Chapter 5
Horava–Lifshitz Gravity
5.1 Introduction
As it is well known, the most complicated problem of the gravity is the development
of its consistent quantum description. Indeed, we have noted in the Chap. 1 that the
Einstein gravity is non-renormalizable since the mass dimension of the gravitational
constant is negative. The natural improvement of situation could consist in adding
the higher-derivative terms which clearly make the UV asymptotics of the propagator
better. However, it is known that in this case the ghosts arise which makes the theory
to be unstable, hence higher-derivative gravity models can be used only as effective
theories for the low-energy domain.
Therefore, in [93], the following idea has been proposed: let us suggest that the
desired extension of gravity involves only second time derivatives, so, the ghosts
will be ruled out, and higher spatial derivatives, therefore the UV behavior of the
propagator will be improved. The similar models for the scalar field, with modified
kinetic terms like
1
2
φ(∂
2
0 + (−1)
z
α
z
)φ have been introduced a long ago within
the condensed matter context in [94] where they were used to describe critical phenomena. In other words, we suggest that the Lorentz symmetry breaking is strong.
Further, such theories with strong difference between spatial and time directions have
been denominated as theories with space-time anisotropy. The number z, defined in
a manner similar to the action above (once more, if the action involves two time
derivatives, it involves 2z spatial derivatives), is called the critical exponent. For the
Lorentz-invariant theories, one has z = 1. To recover the Einstein limit, one must
suppose that the action involves also lower-derivative terms. One can verify that in
such a theory, the dimension of the effective gravitational constant will depend on z,
being actually equal to z − d, in a d-dimensional space-time. Therefore, in (3 + 1)dimensional space-time, the gravity model formulated on the base of the space-time
anisotropy (further such theories became to be called the Horava–Lifshitz (HL) theories) is power-counting renormalizable at z = 3. However, it is clear that for such
a theory, the perturbative calculations will be very involved.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_5
47
Horava–Lifshitz Gravity
5.1 Introduction
As it is well known, the most complicated problem of the gravity is the development
of its consistent quantum description. Indeed, we have noted in the Chap. 1 that the
Einstein gravity is non-renormalizable since the mass dimension of the gravitational
constant is negative. The natural improvement of situation could consist in adding
the higher-derivative terms which clearly make the UV asymptotics of the propagator
better. However, it is known that in this case the ghosts arise which makes the theory
to be unstable, hence higher-derivative gravity models can be used only as effective
theories for the low-energy domain.
Therefore, in [93], the following idea has been proposed: let us suggest that the
desired extension of gravity involves only second time derivatives, so, the ghosts
will be ruled out, and higher spatial derivatives, therefore the UV behavior of the
propagator will be improved. The similar models for the scalar field, with modified
kinetic terms like
1
2
φ(∂
2
0 + (−1)
z
α
z
)φ have been introduced a long ago within
the condensed matter context in [94] where they were used to describe critical phenomena. In other words, we suggest that the Lorentz symmetry breaking is strong.
Further, such theories with strong difference between spatial and time directions have
been denominated as theories with space-time anisotropy. The number z, defined in
a manner similar to the action above (once more, if the action involves two time
derivatives, it involves 2z spatial derivatives), is called the critical exponent. For the
Lorentz-invariant theories, one has z = 1. To recover the Einstein limit, one must
suppose that the action involves also lower-derivative terms. One can verify that in
such a theory, the dimension of the effective gravitational constant will depend on z,
being actually equal to z − d, in a d-dimensional space-time. Therefore, in (3 + 1)dimensional space-time, the gravity model formulated on the base of the space-time
anisotropy (further such theories became to be called the Horava–Lifshitz (HL) theories) is power-counting renormalizable at z = 3. However, it is clear that for such
a theory, the perturbative calculations will be very involved.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
A. Petrov, Introduction to Modified Gravity, SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-52862-1_5
47