Relativistic gravitation theory for the MOND paradigm

Jacob D. Bekenstein

REVIEW

Relativistic gravitation theory for the MOND paradigm

Jacob D. Bekenstein

Astrophysics · Submitted 30 Mar 2004 · 67 references

Pith's verdict: TeVeS is the paper that made relativistic MOND viable, but the MOND limit rests on a hand-chosen free function, and Eqs. 36/50/51 do not close as printed.

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arXiv IDastro-ph/0403694 (version 6)

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CitationJacob D. Bekenstein. "Relativistic gravitation theory for the MOND paradigm." arXiv:astro-ph/0403694 (2004). https://pith.science/paper/PI7X33OZ

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Two things to know. First, this is the paper that made relativistic MOND a real research program, and it earns that. The structural move — a disformal physical metric built from a dynamical 4-vector rather than Sanders' a priori one — is what fixes the lensing failure that killed relativistic AQUAL and PCG. The PPN calculation is clean: beta = gamma = 1 (Eqs. 97-98), so the classical solar-system tests cannot separate it from GR. The lensing deflection (Eq. 109) reduces to an integral over the same potential that governs dynamics, which is the right answer. And the GR limit is shown twice, in cosmology and in quasistatic systems (Secs. III C, III D). The paper is also honest about what it does not do: Sec. VII E leaves the cosmological dark matter problem open, and Sec. VIII C admits the causality analysis freezes the other fields and covers only cosmological and quasistatic backgrounds. Those acknowledgements are appropriate and I do not treat them as new findings. Second, the soft spot is the free function. The MOND limit comes from mu ~ sqrt(y/3) at small y, which is a property of the hand-chosen y(mu) in Eq. 50. Sec. III E says outright there is no theory for F or y and that the choice is 'as an example.' That is honest, but it means the headline claim is conditional: a different F with the same asymptotics gives a different interpolating function mu-tilde(|grad Phi|/a0) in Eq. 64 and different low-acceleration phenomenology. The PPN and lensing results survive; the MOND sector does not. There is also a concrete algebraic issue. Eqs. 36, 50, 51 do not close as printed: at mu = 2, the left side of Eq. 36 with F from Eq. 51 gives 1, while y from Eq. 50 is 0. Same mismatch at mu = 0.5. Either Eq. 50 or Eq. 51 is misprinted, but if the printed F is not the integral of the printed y, the F-dependent estimates in Secs. V and VII need rechecking. This is checkable, not a matter of taste, and worth verifying against the published version. The causality section is more careful than the abstract suggests: the abstract's 'easy to rule out' is stronger than the body, which depends on phi > 0 over a restricted initial-condition range (Sec. VII). Who should read it: anyone working on modified gravity, MOND, or relativistic alternatives to dark matter. It is the reference point for the field. Recommendation: send it to a serious referee. The framework is sound and the checkable derivations hold; the free-function issue and the Eq. 36/50/51 mismatch are exactly what a referee should press on.

Referee report

Significance

This is a significant theoretical advance. Earlier relativistic MOND theories either failed to produce sufficient gravitational lensing (because the physical metric was conformal to the Einstein metric) or exhibited superluminal scalar waves. TeVeS addresses both by using a disformal relation based on a dynamical vector field and by adding a U-dependent term to the scalar kinetic term. The PPN result beta = gamma = 1 (Sec. V, Eqs. 97-98) and the lensing result (Sec. VI, Eq. 109) are important and checkable. The paper is transparent about its limitations: the free function F is chosen 'as an example' (Sec. III E), the cosmological dark matter problem is left open (Sec. VII E), and the preferred-frame PPN parameters are deferred to future work (Sec. V). The construction is elegant and the derivations that are checkable are largely sound.

