4.3 Bumblebee Gravity
41
arising of ghost modes [88]. Again, the ± sign reflect the fact that b
2
> 0. We note
again that the vacua B 0μ are given by the condition B
μ
0 B 0μ = ±b
2 , and these vacua
are not required to be constants, in a curved space-time, which avoids the difficulties
connected with definition of the constant vectors in this case.
First effect to note here is that after Lorentz symmetry breaking, we will have
Nambu–Goldstone modes: if we introduce the vector b μ corresponding to one of the
vacua, i.e. b
μ b μ = ±b
2 , define B μ = b μ + A μ , and rewrite the action (4.1) in terms
of b μ and A μ , the resulting form of the action will be given by the Maxwell term,
plus the axial gauge term proportional to (b
μ A μ )
2 , plus new couplings of the vector
A μ with the curvature, like A
μ A
ν R μν , plus the Carroll-like term b
μ b
ν R μν [77].
Let us discuss some exact solutions for this theory. First, we consider the static
spherically symmetric metric, following the lines of [79]. For the reasons of convenience, we rewrite the metric (3.20) as:
ds
2
= −e
2φ(r ) dt
2
+ e
2ρ(r ) dr
2
+ r
2 d
2
.
(4.7)
Then, we choose the vacuum vector to be purely radial, i.e. b μ = (0, b(r ), 0, 0),
thus one has ∇ μ b ν = 0 if b(r ) = ξ
−1/2 b 0 e
ρ(r ) , ξ is a constant, and the variable φ(r )
becomes irrelevant within modified Einstein equations.
For this metric we find the only non-zero component of the Ricci tensor and the
corresponding scalar curvature to be
R rr =
2ρ
r
; R =
2[1 + 2(r ρ
− 1)e
−2ρ
]
r 2
.
(4.8)
It is convenient to introduce a new dynamical variable =
1−e
−2ρ
r 2 . Its action will
look like:
S =
2
κ
dtdrr
2 e
ρ+φ
(3 + b
2
0 )) +
1 +
b
2
0
2
r
,
(4.9)
where b 0 was defined above.
The equation of motion, after varying with respect to φ, is
(3 + b
2
0 )) +
1 +
b
2
0
2
r
= 0.
(4.10)
Its solution is (r ) = 0 r
L−3 , with 3 − L = (3 + b
2
0 )/(1 + b
2
0 /2), and
g rr = e
2ρ
= (1 − 0 r
L−1
)
−1
,
(4.11)
so, this component is similar to g rr of the Schwarzschild metric, therefore our solution
is characterized by the event horizon. In principle, more results for this metric can be
obtained, f.e. the Hawking temperature [79]. The case when the b μ vacuum vector
possesses not only the radial component but also the temporal one has been also
41
arising of ghost modes [88]. Again, the ± sign reflect the fact that b
2
> 0. We note
again that the vacua B 0μ are given by the condition B
μ
0 B 0μ = ±b
2 , and these vacua
are not required to be constants, in a curved space-time, which avoids the difficulties
connected with definition of the constant vectors in this case.
First effect to note here is that after Lorentz symmetry breaking, we will have
Nambu–Goldstone modes: if we introduce the vector b μ corresponding to one of the
vacua, i.e. b
μ b μ = ±b
2 , define B μ = b μ + A μ , and rewrite the action (4.1) in terms
of b μ and A μ , the resulting form of the action will be given by the Maxwell term,
plus the axial gauge term proportional to (b
μ A μ )
2 , plus new couplings of the vector
A μ with the curvature, like A
μ A
ν R μν , plus the Carroll-like term b
μ b
ν R μν [77].
Let us discuss some exact solutions for this theory. First, we consider the static
spherically symmetric metric, following the lines of [79]. For the reasons of convenience, we rewrite the metric (3.20) as:
ds
2
= −e
2φ(r ) dt
2
+ e
2ρ(r ) dr
2
+ r
2 d
2
.
(4.7)
Then, we choose the vacuum vector to be purely radial, i.e. b μ = (0, b(r ), 0, 0),
thus one has ∇ μ b ν = 0 if b(r ) = ξ
−1/2 b 0 e
ρ(r ) , ξ is a constant, and the variable φ(r )
becomes irrelevant within modified Einstein equations.
For this metric we find the only non-zero component of the Ricci tensor and the
corresponding scalar curvature to be
R rr =
2ρ
r
; R =
2[1 + 2(r ρ
− 1)e
−2ρ
]
r 2
.
(4.8)
It is convenient to introduce a new dynamical variable =
1−e
−2ρ
r 2 . Its action will
look like:
S =
2
κ
dtdrr
2 e
ρ+φ
(3 + b
2
0 )) +
1 +
b
2
0
2
r
,
(4.9)
where b 0 was defined above.
The equation of motion, after varying with respect to φ, is
(3 + b
2
0 )) +
1 +
b
2
0
2
r
= 0.
(4.10)
Its solution is (r ) = 0 r
L−3 , with 3 − L = (3 + b
2
0 )/(1 + b
2
0 /2), and
g rr = e
2ρ
= (1 − 0 r
L−1
)
−1
,
(4.11)
so, this component is similar to g rr of the Schwarzschild metric, therefore our solution
is characterized by the event horizon. In principle, more results for this metric can be
obtained, f.e. the Hawking temperature [79]. The case when the b μ vacuum vector
possesses not only the radial component but also the temporal one has been also