arXiv:2006.12550v1
[gr-qc]
22 Jun 2020
Modified Gravity (MOG), Cosmology and Black Holes
J. W. Moffat
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
and
Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
June 24, 2020
Abstract
A covariant modified gravity (MOG) is formulated by adding to general relativity two new degrees
of freedom, a scalar field gravitational coupling strength
G
= 1
/χ
and a gravitational spin 1 vector field
φ
μ
. The
G
is written as
G
=
G
N
(1 +
α
) where
G
N
is Newton’s constant, and the gravitational source
charge for the vector field is
Q
g
=
√
αG
N
M
, where
M
is the mass of a body. Cosmological solutions of
the theory are derived in a homogeneous and isotropic cosmology. Black holes in MOG are stationary
as the end product of gravitational collapse and are axisymmetric solutions with spherical topology. It
is shown that the scalar field
χ
is constant everywhere for an isolated black hole with asymptotic flat
boundary condition. A consequence of this is that the scalar field loses its monopole moment radiation.
1 Introduction
Dark matter was introduced to explain the stable dynamics of galaxies and galaxy clusters. General rela-
tivity (GR) with only ordinary baryon matter cannot explain the present accumulation of astrophysical and
cosmological data without dark matter. However, dark matter has not been observed in laboratory exper-
iments [1]. Therefore, it is important to consider a modified gravitational theory. The difference between
standard dark matter models and modified gravity is that dark matter models assume that GR is the correct
theory of gravity and a dark matter particle such as WIMPS, axions and fuzzy dark matter are postulated
to belong to the standard particle model. The present work introduces a simplified formulation of modified
gravity (MOG), also called Scalar-Tensor-Vector-Gravity (STVG), that avoids unused generality of the orig-
inal version [2]. The MOG is described by a fully covariant action and field equations, extending GR by the
addition of two gravitational degrees of freedom. The first is
G
= 1
/χ
, where
G
is the coupling strength of
gravity and
χ
is a scalar field. The scalar field
χ
is motivated by the Brans-Dicke gravity theory [3, 4]. The
second degree of freedom is a massive gravitational vector field
φ
µ
. The gravitational coupling of the vector
graviton to matter is universal with the gravitational charge
Q
g
=
√
αG
N
M
, where
α
is a dimensionless
scalar field,
G
N
is Newton’s gravitational constant and
M
is the mass of a body.
We write
G
= 1
/χ
as
G
=
G
N
(1 +
α
) and
χ
is the only scalar field in the theory. The effective running
mass of the spin 1 vector graviton is determined by the parameter
µ
, which fits galaxy rotation curves
and cluster dynamics without exotic dark matter [6, 7, 8, 9].
It has the value
µ
∼
0
.
01
−
0
.
04 kpc
−
1
,
corresponding to
µ
−
1
∼
25
−
100 kpc and an effective mass
m
φ
∼
10
−
26
−
10
−
28
eV. We derive generalized
Friedmann equations and field equations for the scalar field
χ
in a homogeneous and isotropic universe with