4.2 Einstein-aether Gravity
39
Alternatively, as we already noted above, one can introduce the quartic potential. The
approach based on the Lagrange multiplier has been adopted within gravity studies
performed in the paper [82]. In this case, the above constraint is generalized to a
curved space-time as g
μν u μ u ν − 1 = 0, where u μ is the aether vector field.
Our starting point is the action [82]
S = −
1
16πG
d
4 x
√ −g
R + λ(u
μ u μ − 1) + K
αβ
μν ∇ α u
μ
∇ β u
ν
,
(4.2)
where
K
αβ
μν = c 1 g
αβ
g μν + c 2 δ
α
μ δ
β
ν + c 3 δ
α
ν δ
β
μ + c 4 u
α u
β
g μν .
(4.3)
This action involves an above-mentioned constraint introduced with use of the
Lagrange multiplier λ. The c 1 , c 2 , c 3 , c 4 are some dimensionless constants. It is
interesting to note that the term R αβ u
α u
β proposed as the aether term in [77] arises
in this theory (together with some other terms) for the particular case c 3 = −c 2 when
the commutator of covariant derivatives yielding a curvature tensor emerges [83].
The corresponding equations of motion look like [83]:
g αβ u
α u
β
= 1; ∇ α J
α
μ − c 4 ˙
u α ∇ μ u
α
= λu
μ
;
T αβ = −
1
2
g αβ L u + ∇ μ
J
α
(μ u β) − J
μ
(α u β) − J (αβ) u
μ
+
(4.4)
+ c 1 [(∇ μ u α )(∇
μ u ν ) − (∇ α u mu )(∇ β u
μ
)] + c 4 ˙
u α ˙
u β + [u ν ∇ μ J
μν
− c 4 ˙
u
2
]u α u β .
Here ˙
u
μ
= u
α
∇ α u
μ , J
α
μ = K
αβ
μν ∇ β u
ν , and L u is u-dependent part of the Lagrangian.
We note again that the vector u μ has nothing to do with the usual matter, so, the
Einstein-aether theory is an example of a vector-tensor gravity.
So, now our task will consist in finding some solutions for these equations, or,
to be more precise, in checking the consistency of known GR solutions within the
Einstein-aether gravity.
As the simplest example we choose the spherically symmetric static metric, which
is consistent since the vector u μ is time-like, in order to satisfy the constraint. In our
case, it is convenient to choose this metric in the form slightly different from (3.20),
namely,
ds
2
= N (r )dt
2
− B(r )(dr
2
+ r
2 d
2
).
(4.5)
The consistency of this metric within the Einstein-aether gravity has been verified within perturbative methodology for various relations between the parameters
c 1 , c 2 , c 3 , f.e. c 1 + c 2 + c 3 = 0, and c 4 can be chosen to be zero without any problems since it can be removed through a simple change of variables (see details in
[83]) so that the N (r ) and B(r ) turn out to be represented as power series in x = 1/r
providing that they tend to 1 at infinity as it must be, with some lower coefficients
39
Alternatively, as we already noted above, one can introduce the quartic potential. The
approach based on the Lagrange multiplier has been adopted within gravity studies
performed in the paper [82]. In this case, the above constraint is generalized to a
curved space-time as g
μν u μ u ν − 1 = 0, where u μ is the aether vector field.
Our starting point is the action [82]
S = −
1
16πG
d
4 x
√ −g
R + λ(u
μ u μ − 1) + K
αβ
μν ∇ α u
μ
∇ β u
ν
,
(4.2)
where
K
αβ
μν = c 1 g
αβ
g μν + c 2 δ
α
μ δ
β
ν + c 3 δ
α
ν δ
β
μ + c 4 u
α u
β
g μν .
(4.3)
This action involves an above-mentioned constraint introduced with use of the
Lagrange multiplier λ. The c 1 , c 2 , c 3 , c 4 are some dimensionless constants. It is
interesting to note that the term R αβ u
α u
β proposed as the aether term in [77] arises
in this theory (together with some other terms) for the particular case c 3 = −c 2 when
the commutator of covariant derivatives yielding a curvature tensor emerges [83].
The corresponding equations of motion look like [83]:
g αβ u
α u
β
= 1; ∇ α J
α
μ − c 4 ˙
u α ∇ μ u
α
= λu
μ
;
T αβ = −
1
2
g αβ L u + ∇ μ
J
α
(μ u β) − J
μ
(α u β) − J (αβ) u
μ
+
(4.4)
+ c 1 [(∇ μ u α )(∇
μ u ν ) − (∇ α u mu )(∇ β u
μ
)] + c 4 ˙
u α ˙
u β + [u ν ∇ μ J
μν
− c 4 ˙
u
2
]u α u β .
Here ˙
u
μ
= u
α
∇ α u
μ , J
α
μ = K
αβ
μν ∇ β u
ν , and L u is u-dependent part of the Lagrangian.
We note again that the vector u μ has nothing to do with the usual matter, so, the
Einstein-aether theory is an example of a vector-tensor gravity.
So, now our task will consist in finding some solutions for these equations, or,
to be more precise, in checking the consistency of known GR solutions within the
Einstein-aether gravity.
As the simplest example we choose the spherically symmetric static metric, which
is consistent since the vector u μ is time-like, in order to satisfy the constraint. In our
case, it is convenient to choose this metric in the form slightly different from (3.20),
namely,
ds
2
= N (r )dt
2
− B(r )(dr
2
+ r
2 d
2
).
(4.5)
The consistency of this metric within the Einstein-aether gravity has been verified within perturbative methodology for various relations between the parameters
c 1 , c 2 , c 3 , f.e. c 1 + c 2 + c 3 = 0, and c 4 can be chosen to be zero without any problems since it can be removed through a simple change of variables (see details in
[83]) so that the N (r ) and B(r ) turn out to be represented as power series in x = 1/r
providing that they tend to 1 at infinity as it must be, with some lower coefficients