# Scalar–tensor–vector gravity

Scalar–tensor–vector gravity (STVG)[1] is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario. The theory is also often referred to by the acronym MOG (MOdified Gravity).

## Overview

Scalar–tensor–vector gravity theory,[2] also known as MOdified Gravity (MOG), is based on an action principle and postulates the existence of a vector field, while elevating the three constants of the theory to scalar fields. In the weak-field approximation, STVG produces a Yukawa-like modification of the gravitational force due to a point source. Intuitively, this result can be described as follows: far from a source gravity is stronger than the Newtonian prediction, but at shorter distances, it is counteracted by a repulsive fifth force due to the vector field.

STVG has been used successfully to explain galaxy rotation curves,[3] the mass profiles of galaxy clusters,[4] gravitational lensing in the Bullet Cluster,[5] and cosmological observations[6] without the need for dark matter. On a smaller scale, in the Solar System, STVG predicts no observable deviation from general relativity.[7] The theory may also offer an explanation for the origin of inertia.[8]

## Mathematical details

STVG is formulated using the action principle. In the following discussion, a metric signature of ${\displaystyle [+,-,-,-]}$  will be used; the speed of light is set to ${\displaystyle c=1}$ , and we are using the following definition for the Ricci tensor:

${\displaystyle R_{\mu \nu }=\partial _{\alpha }\Gamma _{\mu \nu }^{\alpha }-\partial _{\nu }\Gamma _{\mu \alpha }^{\alpha }+\Gamma _{\mu \nu }^{\alpha }\Gamma _{\alpha \beta }^{\beta }-\Gamma _{\mu \beta }^{\alpha }\Gamma _{\alpha \nu }^{\beta }.}$

We begin with the Einstein-Hilbert Lagrangian:

${\displaystyle {\mathcal {L}}_{G}=-{\frac {1}{16\pi G}}(R+2\Lambda ){\sqrt {-g}},}$

where ${\displaystyle R}$  is the trace of the Ricci tensor, ${\displaystyle G}$  is the gravitational constant, ${\displaystyle g}$  is the determinant of the metric tensor ${\displaystyle g_{\mu \nu }}$ , while ${\displaystyle \Lambda }$  is the cosmological constant.

We introduce the Maxwell-Proca Lagrangian for the STVG vector field ${\displaystyle \phi _{\mu }}$ :

${\displaystyle {\mathcal {L}}_{\phi }=-{\frac {1}{4\pi }}\omega \left[{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }-{\frac {1}{2}}\mu ^{2}\phi _{\mu }\phi ^{\mu }+V_{\phi }(\phi )\right]{\sqrt {-g}},}$

where ${\displaystyle B_{\mu \nu }=\partial _{\mu }\phi _{\nu }-\partial _{\nu }\phi _{\mu },\mu }$  is the mass of the vector field, ${\displaystyle \omega }$  characterizes the strength of the coupling between the fifth force and matter, and ${\displaystyle V_{\phi }}$  is a self-interaction potential.

The three constants of the theory, ${\displaystyle G,\mu ,}$  and ${\displaystyle \omega ,}$  are promoted to scalar fields by introducing associated kinetic and potential terms in the Lagrangian density:

${\displaystyle {\mathcal {L}}_{S}=-{\frac {1}{G}}\left[{\frac {1}{2}}g^{\mu \nu }\left({\frac {\nabla _{\mu }G\nabla _{\nu }G}{G^{2}}}+{\frac {\nabla _{\mu }\mu \nabla _{\nu }\mu }{\mu ^{2}}}-\nabla _{\mu }\omega \nabla _{\nu }\omega \right)+{\frac {V_{G}(G)}{G^{2}}}+{\frac {V_{\mu }(\mu )}{\mu ^{2}}}+V_{\omega }(\omega )\right]{\sqrt {-g}},}$

where ${\displaystyle \nabla _{\mu }}$  denotes covariant differentiation with respect to the metric ${\displaystyle g_{\mu \nu },}$  while ${\displaystyle V_{G},V_{\mu },}$  and ${\displaystyle V_{\omega }}$  are the self-interaction potentials associated with the scalar fields.

The STVG action integral takes the form

${\displaystyle S=\int {({\mathcal {L}}_{G}+{\mathcal {L}}_{\phi }+{\mathcal {L}}_{S}+{\mathcal {L}}_{M})}~d^{4}x,}$

where ${\displaystyle {\mathcal {L}}_{M}}$  is the ordinary matter Lagrangian density.

## Spherically symmetric, static vacuum solution

The field equations of STVG can be developed from the action integral using the variational principle. First a test particle Lagrangian is postulated in the form

${\displaystyle {\mathcal {L}}_{\mathrm {TP} }=-m+\alpha \omega q_{5}\phi _{\mu }u^{\mu },}$

where ${\displaystyle m}$  is the test particle mass, ${\displaystyle \alpha }$  is a factor representing the nonlinearity of the theory, ${\displaystyle q_{5}}$  is the test particle's fifth-force charge, and ${\displaystyle u^{\mu }=dx^{\mu }/ds}$  is its four-velocity. Assuming that the fifth-force charge is proportional to mass, i.e., ${\displaystyle q_{5}=\kappa m,}$  the value of ${\displaystyle \kappa ={\sqrt {G_{N}/\omega }}}$  is determined and the following equation of motion is obtained in the spherically symmetric, static gravitational field of a point mass of mass ${\displaystyle M}$ :

