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).
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
[+,-,-,-]
c=1
R\alpha\beta
\gamma | |
=\partial | |
\alpha\beta |
\gamma | |
-\partial | |
\alpha\gamma |
\gamma | |
+\Gamma | |
\alpha\beta |
\delta | |
\Gamma | |
\gamma\delta |
\gamma | |
-\Gamma | |
\alpha\delta |
\delta | |
\Gamma | |
\gamma\beta |
.
We begin with the Einstein–Hilbert Lagrangian:
{lL} | ||||
|
(R+2Λ)\sqrt{-g},
where
R
G
g
g\alpha\beta
Λ
We introduce the Maxwell-Proca Lagrangian for the STVG covector field
\phi\alpha
l{L} | \omega\left[ | ||||
|
1 | |
4 |
B\alpha\betaB\alpha\beta-
1 | |
2 |
\alpha+V | |
\mu | |
\phi(\phi)\right]\sqrt{-g}, |
where
B\alpha\beta=\partial\alpha\phi\beta-\partial\beta\phi\alpha
\phi\alpha,
\mu
\omega
V\phi
The three constants of the theory,
G,\mu,
\omega,
{lL} | \left[ | ||||
|
1 | |
2 |
g\alpha\beta\left(
\partial\alphaG\partial\betaG | + | |
G2 |
\partial\alpha\mu\partial\beta\mu | |
\mu2 |
-\partial\alpha\omega\partial
+ | |||||
|
V\mu(\mu) | |
\mu2 |
+V\omega(\omega)\right]\sqrt{-g},
where
VG,V\mu,
V\omega
The STVG action integral takes the form
S=\int{({lL}G+{lL}\phi+{lL}S+{lL}
4x, | |
M)}~d |
where
{lL}M
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
{lL}TP=-m+\alpha\omegaq5\phi\muu\mu,
where
m
\alpha
q5
u\mu=dx\mu/ds
q5=\kappam,
\kappa=\sqrt{GN/\omega}
M
\ddot{r}=- | GNM |
r2 |
\left[1+\alpha-\alpha(1+\mur)e-\mu\right],
where
GN
\alpha
\mu
M
\mu= | D |
\sqrt{M |
\alpha= | Ginfty-GN |
GN |
M | |
(\sqrt{M |
+E)2},
where
Ginfty\simeq20GN
D
E
D\simeq252 ⋅ 10
1/2 | |
M | |
\odot |
kpc-1,
E\simeq502 ⋅ 10
1/2 | |
M | |
\odot |
,
where
M\odot
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]
A 2017 article on Forbes by Ethan Siegel states that the Bullet Cluster still "proves dark matter exists, but not for the reason most physicists think". There he argues in favor of dark matter over non-local gravity theories, such as STVG/MOG. Observations show that in "undisturbed" galaxy clusters the reconstructed mass from gravitational lensing is located where matter is distributed, and a separation of matter from gravitation only seems to appear after a collision or interaction has taken place. According to Ethan Siegel: "Adding dark matter makes this work, but non-local gravity would make differing before-and-after predictions that can't both match up, simultaneously, with what we observe."[10]