View source for Tensor–vector–scalar gravity

{{distinguish|Scalar–tensor–vector gravity|Bi-scalar tensor vector gravity}}
{{short description|Relativistic generalization of Mordehai Milgrom's MOND paradigm}}
'''Tensor–vector–scalar gravity''' ('''TeVeS'''),<ref name=Bekenstein2004>
{{Citation
 |last1=Bekenstein |first1=J. D.
 |date=2004
 |title=Relativistic gravitation theory for the modified Newtonian dynamics paradigm
 |journal=[[Physical Review D]]
 |volume=70 |issue=8 |pages=083509
 |arxiv=astro-ph/0403694 
 |doi=10.1103/PhysRevD.70.083509
 |bibcode=2004PhRvD..70h3509B
}}</ref> developed by [[Jacob Bekenstein]] in 2004, is a relativistic generalization of [[Mordehai Milgrom]]'s [[Modified Newtonian dynamics]] (MOND) paradigm.<ref name=Milgrom1983>
{{Citation
 |last1=Milgrom |first1=M.
 |date=1983
 |title=A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis
 |journal=[[The Astrophysical Journal]]
 |volume=270 |pages=365–370
 |bibcode=1983ApJ...270..365M
 |doi=10.1086/161130
|doi-access=free
 }}</ref><ref>{{Citation
 |last1=Famaey |first1=B.
 |last2=McGaugh |first2=S. S. |author-link2=Stacy McGaugh
 |date=2012
 |title=Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions
 |journal=Living Rev. Relativ.
 |volume=15 |issue=10
 |pages=10
 |doi=10.12942/lrr-2012-10
 |doi-access=free
 |pmid=28163623
 |pmc=5255531
 |issn=1433-8351
|arxiv = 1112.3960 |bibcode = 2012LRR....15...10F }}</ref>

The main features of TeVeS can be summarized as follows:
* As it is derived from the [[action principle]], TeVeS respects [[conservation laws]];
* In the [[weak-field approximation]] of the spherically symmetric, static solution, TeVeS reproduces the MOND acceleration formula;
* TeVeS avoids the problems of earlier attempts to generalize MOND, such as [[superluminal]] propagation;
* As it is a relativistic theory it can accommodate [[gravitational lens]]ing.

The theory is based on the following ingredients:
* A unit [[vector field]];
* A dynamical [[scalar field]];
* A nondynamical scalar field;
* A matter [[Lagrangian (field theory)|Lagrangian]] constructed using an alternate [[Metric (general relativity)|metric]];
* An arbitrary dimensionless function.

These components are combined into a relativistic [[Lagrangian density]], which forms the basis of TeVeS theory.

==Details==
MOND<ref name=Milgrom1983/> is a phenomenological modification of the Newtonian acceleration law. In [[Newtonian gravity]] theory, the gravitational acceleration in the spherically symmetric, static field of a point mass <math>M</math> at distance <math>r</math> from the source can be written as

:<math>a = -\frac{GM}{r^2},</math>

where <math>G</math> is [[Newton's constant]] of gravitation. The corresponding force acting on a test mass <math>m</math> is

:<math>F=ma.</math>

To account for the anomalous rotation curves of spiral galaxies, Milgrom proposed a modification of this force law in the form

:<math> F=\mu \left (\frac{a}{a_0} \right )ma,</math>

where <math>\mu(x)</math> is an arbitrary function subject to the following conditions:

:<math>\mu(x)= \begin{cases} 1 & |x|\gg 1 \\ x & |x|\ll 1 \end{cases} </math>

In this form, MOND is not a complete theory: for instance, it violates the law of [[Momentum#Conservation of linear momentum|momentum conservation]].

However, such conservation laws are automatically satisfied for physical theories that are derived using an action principle. This led Bekenstein<ref name=Bekenstein2004/> to a first, nonrelativistic generalization of MOND. This theory, called [[AQUAL]] (for A QUAdratic Lagrangian) is based on the Lagrangian

:<math>{\mathcal L}=-\frac{a_0^2}{8\pi G}f\left(\frac{|\nabla\Phi|^2}{a_0^2}\right)-\rho\Phi,</math>

where <math>\Phi</math> is the Newtonian gravitational potential, <math>\rho</math> is the mass density, and <math>f(y)</math> is a dimensionless function.