Major objections (3)

[Sec. III E, Eqs. (50)-(51)] The printed F(mu) in Eq. (51) does not satisfy the defining relation Eq. (36) with the y(mu) of Eq. (50). At mu = 0.5, the printed F gives -mu F - (1/2) mu^2 F' approximately -2.1, whereas Eq. (50) gives y approximately 0.84. The correct integral appears to be F = (3/8) mu^2 - (3/2) mu + 3/4 + (3/2)/mu + (3/2) ln|1-mu|/mu^2 + C/mu^2. This should be corrected, and Fig. 2, the numerical bound in Eq. (80), and the Friedmann equation (119) should be rechecked against the corrected F. The qualitative claims (F < 0 for 0 < mu < 1, F > 0 for mu > 2) survive with the corrected F, but the numerical estimates may change.
[Sec. V, Eqs. (97)-(98)] The paper computes beta = gamma = 1 but does not compute the preferred-frame PPN parameters alpha_1, alpha_2, alpha_3 or the Nordtvedt parameter. For a theory with a dynamical vector field, these are generically nonzero and are tightly constrained by solar-system experiments (alpha_1 < 10^-4, alpha_2 < 10^-7). The abstract and Sec. I claim that TeVeS 'passes the usual solar system tests of GR,' but this is not supported until these parameters are computed or bounded. The authors acknowledge this in Sec. V ('Future work should look at those coefficients having to do with preferred frame effects'), but the claim should be softened or the calculation done.
[Sec. VIII, Eqs. (147), (167), (169)] The causality analysis concludes that tensor, vector, and scalar perturbations propagate subluminally 'provided phi > 0.' The paper argues that phi > 0 for a range of initial conditions (0.007 < phi_0 << 1, Sec. VII F), but this is not a general proof. The abstract's claim that 'for a range of initial conditions, this last result makes it easy to rule out superluminal propagation' is conditional. The paper should clarify the scope: is phi > 0 guaranteed for all physically reasonable solutions, or only for a subset? If the latter, the causality claim is weaker than it appears.

Minor objections (7)

[Sec. III E] The MOND limit and the identification of a_0 (Eq. 62) depend on the small-y behavior mu approximately sqrt(y/3), which is a property of the chosen y(mu). The paper is transparent that F is chosen 'as an example,' and argues that 'the precise form of mu-tilde makes little difference' (Sec. IV B). It would strengthen the paper to state explicitly the conditions on F under which the MOND limit holds, beyond the two asymptotics.
[Sec. IV C] The discussion of the Pioneer anomaly is speculative ('it is, however, unclear whether the correction in Eq. (69)... has anything to do with the Pioneer anomaly'). This should be softened or removed.
[Sec. III A] The choice of the antisymmetric kinetic term for the vector field is made 'not because of any desire for gauge symmetry... but because this choice precludes appearance of second derivatives of U_alpha in the energy-momentum tensor.' It would be useful to discuss whether this choice is unique or whether other kinetic terms could work.
[Sec. VI A] The derivation of the deflection angle (Eqs. 102-109) drops terms of O(v^2) and O(mu-tilde v^2). The estimates are reasonable, but the paper should state more explicitly the conditions under which the second integral in Eq. (108) is negligible.
[Sec. VII E] The cosmological dark matter problem is acknowledged but not resolved. The paper should state more clearly what range of cosmological observations TeVeS can and cannot address.
[Figs. 1 and 2] The axes and labels could be clearer. Fig. 2 should be rechecked against the corrected F (see major comment 1).
[Throughout] There are several typographical errors: 'indeces' should be 'indices' (Sec. III B), 'sfor' should be 'for' (Sec. II B), 'Nortvedt' should be 'Nordtvedt' (Sec. V), 'Astyrophys' should be 'Astrophys' (ref. [21]), 'noenegligible' should be 'non-negligible' (Sec. VII E), 'rV' should be 'V' (Sec. VIII A), 'd a-tilde/d a-tilde' should be 'd a-tilde/d t-tilde' (Sec. VII D), and 'aquadratic' should be 'a quadratic' (Sec. II C 1). The spelling 'TeVeS' and 'TeVe S' is used inconsistently and should be standardized.

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The paper

The paper claims

A relativistic gravity theory with tensor, vector and scalar fields reproduces MOND at low acceleration, yields the observed gravitational lensing without dark matter, and gives post-Newtonian parameters beta = gamma = 1.

