We first review the nonrelativistic lagrangian theory as a framework for the
MOND equation. Obstructions to a relativistic version of it are discussed
leading up to TeVeS, a relativistic tensor-vector-scalar field theory which
displays both MOND and Newtonian limits. The whys for its particular structure
are discussed and its achievements so far are summarized.Comment: 6 pages, LaTeX, 1 figure, to appear in proceedings of IAP05 in Paris:
  Mass Profiles and Shapes of Cosmological Structures, G. Mamon, F. Combes, C.
  Deffayet and B. Fort (eds), (EDP-Sciences 2005

Bekenstein, Jacob D.

Sanders, Robert H.

English

arXiv

We first review the nonrelativistic Lagrangian theory as a framework for the MOND equation. Obstructions to a relativistic version of it are discussed leading up to TeVeS, a relativistic tensor-vector-scalar field theory which displays both MOND and Newtonian limits. The whys for its particular structure are discussed and its achievements so far are summarized

J.D. Bekenstein

R.H. Sanders

EDP Sciences OAI-PMH repository (1.2.0)

A Primer to Relativistic MOND Theory

Bekenstein, J.D..

Sanders, R.H.

University of Groningen Research Database

  
 University of Groningen
A Primer to Relativistic MOND Theory
Bekenstein, J.D..; Sanders, Robert
Published in:
Physical Review D
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
2005
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Bekenstein, J. D., & Sanders, R. H. (2005). A Primer to Relativistic MOND Theory. Physical Review D,
71(6).
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Mass Profiles and Shapes of Cosmological Structures
G. Mamon, F. Combes, C. Deffayet, B. Fort (eds)
EAS Publications Series, Vol. ?, 2005
A PRIMER TO RELATIVISTIC MOND THEORY
Jacob D. Bekenstein1 and Robert H. Sanders2
Abstract. We first review the nonrelativistic lagrangian theory as a
framework for the MOND equation. Obstructions to a relativistic ver-
sion of it are discussed leading up to TeVeS, a relativistic tensor-vector-
scalar field theory which displays both MOND and Newtonian limits.
The whys for its particular structure are discussed and its achievements
so far are summarized.
1 Introduction
The success of Milgrom’s MOND paradigm (Milgrom 1983) in explaining the
shapes of disk galaxy rotation curves, and correlations between the parameters of
such systems (e.g. Tully-Fisher law) with no appeal to dark haloes is by now well
known—for a modern review see Sanders & McGaugh (2003; henceforth S&McG).
This nonrelativistic scheme is summed up by a novel relation between acceleration
a of each element of an extragalactic system and the local Newtonian field −∇ΦN ,
µ(|a|/a0)a = −∇ΦN , (1.1)
where a0 ≈ 10−10m s−2, and the µ function interpolates between µ(x) = x for
x ≪ 1 and µ(∞) = 1. It is this last proviso which guarantees correspondence
of MOND with Newtonian gravity in the laboratory or the solar system, both of
which are supra a0 situations.
MOND lends itself to reformulation as a gravitational field theory alternative to
Newton’s. In this lagrangian based theory, named AQUAL (Bekenstein & Milgrom
1984; henceforth B&M), the Poisson equation is replaced by
∇ · [µ(|∇Φ|/a0)∇Φ] = 4πGρ, (1.2)
with Φ the physical gravitational potential (so that a = −∇Φ). In spherically
symmetric situations Eq. (1.2) has Eq. (1.1) as a unique solution. But when
1 Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904 Israel
2 Kapteyn Astronomical Institute, P.O. Box 800, 9700AV Groningen, The Netherlands
c© EDP Sciences 2005
DOI: (will be inserted later)
24 Mass Profiles and Shapes of Cosmological Structures
symmetry is lower, an extra term appears on the r.h.s. It is this corrected MOND
equation which is to be used in place of Eq. (1.1): deriving as it does from a
lagrangian, it guarantees that momentum, energy and angular momentum will be
conserved in MOND dynamics. The original MOND formula failed to assure this.
AQUAL has a number of further advantages over plain MOND (B&M, S&McG).
AQUAL is thus a fruitful tool for exploring non dark matter (DM) resolutions
of the missing mass problems. But it has shortcomings. For example, like the
naive MOND formula, it fails to do away with the need for DM in clusters of
galaxies, although its requirements for the dark stuff are certainly lower than in
Newtonian gravity. And since it is basically nonrelativistic, AQUAL is unable to
address a number of key subjects with relativistic overtones: gravitational lensing,
cosmological evolution and the growth of structure, to name a few. How to cast
the MOND idea in a relativistic mold while retaining the benefits of AQUAL ?
2 Relativistic MOND theories leading up to TeVeS
A relativistic version of AQUAL (call it RAQUAL) was offered in its defining
paper (B&M). The basic idea was to replace the physical potential Φ by a scalar
field φ whose equation is a relativistic reflection of Eq. (1.2). This equation is
written on a metric gαβ obeying Einstein’s equations (with a contribution to the
energy momentum tensor from φ). One then defines the physical metric as
g˜αβ ≡ e2φgαβ . (2.1)
This last describes a spacetime which has been locally stretched in every direction
with respect to that described by gαβ. All matter and radiations are supposed to
sense only g˜αβ , and the requisite equations must contain only g˜αβ and never gαβ
by itself. RAQUAL reproduces nonrelativistic AQUAL, and so recovers much of
the successful MOND phenomenology.
However, RAQUAL fails on two counts. It permits superluminal propagation
of φ waves (B&M). And it is unable to give an account of gravitational lensing in
agreement with the basic observation that lensing by galaxy clusters is anomalously
strong compared to what was to be expected in view of their galaxies and gas
content. This last problem is rooted in the conformal relation (2.1) between the
Einstein and physical metrics. Conformally related metrics share the same tracks
for photons (or electromagnetic wave packets). And because φ’s contribution to
the source of Einstein’s equations is weak in clusters of galaxies, the metric relevant
for lensing in RAQUAL is nearly that in GR with no DM.
Breaking the conformal relation has been the central theme in attempts to fix
this problem. For example, it is possible to add to Eq. (2.1) the symmetric tensor
φ,α φ,β multiplied by some scalar (Bekenstein & Sanders 1994). It turns out that
in order for gravitational waves—perturbations of gαβ—to propagate causally, i.e.
within the light cone of g˜αβ, the mentioned scalar factor must be negative, and it is
then found to weaken gravitational lensing, rather than enhancing it as intended.
A way out of the impasse (Sanders 1997) is to introduce a constant vector field
which points in the time direction, and then to stretch spacetime along the vector
Relativistic MOND Theory 25
by a factor e2φ while shrinking it by the same factor orthogonally to the vector.
This does the trick: provided the scalar field φ used comes from a RAQUAL-type
equation, MOND phenomenology is recovered while the lensing is augmented to
the right proportions. However, this “Stratified” theory is not covariant (one needs
the vector to be in the time direction, and this cannot be true in all coordinate
systems). Covariance of equations of a physical theory is so ingrained in modern
physics that we have to look further.
3 The TeVeS field theory
TeVeS is a tensor-vector-scalar covariant field theory (Bekenstein 2004; henceforth
B04) incorporating various motifs already mentioned. Just as in RAQUAL, the
tensor is an Einstein metric gαβ out of which is built the standard Einstein-Hilbert
action. One passes to the physical metric as in the stratified theory, but now using
an everywhere timelike dynamical normalized vector field Uα. In equations
g˜αβ = e
−2φ(gαβ + UαUβ)− e2φUαUβ, (3.1)
together with the normalization requirement gαβUαUβ = −1.
The dynamics of Uα is dictated by the action
Sv = − K
32πG
∫ [
gαβgµνU[α,µ]U[β,ν] − 2(λ/K)(gµνUµUν + 1)
]
(−g)1/2d4x, (3.2)
where the square brackets denote antisymmetrization. The K here is a dimension-
less coupling constant, one of the parameters of the theory. The λ is a Lagrange
multiplier field included to enforce the normalization of the vector field; λ is to
be determined as the equations are solved. The kinetic part of Sv is that of an
Abelian gauge field, but the unit norm constraint deprives Uα of the proverbial
gauge freedom. In addition, Uα is not coupled to a current of any sort, so it would