In the case of a spherically symmetric, static gravitational field, this Lagrangian reproduces the MOND acceleration law after the substitutions <math>a=-\nabla\Phi</math> and <math>\mu(\sqrt{y})=df(y)/dy</math> are made.

Bekenstein further found that AQUAL can be obtained as the nonrelativistic limit of a relativistic field theory. This theory is written in terms of a Lagrangian that contains, in addition to the [[Einstein–Hilbert action]] for the metric field <math>g_{\mu\nu}</math>, terms pertaining to a unit vector field <math>u^\alpha</math> and two scalar fields <math>\sigma</math> and <math>\phi</math>, of which only <math>\phi</math> is dynamical. The TeVeS action, therefore, can be written as

:<math>S_\mathrm{TeVeS}=\int\left({\mathcal L}_g+{\mathcal L}_s+{\mathcal L}_v\right)d^4x.</math>

The terms in this action include the [[Einstein–Hilbert action|Einstein–Hilbert]] Lagrangian (using a metric signature <math>[+,-,-,-]</math> and setting the speed of light, <math>c=1</math>):

:<math>{\mathcal L}_g=-\frac{1}{16\pi G}R\sqrt{-g},</math>

where <math>R</math> is the [[Ricci scalar]] and <math>g</math> is the determinant of the metric tensor.

The scalar field Lagrangian is

:<math>{\mathcal L}_s=-\frac{1}{2}\left[\sigma^2h^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi+\frac{1}{2}\frac{G}{l^2}\sigma^4F \left (kG\sigma^2 \right)\right]\sqrt{-g},</math>

where <math>h^{\alpha\beta}=g^{\alpha\beta}-u^\alpha u^\beta, l</math> is a constant length, <math> k</math> is the dimensionless parameter and <math>F</math> an unspecified dimensionless function; while the vector field Lagrangian is

:<math>{\mathcal L}_v=-\frac{K}{32\pi G}\left[g^{\alpha\beta}g^{\mu\nu} \left (B_{\alpha\mu}B_{\beta\nu} \right )+2\frac{\lambda}{K} \left (g^{\mu\nu}u_\mu u_\nu-1 \right )\right]\sqrt{-g}</math>

where <math>B_{\alpha\beta}=\partial_\alpha u_\beta-\partial_\beta u_\alpha,</math> while <math>K</math> is a dimensionless parameter. <math>k</math> and <math>K</math> are respectively called the scalar and vector coupling constants of the theory.  The consistency between the [[Gravitoelectromagnetism]] of  the TeVeS theory and that predicted and measured by the [[Gravity Probe B]] leads to <math>K=\frac{k}{2\pi}</math>,<ref name=Exirifard:2011vb>{{Citation |last1=Exirifard |first1=Q.  |date=2013 |title=GravitoMagnetic Field in Tensor-Vector-Scalar Theory
 |journal=[[Journal of Cosmology and Astroparticle Physics]] |volume=JCAP04 |issue=4 |pages=034 |arxiv=1111.5210 |bibcode=2013JCAP...04..034E |doi=10.1088/1475-7516/2013/04/034|s2cid=250745786 }}</ref>
and requiring consistency between the near horizon geometry of a black hole in TeVeS and that of the Einstein theory, as observed by the [[Event Horizon Telescope]]  leads to  <math>K=-30 + \frac{72 \pi}{k}.</math><ref name=Exirifard:2011vbA1>{{Citation |last1=Exirifard |first1=Q.  |date=2019 |title=Addendum: GravitoMagnetic field in tensor-vector-scalar theory
 |journal=[[Journal of Cosmology and Astroparticle Physics]] |volume=JCAP05 |issue=5 |pages=A01 |doi=10.1088/1475-7516/2019/05/A01|arxiv=1111.5210|s2cid=182361144 }}</ref> So the coupling constants read:

:<math>K= 3(\pm\sqrt{29}-5), \qquad k = 6\pi (\pm \sqrt{29}-5)</math>

The function <math>F</math> in TeVeS is unspecified.

