# Gravitational waves in Hořava-Lifshitz anisotropic gravity

J. Mestra-Páez<EMAIL_ADDRESS>Departamento de Física, Universidad de
Antofagasta, Aptdo 02800, Chile.  J.M. Peña<EMAIL_ADDRESS>Departamento de
Física, Universidad de Antofagasta, Aptdo 02800, Chile.  A. Restuccia
<EMAIL_ADDRESS>Departamento de Física, Universidad de Antofagasta,
Aptdo 02800, Chile.

###### Abstract

We show that in the anisotropic Hořava-Lifshitz gravity there is a well-
defined wave zone where the physical degrees of freedom propagate according to
a non-relativistic linear evolution equation of high order in spatial
derivatives, which reduces to the wave equation at low energy. This is so,
provided the coupling parameters satisfy some restrictions which we study in
detail. They are imposed to obtain a finite ADM gravitational energy, which
depends manifestly on the terms which break the Lorentz symmetry of the
formulation. The analysis we perform is beyond the linearized approach and
includes all high order terms of the Hamiltonian potential.

††preprint: APS/123-QED

##  I Introduction

The Hořava-Lifshitz gravity theory [1, 2] has been proposed as a candidate of
a renormalizable gravity theory [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16]. Following Lifshitz, the time and the spatial coordinates scale
differently, in a way that the overall coupling of the theory becomes
dimensionless. In this sense the theory is anisotropic and non-relativistic.
It introduces interaction terms with high order spatial derivatives in the
potential that break in a manifest way the relativistic symmetry, but improve
the ultraviolet (UV) behaviour in comparison to GR. The theory is
renormalizable by power counting.

There are different versions of the Hořava-Lifshitz gravity, the projectable
one with interesting applications to cosmology [17, 18] and references
therein, and the non-projectable one where the full number of gravitational
physical degrees of freedom, the transverse-traceless tensorial modes at the
linearized level, become dynamical. Among the non-projectable Hořava-Lifshitz
versions the propagating degrees of freedom differ according to the value of
the dimensionless couplings $\lambda$ on the kinetic term of the action. For
$\lambda\neq 1/3$ the theory propagates, besides the transverse-traceless
tensorial degrees of freedom, a scalar one. Several works have discussed the
problem of strong coupling of this scalar mode [3, 19, 5].

Some restrictions on the couplings of the theory have to be imposed in order
to justify the existence of this scalar mode without violating the well
established gravitational data [20]. For $\lambda=1/3$, the kinetic term on
the action has an additional conformal symmetry. In this case the propagating
degrees of freedom exactly coincide with transverse-traceless tensorial modes
of GR. No additional scalar field is present in the theory. At low energies
the theory depends on two coupling constants $\beta$ and $\alpha$, when
$\beta=1$ and $\alpha=0$ the field equations are exactly the GR equations on a
particular gauge. The restrictions on the coupling constants are in this case
less stringent. In both cases there is a range of values for the coupling
constants for which the theory fits the known gravitational experimental data
satisfied by GR [21].

The linearized theory at low energies coincides with the corresponding
linearized GR formulation [22]. Additionally it satisfies the analogous to the
Einstein quadrupole radiation formula [23]. The coupling to the Maxwell theory
in four dimensions has been recently studied using a Kaluza-Klein approach
from Hořava-Lifshitz theory in five dimensions [24, 25]. The speed of
propagation of the gravitational and electromagnetic physical modes, at low
energies, is the same in agreement with the recent experimental data arising
from the detection of gravitational and electromagnetic waves generated by the
same source [26]. An analysis of the theory including all higher order spatial
derivative terms of the potential was performed in [13].

Although at the linearized level Hořava-Lifshitz theory and GR coincide, the
full theories behave in a different way as can be seen from the static
spherical symmetric solutions of the field equations. In this case, in the
Hořava-Lifshitz theory there are solutions with a throat, connecting an
asymptotically flat manifold with a non-asymptotically flat one [27], see also
[28] where a charged throat is considered. The solutions depend in one
coupling constant $\alpha$ in a way that in the limit when $\alpha\to 0$ the
geometry outside the throat, on the asymtotically flat side, tends to the
geometry of the Schwarzchild solutions in GR outside the black hole. The
asymptotic behaviour of these solutions define the asymptotic flat behaviour
in Hořava-Lifshitz gravity.

Non-linear interaction terms in gravity theories are essential in the
propagation of the physical degrees of freedom. However, in GR in the wave
zone, the propagation of the physical degrees of freedom, at leading order, is
free from such non-linear terms. We may wonder if such is the case for the
Hořava-Lifshitz gravity, since both theories propagate exactly the same
physical degrees of freedom. In particular, we are interested in the effect of
the terms that break the relativistic symmetry on the propagation in the wave
zone.

In [29] we analyzed the wave zone in Hořava-Lifshitz gravity at low energies
that is, when terms in the potential with high order spatial derivatives are
dismissed. We found that the propagation of the physical degrees of freedom is
described by the wave equation as in GR, with speed of propagation given in
terms of the coupling parameter $\beta$.

In this paper we show that there exists a wave zone on the complete Hořava-
Lifshitz theory. We include in our analysis all high order derivative terms in
the potential. We consider the theory at the kinetic conformal point, where
$\lambda=1/3$. We notice that the value $\lambda=1/3$ is protected from
quantum corrections by the presence of a second class constraint which arises
directly from the formulation of the theory [22].

On this zone the constraints of the theory can be solved in powers of $1/r$.
In distinction to GR where the Hamiltonian constraint is of first class, in
this theory there are two second class constraints which allow to obtain the
components $g^{T}$ and $N$ in terms of the transverse traceless tensorial TT
modes. The lapse $N$ is not a Lagrange multiplier in this theory. We analyse
the general coordinate transformations on the wave zone and obtain the field
equations for the physical degrees of freedom described by the TT modes of the
metric. At order $1/r$, the field equations become a linear partial
differential equation, which reduces to the wave equation at low energies.
Although this field equation at order $1/r$ is the same that arises form the
linearized theory, the nontrivial solution for the $g^{T}$ and $N$ fields
provides the existence of a Newtonian background absent in the linearized
theory and relevant in the asymptotic behaviour of the theory, in particular
in the determination of the gravitational energy. In the formulation of the
theory on the wave zone, we follow the approach rigorously established in
[30].

In distinction to the ADM analysis of GR the solution of the constraints and
the dynamical equations involve an elliptic operator of sixth order, yielding
a higher order dispersion relation. The terms which break the Lorentz symmetry
in the action of the theory contribute to the Newtonian background and appear
explicitly in the expression of the gravitational energy. In order to have a
finite gravitational energy some restrictions have to be imposed on the
coupling parameters of the potential. In fact, the solution of the constraints
involve an evolution operator with high order in spatial derivatives, which
may have zero modes. They produce inconsistent contributions to the
gravitational energy. In order to avoid them we impose restrictions to the
coupling parameters.

The relevance of the existence of a wave zone in a theory describing gravity
is directly related to the detection of gravitational waves (GW) in the last
five years. In 2010, they were measured indirectly through the relative
reduction of the distance between the members of a binary pulsar system PSR
B1913+16 by continuos emission of GW [31]. In 2015, it was detected directly
by LIGO-Virgo collaboration the first signal of a GW produced from two black
hole-black hole (BHBH) coalescence GW150914 [32, 33, 34]. After that, there
have been multiple detections of GW due to coalescence of compact-binary
system as BHBH, neutron start-neutron start (NSNS) and neutron star-black hole
(NSBH) [35, 36, 37, 38, 39].

In the case of GW170817 event, also it was detected its electromagnetic
signal, $\gamma$ rays burst  GBR170817A [40], which has been very important to
compare the speeds of both electromagnetic and gravitational signals, it
constrains the difference between the speed of gravity and the speed of light
to be between $-3\times{10}^{-15}$ and $+7\times{10}^{-16}$ times the speed of
light [41, 42]. In the Hořava-Lifshitz at the kinetic conformal point theory
both interactions have the same speed of propagation [24, 25].

A main difference between the wave zone in Hořava-Lifshitz theory and that in
GR is the modified dispersion relation (MDR) involving a polynomial in the
square of the wave number $k$, which at low energy and for $\beta=1$ reduces
to the relativistic relation. Modifications of Dispersion Relations (MDR) in
theories with Lorentz Invariance Violations (LIV) have been discussed in the
literature, for a review of experimental test of LIV in astroparticle physics
scenarios see [43] and its references. The Large Hight Altitude Air Shower
Observatory-(LHAASO) have recently detected Ultra-Hight-Energy-photons (UHE
$\gamma$-rays) in the PeV energy scale [44]. These observations suggest the
existence of subluminal LIV beyond Special Relativity in the photon sector at
the scale of $3,6\times 10^{17}$GeV [45]. In concordance with references [46,
47, 48] in which it is stated that, from data of GRBs observed by the Fermi
Gamma-ray Space Telescope (FGST), to this same energy-scale the speed of light
is energy dependent.

