52
5 Horava–Lifshitz Gravity
We can have also static spherically symmetric solutions described by the Eq.
(3.20). Clearly, the possibility of black holes is of the special interest. We start with
the particular case of the metric (3.20):
ds
2
= − f (r )dt
2
+
dr
2
f (r )
+ r
2 d
2
,
(5.16)
It is clear that the Schwarzschild and Reissner-Nordstrom metrics match this form.
In [101] it has been explicitly shown that, for λ = 1, one has
f (r ) = 1 + ωr
2
−
r (ω 2 r 3 + 4ω M),
(5.17)
The ω is a function of constant parameters of the theory. The essential conclusion is
that at large distances, i.e. r (M/ω)
1/3 , one has f (r ) 1 −
2M
r
+ O(r
−4
), that is,
the Schwarzschild result, i.e. the consistency with the general relativity is achieved.
It has been demonstrated in [101] that the equation f (r ) = 0 has two solutions,
so this black hole has two horizons with r pm = M(1 ±
1 −
1
2ω M 2 ). The naked
singularity is avoided at ω M
2
≥ 1/2.
Now, let us consider the Gödel-type solution (2.26). It has been considered in
details in [97]. First of all, we note that g φφ = D
2
− H
2
= G(r ) (other two components of g i j are 1), and N =
D(r )
√
G(r )
. So, the positiveness of G(r ), and hence satisfying
the causality condition, is necessary to have a consistent (real) value of N !
After some change of variables discussed in [97], we can rewrite this metric as
ds
2
= −
dt
+
2ω
m
e
mx dy
2
+ e
2mx dy
2
+ dr
2
+ dz
2 ,
(5.18)
with G(x) = v
2 e
2mx
> 0, and v
2
= 1 −
4ω
2
m 2 , so, the causality is guaranteed if v
2
> 0.
For this metric, R 1212 = −m
2
v
2 e
2mx , K 12 = −vωe
mx , C
i j
= 0, R = −2m
2 .
To verify the consistency of this solution, we choose the fluid-like matter with
T
μν
= ( p + ρ)u
μ u
ν
+ pg
μν
.
(5.19)
Namely this matter has been used in the original paper [3]. After solving algebraic
equations we find m
2
=
2
3
ω
2 or m
2
=
1
4
ω
2 . However, both these solutions appear
to be not completely satisfactory since they are non-causal (as it has been proved
in [44], the causality is achieved for m
2
≥ 4ω
2
). As for constant parameters of the
theory λ, μ, , they can also be found in terms of m, ω, p, ρ, the explicit values are
given in [97].
Therefore, we have seen that these solutions of GR are consistent within the HL
gravity, at least asymptotically. Again, it is important to note that the HL gravity is
power-counting renormalizable (although up to now there is no examples of fullfledged quantum calculations in the theory). Nevertheless, it must be pointed that it
also displays some difficulties which we will discuss now.
5 Horava–Lifshitz Gravity
We can have also static spherically symmetric solutions described by the Eq.
(3.20). Clearly, the possibility of black holes is of the special interest. We start with
the particular case of the metric (3.20):
ds
2
= − f (r )dt
2
+
dr
2
f (r )
+ r
2 d
2
,
(5.16)
It is clear that the Schwarzschild and Reissner-Nordstrom metrics match this form.
In [101] it has been explicitly shown that, for λ = 1, one has
f (r ) = 1 + ωr
2
−
r (ω 2 r 3 + 4ω M),
(5.17)
The ω is a function of constant parameters of the theory. The essential conclusion is
that at large distances, i.e. r (M/ω)
1/3 , one has f (r ) 1 −
2M
r
+ O(r
−4
), that is,
the Schwarzschild result, i.e. the consistency with the general relativity is achieved.
It has been demonstrated in [101] that the equation f (r ) = 0 has two solutions,
so this black hole has two horizons with r pm = M(1 ±
1 −
1
2ω M 2 ). The naked
singularity is avoided at ω M
2
≥ 1/2.
Now, let us consider the Gödel-type solution (2.26). It has been considered in
details in [97]. First of all, we note that g φφ = D
2
− H
2
= G(r ) (other two components of g i j are 1), and N =
D(r )
√
G(r )
. So, the positiveness of G(r ), and hence satisfying
the causality condition, is necessary to have a consistent (real) value of N !
After some change of variables discussed in [97], we can rewrite this metric as
ds
2
= −
dt
+
2ω
m
e
mx dy
2
+ e
2mx dy
2
+ dr
2
+ dz
2 ,
(5.18)
with G(x) = v
2 e
2mx
> 0, and v
2
= 1 −
4ω
2
m 2 , so, the causality is guaranteed if v
2
> 0.
For this metric, R 1212 = −m
2
v
2 e
2mx , K 12 = −vωe
mx , C
i j
= 0, R = −2m
2 .
To verify the consistency of this solution, we choose the fluid-like matter with
T
μν
= ( p + ρ)u
μ u
ν
+ pg
μν
.
(5.19)
Namely this matter has been used in the original paper [3]. After solving algebraic
equations we find m
2
=
2
3
ω
2 or m
2
=
1
4
ω
2 . However, both these solutions appear
to be not completely satisfactory since they are non-causal (as it has been proved
in [44], the causality is achieved for m
2
≥ 4ω
2
). As for constant parameters of the
theory λ, μ, , they can also be found in terms of m, ω, p, ρ, the explicit values are
given in [97].
Therefore, we have seen that these solutions of GR are consistent within the HL
gravity, at least asymptotically. Again, it is important to note that the HL gravity is
power-counting renormalizable (although up to now there is no examples of fullfledged quantum calculations in the theory). Nevertheless, it must be pointed that it
also displays some difficulties which we will discuss now.