be incorrect to view the vector as mediating a repulsive Coulomb-like force.
The dynamical scalar field φ required in Eq. (3.1) is assigned the action
Ss = − 1
2
∫ [
σ2hαβφ,αφ,β +
1
2
Gℓ−2σ4F (kGσ2)
]
(−g)1/2d4x, (3.3)
which involves an auxiliary nondynamical scalar field σ, a new dimensionless pa-
rameter, k, a scale parameter, ℓ, and an unspecified function F . In the kinetic part
of Ss, φ,αφ,β is contracted not with g
αβ , but with the tensor hαβ ≡ gαβ − UαUβ ;
this is introduced to cope with the problem of superluminal propagation of scalar
perturbations evidenced by RAQUAL. With hαβ the propagation is found to be
subluminal with respect to the physical metric on condition that φ > 0 (B04).
Contrary to appearances, Ss is not a quadratic action. First σ
2 must be expressed
in terms of φ,α by varying Ss with respect to σ
2, and inverting the resulting equa-
tion. When the result is substituted in both terms of Ss, the action becomes
aquadratic in φ,α, resembling RAQUAL’s action, with the scalar h
αβφ,α φ,β re-
placing gαβφ,α φ,β . The form (3.3) is, however, more convenient for calculation.
26 Mass Profiles and Shapes of Cosmological Structures
These pieces of the total action must be supplemented with the matter action
Sm. In accordance with the Einstein equivalence principle (Will 1993) it must be
formed by replacing gαβ 7→ g˜αβ everywhere in the matter action in use in general
relativity (GR), and modifying all covariant derivatives accordingly. In this way
all types of matter respond to gravitational fields in the same way, and it could
be said that matter and radiation delineate the physical metric. Now because g˜αβ
contains φ as well as Uα, the variations of the total action with respect to these
quantities engender variations of Sm which create sources for the equations of φ
and Uα. No direct coupling of these fields to the matter variables is thus required,
which is pleasant as no principles are evident for determining such couplings.
Is TeVeS unique ? Not at all. For example, one could choose to write Ss
and Sv with g˜αβ instead of gαβ. The present approach seems preferable in that
evidently gαβ , φ and Uα are all gravitational fields, so their actions should not
involve the “matter’s metric” g˜αβ. Further, one could add to Ss a kinetic term
for σ such as is included in Phase Coupled Gravity (PCG), a theory devised in
a partially successful attempt to forestall superluminal propagation (Bekenstein
1986, 1990). In addition one could redefine hαβ as gαβ − ζ Uα Uβ (ζ should be
positive but not small in order to obviate superluminality; it could be a constant).
An additional freedom is the choice of function F in Eq. (3.3); each such choice
defines a distinct theory. Some of the freedoms mentioned have been used to make
a variant of TeVeS (Sanders 2005a, 2005b; henceforth S05).
4 Results from TeVeS
TeVeS has GR as a limit for K → 0 and ℓ→∞ (B04). Thus if |K| ≪ 1 and the
size of the system of interest falls well short of ℓ, predictions of TeVeS and GR
should not be that different. To verify that TeVeS can also give MOND in some
other limit one must make a choice for the function F in action (3.3). B04 makes
the choice which is equivalent to the relation
kℓ2hµνφ,µφ,ν = (3/4)k
2G2σ4(kGσ2 − 2)2(1− kGσ2)−1. (4.1)
This is to be regarded as defining a toy theory. A better choice would have the
r.h.s. change sign smoothly through kGσ2 = 1, a transition corresponding to the
passage between a quasistatic system such as a cluster of galaxies (hµνφ,µφ,ν > 0)
and cosmology (hµνφ,µφ,ν < 0) (see B04).
Imagine linearizing the Einstein-like equations of TeVeS about flat spacetime
(gαβ = ηαβ+ · · ·) in a quasistatic situation. The final result for the physical metric
after suitable choice of units is (B04)
ds2 = −(1 + 2Φ)dt2 + (1− 2Φ)δijdxidxj ; Φ ≡ ΞΦN + φ. (4.2)
Here ΦN is the usual Newtonian potential of the baryonic matter and Ξ is a con-
stant depending on K and on the asymptotic (cosmological) value of the scalar
field φ; Ξ is expected to be close to unity. With Φ interpreted as the physical
gravitational potential, metric (4.2) is precisely that used in GR to discuss both
Relativistic MOND Theory 27
dynamics and gravitational lensing. An immediate conclusion is that if GR suc-
ceeds in describing both the dynamics of, say, a galaxy, and its lensing properties
with a single assumed DM distribution, and if TeVeS succeeds in predicting the
dynamics everywhere from the observed baryonic distribution, then TeVeS will
predict the same lensing pattern as GR. Thus the problem that accosted RAQUAL
and variants of it (Sec. 2) is solved: lensing can be described using MOND ideas.
This much was clear earlier from the non-covariant framework (Sanders 1997). Of
course the problem with galaxy cluster dynamics (Sec. 1) still looms. But it is
important to realize that the two mentioned problems are distinct.
Obviously, to determine Φ requires one to calculate φ from the baryonic mass
distribution using the TeVeS analog of Eq. (1.2). In the linearized framework
mentioned and assuming spherical symmetry, one can be immediately integrate
the said equation once to get (we assume k ≪ 1 as well as Ξ ≈ 1)
µ∇Φ = ∇ΦN , µ ≈ −1 +
√
1 + 4|∇Φ|/a0
1 +
√
1 + 4|∇Φ|/a0
; a0 ≈
√
3k
4πℓ
. (4.3)
Thus we get the MOND equation (1.1), as well as a formula for µ showing the
appropriate asymptotic behavior at small and large |∇Φ|/a0, and the value of a0
expressed in terms of the TeVeS parameters. MOND is recovered. Famaey and
Binney (2005) have noted that the Galaxy’s rotation curve is not fully consistent
with this µ. Apparently a better F is needed.
Fig. 1. |∇Φ| (dashed), |∇(Φ − ΦN )| (in units of 10
−8 cms−2) with k = 0.03 (solid) or
k = 0.1 (dotted) from TeVeS plotted vs the distance from the Sun. Observed constraints
on the non-Newtonian part of the acceleration, |∇(Φ − ΦN )|, are (left to right): from
the precession of perihelion of Mercury, and of Icarus, from variation of Kepler’s con-
stant between Earth and Mars, between inner planets and Jupiter, Uranus or Neptune,
respectively. The solid bar is the Pioneer anomaly range.
The approximation inherent in Eq. (4.3) is good for all extragalactic applica-
28 Mass Profiles and Shapes of Cosmological Structures
tions, but begins to fail when |∇Φ| > (π/k)2a0. In such situations, e.g. the solar
system, it is best to start from the first term of the Laurent expansion of the r.h.s.
of Eq. (4.1) about kGσ2 = 1. One then obtains
µ ≈ 1
Ξ + k/4π
(
1− 16π
3
k3
a0
2
|∇Φ|2
)
. (4.4)
The fact that µ asymptotes to (Ξ+k/4π)−1 as |∇Φ| → ∞means that for arbitrarily
strong fields gravity is Newtonian, except that the effective gravitational constant
is GN = (Ξ + k/4π)G. It should be mentioned that the salient post-Newtonian
parameters (Will 1993), important for inner solar system celestial dynamics, agree
with those of GR; in particular β = 1 and γ = 1 (Giannios 2004, B04).
Notice the slow approach of µ to its asymptotic value as |∇Φ| strengthens.
One consequence is a slow increase of the Kepler constant a3/P 2 with distance
from the Sun. Fig. 1 illustrates this effect and compares it with the constraints
set by measurements of the orbits of diverse planets. Although a better form
of F is clearly required, the significant point is that, in such theories, the total
force in the outer Solar System will inevitably deviate from inverse square. This
is of considerable interest in view of the Pioneer anomaly (Anderson et al. 1998,
Turyshev 2005).
References
Anderson, J.D. et al., 1998, Phys.Rev.Lett., 81, 2858
Bekenstein, J.D. & Milgrom, M. 1984, ApJ, 286, 7 (B&M)
Bekenstein, J.D. 1988, in Second Canadian Conference on General Relativity and Rela-
tivistic Astrophysics, eds. A. Coley, C. Dyer & T. Tupper (Singapore: World Scien-
tific), 68.
Bekenstein, J.D. 1990, in Developments in General Relativity, Astrophysics and Quantum
Theory, eds. F. I. Cooperstock, L. P. Horwitz & J. Rosen (Bristol: IOP Publishing),
156
Bekenstein, J. D. & Sanders, R.H. 1994, ApJ, 429, 480
Bekenstein, J.D. 2004, Phys. Rev. D, 70, 083509 (erratum, 2005, Phys. Rev. D, 71,
069901) (B04)
Famaey, B. & Binney, J. 2005, astro-ph/0506723
Giannios, D. 2005 Phys. Rev. D, 71, 103511
Milgrom, M. 1983, ApJ, 270, 365, 379 and 384
Sanders, R.H.1997, ApJ, 480, 492
Sanders R.H. & McGaugh, S.S. 2002, Ann. Rev. Astron. Astrophys., 40, 263 (S&McG)
Sanders, R.H. 2005a, astro-ph/0502222
Sanders, R.H. 2005b, this collection (S05)
Turyshev, S., 2005, this collection
Will, C. 1993, Theory and Experiment in Gravitational Physics (Cambridge: Cambridge
Univ. Press)