##  II Foliations in Hořava gravity

We consider a 3-dimensional foliation of a 4-dimensional manifold $M$. It is a
decomposition of $M$ into a union of disjoint 3-dimensional submanifolds, the
leaves of the foliation, such that the covering of $M$ by charts $U_{i}$
together with homeomorphisms
$\varphi_{i}:U_{i}\iff\mathcal{N}_{i}\subset\mathbb{R}^{4}$ satisfy for
overlapping pairs $U_{i},U_{j}$:

$\varphi_{j}\circ\varphi_{i}^{-1}:\mathbb{R}^{4}\to\mathbb{R}^{4},$ (1)

is a $C^{\infty}$ bijection from $\varphi_{i}(U_{i}\cap U_{j})$ onto
$\varphi_{j}(U_{i}\cap U_{j})$ and

$\varphi_{j}\circ\varphi_{i}^{-1}:(t,x)\to(\tilde{t}(t),\tilde{x}(t,x))\,,$
(2)

where $x$ are coordinates on the 3-dimensional leaves, $\Sigma_{t}$, and $t$
on the codimension $1$ manifold which we take to be $R$. We assume that the
leaves are Riemannian manifolds with metric $g_{ij}$. In the Hořava-Lifshitz
formulation $t$ and $x^{i}$, $i=1,2,3$ have different dimensions and there is
no a priori universal constant of dimension $[T]^{-1}[L]$. Although there are
couplings with this dimension they run with the energy scale, hence only on
the infrared limit (or UV one) we can have such constant. Although there is
not a metric on the $M$ manifold, there are intrinsic geometrical objects
which we define:

$N(t,x)\,dt,$ (3)

with dimension $[T]$ and

$dx^{i}+N^{i}(t,x)dt,$ (4)

with dimension $[L]$. They transform under (2) :

$\displaystyle\tilde{N}(\tilde{t},\tilde{x})d\tilde{t}=N(x,t)dt,$ (5)
$\displaystyle
d\tilde{x}^{i}+\tilde{N}^{i}(\tilde{t},\tilde{x})d\tilde{t}=\frac{\partial\tilde{x}^{i}}{\partial
x^{j}}[dx^{j}+N^{j}(t,x)dt],$ (6)

and allow to have an intrinsic volume element

$\displaystyle
Ndt\wedge(dx^{1}+N^{1}dt)\wedge(dx^{2}+N^{2}dt)\wedge(dx^{3}+N^{3}dt)\sqrt{g}=$
$\displaystyle Ndt\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}\sqrt{g}\,,\qquad$
(7)

where $g$ is the determinant of the 3-dimensional metric $g_{ij}$ on the
leaves. The metric is taken to be a dimensionless tensor under the
diffeomorphisms on the leaves.

From (5,6) we obtain the transformation law of $N$, a dimensionless density,
and $N^{i}$. In the Hořava-Lifshitz formulation of gravity $t$ scales as
$b^{z}$ while $x^{i}$ as $b^{1}$, consequently, the dimension of $t$ is
$[t]=[L]^{z}$ and that of $N^{i}$ is $[N^{i}]=[L]^{1-z}$. We notice that the
above construction is generic, for any value of $z$. In the 4-dimensional
Hořava-Lifshitz gravitational theories $z=3$ in order to have a dimensionless
overall coupling constant. Hence $[N^{i}]=[L]^{-2}$ and as we have determined
$[N]=[L]^{0}$.

The transformation law for $N$ and $N^{i}$ under

$\tilde{t}=\tilde{t}(t)\quad,\qquad\tilde{x}^{i}=\tilde{x}^{i}(t,x),$ (8)

is then

$\displaystyle\tilde{N}(\tilde{t},\tilde{x})\dot{\tilde{t}}(t)=N(t,x)\,,$ (9)
$\displaystyle\tilde{N}^{i}(\tilde{t},\tilde{x})\dot{\tilde{t}}(t)=-\dot{\tilde{x}}^{i}(t,x)+\frac{\partial\tilde{x}^{i}}{\partial
x^{j}}N^{j}(t,x),$ (10)

respectively, $()\dot{}\equiv\frac{\partial}{\partial t}()$. We notice that
$N$ and $N^{i}$ transform as densities under time reparametrization; $N^{i}$
does not transform as a vector field under diffeomorphisms on the leaves. The
first term on the right hand side member of (10) is characteristic of the
transformation law of a Lagrange multiplier. If we rewrite the general finite
transformation of coordinates as

$\tilde{t}=t+f(t)\,,\qquad\tilde{x}^{i}=x^{i}+\xi^{i}(t,x),$ (11)

$f$ and $\xi^{i}$ are $C^{1}$ arbitrary functions, we obtain

$\tilde{N}(\tilde{t},\tilde{x})(1+\dot{f}(t))=N(t,x)\,,$ (12)
$\tilde{N}^{i}(\tilde{t},\tilde{x})(1+\dot{f}(t))=-\dot{\xi}^{i}(t,x)+N^{i}(t,x)+\frac{\partial\xi^{i}}{\partial
x^{j}}N^{j}(t,x)\,,$ (13)

and also

$g_{ij}(t,x)=\frac{\partial\tilde{x}^{l}}{\partial
x^{i}}\frac{\partial\tilde{x}^{m}}{\partial
x^{j}}\tilde{g}_{lm}(\tilde{t},\tilde{x}).$ (14)

###  II.1 The T+L decomposition

In the following analysis we will use the T+L decomposition of symmetric
tensors fields ($f_{ij}$) vanishing at infinity in terms of linear orthogonal
symmetric parts [49],

$f_{ij}=f_{ij}^{TT}+f_{ij}^{T}+2f_{(i,j)}\,.$ (15)

The transverse part
$f_{ij}^{T}\equiv\frac{1}{2}[\delta_{ij}f^{T}-\frac{1}{\Delta}f_{,ij}^{T}]$
satisfies $f_{ij,j}^{T}=0$. The transverse trace-less $f_{ij}^{TT}$ satisfies
$f_{ij,j}^{TT}=0$ and $f_{ii}^{TT}=0$. The remaining term $2f_{(i,j)}$ is its
longitudinal part. $f^{T}=f_{ii}^{T}$ is the trace of the transverse part of
$f_{ij}$. $\frac{1}{\Delta}$ is the inverse of the flat space Laplacian,
defined on the space of functions which vanishes at infinity.

The tensor components are

$\displaystyle f_{i,j}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\left\\{f_{ik,kj}-\frac{1}{2}\left[\frac{1}{\Delta}f_{lm,lm}\right]_{,ij}\right\\},$
(16) $\displaystyle f_{ij}^{T}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[f^{T}\delta_{ij}-\frac{1}{\Delta}f_{,ij}^{T}\right],$
(17) $\displaystyle f^{T}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\left(f_{ll,mm}-f_{lm,lm}\right),$ (18)
$\displaystyle f_{ij}^{TT}$ $\displaystyle=$ $\displaystyle
f_{ij}-f_{ij}^{T}-2f_{(i,j)}\,.$ (19)

##  III The 3+1-Hořava-Lifshitz gravity at the conformal kinetic point

The 3+1 anisotropic Hořava-Lifshitz Hamiltonian at the conformal kinetic point
is given by

$\displaystyle
H=\int_{\Sigma_{t}}d^{3}x\bigg{\\{}N\sqrt{g}\left[\frac{\pi^{ij}\pi_{ij}}{g}-\mathcal{V}\left(g_{ij},N\right)\right]$
$\displaystyle-N_{j}H^{j}-\sigma P_{N}-\mu\pi\bigg{\\}}+\beta E_{ADM}.$ (20)

Where $(N_{i},\sigma,\mu)$ are Lagrange multiplier, $(H^{j},P_{N},\pi)$ are
primary constraints, and the potential
$\mathcal{V}=\mathcal{V}^{(1)}+\mathcal{V}^{(2)}+\mathcal{V}^{(3)}$ is the
most general scalar constructed from the 3-dimensional spacelike metric and
the lapse $N$-function. It is independent of the conjugate momenta and of the
$N_{i}\equiv g_{ij}N^{j}$. The expression of the potential, where only the
interacting terms which will contribute to the wave zone are explicitly given,
is the following

