# 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: $R_{\mu\nu}=\partial_\alpha\Gamma^\alpha_{\mu\nu}-\partial_\nu\Gamma^\alpha_{\mu\alpha}+\Gamma^\alpha_{\mu\nu}\Gamma^\beta_{\alpha\beta}-\Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\alpha\nu}.$

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

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

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 $M$:

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

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:

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

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

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]