The reading

MOND is a rule that says gravity departs from Newton's law when accelerations fall below a fixed tiny value, about one ten-billionth of a centimetre per second squared. It explains flat galaxy rotation curves without dark matter, but for two decades it was only a formula for accelerations, not a full theory: it could not say how light bends, and its first relativistic versions either predicted too little lensing or allowed signals to travel faster than light. Bekenstein's paper supplies the missing theory. Gravity is carried by three dynamical fields: the usual spacetime metric, a scalar field, and a 4-vector field that picks out a preferred time direction locally but is itself dynamical, not imposed by hand. The physical metric that matter and light actually feel is built from all three by a rule that stretches spacetime differently along and perpendicular to the vector. The scalar field's equation is engineered, through a freely chosen function, to reduce to the MOND rule at low acceleration and to Newtonian gravity at high acceleration. The paper then computes the standard post-Newtonian parameters and finds both equal to unity, matching general relativity and the solar-system measurements. It shows that the same potential that governs how stars move also governs how light bends, so a galaxy that MOND describes dynamically will lens light by the observed amount. Finally it examines flat cosmological models and finds the scalar field stays small and positive, which is what keeps all three kinds of waves from travelling faster than light.

Extended reading notes

Core claim

TeVeS, a theory with a metric, a dynamical scalar and a dynamical 4-vector, has a nonrelativistic limit that reproduces the MOND formula at low acceleration and Newtonian gravity at high acceleration, yields post-Newtonian parameters beta = gamma = 1 in agreement with solar-system tests, and predicts gravitational lensing by nonrelativistic systems governed by the same potential that governs particle dynamics, so that lensing agrees with observation to the same extent that MOND describes the dynamics (abstract s3, s6, s7; Sec. V Eqs. 97-98; Sec. VI Eq. 109).

Load-bearing premise

The entire MOND limit rests on the hand-chosen free function y(mu) in Eq. 50, picked so that mu approaches 1 at large argument (Newtonian limit) and mu is approximately sqrt(y/3) at small argument (MOND limit). Nothing derives this function; Sec. III E states there is no theory for F or y and that the choice is made 'as an example.' The MOND function in Eq. 64 and the identification of a0 in Eq. 62 follow from that choice, so a different F would give a different low-acceleration phenomenology while leaving the rest of the formalism intact.

Original abstract

The modified newtonian dynamics (MOND) paradigm of Milgrom can boast of a number of successful predictions regarding galactic dynamics; these are made without the assumption that dark matter plays a significant role. MOND requires gravitation to depart from Newtonian theory in the extragalactic regime where dynamical accelerations are small. So far relativistic gravitation theories proposed to underpin MOND have either clashed with the post-Newtonian tests of general relativity, or failed to provide significant gravitational lensing, or violated hallowed principles by exhibiting superluminal scalar waves or an \textit{a priori} vector field. We develop a relativistic MOND inspired theory which resolves these problems. In it gravitation is mediated by metric, a scalar field and a 4-vector field, all three dynamical. For a simple choice of its free function, the theory has a Newtonian limit for nonrelativistic dynamics with significant acceleration, but a MOND limit when accelerations are small. We calculate the $\beta$ and $\gamma$ PPN coefficients showing them to agree with solar system measurements. The gravitational light deflection by nonrelativistic systems is governed by the same potential responsible for dynamics of particles. To the extent that MOND successfully describes dynamics of a system, the new theory's predictions for lensing by that system's visible matter will agree as well with observations as general relativity's predictions made with a dynamically successful dark halo model. Cosmological models based on the theory are quite similar to those based on general relativity; they predict slow evolution of the scalar field. For a range of initial conditions, this last result makes it easy to rule out superluminal propagation of metric, scalar and vector waves.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 8,367 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.36 · human participants 0.01 · controlled comparison 0.18 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.94 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v6. Results and the main claim are underlined; show every role.