$\displaystyle\mathcal{V}^{(1)}$ $\displaystyle=$ $\displaystyle\beta R+\alpha
a_{i}a^{i},$ (21) $\displaystyle\mathcal{V}^{(2)}$ $\displaystyle=$
$\displaystyle\alpha_{1}R\nabla_{i}a^{i}+\alpha_{2}\nabla_{i}a_{j}\nabla^{i}a^{j}+\beta_{1}R_{ij}R^{ij}$
(22) $\displaystyle+\beta_{2}R^{2}+O^{(2)}(a_{i}a^{i}),$
$\displaystyle\mathcal{V}^{(3)}$ $\displaystyle=$
$\displaystyle\alpha_{3}\nabla^{2}R\nabla_{i}a^{i}+\alpha_{4}\nabla^{2}a_{i}\nabla^{2}a^{i}+\beta_{3}\nabla_{i}R_{jk}\nabla^{i}R^{jk}$
(23) $\displaystyle+\beta_{4}\nabla_{i}R\nabla^{i}R+O^{(3)}(R_{ij},a_{k})\,,$

here, $a_{i}\equiv\frac{1}{N}\partial_{i}N$, is a 3-vector under
transformations (8) with dimension $[L]^{-1}$ [2], and $\alpha$’s and
$\beta$’s are coupling constants. We notice that couplings involved in (23)
are dimensionless. The terms $O^{(2)}$ and $O^{(3)}$ contain products of more
than two fields. They will not contribute to the leading order term neither to
the next order one in the wave zone.

The surface integral,

$E_{ADM}\equiv\oint_{\partial\Sigma_{t}}\left(\partial_{j}g_{ij}-\partial_{i}g_{jj}\right)dS_{i},\,$
(24)

is added in order to ensure the differentiability of the Hamiltonian with
respect to the metric, see [50] where this idea was introduced for RG.
$E_{ADM}$ is the well known ADM-energy in General Relativity.

The primary constraints,

$\displaystyle\pi\equiv g_{ij}\pi^{ij}=0,$ (25) $\displaystyle P_{N}=0,$ (26)

are second class, while the momentum constraints,

$\displaystyle H^{j}\equiv 2\nabla_{i}\pi^{ij}=0,$ (27)

are first class ones.

The conservation of the primary constrains (25, 26) yields the constraints

$\displaystyle H_{P}$ $\displaystyle\equiv$
$\displaystyle\frac{3}{2}\frac{N}{\sqrt{g}}\pi^{ij}\pi_{ij}+\frac{1}{2}N\sqrt{g}\beta
R+N\sqrt{g}\left(\frac{\alpha}{2}-2\beta\right)a_{i}a^{i}$ (28)
$\displaystyle-2\beta N\sqrt{g}\nabla^{i}a_{i}+\\{\pi,\mathcal{U}\\}_{PB}=0,$
$\displaystyle H_{N}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{\sqrt{g}}\left(\pi^{ij}\pi_{ij}-\beta
gR\right)-\alpha\sqrt{g}a_{i}a^{i}$ (29)
$\displaystyle-2\alpha\sqrt{g}\nabla_{i}a^{i}+\\{P_{N},\mathcal{U}\\}_{PB}=0\,,$

where

$\mathcal{U}\equiv-\int_{\Sigma_{t}}d^{3}xN\sqrt{g}(\mathcal{V}^{(2)}+\mathcal{V}^{(3)})\,.$
(30)

These are second class constraints. Conservation of (28, 29) determine
Lagrange multipliers. Then, (25), (26), (27), (28) and (29) is the complete
set of constraints.

The dynamical field equations are

$\displaystyle\dot{g}_{ij}$ $\displaystyle=$
$\displaystyle\frac{2N}{\sqrt{g}}\pi_{ij}+2\nabla_{(i}N_{j)}-\mu g_{ij},$ (31)
$\displaystyle\dot{\pi}^{ij}$ $\displaystyle=$
$\displaystyle\frac{N}{2}\frac{g^{ij}}{\sqrt{g}}\pi^{kl}\pi_{kl}-\frac{2N}{\sqrt{g}}\pi^{ik}\pi^{j}{}_{k}+N\sqrt{g}\beta\left(\frac{R}{2}g^{ij}-R^{ij}\right)$
(32) $\displaystyle-\alpha
N\sqrt{g}\left(a^{i}a^{j}-\frac{1}{2}g^{ij}a_{k}a^{k}\right)-\nabla_{k}\left[2\pi^{k(i}N^{j)}-\pi^{ij}N^{k}\right]$
$\displaystyle+\beta\sqrt{g}\left[\nabla^{(i}\nabla^{j)}N-g^{ij}\nabla^{2}N\right]+\mu\pi^{ij}+\\{\pi^{ij},\mathcal{U}\\}_{PB}\,,$

where the last term in (32) will contribute with high order in spatial
derivatives terms to the wave zone.

The Hamiltonian (20) can be rewritten in terms of the constraints of the
theory, up to total divergence. In fact, (29) can be written as

$\displaystyle
H_{N}\equiv\sqrt{g}\left(\frac{\pi^{ij}\pi_{ij}}{g}-\mathcal{V}\right)-\frac{\sqrt{g}}{N}\nabla_{i}\left[-N\frac{\partial\mathcal{V}}{\partial
a_{i}}\right.$
$\displaystyle\left.+\nabla_{j}\left(N\frac{\partial\mathcal{V}}{\partial\nabla_{j}a_{i}}\right)-\nabla_{k}\nabla_{j}\left(N\frac{\partial\mathcal{V}}{\partial\nabla_{j}\nabla_{k}a_{i}}\right)+\cdots\right]=0,\qquad$
(33)

where the first term arise from variation of the factor $N$ while the second
terms comes from the variation of the potential. It follows then, that

$\displaystyle
N\sqrt{g}\left(\frac{\pi^{ij}\pi_{ij}}{g}-\mathcal{V}\right)=NH_{N}+\sqrt{g}\nabla_{i}\left[-N\frac{\partial\mathcal{V}}{\partial
a_{i}}\right.$
$\displaystyle\left.+\nabla_{j}\left(N\frac{\partial\mathcal{V}}{\partial\nabla_{j}a_{i}}\right)-\nabla_{k}\nabla_{j}\left(N\frac{\partial\mathcal{V}}{\partial\nabla_{j}\nabla_{k}a_{i}}\right)+\,\cdots\right].\qquad$
(34)

Under asymptotically flat conditions:
$g_{ij}=\delta_{ij}+\mathcal{O}\left(\frac{1}{r}\right)$,
$N=1+\mathcal{O}\left(\frac{1}{r}\right)$,
$N_{i}=\mathcal{O}\left(\frac{1}{r}\right)$,
$\pi^{ij}=\mathcal{O}\left(\frac{1}{r}\right)$, and all first spatial
derivatives of fields of order $\mathcal{O}\left(\frac{1}{r^{2}}\right)$, the
total divergence that contributes at infinity arises from the first term in
the parenthesis. It is

$-2\alpha\sqrt{g}\nabla_{i}\left(g^{ij}\nabla_{j}N\right).$ (35)

We may now evaluate the physical Hamiltonian density in a particular
coordinate system. We proceed as in [29]. The Lagrangian evaluated on the
constraints submanifold is given by

$L=\int dtd^{3}x\left(\pi^{ij}\dot{g}_{ij}-\mathcal{H}\right)-E_{ADM},$ (36)

where $\mathcal{H}\equiv-2\alpha\sqrt{g}\nabla_{i}\nabla^{i}N+\text{total
divergence}$. The generic terms ”total divergence” do not contribute to the
Lagrangian under the asymptotically flat conditions. The Lagrangian $L$ can be
expressed in terms of the T+L components as in [29]. We may consider the
coordinate condition

$g_{i}=x^{i}+\left(\frac{1}{4\Delta}\right)g_{,i}^{T}\,,\quad\Delta\equiv\delta^{ij}\partial_{i}\partial_{j},$
(37)

where $g_{i}$ is the longitudinal part of the metric, and we end up, following
[29] with

$\mathcal{H}=-\beta\Delta
g^{T}-2\alpha\sqrt{g}\nabla_{i}\left(g^{ij}\partial_{j}N\right),$ (38)

as the Hamiltonian density. $\Delta g^{T}$ and $\Delta N$ are obtained from
the constraints (28,29) in terms of the $g_{ij}^{TT}$, $\pi^{ijTT}$ physical
components.

The gravitational energy is then given by

$E=\int dt\,d^{3}x\,\mathcal{H}=\oint_{\partial\Sigma_{t}}\left(-\beta
g_{,i}^{T}-2\alpha N_{,i}\right)dS_{i},$ (39)

which is, of course, independent of the coordinate condition (37).

