# Tensor–vector–scalar gravity

Tensor–vector–scalar gravity (TeVeS),[1] developed by Jacob Bekenstein in 2004, is a relativistic generalization of Mordehai Milgrom's Modified Newtonian dynamics (MOND) paradigm.[2][3]

The main features of TeVeS can be summarized as follows:

The theory is based on the following ingredients:

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

## Details

MOND[2] 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 ${\displaystyle M}$  at distance ${\displaystyle r}$  from the source can be written as

${\displaystyle a=-{\frac {GM}{r^{2}}},}$

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

${\displaystyle F=ma.}$

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

${\displaystyle F=\mu \left({\frac {a}{a_{0}}}\right)ma,}$

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

${\displaystyle \mu (x)={\begin{cases}1&|x|\gg 1\\x&|x|\ll 1\end{cases}}}$

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[1] to a first, nonrelativistic generalization of MOND. This theory, called AQUAL (for A QUAdratic Lagrangian) is based on the Lagrangian

${\displaystyle {\mathcal {L}}=-{\frac {a_{0}^{2}}{8\pi G}}f\left({\frac {|\nabla \Phi |^{2}}{a_{0}^{2}}}\right)-\rho \Phi ,}$

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

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

${\displaystyle S_{\mathrm {TeVeS} }=\int \left({\mathcal {L}}_{g}+{\mathcal {L}}_{s}+{\mathcal {L}}_{v}\right)d^{4}x.}$

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

${\displaystyle {\mathcal {L}}_{g}=-{\frac {1}{16\pi G}}R{\sqrt {-g}},}$

where ${\displaystyle R}$  is the Ricci scalar and ${\displaystyle g}$  is the determinant of the metric tensor.

The scalar field Lagrangian is

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

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

${\displaystyle {\mathcal {L}}_{v}=-{\frac {K}{32\pi G}}\left[g^{\alpha \beta }g^{\mu \nu }\left(B_{\alpha \mu }B_{\beta \nu }\right)+2{\frac {\lambda }{K}}\left(g^{\mu \nu }u_{\mu }u_{\nu }-1\right)\right]{\sqrt {-g}}}$

where ${\displaystyle B_{\alpha \beta }=\partial _{\alpha }u_{\beta }-\partial _{\beta }u_{\alpha },}$  while ${\displaystyle K}$  is a dimensionless parameter. ${\displaystyle k}$  and ${\displaystyle K}$  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 ${\displaystyle K={\frac {k}{2\pi }}}$ ,[4] 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 ${\displaystyle K=-30+{\frac {72\pi }{\kappa }}.}$ [5] So the coupling constants read:

${\displaystyle K=3(\pm {\sqrt {29}}-5),\kappa =6\pi (\pm {\sqrt {29}}-5)}$

The function ${\displaystyle F}$  in TeVeS is unspecified.

TeVeS also introduces a "physical metric" in the form

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

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

${\displaystyle S_{m}=\int {\mathcal {L}}\left({\hat {g}}_{\mu \nu },f^{\alpha },f_{|\mu }^{\alpha },\ldots \right){\sqrt {-{\hat {g}}}}d^{4}x,}$

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

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 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[6] 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.[7] A possible resolution may be in the form of massive (around 2eV) neutrinos.[8]

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 theories.[9]

A quantity ${\displaystyle E_{G}}$  [10] 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[11] ${\displaystyle E_{G}=0.392\pm {0.065}}$  (~16%) consistent with GR, GR plus Lambda CDM and the extended form of GR known as ${\displaystyle f(R)}$  theory, but ruling out a particular TeVeS model predicting ${\displaystyle E_{G}=0.22}$ . 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.[12]

## References

1. ^ a b Bekenstein, J. D. (2004), "Relativistic gravitation theory for the modified Newtonian dynamics paradigm", Physical Review D, 70 (8): 083509, arXiv:astro-ph/0403694, Bibcode:2004PhRvD..70h3509B, doi:10.1103/PhysRevD.70.083509
2. ^ a b Milgrom, M. (1983), "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis", The Astrophysical Journal, 270: 365–370, Bibcode:1983ApJ...270..365M, doi:10.1086/161130
3. ^ Famaey, B.; McGaugh, S. S. (2012), "Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions", Living Rev. Relativ., 15 (10): 10, arXiv:1112.3960, Bibcode:2012LRR....15...10F, doi:10.12942/lrr-2012-10, ISSN 1433-8351, PMC 5255531, PMID 28163623
4. ^ Exirifard, Q. (2013), "GravitoMagnetic Field in Tensor-Vector-Scalar Theory", Journal of Cosmology and Astroparticle Physics, JCAP04: 034, arXiv:1111.5210, Bibcode:2013JCAP...04..034E, doi:10.1088/1475-7516/2013/04/034
5. ^ Exirifard, Q. (2019), "Addendum: GravitoMagnetic field in tensor-vector-scalar theory", Journal of Cosmology and Astroparticle Physics, JCAP05: A01, arXiv:1111.5210, doi:10.1088/1475-7516/2019/05/A01
6. ^ Seifert, M. D. (2007), "Stability of spherically symmetric solutions in modified theories of gravity", Physical Review D, 76 (6): 064002, arXiv:gr-qc/0703060, Bibcode:2007PhRvD..76f4002S, doi:10.1103/PhysRevD.76.064002
7. ^ Mavromatos, Nick E.; Sakellariadou, Mairi; Yusaf, Muhammad Furqaan (2009), "Can TeVeS avoid Dark Matter on galactic scales?", Physical Review D, 79 (8): 081301, arXiv:0901.3932, Bibcode:2009PhRvD..79h1301M, doi:10.1103/PhysRevD.79.081301
8. ^ Angus, G. W.; Shan, H. Y.; Zhao, H. S.; Famaey, B. (2007), "On the Proof of Dark Matter, the Law of Gravity, and the Mass of Neutrinos", The Astrophysical Journal Letters, 654 (1): L13–L16, arXiv:astro-ph/0609125, Bibcode:2007ApJ...654L..13A, doi:10.1086/510738
9. ^ Clowe, D.; Bradač, M.; Gonzalez, A. H.; Markevitch, M.; Randall, S. W.; Jones, C.; Zaritsky, D. (2006), "A Direct Empirical Proof of the Existence of Dark Matter", The Astrophysical Journal Letters, 648 (2): L109, arXiv:astro-ph/0608407, Bibcode:2006ApJ...648L.109C, doi:10.1086/508162
10. ^ Zhang, P.; Liguori, M.; Bean, R.; Dodelson, S. (2007), "Probing Gravity at Cosmological Scales by Measurements which Test the Relationship between Gravitational Lensing and Matter Overdensity", Physical Review Letters, 99 (14): 141302, arXiv:0704.1932, Bibcode:2007PhRvL..99n1302Z, doi:10.1103/PhysRevLett.99.141302, PMID 17930657
11. ^ Reyes, R.; Mandelbaum, R.; Seljak, U.; Baldauf, T.; Gunn, J. E.; Lombriser, L.; Smith, R. E. (2010), "Confirmation of general relativity on large scales from weak lensing and galaxy velocities", Nature, 464 (7286): 256–258, arXiv:1003.2185, Bibcode:2010Natur.464..256R, doi:10.1038/nature08857, PMID 20220843
12. ^ Boran, Sibel; Desai, Shantanu; Kahya, Emre; Woodard, Richard (2018). "GW170817 Falsifies Dark Matter Emulators". Physical Review D. 97 (4): 041501. arXiv:1710.06168. Bibcode:2018PhRvD..97d1501B. doi:10.1103/PhysRevD.97.041501.