Massive graviton propagation of the deformed Ho\v{r}ava-Lifshitz gravity without projectability condition

We study graviton propagations of scalar, vector, and tensor modes in the deformed Ho\v{r}ava-Lifshitz gravity ($\lambda R$-model) without projectability condition. The quadratic Lagrangian is invariant under diffeomorphism only for $\lambda=1$ case, which contradicts to the fact that $\lambda$ is irrelevant to a consistent Hamiltonian approach to the $\lambda R$ model. In this case, as far as scalar propagations are concerned, there is no essential difference between deformed Ho\v{r}ava-Lifshitz gravity ($\lambda R$-model) and general relativity. This implies that there are two degrees of freedom for a massless graviton without Ho\v{r}ava scalar, and five degrees of freedom appear for a massive graviton when introducing Lorentz-violating and Fierz-Pauli mass terms. Finally, it is shown that for $\lambda=1$, the vDVZ discontinuity is absent in the massless limit of Lorentz-violating mass terms by considering external source terms.

approach to the dHL gravity, there remains a subtle issue on the equivalence 1 With the projectability condition, the authors [8,10] have argued that ψ is propagating around the Minkowski space but it has a negative kinetic term, showing a ghost instability.
In this case, the Hořava scalar becomes ghost if the sound speed square (c 2 ψ ) is positive. In order to avoid a ghost instability, the sound speed square must be negative, but it is inevitably unstable (gradient instability). Thus, one way to avoid this is to choose the case that the sound speed square is close to zero (c 2 ψ → 0), which implies the limit of λ → 1. Unfortunately, in the limit of λ → 1, the cubic interactions are important at very low energies [18]. This invalidates any linearized analysis and any predictability of quantum gravity is lost due to unsuppressed loop corrections. This strong coupling problem appears for an interacting theory of dHL gravity beyond the linearized theory. This casts serious doubts on the UV completeness of the theory. Also, it was shown that adding the mass term does not cure a ghost instability in the Hořava scalar [19]. However, it was suggested that there are many ways to tame the gradient instability of Hořava scalar [20]. These are included (i) the time scale is required to be longer than either the Jeans time scale or the Hubble time scale (ii) higher spatial derivatives would stabilize this instability when considering the dispersion relation (iii) a phenomenological constraint on the renormalization group flow may resolve the instability.
On the other hand, the authors [21] have tried to extend the theory to make a healthy Hořava-Lifshitz gravity. However, there has been some debate as to whether this theory is really healthy. The authors [22] considered the IR limit of this theory and showed that it suffered from the strong coupling problem, too. To response it, the original authors [23] have claimed that the strong coupling scale might exceed the cut-off scale for the derivative expansions and thus, it seems to be no strong coupling issue. More recently, the authors [24] has argued that the alleged strong coupling problem is genuine and not merely an artifact of a truncation the derivative expansion.
Hence, a current status of the dHL gravity may be summarized: the projectability condition from condensed matter physics may not be appropriate for describing the (quantum) gravity. Instead, if one does not impose the projectability condition, the dHL gravity may lead to general relativity without the strong coupling problem in the IR limit.
Inspired by a recent work of the consistency of the λR-model (IR version of dHL gravity) [17], we will perform a perturbation analysis of the dHL gravity without the projectability condition thoroughly. In this work, without the projectability condition, we investigate massive graviton propagations of scalar, vector, and tensor modes in the perturbation of dHL gravity by introducing Lorentz-violating mass term (18) and Fierz-Pauli mass term (19). A motivation of the introduction of these mass terms is to investigate the strong coupling problem and the vDVZ discontinuity. Even these mass terms violate the full diffeomorphism symmetry without the projectability condition, it provides more DOF through the spontaneous symmetry breaking: less symmetry means more degrees of freedom. Hence, we expect the change that 2 DOF (for massless theory) → 5 DOF (for massive theory) including the Hořava scalar. We will show that the strong coupling problem is not serious for vector and scalar modes when choosing Lorentz-violating mass term [26]. We will confirm that the Hořava scalar survives in the massless limit of Fierz-Pauli mass term (vDVZ discontinuity), but it is absent in the massless limit of Lorentz-violating mass term (no vDVZ discontinuity) [7].