${\displaystyle {\ddot {r}}=-{\frac {G_{N}M}{r^{2}}}\left[1+\alpha -\alpha (1+\mu r)e^{-\mu r}\right],}$

where ${\displaystyle G_{N}}$  is Newton's constant of gravitation. Further study of the field equations allows a determination of ${\displaystyle \alpha }$  and ${\displaystyle \mu }$  for a point gravitational source of mass ${\displaystyle M}$  in the form[9]

${\displaystyle \mu ={\frac {D}{\sqrt {M}}},}$
${\displaystyle \alpha ={\frac {G_{\infty }-G_{N}}{G_{N}}}{\frac {M}{({\sqrt {M}}+E)^{2}}},}$

where ${\displaystyle G_{\infty }\simeq 20G_{N}}$  is determined from cosmological observations, while for the constants ${\displaystyle D}$  and ${\displaystyle E}$  galaxy rotation curves yield the following values:

${\displaystyle D\simeq 6250M_{\odot }^{1/2}\mathrm {kpc} ^{-1},}$
${\displaystyle E\simeq 25000M_{\odot }^{1/2},}$

where ${\displaystyle M_{\odot }}$  is the mass of the Sun. These results form the basis of a series of calculations that are used to confront the theory with observation.

## Observations

STVG/MOG has been applied successfully to a range of astronomical, astrophysical, and cosmological phenomena.

On the scale of the Solar System, the theory predicts no deviation[7] from the results of Newton and Einstein. This is also true for star clusters containing no more than a maximum of a few million solar masses.

The theory accounts for the rotation curves of spiral galaxies,[3] correctly reproducing the Tully-Fisher law.[9]

STVG is in good agreement with the mass profiles of galaxy clusters.[4]

STVG can also account for key cosmological observations, including:[6]

## Where this Theory Succeeds and Fails

The weak field limit of this theory predicts an enhanced gravitational attraction on the boundaries of galaxies, where phenomena related to dark matter use to happen and agrees with General Relativity inward. The behavior of this weak field limit served, for instance, to correctly describe galaxy rotation curves, galactic light bending, and the Bullet Cluster phenomena, without requiring the existence of dark matter. The transition of a standard to enhanced gravitational attraction comes from the interplay between tensor, vector and scalar physical fields. However, this mechanism seems to work solely where gravity is weak. Close to black holes or other compact objects like neutron stars, the gravitational field is very strong and Moffat’s mechanism to retrieve General Relativity breaks.[10][11][12]

## References

1. ^ McKee, M. (25 January 2006). "Gravity theory dispenses with dark matter". New Scientist. Retrieved 2008-07-26.
2. ^ Moffat, J. W. (2006). "Scalar-Tensor-Vector Gravity Theory". Journal of Cosmology and Astroparticle Physics. 3 (3): 4. arXiv:gr-qc/0506021. Bibcode:2006JCAP...03..004M. doi:10.1088/1475-7516/2006/03/004.
3. ^ a b Brownstein, J. R.; Moffat, J. W. (2006). "Galaxy Rotation Curves Without Non-Baryonic Dark Matter". Astrophysical Journal. 636 (2): 721–741. arXiv:astro-ph/0506370. Bibcode:2006ApJ...636..721B. doi:10.1086/498208.
4. ^ a b Brownstein, J. R.; Moffat, J. W. (2006). "Galaxy Cluster Masses Without Non-Baryonic Dark Matter". Monthly Notices of the Royal Astronomical Society. 367 (2): 527–540. arXiv:astro-ph/0507222. Bibcode:2006MNRAS.367..527B. doi:10.1111/j.1365-2966.2006.09996.x.
5. ^ Brownstein, J. R.; Moffat, J. W. (2007). "The Bullet Cluster 1E0657-558 evidence shows Modified Gravity in the absence of Dark Matter". Monthly Notices of the Royal Astronomical Society. 382 (1): 29–47. arXiv:astro-ph/0702146. Bibcode:2007MNRAS.382...29B. doi:10.1111/j.1365-2966.2007.12275.x.
6. ^ a b Moffat, J. W.; Toth, V. T. (2007). "Modified Gravity: Cosmology without dark matter or Einstein's cosmological constant". arXiv:0710.0364 [astro-ph].
7. ^ a b Moffat, J. W.; Toth, V. T. (2008). "Testing modified gravity with globular cluster velocity dispersions". Astrophysical Journal. 680 (2): 1158–1161. arXiv:0708.1935. Bibcode:2008ApJ...680.1158M. doi:10.1086/587926.
8. ^ Moffat, J. W.; Toth, V. T. (2009). "Modified gravity and the origin of inertia". Monthly Notices of the Royal Astronomical Society Letters. 395 (1): L25. arXiv:0710.3415. Bibcode:2009MNRAS.395L..25M. doi:10.1111/j.1745-3933.2009.00633.x.
9. ^ a b Moffat, J. W.; Toth, V. T. (2009). "Fundamental parameter-free solutions in Modified Gravity". Classical and Quantum Gravity. 26 (8): 085002. arXiv:0712.1796. Bibcode:2009CQGra..26h5002M. doi:10.1088/0264-9381/26/8/085002.
10. ^ g. Lopez Armengol, Federico (2018). "Exploring the Discrepancies Between General Relativity and Scalar-Tensor-Vector Gravity Theory". Science Trends. doi:10.31988/SciTrends.8993.
11. ^ Lopez Armengol, Federico G.; Romero, Gustavo E. (2017). "Effects of Scalar-Tensor-Vector Gravity on relativistic jets". Astrophysics and Space Science. 362 (11). arXiv:1611.09918. doi:10.1007/s10509-017-3197-6.