Abstract: We first review the nonrelativistic lagrangian theory as a framework for the MOND equation. Obstructions to a relativistic version of it are discussed leading up to TeVeS, a relativistic tensor-vector-scalar field theory which displays both MOND and Newtonian limits. The whys for its particular structure are discussed and its achievements so far are summarized

ARTS repository - University of Groningen

   University of GroningenA Primer to Relativistic MOND TheoryBekenstein, J.D..; Sanders, R.H.Published in:Physical Review DIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Bekenstein, J. D., & Sanders, R. H. (2005). A Primer to Relativistic MOND Theory. Physical Review D,71(6).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 31-10-2023arXiv:astro-ph/0509519 v1   18 Sep 2005Mass Profiles and Shapes of Cosmological StructuresG. Mamon, F. Combes, C. Deffayet, B. Fort (eds)EAS Publications Series, Vol. ?, 2005A PRIMER TO RELATIVISTIC MOND THEORYJacob D. Bekenstein1 and Robert H. Sanders2Abstract. We first review the nonrelativistic lagrangian theory as aframework for the MOND equation. Obstructions to a relativistic ver-sion of it are discussed leading up to TeVeS, a relativistic tensor-vector-scalar field theory which displays both MOND and Newtonian limits.The whys for its particular structure are discussed and its achievementsso far are summarized.1 IntroductionThe success of Milgrom’s MOND paradigm (Milgrom 1983) in explaining theshapes of disk galaxy rotation curves, and correlations between the parameters ofsuch systems (e.g. Tully-Fisher law) with no appeal to dark haloes is by now wellknown—for a modern review see Sanders & McGaugh (2003; henceforth S&McG).This nonrelativistic scheme is summed up by a novel relation between accelerationa of each element of an extragalactic system and the local Newtonian field −∇ΦN ,µ(|a|/a0)a = −∇ΦN , (1.1)where a0 ≈ 10−10m s−2, and the µ function interpolates between µ(x) = x forx ≪ 1 and µ(∞) = 1. It is this last proviso which guarantees correspondenceof MOND with Newtonian gravity in the laboratory or the solar system, both ofwhich are supra a0 situations.MOND lends itself to reformulation as a gravitational field theory alternative toNewton’s. In this lagrangian based theory, named AQUAL (Bekenstein & Milgrom1984; henceforth B&M), the Poisson equation is replaced by∇ · [µ(|∇Φ|/a0)∇Φ] = 4πGρ, (1.2)with Φ the physical gravitational potential (so that a = −∇Φ). In sphericallysymmetric situations Eq. (1.2) has Eq. (1.1) as a unique solution. But when1 Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904 Israel2 Kapteyn Astronomical Institute, P.O. Box 800, 9700AV Groningen, The Netherlandsc© EDP Sciences 2005DOI: (will be inserted later)24 Mass Profiles and Shapes of Cosmological Structuressymmetry is lower, an extra term appears on the r.h.s. It is this corrected MONDequation which is to be used in place of Eq. (1.1): deriving as it does from alagrangian, it guarantees that momentum, energy and angular momentum will beconserved in MOND dynamics. The original MOND formula failed to assure this.AQUAL has a number of further advantages over plain MOND (B&M, S&McG).AQUAL is thus a fruitful tool for exploring non dark matter (DM) resolutionsof the missing mass problems. But it has shortcomings. For example, like thenaive MOND formula, it fails to do away with the need for DM in clusters ofgalaxies, although its requirements for the dark stuff are certainly lower than inNewtonian gravity. And since it is basically nonrelativistic, AQUAL is unable toaddress a number of key subjects with relativistic overtones: gravitational lensing,cosmological evolution and the growth of structure, to name a few. How to castthe MOND idea in a relativistic mold while retaining the benefits of AQUAL ?2 Relativistic MOND theories leading up to TeVeSA relativistic version of AQUAL (call it RAQUAL) was offered in its definingpaper (B&M). The basic idea was to replace the physical potential Φ by a scalarfield φ whose equation is a relativistic reflection of Eq. (1.2). This equation iswritten on a metric gαβ obeying Einstein’s equations (with a contribution to theenergy momentum tensor from φ). One then defines the physical metric asg̃αβ ≡ e2φgαβ . (2.1)This last describes a spacetime which has been locally stretched in every directionwith respect to that described by gαβ. All matter and radiations are supposed tosense only g̃αβ , and the requisite equations must contain only g̃αβ and never gαβby itself. RAQUAL reproduces nonrelativistic AQUAL, and so recovers much ofthe successful MOND phenomenology.However, RAQUAL fails on two counts. It permits superluminal propagationof φ waves (B&M). And it is unable to give an account of gravitational lensing inagreement with the basic observation that lensing by galaxy clusters is anomalouslystrong compared to what was to be expected in view of their galaxies and gascontent. This last problem is rooted in the conformal relation (2.1) between theEinstein and physical metrics. Conformally related metrics share the same tracksfor photons (or electromagnetic wave packets). And because φ’s contribution tothe source of Einstein’s equations is weak in clusters of galaxies, the metric relevantfor lensing in RAQUAL is nearly that in GR with no DM.Breaking the conformal relation has been the central theme in attempts to fixthis problem. For example, it is possible to add to Eq. (2.1) the symmetric tensorφ,α φ,β multiplied by some scalar (Bekenstein & Sanders 1994). It turns out thatin order for gravitational waves—perturbations of gαβ—to propagate causally, i.e.within the light cone of g̃αβ, the mentioned scalar factor must be negative, and it isthen found to weaken gravitational lensing, rather than enhancing it as intended.A way out of the impasse (Sanders 1997) is to introduce a constant vector fieldwhich points in the time direction, and then to stretch spacetime along the vectorRelativistic MOND Theory 25by a factor e2φ while shrinking it by the same factor orthogonally to the vector.This does the trick: provided the scalar field φ used comes from a RAQUAL-typeequation, MOND phenomenology is recovered while the lensing is augmented tothe right proportions. However, this “Stratified” theory is not covariant (one needsthe vector to be in the time direction, and this cannot be true in all coordinatesystems). Covariance of equations of a physical theory is so ingrained in modernphysics that we have to look further.3 The TeVeS field theoryTeVeS is a tensor-vector-scalar covariant field theory (Bekenstein 2004; henceforthB04) incorporating various motifs already mentioned. Just as in RAQUAL, thetensor is an Einstein metric gαβ out of which is built the standard Einstein-Hilbertaction. One passes to the physical metric as in the stratified theory, but now usingan everywhere timelike dynamical normalized vector field Uα. In equationsg̃αβ = e−2φ(gαβ + UαUβ) − e2φUαUβ, (3.1)together with the normalization requirement gαβUαUβ = −1.The dynamics of Uα is dictated by the actionSv = −K32πG∫[gαβgµνU[α,µ]U[β,ν] − 2(λ/K)(gµνUµUν + 1)](−g)1/2d4x, (3.2)where the square brackets denote antisymmetrization. The K here is a dimension-less coupling constant, one of the parameters of the theory. The λ is a Lagrangemultiplier field included to enforce the normalization of the vector field; λ is tobe determined as the equations are solved. The kinetic part of Sv is that of anAbelian gauge field, but the unit norm constraint deprives Uα of the proverbialgauge freedom. In addition, Uα is not coupled to a current of any sort, so it wouldbe incorrect to view the vector as mediating a repulsive Coulomb-like force.The dynamical scalar field φ required in Eq. (3.1) is assigned the actionSs = −12∫[σ2hαβφ,αφ,β +12Gℓ−2σ4F (kGσ2)](−g)1/2d4x, (3.3)which involves an auxiliary nondynamical scalar field σ, a new dimensionless pa-rameter, k, a scale parameter, ℓ, and an unspecified function F . In the kinetic partof Ss, φ,αφ,β is contracted