dHL gravity
First of all, we introduce the ADM formalism where the metric is parameterized as Then, the Einstein-Hilbert action can be expressed as where G is Newton's constant and extrinsic curvature K ij takes the form Here, a dot denotes a derivative with respect to t( "˙" = ∂ ∂t ). On the other hand, the action of the dHL gravity is given by [6] Here C ij is the Cotton tensor defined by which is obtained from the variation of gravitational Chern-Simons term with coupling 1/η 2 .
The full equations of motion were derived in [27] and [28], but we do not write them due to the length. Taking a limit of Λ W → 0 in L 0 + √ gN µ 4 R, we obtain the λR-model [6] Comparing Eq.(8) with general relativity (2), the speed of light and Newton's constant are determined by Since we consider the z = 3 Hořava-Lifshitz gravity, scaling dimensions are it is better to switch from the c = 1 units to (9) units that impose the scaling dimensions.
Switching back to c = 1 units leads to the case [17] S λR which is suitable for discussing the IR properties (large distances) of strong coupling problem and vDVZ discontinuity.
The deformed Lagrangian which is relevant to our study takes the form [6] L ≡L λR + L h (11) where a characterized parameter ω is given by Actually, the Lagrangian (12) is enough to describe scalar and vector propagations because (13) from the Cotton tensor contributes to tensor propagations only. For λ = 1, taking the limit of ω → ∞ while keeping c 2 = 1 fixed is equivalent to recovering the Einstein gravity (λ = 1R-model). Explicitly, this limit implies κ 2 → 0(µ 4 ∼ κ −2 → ∞) which means that the kinetic term and curvature term µ 4 R dominate over all higher order curvature terms.
The deformed Lagrangian (11) can be redefined to bẽ where L K (L V ) denote the kinetic (potential) Lagrangian with (without) temporal derivative terms.
We wish to consider perturbations of the metric around Minkowski spacetimes, which is a solution to the full Lagrangian (11) where a dimensionless coupling constant η from gravitational Chern-Simons term is included to define the perturbation. The inclusion of η makes sense because the non-interacting limit corresponds to sending η → 0, while keeping the ratio γ = κ/η fixed [1]. This in turn provides the limit of κ → 0(ω → ∞). For λ = 1, this limit yields a one-parameter family of free-field fixed points parameterized by γ.
At quadratic order the λR-action (8) turns out to be with h = h ii . A general Lorentz-violating mass term is given by [26] S LV As was pointed out in [29], S LV 2 provides various phases of massive gravity in general relativity. In this work, we add (18) to the linearized theory of dHL gravity to investigate strong coupling problem and the vDVZ discontinuity. In this work, we choose the case of m 0 = 0, where the lapse field n enters the action linearly and thus, it still acts as a Lagrange multiplier. If one considers a non-zero mass m 0 seriously, it induces a ghost instability [26,30]. At this stage, we would like to mention that for generic backgrounds, m 2 1 = 0 has provided a well-defined case in bi-gravity and massive gravity [31,30]. Also, the generic case could be well behaved in generic backgrounds [31].
We compare (18) with the Lorentz-invariant Fierz-Pauli mass term [32] In order to analyze physical propagations thoroughly, it is convenient to use the cosmological decomposition in terms of scalar, vector, and tensor modes under spatial rotations which implies that [m 2 i ] = 2. The Fierz-Pauli mass term is recovered when all masses are equal except for m 0 as The quadratic action for λR-model is obtained by substituting (20) into the quadratic action (17) as with We have the coupling of 1 2γ 2 in the quadratic action. The higher order action from L h takes the form We find that two modes of scalar ψ and tensor t ij exist in S h 2 only, missing vector modes. Since the spatial slice is conformally flat, the vanishing Cotton tensor and the absence of six derivative terms result in the scalar sector. Also, the Cotton tensor does not contribute to vector modes (V i ,Ḟ i ). The vectors are absent in S h 2 because the vector belongs to gauge degrees of freedom in the massless gravity theory.
Before we proceed, we mention the foliation-preserving diffeomorphism (FDiff) in the dHL gravity with the projectability condition. Considering the anisotropic scaling of temporal and spatial coordinates (t → b z t, x i → bx i ), the time coordinate t plays a privileged role. A quadratic action of S λR 2 + S h 2 should be invariant under FDiff whose transformation is given by which shows that the spacetime symmetry is smaller than the full diffeomorphism (Diff) in the general relativity FDiff (Diff) invariance are dynamical symmetry of dHL gravity with the projectability condition (general relativity) and not just symmetry of the background spacetimes. Hence, it controls the number of propagating degrees of freedom: more symmetry means less degrees of freedom. It is well known that general relativity as a massless gravity theory has two degrees of freedom, while the dHL gravity with the projectability condition may have three.
In this work, we consider the dHL gravity without imposing the projectability condition.
In this case, Diff is more suitable than FDiff. Using the notation of ǫ µ = (ǫ 0 , ǫ i ) and ǫ ν = η νµ ǫ µ , the perturbation of metric transforms as Further, making a decomposition ǫ i into a scalar ξ and a pure vector ζ i as ǫ i = ∂ i ξ + ζ i with ∂ i ζ i = 0, one finds the transformation for scalars On the other hand, the vector and the tensor take the forms For the Diff transformations, gauge invariant combinations are for tensor and vector, respectively and for two scalar modes 2 . We note that Π is not a gauge-invariant scalar mode.
Let us try to express the quadratic action (23) in terms of gauge-invariant quantities with the spatial Laplacian △ = ∂ 2 . However, it is no doubt that for general λ, the quadratic action of λR-model is not expressed in terms of gauge-invariant quantities. This contrasts to the Hamiltonian approach which shows that the value of λ is completely irrelevant for finding two physical degrees of freedom for a massless graviton [17]. In the Hamiltonian approach, they have used Diff as full dynamical symmetry and have chosen a gauge-fixing to identify four degrees of freedom in phase space. However, in this Lagrangian approach, we are working with the quadratic action and not making a gauge-fixing. The Diff could be manifestly realized at the quadratic action only for λ = 1. Then, the λ = 1 case leads to the gauge-invariant action as At this stage, it is unclear why the value of λ is uniquely determined to be 1 in the perturbation theory. Other possibility includes the case that for generic λ, Π and A are separately gauge-invariant scalars. However, this is not the case. An allowable case is that for generic λ, Π is a gauge-invariant scalar and A is a parameter, which is exactly the dHL gravity with the projectability condition. We note that S h 2 in (24) contains only ψ and t ij , which are gauge-invariant quantities.
On the other hand, the mass term (18) leads to which is not obviously invariant under Diff because we could not express whole terms in terms of gauge-invariant quantities. However, these do not give rise to any problem because we are interested in the massless limit of Lorentz-violating mass term and we do not impose any gauge to perform the perturbation analysis around the Minkowski background.

