3.3 Brans–Dicke Gravity
33
The typical cases are:
(i) 4
2
− m
2
= 0 (causal solution!), ρ + E
2
0 = 0. In this case φ is a trigonometric
function.
(ii) ρ = const, φ = const—trivial case reducing to GR.
For φ = φ(t), one arrives at φ = const, and this case is also trivial. In principle,
more involved situations can be studied as well. As for the black hole solutions in
BD gravity, we strongly recommend the classical paper [60]. In principle, many
other solutions for the BD gravity have been studied, including global monopoles,
wormholes etc., but the limited volume of these notes does not allow for their detailed
discussion.
3.4 Galileons
One of the most important examples of the scalar-tensor gravity models is the
galileons theory proposed originally in [61]. Its key idea is as follows: let us consider
the most general scalar-tensor action involves no more than second derivatives of the
metric tensor and no more than the first ones of the scalar field. Effectively, it was a
suggestion of the Lovelock-like construction not only in the gravitational sector but
also in the scalar one. So, we suggest the action to look like
S =
d
4 x
√ −gL(g μν , ∂ λ g μν , ∂ λ ∂ ρ g μν ; φ, ∂ μ φ).
(3.46)
As a result, the equations of motion involve various tensors constructed on the base
of the Riemann curvature and its covariant derivatives, and various derivatives of the
scalar field. In principle we can have the gravity equations of motion with Lovelocklike l.h.s. and non-canonical scalar-dependent r.h.s., and strongly nonlinear equations
of motion for the scalar. We note that there is no ghost problem here since there is no
higher derivatives. In principle, even on the flat background, one can have a theory
of a scalar field with highly nonlinear equation of motion, the so-called K -theory
(see [62] and references therein).
However, the model (3.46) was forgotten for a long time and revitalized only in
2008, in the paper [63] where the concept of galileons was formulated. Its key idea
consists in invariance of the theory with respect to the combination of dilatations and
conformal transformations so that the new scalar π varies as π → π + c + b μ x
μ ,
where c and b μ are constants. These transformations look similarly to the Galilean
ones, therefore the π was called the galileon. So, again, the key idea is that we have
derivative couplings but no higher derivatives in the kinetic term.
There are five terms with the symmetry above. Let us introduce notations
μν
=
∂
μ
∂
ν
π, [A] = A
μ
μ for trace (so,
1
2
[]∂π · ∂π =
1
2
π∂
μ
π∂ μ π), [] = π, etc.), and
use a dot for the usual scalar product like A · B ≡ A μ B
μ . So, we can write our five
terms as:
33
The typical cases are:
(i) 4
2
− m
2
= 0 (causal solution!), ρ + E
2
0 = 0. In this case φ is a trigonometric
function.
(ii) ρ = const, φ = const—trivial case reducing to GR.
For φ = φ(t), one arrives at φ = const, and this case is also trivial. In principle,
more involved situations can be studied as well. As for the black hole solutions in
BD gravity, we strongly recommend the classical paper [60]. In principle, many
other solutions for the BD gravity have been studied, including global monopoles,
wormholes etc., but the limited volume of these notes does not allow for their detailed
discussion.
3.4 Galileons
One of the most important examples of the scalar-tensor gravity models is the
galileons theory proposed originally in [61]. Its key idea is as follows: let us consider
the most general scalar-tensor action involves no more than second derivatives of the
metric tensor and no more than the first ones of the scalar field. Effectively, it was a
suggestion of the Lovelock-like construction not only in the gravitational sector but
also in the scalar one. So, we suggest the action to look like
S =
d
4 x
√ −gL(g μν , ∂ λ g μν , ∂ λ ∂ ρ g μν ; φ, ∂ μ φ).
(3.46)
As a result, the equations of motion involve various tensors constructed on the base
of the Riemann curvature and its covariant derivatives, and various derivatives of the
scalar field. In principle we can have the gravity equations of motion with Lovelocklike l.h.s. and non-canonical scalar-dependent r.h.s., and strongly nonlinear equations
of motion for the scalar. We note that there is no ghost problem here since there is no
higher derivatives. In principle, even on the flat background, one can have a theory
of a scalar field with highly nonlinear equation of motion, the so-called K -theory
(see [62] and references therein).
However, the model (3.46) was forgotten for a long time and revitalized only in
2008, in the paper [63] where the concept of galileons was formulated. Its key idea
consists in invariance of the theory with respect to the combination of dilatations and
conformal transformations so that the new scalar π varies as π → π + c + b μ x
μ ,
where c and b μ are constants. These transformations look similarly to the Galilean
ones, therefore the π was called the galileon. So, again, the key idea is that we have
derivative couplings but no higher derivatives in the kinetic term.
There are five terms with the symmetry above. Let us introduce notations
μν
=
∂
μ
∂
ν
π, [A] = A
μ
μ for trace (so,
1
2
[]∂π · ∂π =
1
2
π∂
μ
π∂ μ π), [] = π, etc.), and
use a dot for the usual scalar product like A · B ≡ A μ B
μ . So, we can write our five
terms as: