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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).



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 detailsEdit

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  , and we are using the following definition for the Ricci tensor:  

We begin with the Einstein-Hilbert Lagrangian:


where   is the trace of the Ricci tensor,   is the gravitational constant,   is the determinant of the metric tensor  , while   is the cosmological constant.

We introduce the Maxwell-Proca Lagrangian for the STVG vector field  :


where  ,   is the mass of the vector field,   characterizes the strength of the coupling between the fifth force and matter, and   is a self-interaction potential.

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


where   denotes covariant differentiation with respect to the metric  , while  ,  , and   are the self-interaction potentials associated with the scalar fields.

The STVG action integral takes the form


where   is the ordinary matter Lagrangian density.

Spherically symmetric, static vacuum solutionEdit

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


where   is the test particle mass,   is a factor representing the nonlinearity of the theory,   is the test particle's fifth-force charge, and   is its four-velocity. Assuming that the fifth-force charge is proportional to mass, i.e.,  , the value of   is determined and the following equation of motion is obtained in the spherically symmetric, static gravitational field of a point mass of mass  :


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



where   is determined from cosmological observations, while for the constants   and   galaxy rotation curves yield the following values:



where   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 alsoEdit


  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.