Massive propagations
Without the projectability condition, we conjecture that out of the 5 DOF of a massive graviton, 2 of these are expressed as transverse and traceless tensor modes t ij , 2 of these are expressed as transverse vector modes F i , and the remaining one is from Hořava scalar ψ.

Tensor modes
The field equation for tensors is given bÿ The requirement that these modes are not tachyonic gives the stability condition In the absence of mass, these modes describe the chiral primordial gravitational waves [12,35]. These circularly polarized modes are possible because the Cotton tensor C ij is present, making parity violation. In the presence of mass term, it may describe massive chiral gravitational waves.

Vector modes
It is clear from Eqs. (23) and (34) that V i enters the action without temporal derivatives, that is, it is a non-dynamical field in the massless theory. A massive vector Lagrangian takes the form with w i = V i −Ḟ i . It is obvious that in the absence of mass terms, w i is a non-propagating vector mode. We integrate V i out using the field equation obtained by varying the action Plugging this expression into Eq.(37) leads to be In order to obtain a canonical action, we introduce a canonical vector fieldF i defined by in the c = 1 units. Then, the Lagrangian (40) takes a canonical form Now let us discuss the strong coupling issue. In order to discuss the strong coupling problem, we first note that 1 8πG = 4c which shows that vector modes at small m is precisely the same as in the Fierz-Pauli case.
The analysis in Ref. [36] suggests that the strong coupling occurs at E ∼ √ mM P l , which is a high scale. In comparison to the Fierz-Pauli case, vector field changes nothing except the speed of light. Its equation of motion is given bÿ which leads to the dispersion relation For m 2 1 > 0 and m 2 2 > 0, it is obvious that there is no ghosts. In the Fierz-Pauli case of m 2 1 = m 2 2 , the massive vector equation reduces to which represents a massive vector with two degrees of freedom. Here = −∂ 2 0 + △ with ∂ 0 = ∂ ∂x 0 with x 0 = ct.