not with gαβ , but with the tensor hαβ ≡ gαβ − UαUβ ;this is introduced to cope with the problem of superluminal propagation of scalarperturbations evidenced by RAQUAL. With hαβ the propagation is found to besubluminal with respect to the physical metric on condition that φ > 0 (B04).Contrary to appearances, Ss is not a quadratic action. First σ2 must be expressedin terms of φ,α by varying Ss with respect to σ2, and inverting the resulting equa-tion. When the result is substituted in both terms of Ss, the action becomesaquadratic in φ,α, resembling RAQUAL’s action, with the scalar hαβφ,α φ,β re-placing gαβφ,α φ,β . The form (3.3) is, however, more convenient for calculation.26 Mass Profiles and Shapes of Cosmological StructuresThese pieces of the total action must be supplemented with the matter actionSm. In accordance with the Einstein equivalence principle (Will 1993) it must beformed by replacing gαβ 7→ g̃αβ everywhere in the matter action in use in generalrelativity (GR), and modifying all covariant derivatives accordingly. In this wayall types of matter respond to gravitational fields in the same way, and it couldbe said that matter and radiation delineate the physical metric. Now because g̃αβcontains φ as well as Uα, the variations of the total action with respect to thesequantities engender variations of Sm which create sources for the equations of φand Uα. No direct coupling of these fields to the matter variables is thus required,which is pleasant as no principles are evident for determining such couplings.Is TeVeS unique ? Not at all. For example, one could choose to write Ssand Sv with g̃αβ instead of gαβ. The present approach seems preferable in thatevidently gαβ , φ and Uα are all gravitational fields, so their actions should notinvolve the “matter’s metric” g̃αβ. Further, one could add to Ss a kinetic termfor σ such as is included in Phase Coupled Gravity (PCG), a theory devised ina partially successful attempt to forestall superluminal propagation (Bekenstein1986, 1990). In addition one could redefine hαβ as gαβ − ζ Uα Uβ (ζ should bepositive but not small in order to obviate superluminality; it could be a constant).An additional freedom is the choice of function F in Eq. (3.3); each such choicedefines a distinct theory. Some of the freedoms mentioned have been used to makea variant of TeVeS (Sanders 2005a, 2005b; henceforth S05).4 Results from TeVeSTeVeS has GR as a limit for K → 0 and ℓ → ∞ (B04). Thus if |K| ≪ 1 and thesize of the system of interest falls well short of ℓ, predictions of TeVeS and GRshould not be that different. To verify that TeVeS can also give MOND in someother limit one must make a choice for the function F in action (3.3). B04 makesthe choice which is equivalent to the relationkℓ2hµνφ,µφ,ν = (3/4)k2G2σ4(kGσ2 − 2)2(1 − kGσ2)−1. (4.1)This is to be regarded as defining a toy theory. A better choice would have ther.h.s. change sign smoothly through kGσ2 = 1, a transition corresponding to thepassage between a quasistatic system such as a cluster of galaxies (hµνφ,µφ,ν > 0)and cosmology (hµνφ,µφ,ν < 0) (see B04).Imagine linearizing the Einstein-like equations of TeVeS about flat spacetime(gαβ = ηαβ + · · ·) in a quasistatic situation. The final result for the physical metricafter suitable choice of units is (B04)ds2 = −(1 + 2Φ)dt2 + (1 − 2Φ)δijdxidxj ; Φ ≡ ΞΦN + φ. (4.2)Here ΦN is the usual Newtonian potential of the baryonic matter and Ξ is a con-stant depending on K and on the asymptotic (cosmological) value of the scalarfield φ; Ξ is expected to be close to unity. With Φ interpreted as the physicalgravitational potential, metric (4.2) is precisely that used in GR to discuss bothRelativistic MOND Theory 27dynamics and gravitational lensing. An immediate conclusion is that if GR suc-ceeds in describing both the dynamics of, say, a galaxy, and its lensing propertieswith a single assumed DM distribution, and if TeVeS succeeds in predicting thedynamics everywhere from the observed baryonic distribution, then TeVeS willpredict the same lensing pattern as GR. Thus the problem that accosted RAQUALand variants of it (Sec. 2) is solved: lensing can be described using MOND ideas.This much was clear earlier from the non-covariant framework (Sanders 1997). Ofcourse the problem with galaxy cluster dynamics (Sec. 1) still looms. But it isimportant to realize that the two mentioned problems are distinct.Obviously, to determine Φ requires one to calculate φ from the baryonic massdistribution using the TeVeS analog of Eq. (1.2). In the linearized frameworkmentioned and assuming spherical symmetry, one can be immediately integratethe said equation once to get (we assume k ≪ 1 as well as Ξ ≈ 1)µ∇Φ = ∇ΦN , µ ≈−1 +√1 + 4|∇Φ|/a01 +√1 + 4|∇Φ|/a0; a0 ≈√3k4πℓ. (4.3)Thus we get the MOND equation (1.1), as well as a formula for µ showing theappropriate asymptotic behavior at small and large |∇Φ|/a0, and the value of a0expressed in terms of the TeVeS parameters. MOND is recovered. Famaey andBinney (2005) have noted that the Galaxy’s rotation curve is not fully consistentwith this µ. Apparently a better F is needed.Fig. 1. |∇Φ| (dashed), |∇(Φ − ΦN )| (in units of 10−8 cms−2) with k = 0.03 (solid) ork = 0.1 (dotted) from TeVeS plotted vs the distance from the Sun. Observed constraintson the non-Newtonian part of the acceleration, |∇(Φ − ΦN )|, are (left to right): fromthe precession of perihelion of Mercury, and of Icarus, from variation of Kepler’s con-stant between Earth and Mars, between inner planets and Jupiter, Uranus or Neptune,respectively. The solid bar is the Pioneer anomaly range.The approximation inherent in Eq. (4.3) is good for all extragalactic applica-28 Mass Profiles and Shapes of Cosmological Structurestions, but begins to fail when |∇Φ| > (π/k)2a0. In such situations, e.g. the solarsystem, it is best to start from the first term of the Laurent expansion of the r.h.s.of Eq. (4.1) about kGσ2 = 1. One then obtainsµ ≈ 1Ξ + k/4π(1 − 16π3k3a02|∇Φ|2). (4.4)The fact that µ asymptotes to (Ξ+k/4π)−1 as |∇Φ| → ∞ means that for arbitrarilystrong fields gravity is Newtonian, except that the effective gravitational constantis GN = (Ξ + k/4π)G. It should be mentioned that the salient post-Newtonianparameters (Will 1993), important for inner solar system celestial dynamics, agreewith those of GR; in particular β = 1 and γ = 1 (Giannios 2004, B04).Notice the slow approach of µ to its asymptotic value as |∇Φ| strengthens.One consequence is a slow increase of the Kepler constant a3/P 2 with distancefrom the Sun. Fig. 1 illustrates this effect and compares it with the constraintsset by measurements of the orbits of diverse planets. Although a better formof F is clearly required, the significant point is that, in such theories, the totalforce in the outer Solar System will inevitably deviate from inverse square. Thisis of considerable interest in view of the Pioneer anomaly (Anderson et al. 1998,Turyshev 2005).ReferencesAnderson, J.D. et al., 1998, Phys.Rev.Lett., 81, 2858Bekenstein, J.D. & Milgrom, M. 1984, ApJ, 286, 7 (B&M)Bekenstein, J.D. 1988, in Second Canadian Conference on General Relativity and Rela-tivistic Astrophysics, eds. A. Coley, C. Dyer & T. Tupper (Singapore: World Scien-tific), 68.Bekenstein, J.D. 1990, in Developments in General Relativity, Astrophysics and QuantumTheory, eds. F. I. Cooperstock, L. P. Horwitz & J. Rosen (Bristol: IOP Publishing),156Bekenstein, J. D. & Sanders, R.H. 1994, ApJ, 429, 480Bekenstein, J.D. 2004, Phys. Rev. D, 70, 083509 (erratum, 2005, Phys. Rev. D, 71,069901) (B04)Famaey, B. & Binney, J. 2005, astro-ph/0506723Giannios, D. 2005 Phys. Rev. D, 71, 103511Milgrom, M. 1983, ApJ, 270, 365, 379 and 384Sanders, R.H.1997, ApJ, 480, 492Sanders R.H. & McGaugh, S.S. 2002, Ann. Rev. Astron. Astrophys., 40, 263 (S&McG)Sanders, R.H. 2005a, astro-ph/0502222Sanders, R.H. 2005b, this collection (S05)Turyshev, S., 2005, this collectionWill, C. 1993, Theory and Experiment in Gravitational Physics (Cambridge: CambridgeUniv. Press)

