
[Sec. III C, Eqs. (20)–(24)] The notation in the field equations is occasionally informal: F' is not a scalar and S^λ_{μν} is not a tensor, yet covariant derivatives of F'S^λ_{μν} are written without a detailed specification. The convention should be stated explicitly to avoid ambiguity.
[Sec. III E, Eq. (33)] In Eq. (33), the free indices μ,ν are not explicitly tied to the spatial divergence index i; the reader must infer that the equation holds for each component of ar S^i_{μν}. A short clarification would improve readability.
[Abstract and Introduction] The phrase 'MOND has limelighted' is nonstandard; consider replacing with 'brought into focus' or 'highlighted.' Also, the PACS numbers line appears empty and should either be populated or removed.

Circularity Check

1 steps flagged · score 6.0 of 10

The nonrelativistic MOND equation is encoded in the free function F rather than independently derived; the metric/lensing and BIMOND/f(Q) mappings are not circular.

self definitional [Sec. III, Eq. (11); Sec. III E, Eq. (40)]
"In the opposite limit, defined by ℓ_M → 0 with G/ℓ_M fixed, scale invariance of the nonrelativistic, deep-MOND limit dictates (see Sec. III E) F (z) → F (0) + α z^{3/2} for z → 0. (11)... ⃗∇ · [F ′(|⃗∇φ|^2/a_0^2)⃗∇φ] = 4πGρ. (40) This is the nonlinear Poisson MOND theory of Ref. [14] (dubbed AQUAL – for aquadratic Lagrangian)."

Because z = R/a_0^2 reduces to (⃗∇φ)^2/a_0^2 in the static weak-field limit, the assumed deep-MOND asymptotic F′(z) ∝ z^{1/2}, when inserted into Eq. (40), immediately yields the deep-MOND AQUAL equation. The paper even fixes the normalization of R precisely so that the argument of F becomes (⃗∇φ/a_0)^2. Thus the advertised nonrelativistic MOND equation is not an independent output of a general Lagrangian; it is the defining behavior of F chosen in Eq. (7). The construction is candidly a proof of concept, but the central MOND content is an input encoded in F rather than a derived prediction.

full rationale

The formal derivation is mostly self-contained: the weak-field reduction, the identification of the metric form g_μν ≈ η_μν − 2φδ_μν, the lensing statement, and the mathematical equivalences with constrained BIMOND and f(Q) theories follow from the definitions and are not circular. The circularity is concentrated in the MOND limit: Eq. (11) stipulates the deep-MOND behavior of F, and the normalization of R is chosen so that the nonrelativistic argument becomes (⃗∇φ/a_0)^2; Eq. (40) then simply restates that choice as the AQUAL equation. The use of the author's earlier BIMOND field equation (Ref. [17]) is a normal citation of a prior variational derivation and is not, by itself, load-bearing circularity. The paper's open caveat about modifying matter actions (Sec. III F: 'questions related to modifying the matter actions remain open') is a completeness limitation rather than a circular step.

Assumptions & free parameters 3 free parameters · 6 assumptions · 3 invented entities

The central construction rests on MOND inputs (a0 and an arbitrary interpolation function F), on the hypothesis that general covariance is emergent, and on a preferred inertial frame. The deep-MOND limit is imposed by the asymptotic form of F, while the equivalences with BIMOND and f(Q) are derived. These are the main unpaid inputs in the argument.

free parameters (3)

MOND acceleration a0 = 1.2e-8 cm/s^2
Determined from galactic rotation phenomenology; enters the MOND length ℓ_M = c^2/a0 and sets the scale of R/a0^2 in the action.
Interpolation function F(z) = Not fully specified; F(z) -> z + ζ for z >> 1 and F'(z) ∝ z^(1/2) for z << 1, with α = 2/3 for the standard a0…
Free function governing the transition between deep-MOND and GR limits; its asymptotic forms are chosen by hand to reproduce known MOND behavior.
Constant ζ (or F(∞)) = Order 1, not specified
Constant part of F at large z produces the cosmological constant Λ = -a0^2 F(∞); its order is chosen to match the observed acceleration scale.