Scalar mode
The scalar Lagrangian with four different masses takes the form From the Lagrangian (49), we find that there exist a Lagrange multiplier A and a nondynamical field B. Their variations with respect to A are given by which implies that E can be expressed in terms of ψ On the other hand, the variation with respect to B leads to Using Eqs. (51) and (52), we can express B in terms of ψ as Hence, we rewrite Π in terms of Hořava scalar ψ as Plugging E, B, and Π into (49) leads to a very complicated Lagrangian for ψ It seems difficult to derive the equation of motion for the Hořava scalar. However, in the limit of λ → 1, we have a simplified Lagrangian for ψ which is identified with the quadratic Lagrangian for general relativity with the LV mass term [26]. The equation of motion for ψ is given by For m 2 1 > m 2 4 , m 2 2 > m 2 3 , 4m 2 2 > m 2 4 , there are no ghost because all terms in the equation have correct signs (△ is negative-definite). This contrasts to that of dHL gravity with the projectability condition [19], which indicates that there is no ghost free, massive propagation for the Hořava scalar ψ.
In order to discuss the strong coupling issue, let us remind the relation of γ ∝ 1/M P l in the c = 1 units. Considering the Lagrangian (56), we could define the normalization factor relating ψ and its canonically normalized field ψ c as Also, considering Eqs.(51) and (53), one finds that which are in a complete analogy to vector modes. All these ensure that the strong coupling scale in the scalar sector is the same as in vector sector (E ∼ √ mM P l ).
However, in the case of Fierz-Pauli mass (22), this picture is changed. In this case, we and which implies the low energy scale of strong coupling (E ∼ (m 4 M P l ) 1/5 ) [36]. Moreover, the equation (57) reduces to a simpler equation which is nothing but the massive Klein-Gordon equation Finally, we confirm that a massive graviton takes five degrees of freedom for both the Lorentz-violating and Fierz-Pauli mass terms.

vDVZ discontinuity
Since the Lagrangian (55) takes a complicated form, it is a formidable task to prove that for generic λ, there is no vDVZ discontinuity in the massless limit. Instead, we will show that for λ = 1 case, there is no the vDVZ discontinuity, in comparison to the Fierz-Pauli case. In order to show it, we have to introduce the external source term A covariant form of the source conservation-law ∂ µ T µν = 0 is slightly modified to havė On later, a will be determined to be a = c 2 . Then, we could express the above in terms of gauge-invariant modes as Choosing a = c 2 , we note that c 2 A −Π is nothing but a gauge-invariant scalar Φ under Diff transformations. We wish to study the vDVZ discontinuity by making use of (67).

λ = 1R-model
First of all, we consider the quadratic action of S λ=1R 2 in (32) together with external source in (67) to find massless propagations in general relativity. Using propagators with source are derived as [26] where Φ = A−Π is a gauge-invariant scalar which plays the role of Newtonian potential. The above shows clearly that tensor modes t ij are propagating on the Minkowski background, while vector and all scalars are non-propagating because there is no kinetic terms ̟ 2 (second order temporal derivative terms). It confirms that the λ = 1R-model has two propagating degrees of freedom for a massless graviton, which is equivalent to the general relativity.
All propagators (69)-(72) are those of the dHL gravity in the limit of ω → ∞ and in the massless limit.

Tensor modes
The tensor equation takes a relatively simple form We find that there is no the vDVZ discontinuity because a single mass term m 2 2 (≥ 0) is present. It is obvious that for T ij = 0, the above equation reduces to Eq.(35). In the massless limit of m 2 2 → 0, Eq.(73) has described chiral gravitational waves without ghost propagating on the Minkowski background [35]. In the massive case, (73) may describe massive chiral gravitational waves.

Vector modes
From (37) and (67), the vector equations are derived as From equation (74), we find Plugging this into equation (75) leads to We define the massless limit of the Lorentz-violating mass term as [26] MLLV : m 2 i → 0, while the massless limit of the Lorentz-invariant Fierz-Pauli mass term is defined by The gauge-invariant vector takes the form Under the MLLV of (78), the gauge-invariant vector reduces to (70) which shows that the vector is non-propagating in the massless limit. Also, we note that nothing changes for the Fierz-Pauli case of m 2 1 = m 2 2 because the gauge-invariant vector takes the form which leads again to Eq.(81) in the MLFP of Eq.(79).

