Tensor–vector–scalar gravity

MOND 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 $M$ at distance $r$ from the source can be written as

:$a\; =\; -\backslash frac\{GM\}\{r^2\},$

where $G$ is Newton's constant of gravitation. The corresponding force acting on a test mass $m$ is

:$F=ma.$

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

:$F=\backslash mu\; \backslash left\; (\backslash frac\{a\}\{a\_0\}\; \backslash right\; )ma,$

where $\backslash mu(x)$ is an arbitrary function subject to the following conditions:

:

In this form, MOND is not a complete theory: for instance, it violates the law of momentum conservation.

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

:$\{\backslash mathcal\; L\}=-\backslash frac\{a\_0^2\}\{8\backslash pi\; G\}f\backslash left(\backslash frac\{|\backslash nabla\backslash Phi|^2\}\{a\_0^2\}\backslash right)-\backslash rho\backslash Phi,$

where $\backslash Phi$ is the Newtonian gravitational potential, $\backslash rho$ is the mass density, and $f(y)$ is a dimensionless function.

In the case of a spherically symmetric, static gravitational field, this Lagrangian reproduces the MOND acceleration law after the substitutions $a=-\backslash nabla\backslash Phi$ and $\backslash mu(\backslash sqrt\{y\})=df(y)/dy$ 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 $g\_\{\backslash mu\backslash nu\}$, terms pertaining to a unit vector field $u^\backslash alpha$ and two scalar fields $\backslash sigma$ and $\backslash phi$, of which only $\backslash phi$ is dynamical. The TeVeS action, therefore, can be written as

:$S\_\backslash mathrm\{TeVeS\}=\backslash int\backslash left(\{\backslash mathcal\; L\}\_g+\{\backslash mathcal\; L\}\_s+\{\backslash mathcal\; L\}\_v\backslash right)d^4x.$

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

:$\{\backslash mathcal\; L\}\_g=-\backslash frac\{1\}\{16\backslash pi\; G\}R\backslash sqrt\{-g\},$

where $R$ is the Ricci scalar and $g$ is the determinant of the metric tensor.

The scalar field Lagrangian is

:$\{\backslash mathcal\; L\}\_s=-\backslash frac\{1\}\{2\}\backslash left[\backslash sigma^2h^\{\backslash alpha\backslash beta\}\backslash partial\_\backslash alpha\backslash phi\backslash partial\_\backslash beta\backslash phi+\backslash frac\{1\}\{2\}\backslash frac\{G\}\{l^2\}\backslash sigma^4F\; \backslash left\; (kG\backslash sigma^2\; \backslash right)\backslash right]\backslash sqrt\{-g\},$

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

:$\{\backslash mathcal\; L\}\_v=-\backslash frac\{K\}\{32\backslash pi\; G\}\backslash left[g^\{\backslash alpha\backslash beta\}g^\{\backslash mu\backslash nu\}\; \backslash left\; (B\_\{\backslash alpha\backslash mu\}B\_\{\backslash beta\backslash nu\}\; \backslash right\; )+2\backslash frac\{\backslash lambda\}\{K\}\; \backslash left\; (g^\{\backslash mu\backslash nu\}u\_\backslash mu\; u\_\backslash nu-1\; \backslash right\; )\backslash right]\backslash sqrt\{-g\}$

where $B\_\{\backslash alpha\backslash beta\}=\backslash partial\_\backslash alpha\; u\_\backslash beta-\backslash partial\_\backslash beta\; u\_\backslash alpha,$ while $K$ is a dimensionless parameter. $k$ and $K$ 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 $K=\backslash frac\{k\}\{2\backslash pi\}$,{{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 }}

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 $K=-30\; +\; \backslash frac\{72\; \backslash pi\}\{k\}.${{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 }} So the coupling constants read:

:$K=\; 3(\backslash pm\backslash sqrt\{29\}-5),\; \backslash qquad\; k\; =\; 6\backslash pi\; (\backslash pm\; \backslash sqrt\{29\}-5)$

The function $F$ in TeVeS is unspecified.

TeVeS also introduces a "physical metric" in the form

:$\{\backslash hat\; g\}^\{\backslash mu\backslash nu\}=e^\{2\backslash phi\}g^\{\backslash mu\backslash nu\}-2u^\backslash alpha\; u^\backslash beta\backslash sinh(2\backslash phi).$

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

:$S\_m=\backslash int\{\backslash mathcal\; L\}\; \backslash left\; (\{\backslash hat\; g\}\_\{\backslash mu\backslash nu\},f^\backslash alpha,f^\backslash alpha\_\{|\backslash mu\},\backslash ldots\; \backslash right)\backslash sqrt\{-\{\backslash hat\; g\}\}d^4x,$

where covariant derivatives with respect to $\{\backslash hat\; g\}\_\{\backslash mu\backslash nu\}$ are denoted by $|.$

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.

In addition to its ability to account for the 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{{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 }} shows that with Bekenstein's proposed parameters, a TeVeS star is highly unstable, on the scale of approximately 10⁶ seconds (two weeks). The ability of the theory to simultaneously account for galactic dynamics and lensing is also challenged.{{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 }} A possible resolution may be in the form of massive (around 2eV) neutrinos.{{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 }}

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.

{{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

}}

A quantity $E\_G$

{{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

}} 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{{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

|volume=464 |issue=7286 |pages=256–258

|arxiv=1003.2185

|bibcode=2010Natur.464..256R

|doi=10.1038/nature08857

|pmid=20220843

|s2cid=205219902 }} $E\_G=0.392\backslash pm\{0.065\}$ (~16%) consistent with GR, GR plus Lambda CDM and the extended form of GR known as $f(R)$ theory, but ruling out a particular TeVeS model predicting $E\_G=0.22$. 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.{{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 }}

• Gauge vector–tensor gravity
• Modified Newtonian dynamics
• Nonsymmetric gravitational theory
• Scalar–tensor–vector gravity

{{Reflist}}

• {{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)

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Category:Theories of gravity

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