In order to obtain $g^{T}$ and $N$ from the constraints (28,29) a condition on
the invertibility of the higher order derivative operator must be imposed. We
discuss this point in the section (VI.3). The invertibility condition,
expressed as the absence of the zero modes, is assured once several conditions
on the coupling parameters are imposed.

##  IV The Wave zone definition

The ”wave zone” is an asymptotic region in space within which the degrees of
freedom of the theory represent free radiation. That is, the leading order
satisfies a wave equation and represents traveling spherical waves that escape
to infinity without being affected by the source.

In the wave zone the self-interactions do not affect the dominant order. In
the analysis the non-linearities are not eliminated a priori, indeed there can
exist non-trivial static terms of the same order of the leading canonical one,
but nevertheless they do not affect its propagation as free-radiation. This
approach is different from the first-order perturbation analysis where non-
linear effects are eliminated from the beginning. The existence of a wave zone
is a very important property of non-linear theories describing long range
interactions.

In linear theories like classical electrodynamics, non-linearities may be due
to sources and they do not occur if we take distances far enough from them.
Then in this case, if radiation has a wave-number $k$, the wave zone coincide
with the ”far zone” $kr\gg 1$. That is, within spatial distance between source
and the wave zone, $r$, there fit a large number of wavelengths and it
guarantees that gradients and temporal derivatives of canonical modes are
$\mathcal{O}(1/r)$. In general, we can apply the time-Fourier-transformation
to the wave equation, it becomes a Helmholtz equation $\Delta u+k^{2}u=0$ and
one can write its solution in the form
$u=\frac{e^{ikr}}{r}\sum_{n=0}\frac{f_{n}(\theta,\phi)}{r^{n}}$. Then it is
straightforward to prove that if the $f_{0}$-coefficient of the $1/r$\- term
is zero then $u=0$, that is, in the wave zone the only non-zero solution of
the wave equation contains always the $\mathcal{O}(1/r)$ contribution. This is
different from the solution of the Laplace equation. In fact, even if the
monopole, dipole, etc, contributions are zero, the multipolar contributions
can be nonzero.

In general, for non-linear theories the propagating fields may act as
effective sources of itself. This self-interaction phenomena break-up the
desired free-radiation behavior. Therefore the far zone condition is not
sufficient to define the wave zone. The additional requirements have to
guarantee that the field amplitudes are small such that the non-linear terms
are ”negligible”. Here ”negligible” means that it does not affect the free-
wave propagation.

For GR there exists a well-defined wave zone, in which the background-
curvature does not affect the radiation of canonical modes. The gauge
invariant fields have the following behaviour: The oscillatory part of
$g_{ij}^{TT}\sim\pi^{ijTT}=\mathcal{O}(1/r)$, the oscillatory part of
$g_{ij}^{T}\sim\mathcal{O}(1/r^{2})$, the static part of
$g_{ij}^{T}=\mathcal{O}(1/r)$ and of $\pi^{ijL}=\mathcal{O}(1/r^{2})$. The
remaining parts of $g_{ij}$ and $\pi^{ij}$ and ADM variables $(N,N^{i})$ are
gauge dependent and they do not affect the propagation of canonical modes
[30].

The wave zone split the space into three regions: the interior, wave zone and
the exterior. The ”interior” region bounded by the sphere of radius $R$
centered at the origin, $B_{R}(0)$, contains the sources and possibly space-
time singularities; here the phenomena of self-interaction are not negligible
even at points where sources are not located. The wave zone,
$B_{R_{1}}(0)/B_{R}(0)$ with $R<r<R_{1}$, where the curvature does not affect
radiation. Finally the ”exterior” region, $\mathbb{R}^{3}/B_{R_{1}}(0)$, where
the propagating fields decay rapidly. This represents an asymptotically flat
space-time.

In this work we prove that in the 3+1 non-projectable Hořava-Lifshitz theory
at the kinetic-conformal point, there exists a wave zone as it does in GR.
That is, the $(g_{ij}^{TT},\pi^{ijTT})\sim f(\theta,\phi)\exp
i(kx-\omega(k)t)/r$, with $f(\theta,\phi)=\mathcal{O}(1)$, propagate without
being affected by the static parts in the same way as it happens in the
linearized theory. Due to the anisotropic character of this theory the
dispersion relation contains, besides the terms $\omega^{2}$ and $k^{2}$,
$k^{4}$ and $k^{6}$ terms associate to higher order derivative terms, this
represents a propagating dispersive signal. At low energies the dispersion
reduces to the usual non-dispersive wave and differs from GR in the speed of
propagation. The speed of propagation at low energies is equal to
$\sqrt{\beta}$ and the recent gravitational waves observations fixed it to $1$
the GR value, up to an error of the order of one part in $10^{15}$[41, 42, 21,
20].

We define the wave zone in Hořava-Lifshitz theory, as in GR, to be the space
like domain far away from the sources satisfying the conditions:

  1. $i)$

$kr\gg 1$. $k$ is the wave number, $r=(x_{i}x^{i})^{1/2}$.

  2. $ii)$

$\displaystyle|g_{ij}-\delta_{ij}|\lesssim A/r,$ (40)
$\displaystyle\quad|N-1|\lesssim A/r,$ (41) $\displaystyle\quad|N^{i}|\lesssim
A/r,$ (42)

where $A(t,\theta,\phi)$ represents functions of time and angles such that $A$
and all its derivatives are bounded.

  3. $iii)$

$\displaystyle|\partial g_{ij}/\partial(kr)|^{2}\ll|g_{ij}-\delta_{ij}|\,,$
(43) $\displaystyle|\partial N_{i}/\partial(kr)|^{2}\ll|N_{i}|\,,$ (44)
$\displaystyle|\partial N/\partial(kr)|^{2}\ll|N-1|\,,$ (45)

are fulfilled. Here we used the short notation $\lesssim A/r$ $\equiv$
$\mathcal{O}(A/r)$, thus ”$\lesssim A/r$” means the left part decrease at
least as $A/r$. And $(g,\delta)$ are symbolic representations of any component
of the metric and the euclidean tridimensional metric tensor.

The three conditions can be satisfied by choosing a region with large enough
distance from the source. The first condition is the same one as in linear
theories and indicates that the wave zone is quite far from the sources (in
distance measured in units of wavelength) and guarantees that the gradients of
the canonical modes are $\mathcal{O}(1/r)$ and exclude the wave number
$k\rightarrow 0$ in the wave zone. The second and third conditions are
necessary to ensure that, in the wave zone, there are no self-interactions and
the canonical variables propagate freely. The second condition is imposed in
order to ensure that not only the perturbations of the dynamic modes are
$\mathcal{O}(1/r)$ but that the Newtonian parts are also $\mathcal{O}(1/r)$,
and guarantees that the terms of higher order than the leading one in
$|g_{ij}-\delta_{ij}|$ remain negligible compared to the leading order. The
third condition is taken in order to ensure that the non-linear terms
containing spatial derivatives are small compared to the leading order. This
implies that for a wave-number $k>k_{min}$ the radiation can be treated as
free radiation if $k_{min}\gg k_{max}(A/r)^{1/2}$, here $|g-\delta|\sim A/r$
and $k_{max}$ is the maximum frequency. It ensures that the interference of
two or more sub-leading $\mathcal{O}(1/r^{2})$ modes do not generate an effect
comparable to leading $\mathcal{O}(1/r)$.

Conditions (40,41,42) are preserved under a change of coordinates (8) if and
only if

$\displaystyle|\dot{\xi}^{i}|\lesssim\frac{A^{i}}{r}\ll
1,\quad|{\xi^{i}}_{,j}|\lesssim\frac{{A^{i}}_{j}}{r}\ll
1,\quad|\dot{f}(t)|\lesssim\frac{A}{r}\ll 1.$

We then have, besides (12,13,14),

$\tilde{N}(1+\dot{f})=N,$ (47)
$\tilde{N}^{i}(\tilde{t},\tilde{x})(1+\dot{f}(t))=N^{i}(t,x)-\dot{\xi}^{i}(t,x)+\mathcal{O}(1/r^{2})\,,$
(48)
$\tilde{g}_{ij}(\tilde{t},\tilde{x})-g_{ij}(t,x)=-(\xi_{i,j}(x,t)+\xi_{j,i}(x,t))+\mathcal{O}(1/r^{2}).$
(49)

We remark that $\frac{\mathcal{O}(1/{r^{2}})}{1/{r}}\ll 1$.