   University of GroningenA Primer to Relativistic MOND TheoryBekenstein, J.D..; Sanders, R.H.Published in:Physical Review DIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Bekenstein, J. D., & Sanders, R. H. (2005). A Primer to Relativistic MOND Theory. Physical Review D,71(6).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 13-02-2023arXiv:astro-ph/0509519 v1   18 Sep 2005Mass Profiles and Shapes of Cosmological StructuresG. Mamon, F. Combes, C. Deffayet, B. Fort (eds)EAS Publications Series, Vol. ?, 2005A PRIMER TO RELATIVISTIC MOND THEORYJacob D. Bekenstein1 and Robert H. Sanders2Abstract. We first review the nonrelativistic lagrangian theory as aframework for the MOND equation. Obstructions to a relativistic ver-sion of it are discussed leading up to TeVeS, a relativistic tensor-vector-scalar field theory which displays both MOND and Newtonian limits.The whys for its particular structure are discussed and its achievementsso far are summarized.1 IntroductionThe success of Milgrom’s MOND paradigm (Milgrom 1983) in explaining theshapes of disk galaxy rotation curves, and correlations between the parameters ofsuch systems (e.g. Tully-Fisher law) with no appeal to dark haloes is by now wellknown—for a modern review see Sanders & McGaugh (2003; henceforth S&McG).This nonrelativistic scheme is summed up by a novel relation between accelerationa of each element of an extragalactic system and the local Newtonian field −∇ΦN ,µ(|a|/a0)a = −∇ΦN , (1.1)where a0 ≈ 10−10m s−2, and the µ function interpolates between µ(x) = x forx ≪ 1 and µ(∞) = 1. It is this last proviso which guarantees correspondenceof MOND with Newtonian gravity in the laboratory or the solar system, both ofwhich are supra a0 situations.MOND lends itself to reformulation as a gravitational field theory alternative toNewton’s. In this lagrangian based theory, named AQUAL (Bekenstein & Milgrom1984; henceforth B&M), the Poisson equation is replaced by∇ · [µ(|∇Φ|/a0)∇Φ] = 4πGρ, (1.2)with Φ the physical gravitational potential (so that a = −∇Φ). In sphericallysymmetric situations Eq. (1.2) has Eq. (1.1) as a unique solution. But when1 Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904 Israel2 Kapteyn Astronomical Institute, P.O. Box 800, 9700AV Groningen, The Netherlandsc© EDP Sciences 2005DOI: (will be inserted later)24 Mass Profiles and Shapes of Cosmological Structuressymmetry is lower, an extra term appears on the r.h.s. It is this corrected MONDequation which is to be used in place of Eq. (1.1): deriving as it does from alagrangian, it guarantees that momentum, energy and angular momentum will beconserved in MOND dynamics. The original MOND formula failed to assure this.AQUAL has a number of further advantages over plain MOND (B&M, S&McG).AQUAL is thus a fruitful tool for exploring non dark matter (DM) resolutionsof the missing mass problems. But it has shortcomings. For example, like thenaive MOND formula, it fails to do away with the need for DM in clusters ofgalaxies, although its requirements for the dark stuff are certainly lower than inNewtonian gravity. And since it is basically nonrelativistic, AQUAL is unable toaddress a number of key subjects with relativistic overtones: gravitational lensing,cosmological evolution and the growth of structure, to name a few. How to castthe MOND idea in a relativistic mold while retaining the benefits of AQUAL ?2 Relativistic MOND theories leading up to TeVeSA relativistic version of AQUAL (call it RAQUAL) was offered in its definingpaper (B&M). The basic idea was to replace the physical potential Φ by a scalarfield φ whose equation is a relativistic reflection of Eq. (1.2). This equation iswritten on a metric gαβ obeying Einstein’s equations (with a contribution to theenergy momentum tensor from φ). One then defines the physical metric asg̃αβ ≡ e2φgαβ . (2.1)This last describes a spacetime which has been locally stretched in every directionwith respect to that described by gαβ. All matter and radiations are supposed tosense only g̃αβ , and the requisite equations must contain only g̃αβ and never gαβby itself. RAQUAL reproduces nonrelativistic AQUAL, and so recovers much ofthe successful MOND phenomenology.However, RAQUAL fails on two counts. It permits superluminal propagationof φ waves (B&M). And it is unable to give an account of gravitational lensing inagreement with the basic observation that lensing by galaxy clusters is anomalouslystrong compared to what was to be expected in view of their galaxies and gascontent. This last problem is rooted in the conformal relation (2.1) between theEinstein and physical metrics. Conformally related metrics share the same tracksfor photons (or electromagnetic wave packets). And because φ’s contribution tothe source of Einstein’s equations is weak in clusters of galaxies, the metric relevantfor lensing in RAQUAL is nearly that in GR with no DM.Breaking the conformal relation has been the central theme in attempts to fixthis problem. For example, it is possible to add to Eq. (2.1) the symmetric tensorφ,α φ,β multiplied by some scalar (Bekenstein & Sanders 1994). It turns out thatin order for gravitational waves—perturbations of gαβ—to propagate causally, i.e.within the light cone of g̃αβ, the mentioned scalar factor must be negative, and it isthen found to weaken gravitational lensing, rather than enhancing it as intended.A way out of the impasse (Sanders 1997) is to introduce a constant vector fieldwhich points in the time direction, and then to stretch spacetime along the vectorRelativistic MOND Theory 25by a factor e2φ while shrinking it by the same factor orthogonally to the vector.This does the trick: provided the scalar field φ used comes from a RAQUAL-typeequation, MOND phenomenology is recovered while the lensing is augmented tothe right proportions. However, this “Stratified” theory is not covariant (one needsthe vector to be in the time direction, and this cannot be true in all coordinatesystems). Covariance of equations of a physical theory is so ingrained in modernphysics that we have to look further.3 The TeVeS field theoryTeVeS is a tensor-vector-scalar covariant field theory (Bekenstein 2004; henceforthB04) incorporating various motifs already mentioned. Just as in RAQUAL, thetensor is an Einstein metric gαβ out of which is built the standard Einstein-Hilbertaction. One passes to the physical metric as in the stratified theory, but now usingan everywhere timelike dynamical normalized vector field Uα. In equationsg̃αβ = e−2φ(gαβ + UαUβ) − e2φUαUβ, (3.1)together with the normalization requirement gαβUαUβ = −1.The dynamics of Uα is dictated by the actionSv = −K32πG∫[gαβgµνU[α,µ]U[β,ν] − 2(λ/K)(gµνUµUν + 1)](−g)1/2d4x, (3.2)where the square brackets denote antisymmetrization. The K here is a dimension-less coupling constant, one of the parameters of the theory. The λ is a Lagrangemultiplier field included to enforce the normalization of the vector field; λ is tobe determined as the equations are solved. The kinetic part of Sv is that of anAbelian gauge field, but the unit norm constraint deprives Uα of the proverbialgauge freedom. In addition, Uα is not coupled to a current of any sort, so it wouldbe incorrect to view the vector as mediating a repulsive Coulomb-like force.The dynamical scalar field φ required in Eq. (3.1) is assigned the actionSs = −12∫[σ2hαβφ,αφ,β +12Gℓ−2σ4F (kGσ2)](−g)1/2d4x, (3.3)which involves an auxiliary nondynamical scalar field σ, a new dimensionless pa-rameter, k, a scale parameter, ℓ, and an unspecified function F . In the kinetic partof Ss, φ,αφ,β is contracted not with gαβ , but with the tensor hαβ ≡ gαβ − UαUβ ;this is introduced to cope with the problem of superluminal propagation of scalarperturbations evidenced by RAQUAL. With hαβ the propagation is found to