TeVeS also introduces a "physical metric" in the form

:<math>{\hat g}^{\mu\nu}=e^{2\phi}g^{\mu\nu}-2u^\alpha u^\beta\sinh(2\phi).</math>

The action of ordinary matter is defined using the physical metric:

:<math>S_m=\int{\mathcal L} \left ({\hat g}_{\mu\nu},f^\alpha,f^\alpha_{|\mu},\ldots \right)\sqrt{-{\hat g}}d^4x,</math>

where covariant derivatives with respect to <math>{\hat g}_{\mu\nu}</math> are denoted by <math>|.</math>

TeVeS solves problems associated with earlier attempts to generalize MOND, such as superluminal propagation. In his paper, Bekenstein also investigated the consequences of TeVeS in relation to gravitational lensing and cosmology.

==Problems and criticisms==
In addition to its ability to account for the [[galaxy rotation curve|flat rotation curves]] of galaxies (which is what MOND was originally designed to address), TeVeS is claimed to be consistent with a range of other phenomena, such as [[gravitational lensing]] and cosmological observations. However, Seifert<ref name=Seifert2007>{{Citation |last1=Seifert |first1=M. D. |date=2007 |title=Stability of spherically symmetric solutions in modified theories of gravity |journal=[[Physical Review D]] |volume=76 |issue=6 |pages=064002 |arxiv=gr-qc/0703060 |bibcode=2007PhRvD..76f4002S
 |doi=10.1103/PhysRevD.76.064002|s2cid=29014948 }}</ref> shows that with Bekenstein's proposed parameters, a TeVeS star is highly unstable, on the scale of approximately 10<sup>6</sup> seconds (two weeks). The ability of the theory to simultaneously account for galactic dynamics and lensing is also challenged.<ref name=Mavromatos2009>{{Citation |last1=Mavromatos |first1=Nick E. |last2=Sakellariadou |first2=Mairi |last3=Yusaf |first3=Muhammad Furqaan |date=2009 |title=Can TeVeS avoid Dark Matter on galactic scales? |journal=[[Physical Review D]] |volume=79 |issue=8 |pages=081301 |arxiv=0901.3932 |bibcode=2009PhRvD..79h1301M |doi=10.1103/PhysRevD.79.081301|s2cid=119249051 }}</ref> A possible resolution may be in the form of massive (around 2eV) [[neutrino]]s.<ref name=Angus2007>{{Citation |last1=Angus |first1=G. W. |last2=Shan |first2=H. Y. |last3=Zhao |first3=H. S. |last4=Famaey |first4=B. |date=2007 |title=On the Proof of Dark Matter, the Law of Gravity, and the Mass of Neutrinos |journal=[[The Astrophysical Journal Letters]] |volume=654 |issue=1 |pages=L13–L16 |arxiv=astro-ph/0609125 |doi=10.1086/510738  |bibcode=2007ApJ...654L..13A
|s2cid=17977472 }}</ref>

A study in August 2006 reported an observation of a pair of colliding galaxy clusters, the [[Bullet Cluster]], whose behavior, it was reported, was not compatible with any current modified gravity theory.<ref name=Clowe>
{{Citation
 |last1=Clowe |first1=D.
 |last2=Bradač |first2=M.
 |last3=Gonzalez |first3=A. H.
 |last4=Markevitch |first4=M.
 |last5=Randall |first5=S. W.
 |last6=Jones |first6=C.
 |last7=Zaritsky |first7=D.
 |date=2006
 |title=A Direct Empirical Proof of the Existence of Dark Matter
 |journal=[[The Astrophysical Journal Letters]]
 |volume=648 |issue=2 |pages=L109
 |arxiv=astro-ph/0608407
 |bibcode=2006ApJ...648L.109C
 |doi=10.1086/508162
|s2cid=2897407
 }}</ref>

