Scalar–tensor–vector gravity

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]

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.

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.

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 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.

• Modified Newtonian dynamics
• Nonsymmetric gravitational theory
• Tensor–vector–scalar gravity