Scalar–tensor–vector gravity

Not to be confused with Tensor–vector–scalar gravity or Bi-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 $[+,-,-,-]$ will be used; the speed of light is set to $c=1$ , and we are using the following definition for the Ricci tensor: $R_{{\mu \nu }}=\partial _{\alpha }\Gamma _{{\mu \nu }}^{\alpha }-\partial _{\nu }\Gamma _{{\mu \alpha }}^{\alpha }+\Gamma _{{\mu \nu }}^{\alpha }\Gamma _{{\alpha \beta }}^{\beta }-\Gamma _{{\mu \beta }}^{\alpha }\Gamma _{{\alpha \nu }}^{\beta }.$

We begin with the Einstein-Hilbert Lagrangian:

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

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

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

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

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

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

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

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

The STVG action integral takes the form

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

where ${{\mathcal L}}_{M}$ 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

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

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

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

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

$\mu ={\frac {D}{{\sqrt {M}}}},$

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

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

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

$E\simeq 25000M_{\odot }^{{1/2}},$

where $M_\odot$ 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]

The acoustic peaks in the cosmic microwave background radiation;
The accelerating expansion of the universe that is apparent from type Ia supernova observations;
The matter power spectrum of the universe that is observed in the form of galaxy-galaxy correlations.

References

↑ McKee, M. (25 January 2006). "Gravity theory dispenses with dark matter". New Scientist. Retrieved 2008-07-26.
↑ Moffat, J. W. (2006). "Scalar-Tensor-Vector Gravity Theory". Journal of Cosmology and Astroparticle Physics. 3: 4. arXiv:gr-qc/0506021. Bibcode:2006JCAP...03..004M. doi:10.1088/1475-7516/2006/03/004.
1 2 Brownstein, J. R.; Moffat, J. W. (2006). "Galaxy Rotation Curves Without Non-Baryonic Dark Matter". Astrophysical Journal. 636: 721–741. arXiv:astro-ph/0506370. Bibcode:2006ApJ...636..721B. doi:10.1086/498208.
1 2 Brownstein, J. R.; Moffat, J. W. (2006). "Galaxy Cluster Masses Without Non-Baryonic Dark Matter". Monthly Notices of the Royal Astronomical Society. 367: 527–540. arXiv:astro-ph/0507222. Bibcode:2006MNRAS.367..527B. doi:10.1111/j.1365-2966.2006.09996.x.
↑ 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: 29–47. arXiv:astro-ph/0702146. Bibcode:2007MNRAS.382...29B. doi:10.1111/j.1365-2966.2007.12275.x.
1 2 Moffat, J. W.; Toth, V. T. (2007). "Modified Gravity: Cosmology without dark matter or Einstein's cosmological constant". arXiv:0710.0364 [astro-ph].
1 2 Moffat, J. W.; Toth, V. T. (2008). "Testing modified gravity with globular cluster velocity dispersions". Astrophysical Journal. 680: 1158–1161. arXiv:0708.1935. Bibcode:2008ApJ...680.1158M. doi:10.1086/587926.
↑ Moffat, J. W.; Toth, V. T. (2009). "Modified gravity and the origin of inertia". Monthly Notices of the Royal Astronomical Society Letters. 395: L25. arXiv:0710.3415. Bibcode:2009MNRAS.395L..25M. doi:10.1111/j.1745-3933.2009.00633.x.
1 2 Moffat, J. W.; Toth, V. T. (2009). "Fundamental parameter-free solutions in Modified Gravity". Classical and Quantum Gravity. 26: 085002. arXiv:0712.1796. Bibcode:2009CQGra..26h5002M. doi:10.1088/0264-9381/26/8/085002.

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