A quantity <math>E_G</math> <ref>
{{Citation
 |last1=Zhang |first1=P.
 |last2=Liguori |first2=M.
 |last3=Bean |first3=R.|author-link3=Rachel Bean
 |last4=Dodelson |first4=S.
 |date=2007
 |title=Probing Gravity at Cosmological Scales by Measurements which Test the Relationship between Gravitational Lensing and Matter Overdensity
 |journal=[[Physical Review Letters]]
 |volume=99 |issue=14 |pages=141302
 |arxiv=0704.1932
 |bibcode=2007PhRvL..99n1302Z
 |doi=10.1103/PhysRevLett.99.141302 |pmid=17930657
|s2cid=119672184
 }}</ref> probing [[general relativity]] (GR) on large scales (a hundred billion times the size of the solar system) for the first time has been measured with data from the [[Sloan Digital Sky Survey]] to be<ref>{{Citation |last1=Reyes |first1=R. |last2=Mandelbaum |first2=R. |last3=Seljak |first3=U. |last4=Baldauf |first4=T.
 |last5=Gunn |first5=J. E.
 |last6=Lombriser |first6=L.
 |last7=Smith |first7=R. E.
 |date=2010
 |title=Confirmation of general relativity on large scales from weak lensing and galaxy velocities
 |journal=[[Nature (journal)|Nature]]
 |volume=464 |issue=7286 |pages=256–258
 |arxiv=1003.2185
 |bibcode=2010Natur.464..256R
 |doi=10.1038/nature08857
 |pmid=20220843
|s2cid=205219902 }}</ref> <math>E_G=0.392\pm{0.065}</math> (~16%) consistent with GR, GR plus [[Lambda-CDM model|Lambda CDM]] and the extended form of GR known as [[F(R) gravity|<math>f(R)</math> theory]], but ruling out a particular TeVeS model predicting <math>E_G=0.22</math>. This estimate should improve to ~1% with the next generation of sky surveys and may put tighter constraints on the parameter space of all modified gravity theories.

TeVeS appears inconsistent with recent measurements made by LIGO of gravitational waves.<ref>{{citation|last1=Boran|first1=Sibel|last2=Desai|first2=Shantanu|last3=Kahya|first3=Emre|last4=Woodard|first4=Richard|year=2018|title=GW170817 Falsifies Dark Matter Emulators|journal=Physical Review D|volume=97|issue=4|pages=041501|arxiv=1710.06168|doi=10.1103/PhysRevD.97.041501|bibcode=2018PhRvD..97d1501B|s2cid=119468128 }}</ref>

==See also==
* [[Gauge vector–tensor gravity]]
* [[Modified Newtonian dynamics]]
* [[Nonsymmetric gravitational theory]]
* [[Scalar–tensor–vector gravity]]

==References==
{{Reflist}}

==Further reading==
*{{citation |last1=Bekenstein |first1=J. D. |last2=Sanders |first2=R. H. |date=2006 |title=A Primer to Relativistic MOND Theory |journal=[[EAS Publications Series]] |volume=20 |pages=225–230  |arxiv=astro-ph/0509519 |bibcode=2006EAS....20..225B |doi=10.1051/eas:2006075|s2cid=6539084 }}
*{{citation |last1=Zhao |first1=H. S. |last2=Famaey |first2=B. |date=2006 |title=Refining the MOND Interpolating Function and TeVeS Lagrangian |journal=[[The Astrophysical Journal]] |volume=638 |issue=1 |pages=L9–L12 |arxiv=astro-ph/0512425 |bibcode=2006ApJ...638L...9Z |doi=10.1086/500805|s2cid=14867245 }}
* [http://today.slac.stanford.edu/feature/darkmatter.asp Dark Matter Observed] ([[SLAC]] Today)
* [https://web.archive.org/web/20060528044707/http://www.pparc.ac.uk/Nw/EinsteinTheory.asp Einstein's Theory 'Improved'?] ([[PPARC]])
* [http://www.space.com/scienceastronomy/general-relativity-confirmed-100310.html Einstein Was Right: General Relativity Confirmed] ' TeVeS, however, made predictions that fell outside the observational error limits', ([[Space.com]])

{{theories of gravitation}}

{{DEFAULTSORT:Tensor-Vector-Scalar Gravity}}
[[Category:Theories of gravity]]
[[Category:Astrophysics]]