54
5 Horava–Lifshitz Gravity
It was argued in [104] that if we introduce u μ =
∂ μ φ
√
X
, with X = g μν ∂
μ
φ∂
ν
φ, we
can add some terms to our action to get a consistent theory! Actually, we have
S = −
1
κ 2
d
4 x
√ −g(R 4 + (λ − 1)(∇ μ u
μ
)
2
+ αu
μ
(∇ μ u
ν
)u
λ
(∇ λ u ν ) + · · · ),
(5.21)
and this action, for splitting φ → t + χ, yields reasonable dispersion relations for χ
like ω
2
= Cp
2 , with C is a some number. In [104], also some cosmological impacts
of this term were studied. An aside result is an emergence of Einstein-aether action.
So, the consistent extension of the HL gravity is found.
Another approach is based on use of so-called projectable version of the HL
gravity, where the lapse N is suggested to be a function of a time only, N = N (t).
However, it turns out to be that although in this case the theory is strongly simplified,
the scalar excitation is still unstable and cannot be ruled out [105].
5.5 Conclusions
Let us make some conclusions regarding the HL gravity. As we already noted, the
key idea of the HL gravity is that the usual general covariance is an essentially lowenergy phenomenon but not a fundamental feature of the nature. In a certain sense,
it can be said that the HL concept was developed to “sacrifice” general covariance in
order to conciliate desired renormalizability with absence of ghosts. In this context, it
should be noted that breaking of general covariance in gravity is discussed as well in
“usual” Lorentz-breaking gravity models without strong space-time asymmetry [76].
We demonstrated how the known GR solutions are modified within the HL context.
Within the cosmological context, accelerated and bouncing solutions are possible,
thus the HL gravity is a good candidate to solve the dark energy problem. We demonstrated that there are black hole solutions behaving like usual Schwarzschild BHs
at large distances. Also, we demonstrated that the Gödel-type solutions consistent
within the HL gravity are non-causal, but one should note that the Gödel solution
itself is non-causal.
However, quantum description of the HL gravity is rather problematic. One of
the reasons is a very complicated structure of the classical action potentially implying a very large number of divergent contributions, therefore while the HL is power
counting renormalizable, we cannot yet be sure that it is multiplicatively renormalizable. Another difficulty is the question about an extra degree of freedom. While it
was in principle solved in [104], where the “healthy extension” of HL gravity was
introduced, the problem now consists in obtaining physically measurable results on
the base of this extension. Therefore, even in this case we have more questions than
answers. To close the discussion, we recommend an excellent review on HL gravity
presented in [106].
5 Horava–Lifshitz Gravity
It was argued in [104] that if we introduce u μ =
∂ μ φ
√
X
, with X = g μν ∂
μ
φ∂
ν
φ, we
can add some terms to our action to get a consistent theory! Actually, we have
S = −
1
κ 2
d
4 x
√ −g(R 4 + (λ − 1)(∇ μ u
μ
)
2
+ αu
μ
(∇ μ u
ν
)u
λ
(∇ λ u ν ) + · · · ),
(5.21)
and this action, for splitting φ → t + χ, yields reasonable dispersion relations for χ
like ω
2
= Cp
2 , with C is a some number. In [104], also some cosmological impacts
of this term were studied. An aside result is an emergence of Einstein-aether action.
So, the consistent extension of the HL gravity is found.
Another approach is based on use of so-called projectable version of the HL
gravity, where the lapse N is suggested to be a function of a time only, N = N (t).
However, it turns out to be that although in this case the theory is strongly simplified,
the scalar excitation is still unstable and cannot be ruled out [105].
5.5 Conclusions
Let us make some conclusions regarding the HL gravity. As we already noted, the
key idea of the HL gravity is that the usual general covariance is an essentially lowenergy phenomenon but not a fundamental feature of the nature. In a certain sense,
it can be said that the HL concept was developed to “sacrifice” general covariance in
order to conciliate desired renormalizability with absence of ghosts. In this context, it
should be noted that breaking of general covariance in gravity is discussed as well in
“usual” Lorentz-breaking gravity models without strong space-time asymmetry [76].
We demonstrated how the known GR solutions are modified within the HL context.
Within the cosmological context, accelerated and bouncing solutions are possible,
thus the HL gravity is a good candidate to solve the dark energy problem. We demonstrated that there are black hole solutions behaving like usual Schwarzschild BHs
at large distances. Also, we demonstrated that the Gödel-type solutions consistent
within the HL gravity are non-causal, but one should note that the Gödel solution
itself is non-causal.
However, quantum description of the HL gravity is rather problematic. One of
the reasons is a very complicated structure of the classical action potentially implying a very large number of divergent contributions, therefore while the HL is power
counting renormalizable, we cannot yet be sure that it is multiplicatively renormalizable. Another difficulty is the question about an extra degree of freedom. While it
was in principle solved in [104], where the “healthy extension” of HL gravity was
introduced, the problem now consists in obtaining physically measurable results on
the base of this extension. Therefore, even in this case we have more questions than
answers. To close the discussion, we recommend an excellent review on HL gravity
presented in [106].