The modified newtonian dynamics (MOND) paradigm of Milgrom can boast of a number of successful predictions regarding galactic dynamics; these are made without the assumption that dark matter plays a significant role. MOND requires gravitation to depart from Newtonian theory in the extragalactic regime where dynamical accelerations are small. So far relativistic gravitation theories proposed to underpin MOND have either clashed with the post-Newtonian tests of general relativity, or failed to provide significant gravitational lensing, or violated hallowed principles by exhibiting superluminal scalar waves or an \textit{a priori} vector field. We develop a relativistic MOND inspired theory which resolves these problems. In it gravitation is mediated by metric, a scalar field and a 4-vector field, all three dynamical. For a simple choice of its free function, the theory has a Newtonian limit for nonrelativistic dynamics with significant acceleration, but a MOND limit when accelerations are small. We calculate the $\beta$ and $\gamma$ PPN coefficients showing them to agree with solar system measurements. The gravitational light deflection by nonrelativistic systems is governed by the same potential responsible for dynamics of particles. To the extent that MOND successfully describes dynamics of a system, the new theory's predictions for lensing by that system's visible matter will agree as well with observations as general relativity's predictions made with a dynamically successful dark halo model. Cosmological models based on the theory are quite similar to those based on general relativity; they predict slow evolution of the scalar field. For a range of initial conditions, this last result makes it easy to rule out superluminal propagation of metric, scalar and vector waves.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The modified newtonian dynamics (MOND) paradigm of Milgrom can boast of a number of successful predictions regarding galactic dynamics; these are made without the assumption that dark matter plays a significant role.	background 1.00	0.04	0.03	asserted 0.99
s1	MOND requires gravitation to depart from Newtonian theory in the extragalactic regime where dynamical accelerations are small.	background 1.00	0.05	0.03	asserted 1.00
s2	So far relativistic gravitation theories proposed to underpin MOND have either clashed with the post-Newtonian tests of general relativity, or failed to provide significant gravitational lensing, or violated hallowed principles by exhibiting superluminal scalar waves or an \textit{a priori} vector field.	gap 0.59	0.04	0.04	asserted 1.00
s3	We develop a relativistic MOND inspired theory which resolves these problems.	contribution 1.00	0.94	0.53	asserted 1.00
s4	In it gravitation is mediated by metric, a scalar field and a 4-vector field, all three dynamical.	contribution 0.51	0.78	0.10	asserted 1.00
s5	For a simple choice of its free function, the theory has a Newtonian limit for nonrelativistic dynamics with significant acceleration, but a MOND limit when accelerations are small.	contribution 0.51	0.74	0.18	asserted 0.98
s6	We calculate the $\beta$ and $\gamma$ PPN coefficients showing them to agree with solar system measurements.	result 0.93	0.96	0.09	asserted 1.00
s7	The gravitational light deflection by nonrelativistic systems is governed by the same potential responsible for dynamics of particles.	result 0.98	0.79	0.10	asserted 1.00
s8	To the extent that MOND successfully describes dynamics of a system, the new theory's predictions for lensing by that system's visible matter will agree as well with observations as general relativity's predictions made with a dynamically successful dark halo model.	result 0.57	0.54	0.21	asserted 0.72
s9	Cosmological models based on the theory are quite similar to those based on general relativity; they predict slow evolution of the scalar field.	result 0.95	0.73	0.06	asserted 0.99
s10	For a range of initial conditions, this last result makes it easy to rule out superluminal propagation of metric, scalar and vector waves.	result 0.60	0.78	0.06	asserted 0.79

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

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How later papers used it

253 later papers cite it

Evidence: substantial. 253 of the citing papers share no author with it, and 23 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 364 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 2 used its method or tool, 1 used its data or values and 1 compared its own results with it.

Uses its method or tool 2 of 3 papers with this label

Used this paper's approximation to solve the equations analytically.

Incompatibility of Rotation Curves with Gravitational Lensing for TeVeS (2009) · 0.92 · paraphrased: the licence does not allow quoting
Referred to Appendix C of this paper for a method to obtain the determinant of the disformal metric.

Disformal Scalar Fields and the Dark Sector of the Universe (2010) · 0.88 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 3 papers with this label

Took this paper's phenomenological value k of about 0.03.

The LTP Experiment on LISA Pathfinder: Operational Definition of TT Gauge in Space (2011) · 0.94 · paraphrased: the licence does not allow quoting

Compares with it 1 of 8 papers with this label

Compared its own interpolating function, fixed via a relation in its numerical calculations, with the mu-function definition in this paper.

Data Analysis Methods for Testing Alternative Theories of Gravity with LISA Pathfinder (2014) · 0.86 · paraphrased: the licence does not allow quoting

How the literature uses it

177 arXiv papers cite this in 262 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 123 sentences without a confident label

Contests it 3 papers

In its original formulation [ref] , TeVeS suffers from several theoretical and experimental difficulties [ref] , notably a serious instability [ref] .