besubluminal with respect to the physical metric on condition that φ > 0 (B04).Contrary to appearances, Ss is not a quadratic action. First σ2 must be expressedin terms of φ,α by varying Ss with respect to σ2, and inverting the resulting equa-tion. When the result is substituted in both terms of Ss, the action becomesaquadratic in φ,α, resembling RAQUAL’s action, with the scalar hαβφ,α φ,β re-placing gαβφ,α φ,β . The form (3.3) is, however, more convenient for calculation.26 Mass Profiles and Shapes of Cosmological StructuresThese pieces of the total action must be supplemented with the matter actionSm. In accordance with the Einstein equivalence principle (Will 1993) it must beformed by replacing gαβ 7→ g̃αβ everywhere in the matter action in use in generalrelativity (GR), and modifying all covariant derivatives accordingly. In this wayall types of matter respond to gravitational fields in the same way, and it couldbe said that matter and radiation delineate the physical metric. Now because g̃αβcontains φ as well as Uα, the variations of the total action with respect to thesequantities engender variations of Sm which create sources for the equations of φand Uα. No direct coupling of these fields to the matter variables is thus required,which is pleasant as no principles are evident for determining such couplings.Is TeVeS unique ? Not at all. For example, one could choose to write Ssand Sv with g̃αβ instead of gαβ. The present approach seems preferable in thatevidently gαβ , φ and Uα are all gravitational fields, so their actions should notinvolve the “matter’s metric” g̃αβ. Further, one could add to Ss a kinetic termfor σ such as is included in Phase Coupled Gravity (PCG), a theory devised ina partially successful attempt to forestall superluminal propagation (Bekenstein1986, 1990). In addition one could redefine hαβ as gαβ − ζ Uα Uβ (ζ should bepositive but not small in order to obviate superluminality; it could be a constant).An additional freedom is the choice of function F in Eq. (3.3); each such choicedefines a distinct theory. Some of the freedoms mentioned have been used to makea variant of TeVeS (Sanders 2005a, 2005b; henceforth S05).4 Results from TeVeSTeVeS has GR as a limit for K → 0 and ℓ → ∞ (B04). Thus if |K| ≪ 1 and thesize of the system of interest falls well short of ℓ, predictions of TeVeS and GRshould not be that different. To verify that TeVeS can also give MOND in someother limit one must make a choice for the function F in action (3.3). B04 makesthe choice which is equivalent to the relationkℓ2hµνφ,µφ,ν = (3/4)k2G2σ4(kGσ2 − 2)2(1 − kGσ2)−1. (4.1)This is to be regarded as defining a toy theory. A better choice would have ther.h.s. change sign smoothly through kGσ2 = 1, a transition corresponding to thepassage between a quasistatic system such as a cluster of galaxies (hµνφ,µφ,ν > 0)and cosmology (hµνφ,µφ,ν < 0) (see B04).Imagine linearizing the Einstein-like equations of TeVeS about flat spacetime(gαβ = ηαβ + · · ·) in a quasistatic situation. The final result for the physical metricafter suitable choice of units is (B04)ds2 = −(1 + 2Φ)dt2 + (1 − 2Φ)δijdxidxj ; Φ ≡ ΞΦN + φ. (4.2)Here ΦN is the usual Newtonian potential of the baryonic matter and Ξ is a con-stant depending on K and on the asymptotic (cosmological) value of the scalarfield φ; Ξ is expected to be close to unity. With Φ interpreted as the physicalgravitational potential, metric (4.2) is precisely that used in GR to discuss bothRelativistic MOND Theory 27dynamics and gravitational lensing. An immediate conclusion is that if GR suc-ceeds in describing both the dynamics of, say, a galaxy, and its lensing propertieswith a single assumed DM distribution, and if TeVeS succeeds in predicting thedynamics everywhere from the observed baryonic distribution, then TeVeS willpredict the same lensing pattern as GR. Thus the problem that accosted RAQUALand variants of it (Sec. 2) is solved: lensing can be described using MOND ideas.This much was clear earlier from the non-covariant framework (Sanders 1997). Ofcourse the problem with galaxy cluster dynamics (Sec. 1) still looms. But it isimportant to realize that the two mentioned problems are distinct.Obviously, to determine Φ requires one to calculate φ from the baryonic massdistribution using the TeVeS analog of Eq. (1.2). In the linearized frameworkmentioned and assuming spherical symmetry, one can be immediately integratethe said equation once to get (we assume k ≪ 1 as well as Ξ ≈ 1)µ∇Φ = ∇ΦN , µ ≈−1 +√1 + 4|∇Φ|/a01 +√1 + 4|∇Φ|/a0; a0 ≈√3k4πℓ. (4.3)Thus we get the MOND equation (1.1), as well as a formula for µ showing theappropriate asymptotic behavior at small and large |∇Φ|/a0, and the value of a0expressed in terms of the TeVeS parameters. MOND is recovered. Famaey andBinney (2005) have noted that the Galaxy’s rotation curve is not fully consistentwith this µ. Apparently a better F is needed.Fig. 1. |∇Φ| (dashed), |∇(Φ − ΦN )| (in units of 10−8 cms−2) with k = 0.03 (solid) ork = 0.1 (dotted) from TeVeS plotted vs the distance from the Sun. Observed constraintson the non-Newtonian part of the acceleration, |∇(Φ − ΦN )|, are (left to right): fromthe precession of perihelion of Mercury, and of Icarus, from variation of Kepler’s con-stant between Earth and Mars, between inner planets and Jupiter, Uranus or Neptune,respectively. The solid bar is the Pioneer anomaly range.The approximation inherent in Eq. (4.3) is good for all extragalactic applica-28 Mass Profiles and Shapes of Cosmological Structurestions, but begins to fail when |∇Φ| > (π/k)2a0. In such situations, e.g. the solarsystem, it is best to start from the first term of the Laurent expansion of the r.h.s.of Eq. (4.1) about kGσ2 = 1. One then obtainsµ ≈ 1Ξ + k/4π(1 − 16π3k3a02|∇Φ|2). (4.4)The fact that µ asymptotes to (Ξ+k/4π)−1 as |∇Φ| → ∞ means that for arbitrarilystrong fields gravity is Newtonian, except that the effective gravitational constantis GN = (Ξ + k/4π)G. It should be mentioned that the salient post-Newtonianparameters (Will 1993), important for inner solar system celestial dynamics, agreewith those of GR; in particular β = 1 and γ = 1 (Giannios 2004, B04).Notice the slow approach of µ to its asymptotic value as |∇Φ| strengthens.One consequence is a slow increase of the Kepler constant a3/P 2 with distancefrom the Sun. Fig. 1 illustrates this effect and compares it with the constraintsset by measurements of the orbits of diverse planets. Although a better formof F is clearly required, the significant point is that, in such theories, the totalforce in the outer Solar System will inevitably deviate from inverse square. Thisis of considerable interest in view of the Pioneer anomaly (Anderson et al. 1998,Turyshev 2005).ReferencesAnderson, J.D. et al., 1998, Phys.Rev.Lett., 81, 2858Bekenstein, J.D. & Milgrom, M. 1984, ApJ, 286, 7 (B&M)Bekenstein, J.D. 1988, in Second Canadian Conference on General Relativity and Rela-tivistic Astrophysics, eds. A. Coley, C. Dyer & T. Tupper (Singapore: World Scien-tific), 68.Bekenstein, J.D. 1990, in Developments in General Relativity, Astrophysics and QuantumTheory, eds. F. I. Cooperstock, L. P. Horwitz & J. Rosen (Bristol: IOP Publishing),156Bekenstein, J. D. & Sanders, R.H. 1994, ApJ, 429, 480Bekenstein, J.D. 2004, Phys. Rev. D, 70, 083509 (erratum, 2005, Phys. Rev. D, 71,069901) (B04)Famaey, B. & Binney, J. 2005, astro-ph/0506723Giannios, D. 2005 Phys. Rev. D, 71, 103511Milgrom, M. 1983, ApJ, 270, 365, 379 and 384Sanders, R.H.1997, ApJ, 480, 492Sanders R.H. & McGaugh, S.S. 2002, Ann. Rev. Astron. Astrophys., 40, 263 (S&McG)Sanders, R.H. 2005a, astro-ph/0502222Sanders, R.H. 2005b, this collection (S05)Turyshev, S., 2005, this collectionWill, C. 1993, Theory and Experiment in Gravitational Physics (Cambridge: CambridgeUniv. Press)