We then conclude that the longitudinal part of $g_{ij}$ is a gauge dependent
field. It can be fixed by choosing particular coordinates on the leaves. We
notice that the admissible coordinate transformations are not necessarily
infinitesimal ones. In fact, one may have a parameter $\xi\sim\log r$.
Besides, one may have parameters with dependence $1/r$ and also $e^{ikr}/r$ or
$\mathcal{O}(1/r^{2})$. Consequently, the continuous transformations on the
wave-zone are an extension of the infinitesimal ones. These ones have the
usual form

$\displaystyle\tilde{N}({t},{x})-N(t,x)$ $\displaystyle=$
$\displaystyle-\xi^{k}\partial_{k}N(t,x)-\dot{f}(t)N(t,x)$ (50)
$\displaystyle-f(t)\dot{N}(t,x)\,,$
$\displaystyle\tilde{N}^{i}({t},{x})-N^{i}({t},{x})$ $\displaystyle=$
$\displaystyle-\xi^{k}\partial_{k}N^{i}({t},{x})+\partial_{j}\xi^{i}N^{j}({t},{x})-\dot{\xi}^{i}$
(51) $\displaystyle-\dot{f}(t)N^{i}(t,x)-f(t)\dot{N}^{i}(t,x)\,,$
$\displaystyle\tilde{g}_{ij}({t},{x})-{g}_{ij}({t},{x})$ $\displaystyle=$
$\displaystyle-\xi^{k}\partial_{k}g_{ij}({t},{x})-$ (52) $\displaystyle
g_{ki}\partial_{j}\xi^{k}-g_{kj}\partial_{i}\xi^{k}-f\dot{g}_{ij}({t},{x})\,,\qquad$

where in this case, $f(t)$ and $\xi^{i}(t,x)$ are infinitesimal parameters. We
notice that these transformations contain terms which are of order
$\mathcal{O}(1/r^{2})$, hence in the wave-zone at order $\mathcal{O}(1/r)$
they can be eliminated.

##  V Behavior in the wave zone

In the wave zone the metric, lapse and shift fields have the behavior

$\displaystyle g_{ij}-\delta_{ij}\sim N_{i}\sim N-1$ $\displaystyle\lesssim$
$\displaystyle\frac{B_{ij}}{r}+\frac{A_{ij}e^{ikr}}{r},$ (53) $\displaystyle
N_{i,j}\sim g_{ij,l}\sim\Gamma^{i}_{\hskip
2.84544ptjl}\lesssim\frac{B}{r^{2}}+k\frac{Ae^{ikr}}{r}\,,$ (54)

where $A$ and $B$ are a generic functions of time and angles with bounded
derivatives.

###  V.1  Laplacian’s solutions in the wave zone

Let $\varphi\in C^{2}(\Omega)$ be a solution of Poisson equation
$\Delta\varphi=-4\pi\rho$ in $\Omega\equiv\mathbb{R}^{3}/B_{R}(0)$ such that
$\varphi\rightarrow 0$ when $r\rightarrow\infty$. Expanding
$\varphi(r,\theta,\phi)$ into spherical harmonics $Y_{lm}(\theta,\phi)$ a
standard result gives,

$\displaystyle\varphi(r,\theta,\phi)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{l=\infty}\sum_{m=-l}^{m=l}\chi_{lm}(r)r^{l}Y_{lm}(\theta,\phi),$
(55) $\displaystyle\chi_{lm}(r)$ $\displaystyle\equiv$
$\displaystyle\int_{r}^{\infty}\frac{M_{lm}(r^{\prime})}{r^{\prime
2l+2}}dr^{\prime},$ (56) $\displaystyle M_{lm}(r)$ $\displaystyle\equiv$
$\displaystyle\int_{B_{r}(0)}\rho(r^{\prime},\theta^{\prime},\phi^{\prime})r^{\prime
l}Y^{\prime}_{lm}(r^{\prime},\theta^{\prime},\phi^{\prime})d^{3}r^{\prime}.\quad$
(57)

Following Arnowitt, Desser and Misner [30] in the wave zone we have:

  * $i$)

If in the region $r>R$ the source $\rho$ has a oscillatory-asymptotic-behavior
$\rho\sim Y_{lm}e^{ikr}/r^{n}$ then,

$\varphi\sim\left(1/k^{2}\right)Y_{lm}\frac{e^{ikr}}{r^{n}}\left[1+\mathcal{O}(1/kr)\right]+\sum_{l=0}^{l=\infty}\sum_{m=-l}^{m=l}c_{lm}Y_{lm}\frac{1}{r^{l+1}}.$
(58)

Here the $c_{lm}$ do not depend on the spatial coordinates and they are
determined from the behavior of $\rho$ in the region $r\leq R$. Hence, an
oscillatory source produce in $r>R$ a source-independent oscillatory solution
of the same order of $\rho$ plus a static source-depend part $\lesssim 1/r$.
By static we mean non-oscillatory on the space dependence.

  * $ii$)

If in the region $r>R$,

$\psi\sim B_{m}(t,\theta,\phi)/r^{m}+A_{n}(t,\theta,\phi)e^{ikr}/r^{n},$ (59)

with $|B_{m}(t,\theta,\phi)|$ and $|A_{n}(t,\theta,\phi)|$ bounded, then

$\displaystyle\frac{1}{\Delta}\psi,_{j}$ $\displaystyle\sim$
$\displaystyle\frac{\langle B\rangle}{r^{m-1}}+\frac{1}{ik}\frac{\langle
A\rangle e^{ikr}}{r^{n}}$ (60)
$\displaystyle+\sum_{l=1}^{\infty}\sum_{p=-l}^{p=l}c_{lp}Y_{lp}\frac{1}{r^{l+1}},\quad(m\geq
2).$ $\displaystyle\frac{1}{\Delta}\psi,_{ij}$ $\displaystyle\sim$
$\displaystyle\frac{\langle B\rangle}{r^{m}}+\frac{\langle A\rangle
e^{ikr}}{r^{n}}$ (61)
$\displaystyle+\sum_{l=2}^{\infty}\sum_{p=-l}^{p=l}c_{lp}Y_{lp}\frac{1}{r^{l+1}},\quad(m\geq
1).$

$\langle A\rangle$, etc, mean generic angular integrals of $A$ and other
angular functions. For $m\geq 3$ the expressions (60) and (61) may also have
contributions of the form $r^{-m+1}\log r$ and $r^{-m}\log r$ respectively,
which are source independent.

##  VI The wave zone in Hořava-Lifshitz gravity at the kinetic conformal point

We now analyze the solution of the field equations in the wave zone and show
that the equations of the canonical transverse-traceless modes describe free
propagating fields.

###  VI.1 Solution of the primary constraints

Taking a derivative with respect to the $j$-coordinate of the T+L
decomposition of the momentum constraint in (27) we obtain

$\Delta\pi^{j}_{\hskip
2.84544pt,j}=-\frac{1}{2}\left(\pi^{lm}\Gamma^{j}_{\hskip
2.84544ptlm}\right)_{,j}.$ (62)

If we use the estimating of previous section, the result (60), and the same
argument of finite momenta used in the wave zone for RG (see the appendix C of
[30])), then

$\pi^{j}_{\hskip
2.84544pt,j}\lesssim\frac{B}{r^{2}}+k\frac{\tilde{A}e^{ikr}}{r^{2}}\,,$ (63)

where $B$ is source dependent while $\tilde{A}$ is source independent. Taking
now a derivative with respect to the $i$ coordinate of the T+L decomposition
of the momentum constraint (27) we get

$\Delta\pi^{j}_{\hskip 2.84544pt,i}=\left(\pi^{l}_{\hskip
2.84544pt,l}\right)_{,ij}-\left(\pi^{lm}\Gamma^{j}_{\hskip
2.84544ptlm}\right)_{,i}\,.$ (64)

Thus, we can use (60,61) to finally obtain

$\pi^{j}_{\hskip
2.84544pt,i}\lesssim\frac{B_{i}^{j}}{r^{2}}+k\frac{\hat{A}_{i}^{j}e^{ikr}}{r^{2}}\,,$
(65)

for new functions $B_{i}^{j}$ and $\tilde{A}_{i}^{j}$.

The constraint (25) expressed on the T+L decomposition becomes
$\pi^{T}+2\pi^{i}_{\hskip 2.84544pt,i}=0$, then

$\displaystyle\pi^{T}\lesssim\frac{B}{r^{2}}+k\frac{\hat{A}e^{ikr}}{r^{2}},\quad\pi^{ijT}\lesssim\frac{B^{ij}}{r^{2}}+k\frac{\hat{A}^{ij}e^{ikr}}{r^{2}},$
(66)

where we have used the estimate given by (61).