assumptions (6)

standard math Schrodinger identity R = 2R + divergence and the standard Einstein-Hilbert action form
Used in Sec. III, Eqs. (4)-(6), to replace R by 2R and motivate the noncovariant Lagrangian (7).
domain assumption MOND phenomenology, with a0 taken from galaxy data, is the correct low-acceleration description
The paper assumes the MOND paradigm rather than dark matter as the explanation for galactic dynamics; this motivates the entire construction.
ad hoc to paper General covariance is emergent and may be broken at accelerations below a0
The central hypothesis of the paper; motivations are offered in Secs. I and II, but no derivation from a deeper theory is given.
ad hoc to paper A preferred inertial frame exists for evaluating the noncovariant connection terms
The field equations are stated to hold in the "supposed preferred inertial frame" (Sec. III C), and the action requires choosing connections in that frame.
ad hoc to paper The matter action remains the standard minimally coupled action while the gravity action is noncovariant
Action (8) modifies only the gravity sector; the author notes this may be inconsistent and that matter-action modifications remain an open question.
domain assumption Asymptotic flatness and gauge choices select q=0 and h_ij=0 solutions in the nonrelativistic limit
In Sec. III E, the GR-like metric form g = η - 2φδ follows from boundary conditions and the specific quadratic argument R; other F choices can produce extra potentials.

invented entities (3)

Preferred inertial frame at low accelerations, possibly defined by the quantum vacuum
purpose: Makes absolute accelerations meaningful and gives physical content to the noncovariant connection-dependent Lagrangian
The paper draws on prior Unruh and de Sitter vacuum arguments from Ref. [9], but provides no new falsifiable handle; the existence of such a frame is speculative.
Flat constrained auxiliary metric in BIMOND and f(Q)
purpose: Serves as the gauge-fixing device that makes the noncovariant theory a fixed-gauge expression of BIMOND and f(Q)
A mathematical auxiliary field with no independent physical evidence; it is introduced to establish the equivalence.
Stueckelberg fields x^μ(ξ) added in the covariantized formulation
purpose: Restore general covariance a la Stueckelberg to the fixed-gauge noncovariant theory, adding four gravitational degrees of freedom
Auxiliary fields introduced for covariance; their physical status is not established and the paper does not give them an independent signature.

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Pith. "Pith review of Noncovariance at low accelerations as a route to MOND." pith.science (2026). https://pith.science/paper/LHVJQ3NS

@misc{pith2026190801691,
  author       = {Pith},
  title        = {Pith review of: Noncovariance at low accelerations as a route to MOND},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHVJQ3NS}},
  note         = {Machine review of arXiv:1908.01691}
}

abstract

MOND has limelighted the fact that Newtonian dynamics (ND) and general relativity (GR) have not been verified at accelerations below MOND's $a_0$. In particular, we do not know that all the principles underlying ND or GR apply below $a_0$. I discuss possible breakdown of general covariance (GC) in this limit. This resonates well with MOND, which hinges on accelerations. Relaxing GC affords more freedom in constructing MOND theories. I exemplify this with a simplified theory whose gravitational Lagrangian is $\mathcal{L}_M\propto \ell_M^{-2}\mathcal{F}(\ell_M^{2}\mathcal{R})$, where $\mathcal{R}= g^{\mu\nu} (\Gamma^\gamma_{\mu\nu}\Gamma^\lambda_{\lambda\gamma}-\Gamma^\gamma_{\mu\lambda} \Gamma^\lambda_{\nu\gamma})/2$. $\Gamma^\gamma_{\mu\nu}$ is the Levi-Civita connection of a metric, $g_{\mu\nu}$, and $\ell_M=c^2/a_0$ is the MOND length. Requiring $\mathcal{F}(z)\rightarrow z+\zeta$, for $z\gg 1$ gives GR with a cosmological constant $\zeta c^{-4}a_0^2$ for high accelerations. In the MOND limit $\mathcal{F}'(z\ll 1)\propto z^{1/2}$. In the nonrelativistic limit the metric is of the form $g_{\mu\nu}\approx \eta_{\mu\nu}-2\phi\delta_{\mu\nu}$, as in GR, but the potential $\phi$ solves a MOND, nonlinear Poisson analog. This form of $g_{\mu\nu}$ also produces gravitational lensing as in GR only with the MOND potential. I show that this theory is a fixed-gauge expression of BIMOND, with the auxiliary metric constrained to be flat. The latter theory is thus a covariantized version of the former a-la St\"{u}ckelberg. This theory is also a special case of so-called $f(\mathcal{Q})$ theories -- aquadratic generalizations of `symmetric, teleparallel GR', which are, in turn, also equivalent to constrained BIMOND-type theories. (Abridged.)

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In the MOND limit F ′(z ≪ 1) ∝ z1/ 2. In the nonrelativistic limit the metric has a solution of the fo rm gµν ≈ ηµν − 2φδµν , as in GR, but the potential φ solves a MOND, nonlinear Poisson analog. This form of gµν also produces gravitational lensing as in GR, only with the MOND potential. I show that thi s theory is a ﬁxed-gauge expression of bimetric MON...