University of Groningen

   University of GroningenA Primer to Relativistic MOND TheoryBekenstein, J.D..; Sanders, R.H.Published in:Physical Review DIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Bekenstein, J. D., & Sanders, R. H. (2005). A Primer to Relativistic MOND Theory. Physical Review D,71(6).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019arXiv:astro-ph/0509519 v1   18 Sep 2005Mass Profiles and Shapes of Cosmological StructuresG. Mamon, F. Combes, C. Deffayet, B. Fort (eds)EAS Publications Series, Vol. ?, 2005A PRIMER TO RELATIVISTIC MOND THEORYJacob D. Bekenstein1 and Robert H. Sanders2Abstract. We first review the nonrelativistic lagrangian theory as aframework for the MOND equation. Obstructions to a relativistic ver-sion of it are discussed leading up to TeVeS, a relativistic tensor-vector-scalar field theory which displays both MOND and Newtonian limits.The whys for its particular structure are discussed and its achievementsso far are summarized.1 IntroductionThe success of Milgrom’s MOND paradigm (Milgrom 1983) in explaining theshapes of disk galaxy rotation curves, and correlations between the parameters ofsuch systems (e.g. Tully-Fisher law) with no appeal to dark haloes is by now wellknown—for a modern review see Sanders & McGaugh (2003; henceforth S&McG).This nonrelativistic scheme is summed up by a novel relation between accelerationa of each element of an extragalactic system and the local Newtonian field −∇ΦN ,µ(|a|/a0)a = −∇ΦN , (1.1)where a0 ≈ 10−10m s−2, and the µ function interpolates between µ(x) = x forx ≪ 1 and µ(∞) = 1. It is this last proviso which guarantees correspondenceof MOND with Newtonian gravity in the laboratory or the solar system, both ofwhich are supra a0 situations.MOND lends itself to reformulation as a gravitational field theory alternative toNewton’s. In this lagrangian based theory, named AQUAL (Bekenstein & Milgrom1984; henceforth B&M), the Poisson equation is replaced by∇ · [µ(|∇Φ|/a0)∇Φ] = 4πGρ, (1.2)with Φ the physical gravitational potential (so that a = −∇Φ). In sphericallysymmetric situations Eq. (1.2) has Eq. (1.1) as a unique solution. But when1 Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904 Israel2 Kapteyn Astronomical Institute, P.O. Box 800, 9700AV Groningen, The Netherlandsc© EDP Sciences 2005DOI: (will be inserted later)24 Mass Profiles and Shapes of Cosmological Structuressymmetry is lower, an extra term appears on the r.h.s. It is this corrected MONDequation which is to be used in place of Eq. (1.1): deriving as it does from alagrangian, it guarantees that momentum, energy and angular momentum will beconserved in MOND dynamics. The original MOND formula failed to assure this.AQUAL has a number of further advantages over plain MOND (B&M, S&McG).AQUAL is thus a fruitful tool for exploring non dark matter (DM) resolutionsof the missing mass problems. But it has shortcomings. For example, like thenaive MOND formula, it fails to do away with the need for DM in clusters ofgalaxies, although its requirements for the dark stuff are certainly lower than inNewtonian gravity. And since it is basically nonrelativistic, AQUAL is unable toaddress a number of key subjects with relativistic overtones: gravitational lensing,cosmological evolution and the growth of structure, to name a few. How to castthe MOND idea in a relativistic mold while retaining the benefits of AQUAL ?2 Relativistic MOND theories leading up to TeVeSA relativistic version of AQUAL (call it RAQUAL) was offered in its definingpaper (B&M). The basic idea was to replace the physical potential Φ by a scalarfield φ whose equation is a relativistic reflection of Eq. (1.2). This equation iswritten on a metric gαβ obeying Einstein’s equations (with a contribution to theenergy momentum tensor from φ). One then defines the physical metric asg˜αβ ≡ e2φgαβ . (2.1)This last describes a spacetime which has been locally stretched in every directionwith respect to that described by gαβ. All matter and radiations are supposed tosense only g˜αβ , and the requisite equations must contain only g˜αβ and never gαβby itself. RAQUAL reproduces nonrelativistic AQUAL, and so recovers much ofthe successful MOND phenomenology.However, RAQUAL fails on two counts. It permits superluminal propagationof φ waves (B&M). And it is unable to give an account of gravitational lensing inagreement with the basic observation that lensing by galaxy clusters is anomalouslystrong compared to what was to be expected in view of their galaxies and gascontent. This last problem is rooted in the conformal relation (2.1) between theEinstein and physical metrics. Conformally related metrics share the same tracksfor photons (or electromagnetic wave packets). And because φ’s contribution tothe source of Einstein’s equations is weak in clusters of galaxies, the metric relevantfor lensing in RAQUAL is nearly that in GR with no DM.Breaking the conformal relation has been the central theme in attempts to fixthis problem. For example, it is possible to add to Eq. (2.1) the symmetric tensorφ,α φ,β multiplied by some scalar (Bekenstein & Sanders 1994). It turns out thatin order for gravitational waves—perturbations of gαβ—to propagate causally, i.e.within the light cone of g˜αβ, the mentioned scalar factor must be negative, and it isthen found to weaken gravitational lensing, rather than enhancing it as intended.A way out of the impasse (Sanders 1997) is to introduce a constant vector fieldwhich points in the time direction, and then to stretch spacetime along the vectorRelativistic MOND Theory 25by a factor e2φ while shrinking it by the same factor orthogonally to the vector.This does the trick: provided the scalar field φ used comes from a RAQUAL-typeequation, MOND phenomenology is recovered while the lensing is augmented tothe right proportions. However, this “Stratified” theory is not covariant (one needsthe vector to be in the time direction, and this cannot be true in all coordinatesystems). Covariance of equations of a physical theory is so ingrained in modernphysics that we have to look further.3 The TeVeS field theoryTeVeS is a tensor-vector-scalar covariant field theory (Bekenstein 2004; henceforthB04) incorporating various motifs already mentioned. Just as in RAQUAL, thetensor is an Einstein metric gαβ out of which is built the standard Einstein-Hilbertaction. One passes to the physical metric as in the stratified theory, but now usingan everywhere timelike dynamical normalized vector field Uα. In equationsg˜αβ = e−2φ(gαβ + UαUβ)− e2φUαUβ, (3.1)together with the normalization requirement gαβUαUβ = −1.The dynamics of Uα is dictated by the actionSv = − K32πG∫ [gαβgµνU[α,µ]U[β,ν] − 2(λ/K)(gµνUµUν + 1)](−g)1/2d4x, (3.2)where the square brackets denote antisymmetrization. The K here is a dimension-less coupling constant, one of the parameters of the theory. The λ is a Lagrangemultiplier field included to enforce the normalization of the vector field; λ is tobe determined as the equations are solved. The kinetic part of Sv is that of anAbelian gauge field, but the unit norm constraint deprives Uα of the proverbialgauge freedom. In addition, Uα is not coupled to a current of any sort, so it wouldbe incorrect to view the vector as mediating a repulsive Coulomb-like force.The dynamical scalar field φ required in Eq. (3.1) is assigned the actionSs = − 12∫ [σ2hαβφ,αφ,β +12Gℓ−2σ4F (kGσ2)](−g)1/2d4x, (3.3)which involves an auxiliary nondynamical scalar field σ, a new dimensionless pa-rameter, k, a scale parameter, ℓ, and an unspecified function F . In the kinetic partof Ss, φ,αφ,β is contracted not with gαβ , but with the tensor hαβ ≡ gαβ − UαUβ ;this is introduced to cope with the problem of superluminal propagation of scalarperturbations evidenced by RAQUAL. With hαβ the propagation is found to besubluminal with respect to the physical metric on condition that φ > 0 (B04).Contrary to appearances, Ss is not a quadratic action. First σ2 must be expressedin terms of φ,α by varying Ss with respect to σ2, and inverting the resulting equa-tion. When the result is