4.3 Bumblebee Gravity
43
situation, we suggest that the field B μ is one of the vacua which, for the quartic potential V =
λ
2
(B
μ B μ ± b
2
)
2 , will yield vanishing of the potential and its derivative. So,
it remains to find the vacuum for which the stress tensor B μν = ∂ μ B ν − ∂ ν B μ would
vanish as well (the part proportional to Christoffel symbols vanishes identically).
It is clear that the case of the constant B μ is an excellent example. Some interesting cases of such vacua, for the metric in the form (1.8), are: B μ = (ab, 0, 0, 0),
B μ = (0, ab, 0, 0), B μ = (0, 0, 0, ab) (we note that the Gödel metric is characterized
by the constant parameter a).
It remains to check consistency of these solutions with the equation of motion for
the bumblebee field:
∇ μ B
μν
= 2V
(B
2
)B
ν
.
(4.16)
These equations are satisfied immediately. Indeed, the l.h.s. is zero since B
μν
= 0
for these solutions, and its covariant derivative is also zero, and the r.h.s. is zero
for the quartic potential, if B μ is one of the vacua. Therefore, we conclude that the
Gödel solution is consistent in the bumblebee gravity. More detailed discussion on
this solution can be found in [90]. It is clear that a more generic Gödel-type solution
(2.26) can be analyzed along the same lines.
An interesting discussion of the bumblebee field is presented also in [91]. The
starting point is the generalized bumblebee Lagrangian
L = R − ζ ¯
g
αγ
¯
g
βδ B αβ B γδ − V (B
2
),
(4.17)
where V is a some potential of the bumblebee field, ζ is a coupling constant, and
¯
g
αγ
= g
αγ
+ β B
α B
γ is the effective metric.
Then, we carry out background-quantum splitting for gravitational and bumblebee
fields by the formulas g αβ = η αβ + h αβ and B α = ¯
B α + A α , where ¯
B α is one of
vacua, i.e. V ( ¯
B
2
) = V
( ¯
B
2
) = 0.
As a result, we arrive at the linearized equations of motion for the fluctuations
h αβ , A α :
G αβ | = V
( ¯
B
2
) ¯
B α ¯
B β B
2
|,
¯
η
αδ
¯
η
βγ
∂ β F γδ [A] =
1
2ζ
V
( ¯
B
2
) ¯
B
α B
2
|.
(4.18)
where | symbol is for a part linear in fluctuations h αβ , A α , f.e. B
2
| = 2 ¯
B
α A α −
¯
B
α ¯
B
β h αβ , and ¯
η
αδ
= η
αδ
+ β ¯
B
α ¯
B
δ . The F γδ [A] = ∂ γ A δ − ∂ δ A γ as usual.
We can introduce background-dependent densities
ρ m = −V
( ¯
B
2
) ¯
B
2 B
2
|,
ρ e = ±
V
( ¯
B
2
)
| ¯
B 2 |
2ζ
B
2
|
(4.19)
43
situation, we suggest that the field B μ is one of the vacua which, for the quartic potential V =
λ
2
(B
μ B μ ± b
2
)
2 , will yield vanishing of the potential and its derivative. So,
it remains to find the vacuum for which the stress tensor B μν = ∂ μ B ν − ∂ ν B μ would
vanish as well (the part proportional to Christoffel symbols vanishes identically).
It is clear that the case of the constant B μ is an excellent example. Some interesting cases of such vacua, for the metric in the form (1.8), are: B μ = (ab, 0, 0, 0),
B μ = (0, ab, 0, 0), B μ = (0, 0, 0, ab) (we note that the Gödel metric is characterized
by the constant parameter a).
It remains to check consistency of these solutions with the equation of motion for
the bumblebee field:
∇ μ B
μν
= 2V
(B
2
)B
ν
.
(4.16)
These equations are satisfied immediately. Indeed, the l.h.s. is zero since B
μν
= 0
for these solutions, and its covariant derivative is also zero, and the r.h.s. is zero
for the quartic potential, if B μ is one of the vacua. Therefore, we conclude that the
Gödel solution is consistent in the bumblebee gravity. More detailed discussion on
this solution can be found in [90]. It is clear that a more generic Gödel-type solution
(2.26) can be analyzed along the same lines.
An interesting discussion of the bumblebee field is presented also in [91]. The
starting point is the generalized bumblebee Lagrangian
L = R − ζ ¯
g
αγ
¯
g
βδ B αβ B γδ − V (B
2
),
(4.17)
where V is a some potential of the bumblebee field, ζ is a coupling constant, and
¯
g
αγ
= g
αγ
+ β B
α B
γ is the effective metric.
Then, we carry out background-quantum splitting for gravitational and bumblebee
fields by the formulas g αβ = η αβ + h αβ and B α = ¯
B α + A α , where ¯
B α is one of
vacua, i.e. V ( ¯
B
2
) = V
( ¯
B
2
) = 0.
As a result, we arrive at the linearized equations of motion for the fluctuations
h αβ , A α :
G αβ | = V
( ¯
B
2
) ¯
B α ¯
B β B
2
|,
¯
η
αδ
¯
η
βγ
∂ β F γδ [A] =
1
2ζ
V
( ¯
B
2
) ¯
B
α B
2
|.
(4.18)
where | symbol is for a part linear in fluctuations h αβ , A α , f.e. B
2
| = 2 ¯
B
α A α −
¯
B
α ¯
B
β h αβ , and ¯
η
αδ
= η
αδ
+ β ¯
B
α ¯
B
δ . The F γδ [A] = ∂ γ A δ − ∂ δ A γ as usual.
We can introduce background-dependent densities
ρ m = −V
( ¯
B
2
) ¯
B
2 B
2
|,
ρ e = ±
V
( ¯
B
2
)
| ¯
B 2 |
2ζ
B
2
|
(4.19)