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¯S i µν ],i = 8πGρδµ 0δν 0. (33) In our approximation, 18 Γ 0 00 = 0, Γ i 00 = Γ 0 0i = Γ 0 i0 = − 1 2g00,i =φ,i, Γ i 0j = (qi,j −qj,i )/2, Γ 0 ij = −(qi,j +qj,i )/2, Γ i jk = 1 2 (gij,k +gik,j −gjk,i ) = 1 2 (hij,k +hik,j −hjk,i ) +φ,iδjk −φ,jδik −φ,kδij. (34) ¯S i 00 = 2φ,i + 1 2 (h j i,j − ¯h,i ), ¯S i 0j = 1 2 (qi,j −qj,i ) ¯S i jk = 1 2 (hij,k +hik,j...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

symmetric, tel eparallel GR

In the MOND limit F ′(z ≪ 1) ∝ z1/ 2. In the nonrelativistic limit the metric has a solution of the fo rm gµν ≈ ηµν − 2φδµν , as in GR, but the potential φ solves a MOND, nonlinear Poisson analog. This form of gµν also produces gravitational lensing as in GR, only with the MOND potential. I show that thi s theory is a ﬁxed-gauge expression of bimetric MON...

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[2] [2]

Famaey and S

B. Famaey and S. McGaugh, Living Rev. Relativity 15, 10 (2 012)

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[3] [3]

contraction factors

d4x +IM (gµν ,ψ i). (8) According to the basic tenets of MOND, general relativity (possibly with a cosmological constant) should be restored in the limit ℓM → ∞ (a0 → 0). We thus require F (z) → F (∞) +z for z → ∞, (9) where F (∞) is a dimensionless constant. So, in this limit I → − 1 16πG ∫ g 1/ 2 [2R + 2a 2 0F (∞)] d4x +IM (gµν ,ψ i). (10) This is the a...

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[4] [4]

The covariant derivative of F ′S λ µν is understood as the standard expression in terms of the derivatives and the connection

d4x + 1 2 ∫ hµν T µνd4x, (27) 16 Note that F ′ is not a scalar, and S λ µν is not a tensor. The covariant derivative of F ′S λ µν is understood as the standard expression in terms of the derivatives and the connection. 17 Despite the appearance of h2 in them, ( hµν ,α /a0)2 are of zeroth order. The WFL corresponds to hµν ≪ 1, but hµν ,α /a0 are not necess...

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[5] [5]

(21), with Γ α βγ ≈ 1 2η ασ (hβσ,γ +hγσ,β −hβγ,σ )

¯S λ µν ],λ = 8πGTµν, (29) where ¯S λ µν is the WFL of the expression in eq. (21), with Γ α βγ ≈ 1 2η ασ (hβσ,γ +hγσ,β −hβγ,σ ). (30) Thus, Γ ν =h,ν /2, Γ µ =η µν Γ ν, Γ ∗ µ =h µα ,α − (1/2)h, µ , h =h α α. (31) ¯S λ µν,λ = − ¯Gµν (hαβ ) is (minus) the WFL of the Einstein tensor; it satisﬁes the Bianchi ide ntity ¯S λ µν,λ, ν = 0. E. Nonrelativistic limit...

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[6] [6]

¯S i µν ],i = 8πGρδµ 0δν 0. (33) In our approximation, 18 Γ 0 00 = 0, Γ i 00 = Γ 0 0i = Γ 0 i0 = − 1 2g00,i =φ,i, Γ i 0j = (qi,j −qj,i )/2, Γ 0 ij = −(qi,j +qj,i )/2, Γ i jk = 1 2 (gij,k +gik,j −gjk,i ) = 1 2 (hij,k +hik,j −hjk,i ) +φ,iδjk −φ,jδik −φ,kδij. (34) ¯S i 00 = 2φ,i + 1 2 (h j i,j − ¯h,i ), ¯S i 0j = 1 2 (qi,j −qj,i ) ¯S i jk = 1 2 (hij,k +hik,j...

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[7] [7]

modiﬁed gravity

discusses at length when and how such an eﬀect follows from the b asic assumptions of MOND. But the fact is that this eﬀect is present in all MOND theories proposed to date. Some observations pointing to the EFE in action in galactic systems ar e described, e.g., in Refs. [48–53] One dramatic manifestation of the eﬀect is that an external accele ration ≫a...

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[8] [8]

(39) As was the case in BIMOND, the choice of the speciﬁc quadratic argu ment R results in ¯S i jk depending (linearly) only on hij, and not on φ, as shown in eq

¯S i jk],i = 0. (39) As was the case in BIMOND, the choice of the speciﬁc quadratic argu ment R results in ¯S i jk depending (linearly) only on hij, and not on φ, as shown in eq. (35). This means that hij = 0 is a solution if we impose that hij → 0 fast enough at inﬁnity. This is not the case for general choices of the qu adratic argument of F , since the...

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