substituted in both terms of Ss, the action becomesaquadratic in φ,α, resembling RAQUAL’s action, with the scalar hαβφ,α φ,β re-placing gαβφ,α φ,β . The form (3.3) is, however, more convenient for calculation.26 Mass Profiles and Shapes of Cosmological StructuresThese pieces of the total action must be supplemented with the matter actionSm. In accordance with the Einstein equivalence principle (Will 1993) it must beformed by replacing gαβ 7→ g˜αβ everywhere in the matter action in use in generalrelativity (GR), and modifying all covariant derivatives accordingly. In this wayall types of matter respond to gravitational fields in the same way, and it couldbe said that matter and radiation delineate the physical metric. Now because g˜αβcontains φ as well as Uα, the variations of the total action with respect to thesequantities engender variations of Sm which create sources for the equations of φand Uα. No direct coupling of these fields to the matter variables is thus required,which is pleasant as no principles are evident for determining such couplings.Is TeVeS unique ? Not at all. For example, one could choose to write Ssand Sv with g˜αβ instead of gαβ. The present approach seems preferable in thatevidently gαβ , φ and Uα are all gravitational fields, so their actions should notinvolve the “matter’s metric” g˜αβ. Further, one could add to Ss a kinetic termfor σ such as is included in Phase Coupled Gravity (PCG), a theory devised ina partially successful attempt to forestall superluminal propagation (Bekenstein1986, 1990). In addition one could redefine hαβ as gαβ − ζ Uα Uβ (ζ should bepositive but not small in order to obviate superluminality; it could be a constant).An additional freedom is the choice of function F in Eq. (3.3); each such choicedefines a distinct theory. Some of the freedoms mentioned have been used to makea variant of TeVeS (Sanders 2005a, 2005b; henceforth S05).4 Results from TeVeSTeVeS has GR as a limit for K → 0 and ℓ→∞ (B04). Thus if |K| ≪ 1 and thesize of the system of interest falls well short of ℓ, predictions of TeVeS and GRshould not be that different. To verify that TeVeS can also give MOND in someother limit one must make a choice for the function F in action (3.3). B04 makesthe choice which is equivalent to the relationkℓ2hµνφ,µφ,ν = (3/4)k2G2σ4(kGσ2 − 2)2(1− kGσ2)−1. (4.1)This is to be regarded as defining a toy theory. A better choice would have ther.h.s. change sign smoothly through kGσ2 = 1, a transition corresponding to thepassage between a quasistatic system such as a cluster of galaxies (hµνφ,µφ,ν > 0)and cosmology (hµνφ,µφ,ν < 0) (see B04).Imagine linearizing the Einstein-like equations of TeVeS about flat spacetime(gαβ = ηαβ+ · · ·) in a quasistatic situation. The final result for the physical metricafter suitable choice of units is (B04)ds2 = −(1 + 2Φ)dt2 + (1− 2Φ)δijdxidxj ; Φ ≡ ΞΦN + φ. (4.2)Here ΦN is the usual Newtonian potential of the baryonic matter and Ξ is a con-stant depending on K and on the asymptotic (cosmological) value of the scalarfield φ; Ξ is expected to be close to unity. With Φ interpreted as the physicalgravitational potential, metric (4.2) is precisely that used in GR to discuss bothRelativistic MOND Theory 27dynamics and gravitational lensing. An immediate conclusion is that if GR suc-ceeds in describing both the dynamics of, say, a galaxy, and its lensing propertieswith a single assumed DM distribution, and if TeVeS succeeds in predicting thedynamics everywhere from the observed baryonic distribution, then TeVeS willpredict the same lensing pattern as GR. Thus the problem that accosted RAQUALand variants of it (Sec. 2) is solved: lensing can be described using MOND ideas.This much was clear earlier from the non-covariant framework (Sanders 1997). Ofcourse the problem with galaxy cluster dynamics (Sec. 1) still looms. But it isimportant to realize that the two mentioned problems are distinct.Obviously, to determine Φ requires one to calculate φ from the baryonic massdistribution using the TeVeS analog of Eq. (1.2). In the linearized frameworkmentioned and assuming spherical symmetry, one can be immediately integratethe said equation once to get (we assume k ≪ 1 as well as Ξ ≈ 1)µ∇Φ = ∇ΦN , µ ≈ −1 +√1 + 4|∇Φ|/a01 +√1 + 4|∇Φ|/a0; a0 ≈√3k4πℓ. (4.3)Thus we get the MOND equation (1.1), as well as a formula for µ showing theappropriate asymptotic behavior at small and large |∇Φ|/a0, and the value of a0expressed in terms of the TeVeS parameters. MOND is recovered. Famaey andBinney (2005) have noted that the Galaxy’s rotation curve is not fully consistentwith this µ. Apparently a better F is needed.Fig. 1. |∇Φ| (dashed), |∇(Φ − ΦN )| (in units of 10−8 cms−2) with k = 0.03 (solid) ork = 0.1 (dotted) from TeVeS plotted vs the distance from the Sun. Observed constraintson the non-Newtonian part of the acceleration, |∇(Φ − ΦN )|, are (left to right): fromthe precession of perihelion of Mercury, and of Icarus, from variation of Kepler’s con-stant between Earth and Mars, between inner planets and Jupiter, Uranus or Neptune,respectively. The solid bar is the Pioneer anomaly range.The approximation inherent in Eq. (4.3) is good for all extragalactic applica-28 Mass Profiles and Shapes of Cosmological Structurestions, but begins to fail when |∇Φ| > (π/k)2a0. In such situations, e.g. the solarsystem, it is best to start from the first term of the Laurent expansion of the r.h.s.of Eq. (4.1) about kGσ2 = 1. One then obtainsµ ≈ 1Ξ + k/4π(1− 16π3k3a02|∇Φ|2). (4.4)The fact that µ asymptotes to (Ξ+k/4π)−1 as |∇Φ| → ∞means that for arbitrarilystrong fields gravity is Newtonian, except that the effective gravitational constantis GN = (Ξ + k/4π)G. It should be mentioned that the salient post-Newtonianparameters (Will 1993), important for inner solar system celestial dynamics, agreewith those of GR; in particular β = 1 and γ = 1 (Giannios 2004, B04).Notice the slow approach of µ to its asymptotic value as |∇Φ| strengthens.One consequence is a slow increase of the Kepler constant a3/P 2 with distancefrom the Sun. Fig. 1 illustrates this effect and compares it with the constraintsset by measurements of the orbits of diverse planets. Although a better formof F is clearly required, the significant point is that, in such theories, the totalforce in the outer Solar System will inevitably deviate from inverse square. Thisis of considerable interest in view of the Pioneer anomaly (Anderson et al. 1998,Turyshev 2005).ReferencesAnderson, J.D. et al., 1998, Phys.Rev.Lett., 81, 2858Bekenstein, J.D. & Milgrom, M. 1984, ApJ, 286, 7 (B&M)Bekenstein, J.D. 1988, in Second Canadian Conference on General Relativity and Rela-tivistic Astrophysics, eds. A. Coley, C. Dyer & T. Tupper (Singapore: World Scien-tific), 68.Bekenstein, J.D. 1990, in Developments in General Relativity, Astrophysics and QuantumTheory, eds. F. I. Cooperstock, L. P. Horwitz & J. Rosen (Bristol: IOP Publishing),156Bekenstein, J. D. & Sanders, R.H. 1994, ApJ, 429, 480Bekenstein, J.D. 2004, Phys. Rev. D, 70, 083509 (erratum, 2005, Phys. Rev. D, 71,069901) (B04)Famaey, B. & Binney, J. 2005, astro-ph/0506723Giannios, D. 2005 Phys. Rev. D, 71, 103511Milgrom, M. 1983, ApJ, 270, 365, 379 and 384Sanders, R.H.1997, ApJ, 480, 492Sanders R.H. & McGaugh, S.S. 2002, Ann. Rev. Astron. Astrophys., 40, 263 (S&McG)Sanders, R.H. 2005a, astro-ph/0502222Sanders, R.H. 2005b, this collection (S05)Turyshev, S., 2005, this collectionWill, C. 1993, Theory and Experiment in Gravitational Physics (Cambridge: CambridgeUniv. Press)

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Abstract: We first review the nonrelativistic lagrangian theory as a framework for the MOND equation. Obstructions to a relativistic version of it are discussed leading up to TeVeS, a relativistic tensor-vector-scalar field theory which displays both MOND and Newtonian limits. The whys for its particular structure are discussed and its achievements so far are summarized.

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A Primer to Relativistic MOND Theory

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