We note that the TT term of the T+L decomposition is the dominant one among
the terms with an oscillatory behavior,

$\pi^{ijTT}\lesssim\frac{B^{ij}}{r}+k\frac{\hat{A}^{ij}e^{ikr}}{r}.$ (67)

In the following section we will prove that the static parts of
$\pi^{ij}\lesssim B^{ij}/r^{2}$, where $B^{ij}$ does not depend on time.

###  VI.2 Solution of the secondary constraints

Now we calculate the constraints $H_{N}=0$ and $H_{P}=0$. We include the
$\mathcal{V}$-potential terms up to $\mathcal{O}(1/r)$. That is the relevant
terms on the wave zone. We get for the terms of the form
$\pi^{ij}\pi_{ij}\lesssim B/r^{2}+Ae^{ikr}/r^{2}$, where $B=B^{ij}B_{ij}$ . Up
to order $\mathcal{O}(1/r)$ the second class constraints $H_{N}=0$ and
$H_{P}=0$ become

$\displaystyle-\beta R+2\alpha\Delta N-\alpha_{1}\Delta
R-2\alpha_{2}\Delta^{2}N-\alpha_{3}\Delta^{2}R$
$\displaystyle+2\alpha_{4}\Delta^{3}N\lesssim\frac{\tilde{B}}{r^{2}}+k^{2}P_{2}(k^{2})\frac{\hat{A}e^{ikr}}{r^{2}},\qquad$
(68)

$\displaystyle\frac{\beta}{2}R-2\beta\Delta
N-2\alpha_{1}\Delta^{2}N-2\alpha_{3}\Delta^{3}N$
$\displaystyle-\beta_{1}R^{ij}_{\,,\,ij}+\beta_{3}\Delta
R^{ij}_{\,,\,ij}-(\beta_{1}+4\beta_{2})\Delta R$
$\displaystyle+(\beta_{3}+4\beta_{4})\Delta^{2}R\lesssim\frac{\hat{B}}{r^{2}}+k^{2}P_{2}(k^{2})\frac{\hat{A}e^{ikr}}{r^{2}},\quad$
(69)

respectively, where $\tilde{B}$ and $\hat{B}$ are proportional to $B$. We have
used $P_{2}(k^{2})$ to denote a generic second order polynomial on $k^{2}$
with real coefficients.

If we use the gauge condition $g_{ij,j}=0$, we get

$\displaystyle R_{ik}+\frac{1}{2}\Delta g_{ik}^{TT}+\frac{1}{2}\Delta
g_{ik}^{T}+\frac{1}{2}g_{,ik}^{T}\lesssim\frac{\mathcal{B}}{r^{4}}+k^{2}\frac{Ae^{ikr}}{r^{2}},\quad$
(70) $\displaystyle R+\Delta
g^{T}\lesssim\frac{\mathcal{B}}{r^{4}}+k^{2}\frac{Ae^{ikr}}{r^{2}},\quad$ (71)
$\displaystyle
R^{ij}_{\,,\,ij}+\frac{1}{2}\Delta^{2}g^{T}\lesssim\frac{\mathcal{B}}{r^{6}}+k^{4}\frac{Ae^{ikr}}{r^{2}},\quad$
(72) $\displaystyle\Delta
R^{ij}_{\,,\,ij}+\frac{1}{2}\Delta^{3}g^{T}\lesssim\frac{\mathcal{B}}{r^{8}}+k^{6}\frac{Ae^{ikr}}{r^{2}}.\quad$
(73)

Then from (VI.2) and (VI.2 ) we obtain the coupled system of two sixth order
partial differential equations for $N$ and $g^{T}$:

$\displaystyle\left(\beta\Delta+\alpha_{1}\Delta^{2}+\alpha_{3}\Delta^{3}\right)g^{T}+2\left(\alpha\Delta-\alpha_{2}\Delta^{2}+\alpha_{4}\Delta^{3}\right)N$
$\displaystyle\lesssim\frac{\tilde{B}}{r^{2}}+k^{2}P_{2}(k^{2})\frac{\hat{A}e^{ikr}}{r^{2}},\qquad$
(74)
$\displaystyle\left[-\frac{\beta}{2}\Delta+\frac{1}{2}(3\beta_{1}+8\beta_{2})\Delta^{2}-\frac{1}{2}(3\beta_{3}+8\beta_{4})\Delta^{3}\right]g^{T}$
$\displaystyle-2\left[\beta\Delta+\alpha_{1}\Delta^{2}+\alpha_{3}\Delta^{3}\right]N$
$\displaystyle\lesssim\frac{\hat{B}}{r^{2}}+k^{2}P_{2}(k^{2})\frac{\tilde{A}e^{ikr}}{r^{2}}.\qquad$
(75)

From (VI.2) and (VI.2), it follows that the $B$ factor must vanish, and the
non-oscillatory contribution is of order $O(1/r^{3})$. It means that the non-
oscillatory part of the momentum is of order $O(1/r^{2})$. This point is very
relevant for the next arguments.

By assumption the fields decay as in (53) on the wave zone. From the equations
(VI.2, VI.2) it then follows that the non-oscillatory part of the solution for
$g^{T}$ and $N$ can contribute to this order, since on the right hand side
there may be terms of order $1/r^{3}$. On the other side, the oscillatory
terms of the form $e^{ikr}/r$ satisfies, on the wave zone

$(\Delta+k^{2})\frac{e^{ikr}}{r}=0\,.$ (76)

Consequently, for such terms the equations (VI.2, VI.2), considering terms
$g_{o}^{T}\frac{e^{ikr}}{r}$ and $N_{o}\frac{e^{ikr}}{r}$ imply

$\displaystyle D_{1}(k^{2})g_{o}^{T}+D_{2}(k^{2})N_{o}=0\,,$ (77)
$\displaystyle D_{2}(k^{2})g_{o}^{T}+D_{3}(k^{2})N_{o}=0\,,$ (78)

where we define polynomials

$\displaystyle D_{1}(k^{2})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{8}\left[-(3\beta_{3}+4\beta_{4})k^{6}-(3\beta_{1}+8\beta_{2})k^{4}-\beta
k^{2}\right]\,,$ $\displaystyle D_{2}(k^{2})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left[-\alpha_{3}k^{6}+\alpha_{1}k^{4}-\beta
k^{2}\right]\,,$ (80) $\displaystyle D_{3}(k^{2})$ $\displaystyle\equiv$
$\displaystyle-\alpha_{4}k^{6}-\alpha_{2}k^{4}-\alpha k^{2}\,,$ (81)

equations (77, 78) yield $g_{o}^{T}=N_{o}=0$ for these zero modes, except when

$D(k^{2})\equiv D_{1}(k^{2})D_{3}(k^{2})-D_{2}^{2}(k^{2})=0.$ (82)

In the case where the zero modes vanish, the behaviour of $g^{T}$ and $N$ on
the wave zone, taking into account the asymptotic flatness condition
$N-1\rightarrow 0$, is

$N-1\sim g^{T}\lesssim\frac{B}{r}+\frac{\hat{A}e^{ikr}}{r^{2}},$ (83)

hence,

$\displaystyle
a_{i}\lesssim\frac{B_{i}}{r^{2}}+k\frac{\hat{A}_{i}e^{ikr}}{r^{2}},$ (84)
$\displaystyle
a_{i}a^{i}\lesssim\frac{B}{r^{4}}+k^{2}\frac{\hat{A}e^{ikr}}{r^{4}},$ (85)
$\displaystyle\nabla^{i}a_{i}\lesssim\frac{B}{r^{3}}+k^{2}\frac{\hat{A}e^{ikr}}{r^{2}}.$
(86)

On the other side, there are a finite number of solutions of (82) for which
there are nontrivial solutions for $g_{0}^{T}$ and $N_{0}$ and hence
contributions of the form $\frac{e^{ikr}}{r}$ for $g^{T}$ and $N$. In the low
energy case (82) reduces to a term proportional to $k^{2}$ with factor
$-\frac{\beta^{2}}{4}(\beta-\frac{\alpha}{2})$ which is different from zero
for the values of $\beta$ and $\alpha$ determined from experimental data. At
low energies [29], we have always $g_{0}^{T}=N_{0}=0$. In the next section we
analyze the general solution of (82).

