Scalar–tensor–vector gravity

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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]

Mathematical details[edit]

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: 

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[edit]

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]



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.


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]

See also[edit]


  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: 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: 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: 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: 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: 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: 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: 085002. arXiv:0712.1796. Bibcode:2009CQGra..26h5002M. doi:10.1088/0264-9381/26/8/085002.