Nonlocal metric formulations of MOND with sufficient lensing · Introduction · contest 0.77
Possible relativistic generalizations of MOND [ref] seem to be disfavored by LIGO/Virgo observations [ref] and are yet unsuccessful at explaining the structure of the Universe on large scales [ref] .

Two-body problem in theories with kinetic screening · Introduction · contest 0.83
Even the so far most elaborate theory of mond, known as Tensor-Vector-Scalar gravity [ref] has yet to explain the observations of dm made in the bullet cluster.

The impact of non-perturbative effects in dark matter production and detection · Characteristics and possible candidates · contest 0.78

Uses its method or tool 3 papers

In order solve the equations analytically, we have used the following approximation [ref] : e^2ϕ ≃e^2ϕ_c[1-kGm_s(<r)/2πr +k^2G^2m_s^2(<r)/8π^2r^2+O(r^-3)] .

Incompatibility of Rotation Curves with Gravitational Lensing for TeVeS · The TeVeS Model · 0.92
In order to solve the equations numerically, the following approximation [ref] is used [ref] :

Confronting MOND and TeVeS with strong gravitational lensing over galactic scales: an extended survey · Theory of MOND and TeVeS · 1.00
The determinant of the disformal metric is (for a method to obtain the result see Appendix C of Ref. [ref] ) √(-g̅) = A^2√(1+2B/AX)√(-g) .

Disformal Scalar Fields and the Dark Sector of the Universe · A variation of the model · 0.88

Uses its data or values 2 papers

The values of $K$ and $k$ are constrained [ref] from solar system tests on gravity to be $≲0.1$, and by rotation curves to be $≳0.001$.

Confronting MOND and TeVeS with strong gravitational lensing over galactic scales: an extended survey · Deflection Angle · 0.88
We see the correction falls off as $1/r^2$ moving away from the SP and by taking the phenomenological value $k≃0.03$ [ref] in the nearby of the bubble is amounts to order $0.25%$.

The LTP Experiment on LISA Pathfinder: Operational Definition of TT Gauge in Space · MOND · 0.94

Compares with it 5 papers

Unlike in TeVeS [ref] , there is no need in the present theory to invoke a disformal coupling of the photon field to gravity in order to get the light deflection viable.

Modified gravity approach based on a preferred time foliation · Non-relativistic limit · 0.88
While in eq.([ref]) holds, the corresponding equation in MOND according to [ref] is

A Modified Mass Concept could explain both the Dark Matter and the Dark Energy Phenomenon · vs. MOND · 1.00
This is a clear advantage of the GVT theory over the Bekenstein's Tensor-Vector-Scaler theory [ref] .

GravitoMagnetic Force in Modified Newtonian Dynamics · Conclusion and outlook · 0.95

Cites it as support 2 papers

Further on, TeVeS has been shown to be free of acausal propagation of perturbations and it is in agreement with solar system tests [ref] .

Can TeVeS be a viable theory of gravity? · Introduction · 0.98
The desiderata and guiding principles that [ref] lists are either directly imported from GR (as in the case of the Covariance Principle and the weak equivalence principle), or generic demands on garden-variety field theories (as in the case of causality constraints).

Methodological Reflections on the MOND/Dark Matter Debate · Kuhn—Against Facile Invocations of Paradigms · 0.98

Cites it as background 110 papers

We consider the Tensor-Vector-Scalar theory of gravity [ref] (TeVeS) as an alternative to GR.

Can TeVeS be a viable theory of gravity? · Introduction · 0.98
TeVeS [ref] is a bi-metric model in which matter and radiation do not feel the Einstein metric, $g_αβ$, that appears in the canonical kinetic term of the (effective) action, but a modified “physical” metric, $g̃_αβ$, related to the Einstein metric by g̃_αβ = e^-2ϕg_αβ-U_αU_β(e^2ϕ-e^-2ϕ) , where $U_μ, ϕ$ denote the TeVeS vector and scalar field, respectively.

Incompatibility of Rotation Curves with Gravitational Lensing for TeVeS · The TeVeS Model · 0.97
A generally covariant realization of the AQUAL model is the TeVeS theory [ref] .

GravitoMagnetic Field in Tensor-Vector-Scalar Theory · Introduction · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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