###  VI.3 Constraints resolution

The polynomial of equation (82) has $D(k^{2})=k^{4}Q(k^{2})$ where we defined
the quartic polynomial in $k^{2}$ as

$\displaystyle Q(k^{2})$ $\displaystyle\equiv$ $\displaystyle
ak^{8}+bk^{6}+ck^{4}+dk^{2}+e,$ (87) $\displaystyle a$ $\displaystyle\equiv$
$\displaystyle\alpha_{4}\left(3\beta_{3}+4\beta_{4}\right)-2\alpha_{3}^{2},$
(88) $\displaystyle b$ $\displaystyle\equiv$
$\displaystyle\alpha_{2}\left(3\beta_{3}+4\beta_{4}\right)+\alpha_{4}\left(3\beta_{1}+8\beta_{2}\right)$
(89) $\displaystyle+4\alpha_{1}\alpha_{3},$ $\displaystyle c$
$\displaystyle\equiv$
$\displaystyle\alpha\left(3\beta_{3}+4\beta_{4}\right)+\alpha_{2}\left(3\beta_{1}+8\beta_{2}\right)$
(90) $\displaystyle+\beta\left(\alpha_{4}-4\alpha_{3}\right)-2\alpha_{1}^{2},$
$\displaystyle d$ $\displaystyle\equiv$
$\displaystyle\alpha\left(3\beta_{1}+8\beta_{2}\right)+\beta\left(4\alpha_{1}+\alpha_{2}\right),$
(91) $\displaystyle e$ $\displaystyle\equiv$
$\displaystyle\beta\left(\alpha-2\beta\right).$ (92)

For the theory to be power counting renormalizable we demand $a\neq 0$ [13].
Experimental tests at low energies require that $e\leq 0$. Note that $k^{2}=0$
is a root of (82), this root is not considered in the analysis in the wave
zone due to the condition $kr\gg 1$.

We are interested in the roots of $Q(k^{2})=0$, we follow the Descartes
approach. The change of variable $k^{2}=s-\frac{b}{4a}$ yields the depressed
polynomial associate to $Q(k^{2})=0$

$\displaystyle P(s)\equiv a\left(s^{4}+ps^{2}+qs+r\right)=0,$ (93)
$\displaystyle p\equiv\frac{c}{a}-\frac{3}{8}\frac{b^{2}}{a^{2}}\,,$ (94)
$\displaystyle
q\equiv\frac{1}{8}\frac{b^{3}}{c^{3}}+\frac{d}{a}-\frac{bc}{2a^{2}}\,,$ (95)
$\displaystyle
r\equiv-\frac{3}{256}\frac{b^{4}}{a^{4}}+\frac{1}{16}\frac{cb^{2}}{a^{3}}-\frac{bd}{a^{3}}+\frac{e}{a}\,.$
(96)

We factorized

$\displaystyle P(s)=a\left(s^{2}+fs+g\right)\left(s^{2}-fs+h\right)$ (97)
$\displaystyle g+h=f^{2}+p,$ (98) $\displaystyle f(h-g)=q,$ (99)
$\displaystyle gh=r,$ (100)

where $f^{2}$ satisfies solvent cubic polynomial

$R(f^{2})\equiv f^{6}+2pf^{4}+(p^{2}-4r)f^{2}-q^{2}=0.$ (101)

There always exist a real positive solution for (101). In fact, if $q^{2}>0$,
then (101) always has a strictly positive solution. If $q=0$ we can chose
$f^{2}=0$ as root and $P(s)$ is a bi-quadratic polynomial its roots can be
determined using the general quadratic formula for $s^{2}$. If $f\neq 0$ from
(97) we can calculate $g$ and $h$ via

$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(f^{2}+p+\frac{q}{f}\right),$ (102)
$\displaystyle g$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(f^{2}+p-\frac{q}{f}\right).$ (103)

Finally, we can express (87) in a factorized form

$Q(k^{2})=a\left(k^{4}+Ak^{2}+B\right)\left(k^{4}+Ck^{2}+D\right).$ (104)

where the constants $A,B,C,D$ are related to the coupling parameters by

$\displaystyle A=f+\frac{1}{2}\frac{b}{a},\quad
B=\frac{1}{4}\frac{bf}{a}+\frac{1}{16}\frac{b^{2}}{a^{2}}+g,$ (105)
$\displaystyle C=-f+\frac{1}{2}\frac{b}{a},\quad
D=-\frac{1}{4}\frac{bf}{a}+\frac{1}{16}\frac{b^{2}}{a^{2}}+h\,.$ (106)

If $B$ and $D$ have nonzero imaginary part, then there are not zero modes. If
they are real, then there are not zero modes iff the coupling parameters
satisfy the following conditions

$a<0,\quad B>0,\quad D>0,\quad-2\sqrt{B}<A,\quad-2\sqrt{D}<C.$ (107)

The zero modes have, generically, a divergent contribution to the total
gravitational energy. In fact the energy density obtained from the boundary
terms in the Hamiltonian, evaluated at the zero modes decay as $1/r$.

We have thus shown that the conditions (107) must be satisfied in order to
have a consistent formulation of Hořava-Lifshitz gravity.

###  VI.4 The dynamical equations on the wave zone

The T+L components of Equation (31) in the gauge $g_{ij,j}=0$, becomes

$\displaystyle\dot{g}_{ij}^{TT}-2\pi_{ij}^{TT}+\left(\mu g_{ij}\right)^{TT}$
$\displaystyle\lesssim$
$\displaystyle\frac{B}{r^{2}}+k^{2}\frac{\hat{A}e^{ikr}}{r^{2}},$ (108)
$\displaystyle-2N_{(i,j)}+\left(\mu g_{ij}\right)^{L}$ $\displaystyle\lesssim$
$\displaystyle\frac{B}{r^{2}}+k^{2}\frac{\hat{A}e^{ikr}}{r^{2}},$ (109)
$\displaystyle\dot{g}_{ij}^{T}+\left(\mu g_{ij}\right)^{T}$
$\displaystyle\lesssim$
$\displaystyle\frac{B}{r^{2}}+k^{2}\frac{\hat{A}e^{ikr}}{r^{2}},$ (110)

where $\mu g_{ij}=\left(\mu g_{ij}\right)^{TT}+\left(\mu
g_{ij}\right)^{T}+\left(\mu g_{ij}\right)^{L}$ and $\left(\mu
g_{ij}\right)^{T}=\mu\delta_{ij}-\frac{1}{\Delta}\mu_{,ij}+\mathcal{O}\left(1/r^{2}\right)$,
$\left(\mu
g_{ij}\right)^{L}=\frac{1}{\Delta}\mu_{,ij}+\mathcal{O}\left(1/r^{2}\right)$
and $\left(\mu g_{ij}\right)^{TT}=\mathcal{O}\left(1/r^{2}\right)$. Then from
(110) we get $\mu\lesssim\frac{B}{r^{2}}+k^{2}\frac{\hat{A}e^{ikr}}{r^{2}}$
and consequently

$\dot{g}_{ij}^{TT}-2\pi_{ij}^{TT}\lesssim\frac{B}{r^{2}}+k^{2}\frac{\hat{A}e^{ikr}}{r^{2}}\,,$
(111) $N_{(i,j)}\lesssim\frac{B}{r^{2}}+k^{2}\frac{\hat{A}e^{ikr}}{r^{2}}\,.$
(112)

We then have $N_{i}\lesssim\frac{B}{r}+k\frac{\hat{A}e^{ikr}}{r^{2}}$.

Besides we get the following contributions in (32)

$\displaystyle\beta N\frac{\delta R}{\delta g_{ij}}$ $\displaystyle=$
$\displaystyle\frac{\beta}{2}\Delta g^{T}+\mathcal{O}(1/r^{2})\,,$ (113)
$\displaystyle\beta_{1}N\frac{\delta}{\delta g_{ij}}(R_{ij}R^{ij})$
$\displaystyle=$
$\displaystyle\frac{\beta_{1}}{2}\Delta^{2}g_{ij}^{TT}+\mathcal{O}(1/r^{2})\,,$
(114) $\displaystyle\beta_{3}N\frac{\delta}{\delta
g_{ij}}(\nabla_{i}R_{jk}\nabla^{i}R^{jk})$ $\displaystyle=$
$\displaystyle-\frac{\beta_{3}}{2}\Delta^{3}g_{ij}^{TT}+\mathcal{O}(1/r^{2})\,.\qquad$
(115)

Then we obtain,

$\dot{\pi}_{ij}^{TT}=\frac{1}{2}\left(\beta\Delta+\beta_{1}\Delta^{2}-\beta_{3}\Delta^{3}\right)g_{ij}^{TT}+\mathcal{O}(1/r^{2})\,.$
(116)

Equations (111) and (116) show that in the wave zone the TT-part of the fields
up to order $\mathcal{O}(1/r)$ satisfies the sixth order partial differential
equation

$\displaystyle\ddot{g}_{ij}^{TT}-\left(\beta\Delta+\beta_{1}\Delta^{2}-\beta_{3}\Delta^{3}\right)g_{ij}^{TT}$
$\displaystyle=$ $\displaystyle 0,$ (117)
$\displaystyle\ddot{\pi}_{ij}^{TT}-\left(\beta\Delta+\beta_{1}\Delta^{2}-\beta_{3}\Delta^{3}\right)\pi_{ij}^{TT}$
$\displaystyle=$ $\displaystyle 0.$ (118)

The terms with spatial derivative greater than 2, are due to the contributions
of the potential terms $\mathcal{V}^{(2)}$ and $\mathcal{V}^{(3)}$.

