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 , 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 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

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 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. Bibcode:2006JCAP...03..004M. arXiv:gr-qc/0506021Freely accessible. 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. Bibcode:2006ApJ...636..721B. arXiv:astro-ph/0506370Freely accessible. 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. Bibcode:2006MNRAS.367..527B. arXiv:astro-ph/0507222Freely accessible. 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. Bibcode:2007MNRAS.382...29B. arXiv:astro-ph/0702146Freely accessible. 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:0710.0364Freely accessible [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. Bibcode:2008ApJ...680.1158M. arXiv:0708.1935Freely accessible. 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. Bibcode:2009MNRAS.395L..25M. arXiv:0710.3415Freely accessible. 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. Bibcode:2009CQGra..26h5002M. arXiv:0712.1796Freely accessible. doi:10.1088/0264-9381/26/8/085002.