Scalar modes with λ = 1
In the scalar sector, the field equations are obtained by variation of (49)+(67) with respect to A, B, E, and ψ as where Φ = c 2 A − 2Ḃ +Ë is the Newtonian potential [26,30]. Eq.(83) provides while Eq. (84) gives Eq.(85) leads to T 00 Once that time derivatives of ψ and its source T 00 are neglected, the above equation could be solved to give where the n i and d i are polynomials in the masses.
The physics relevant to the vDVZ discontinuity is captured by expanding ψ in powers in the c = 1 units. In the MLLV of (78) with c 1 (m 2 i ) = c 2 (m 2 i ) = 0, we obtain the same form as in (71) which implies that there is no discontinuity at small distances. Also we find that E and B are finite in the massless limit. Therefore, there is no vDVZ discontinuity in the scalar sector.
However, for the Fierz-Pauli case, we find from (90) and (79) that ψ leads to which takes a further form in the c = 1 units This confirms the presence of the vDVZ discontinuity of the Fierz-Pauli case, as in Einstein gravity because the last term implies that ψ is a propagating degree of freedom. Consequently, we have shown that without the projectability condition, the Hořava scalar ψ is related to a scalar degree of freedom appeared in the massless limit of a massive graviton [7].

Discussions
We have studied graviton propagations of scalar, vector, and tensor modes in the dHL gravity (λR-model) without projectability condition. It is emphasized that the quadratic Lagrangian is invariant under diffeomorphism only for λ = 1 case. This contradicts to the fact that λ is irrelevant to a consistent Hamiltonian approach to the λR-model [17]. As far as scalar propagations are concerned, there is no essential difference between dHL gravity (λR-model) and general relativity. This implies that there are two degrees of freedom for a massless graviton without Hořava scalar, and five degrees of freedom including Hořava scalar appear for a massive graviton when introducing Lorentz-violating and Fierz-Pauli mass terms. Importantly, the strong coupling problem is not serious for vector and scalar modes when choosing Lorentz-violating mass term, as was claimed in general relativity [26].
It is shown that for λ = 1, the vDVZ discontinuity is absent in the massless limit of Lorentzviolating mass terms by considering external source terms. The dHL with λ = 1 recovers nicely the general relativity with (without) mass term in the linearized level. At this stage, we wish to distinguish the massless limit of massive dHL gravity from the λ → 1 limit of dHL gravity. The former case provides one scalar (ψ) propagation as was shown in (95) when choosing the Fierz-Pauli mass term, while it provides no scalar propagation as was shown in (93) when choosing the Lorentz-violating mass term. The latter provides no scalar propagation, as was shown in (71).
On the other hand, the other case was the dHL gravity with the projectability condition, the S λR 2 is invariant under FDiff transformation for generic λ. In this case, the lapse perturbation A(t) is not a Lagrange multiplier but a parameter [19]. The gauge transformations for modes is the same as in (28) and (29) except B →B = B +ξ and thus, gauge-invariant scalars are ψ and Π = 2B −Ė. However, this case gives rise to some difficulty in performing a consistent Hamiltonian analysis because the lapse perturbation A plays no role. For 1/3 < λ < 1, the Hořava scalar ψ suffers from the ghost instability. Adding the Lorentzviolating mass term did not cure the ghost instability [19]. In order to avoid the ghost instability, one requires that the sound speed square c 2 ψ be negative, leading to the gradient instability for λ > 1. To resolve this gradient instability, one has to impose the limit of λ → 1, which leads to the strong coupling problem [10,18]. However, it was suggested that there are many ways to tame the gradient instability of Hořava scalar [20].
Finally, we ask whether the projectability condition is really essential for being unable to rescue the dHL gravity from a doubtable modified gravity seriously. If one abandons an original constructing principle of the projectability inspired by condensed matter physics, one has likely found general relativity in the IR limit. However, we wish to point out that although the λR-model has contributed to making a progress toward a consistent Hamiltonian approach to the dHL gravity, there remains a subtle issue on the equivalence between a gauge-fixed version of general relativity (λR-model) and general relativity (see footnote 1).