We can obtain the low energy limit of Equations (117) and (118) if we assume
that $k_{max}\ll K_{UV}$, where $K_{UV}$ is the ultra-violet cut-off. Then the
Laplacian acting on the TT modes is the leading term among the spatial
derivatives ones. In this case (117) and (118) reduce to the wave equation for
the TT modes

$\displaystyle\ddot{g}^{ijTT}-\beta\Delta g^{ijTT}$ $\displaystyle=$
$\displaystyle 0\,,$ (119)
$\displaystyle\ddot{\pi}^{ijTT}-\beta\Delta\pi^{ijTT}$ $\displaystyle=$
$\displaystyle 0\,.$ (120)

The canonical modes travel with a speed $\sqrt{\beta}$, and it only differs
from the wave zone of General Relativity by the value of the velocity of
propagation. Gravitational Wave experiments restrict this value to be between
$-3\times{10}^{-15}$ and $+7\times{10}^{-16}$ times the speed of light [41,
42].

##  VII Solution of the Hořava-Lifshitz wave equation at the kinetic conformal
point

The solution to (117) can be written in terms of the spherical Hankel
functions $h_{l}^{(1)}(kr)$ and $h_{l}^{(2)}(kr)$. In fact,

$u(r,k)\equiv\sum_{l,m}\left[A_{lm}^{(1)}h_{l}^{(1)}(kr)+A_{lm}^{(2)}h_{l}^{(2)}(kr)\right]Y_{lm}(\theta,\phi)\,,$
(121)

satisfies

$-\Delta u(r,k)=k^{2}u(r,k)\,,$ (122)

hence

$\displaystyle\left[\beta(-\Delta)-\beta_{1}(-\Delta)^{2}-\beta_{3}(-\Delta)^{3}\right]u(r,k)$
$\displaystyle=\left(\beta
k^{2}-\beta_{1}k^{4}-\beta_{3}k^{6}\right)u(r,k)\,.$ (123)

Finally, $\Psi(r,t)=e^{i\omega t}u(r,k)$ satisfies eq. (117) provided

$\beta k^{2}-\beta_{1}k^{4}-\beta_{3}k^{6}=\omega^{2}\,.$ (124)

For a given $l$,

$h_{l}^{(1)}(kr)=\left[(-i)^{l+1}\sum_{s=0}^{l}\frac{i^{s}}{s!(2kr)^{s}}\frac{(l+s)!}{(l-s)!}\right]\frac{e^{ikr}}{r}\,,$
(125)

and $h_{l}^{(2)}(kr)=\overline{h_{l}^{(1)}}(kr)$ where $\overline{h}$ denotes
complex conjugation.

In the wave zone

$h_{l}^{(1)}(kr)\approx(-i)^{l+1}\frac{e^{ikr}}{r}\,,\quad
h_{l}^{(2)}(kr)\approx(i)^{l+1}\frac{e^{-ikr}}{r}\,,$ (126)

hence

$\displaystyle
u(r,k)=\sum_{l,m}\left[A_{lm}^{(1)}(-i)^{l+1}Y_{lm}(\theta,\phi)\right]\frac{e^{ikr}}{r}$
$\displaystyle+\sum_{l,m}\left[A_{lm}^{(2)}(i)^{l+1}Y_{lm}(\theta,\phi)\right]\frac{e^{-ikr}}{r}\,.$
(127)

Finally, the outgoing solution to (117) in the wave zone is

$\Psi=\frac{e^{-i\omega
t+ikr}}{r}\sum_{l,m}A_{lm}^{(1)}(-i)^{l+1}Y_{lm}(\theta,\phi)\,.$ (128)

where $k$ is the solution of the dispersion relation (124).

In order to have a positive left hand member of (124) the couplings must
satisfy [13]

$\alpha\neq
2\beta\,,\quad\beta>0\,,\quad\beta_{3}<0\,,\quad\beta_{1}<2\sqrt{\beta|\beta_{3}|}.$
(129)

In this case given $\omega^{2}$ there always exists a unique $k^{2}$
satisfying the dispersion relation. Consequently, given $\omega$ there exists
an outgoing solution with dependence on the radial coordinate ${e^{-i\omega
t+ikr}}/{r}$. The relation between $\omega$ and $k$ can be rewritten as

$\frac{\omega^{2}}{k^{2}}=\beta-\beta_{1}k^{2}-\beta_{3}k^{4}\,,$ (130)

it tells us that at low energies the speed of propagation of the gravitational
wave is $\sqrt{\beta}$ while at high energies the speed depends on the energy
of the wave. Under the above assumptions on the coupling constants, the right
hand member of (130) is always positive.

##  VIII Discussion

We showed that in the Hořava-Lifshitz theory at the kinetic conformal there
exists a wave zone on which the physical degrees of freedom, the transverse-
traceless (TT) tensorial modes propagate as free waves. On that region of
space, provided the set of restrictions on the coupling parameters (107) is
assumed, the other components of the metric on the spacelike leaves of the
foliation as well as the lapse function, are of the same order in powers of
$1/r$, $r$ being the distance to a bounded source. They constitute a static
Newtonian background which does not interact with the propagation of the wave
modes. At larger distance from the source than the wave zone the TT components
of the metric decay with higher powers of $1/r$, while the T modes vary in a
way that the energy of the system is finite. The set of restrictions, equation
(107), is a necessary and sufficient condition for the non-existence of zero
modes to the solutions of the constraints. The zero modes would have a
divergent contribution to the gravitational energy and must be avoided. The
wave solution describing the propagation of the physical degrees of freedom
satisfies a linear partial differential equation quadratic on time derivatives
but cubic on the Laplace operator. At low energies, the infrared limit of the
renormalization flow of the Hořava-Lifshitz theory, the equation reduces to
the wave equation, with a speed of propagation characterized by the square
root of a coupling constant, which should take at this limit a value very near
to the speed of light $c$. At higher energy level the high spacelike
derivative terms in the potential of the Hamiltonian become relevant and the
wave solutions are characterized by a dispersion relation which means that the
speed of propagation of the physical modes becomes energy dependent. The
solution of the dispersion relation requires additional restrictions on the
coupling parameters, they must satisfy (129).

The Hořava-Lifshitz theory at the kinetic conformal point in the low energy
limit depends on two coupling constants $\beta$ and $\alpha$, which may have a
bounded range of values in order that the predictions of the theory become
consistent with the known experimental data, for $\beta=1$ and $\alpha=0$ at
low energies the theory is the same as GR in a particular gauge. For
$\alpha\neq 0$ the static solution with spherical symmetry does not describe a
black hole but a throat connecting an asymptotic flat space and a space with
an essential singularity [13]. That is, the physics of the theory is different
from GR, as predicted in [1], but in the limit $\beta\rightarrow 1$ and
$\alpha\rightarrow 0$ the geometry external to the throat is very similar to
the geometry external to the black hole of the Schwarzschild solution in GR,
however only in the limit $\beta=1$ and $\alpha=0$ there is a black hole. An
analogous relation to GR occurs with the gravitation radiation. We have shown
that the gravitation radiation predicted by the theory at low energies has
similar properties as the radiation in GR, but at higher energies the wave
solutions satisfy a dispersion relation characteristic of the Hořava-Lifshitz
theory. The higher order spacelike derivatives terms in the evolution equation
are related to the higher order derivative terms in the potential that
determine, at the UV regime, the power counting renormalizability of the
theory, the main virtue of the Hořava proposal.

An interesting study would be to analyze the radiative properties of the
gravitation-electromagnetic interaction on the wave zone. It is known that the
gravitation and electromagnetic degrees of freedom propagate, in a
perturbative approach, with the same speed but their interaction behavior in
the wave zone is unknown [24, 25].

###### Acknowledgements.

J. Mestra-Páez acknowledge financial support from Beca Doctorado Nacional 2019
CONICYT, Chile. N° BECA: 21191442. J. M. P is supported by the projects
ANT1956 and SEM18-02 of the University of Antofagasta, Chile.

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