The Hamiltonian dynamics of Hořava gravity

Abstract

We consider the Hamiltonian formulation of Hořava gravity in arbitrary dimensions, which has been proposed as a renormalizable gravity model for quantum gravity without the ghost problem. We study the full constraint analysis of the non-projectable Hořava gravity whose potential, $\mathcal{V}(R)$, is an arbitrary function of the (intrinsic) Ricci scalar R but without the extension terms which depend on the proper acceleration $a_i$. We find that there exist generally three distinct cases of this theory, A, B, and C, depending on (i) whether the Hamiltonian constraint generates new (second-class) constraints or just fixes the associated Lagrange multipliers, or (ii) whether the IR Lorentz-deformation parameter ${\lambda }$ is at the conformal point or not. It is found that, for Cases A and C, the dynamical degrees of freedom are the same as in general relativity, while, for Case B, there is one additional phase-space degree of freedom, representing an extra (odd) scalar graviton mode. This would achieve the dynamical consistency of a restricted model at the fully non-linear level and be a positive result in resolving the long-standing debates about the extra graviton modes of the Hořava gravity. Several exact solutions are also studied as some explicit examples of the new constraints. The structure of the newly obtained, “extended” constraint algebra seems to be generic to Hořava gravity and its general proof would be a challenging problem. Some other challenging problems, which include the path integral quantization and the Dirac bracket quantization are discussed also.

1 Introduction

11 years ago, Hořava proposed a renormalizable, higher-derivative gravity theory, without the ghost problem in the usual tensor graviton modes, which reduces to Einstein gravity in low energy (IR) at the full action level but with improved high-energy (UV) behaviors, by abandoning the Lorentz symmetry from non-equal-footing treatment of space and time [1, 2]. However, due to absence of the full diffeomorphisms (Diff), extra graviton modes can be expected generally and there have been questions about the recovery of general relativity (GR) in IR, and more generally, the consistency of Hořava gravity [3,4,5,6,7,8,9,10,11,12,13,14,15,16].

In this paper, we reconsider the Hamiltonian formulation of the non-projectable Hořava gravity whose potential, $\mathcal{V}(R)$, is an arbitrary function of the (intrinsic) Ricci scalar R in arbitrary dimensions but without the extension terms which depend on the proper acceleration $a_i={\partial }_i N/N$, for simplicity. We study the full constraint analysis and find that there exist generally three distinct cases of this theory, A, B, and C, depending on (i) whether the Hamiltonian constraint generates new (second-class) constraints, or just fixes the associated Lagrange multipliers, according to Dirac’s method [17], or (ii) whether the IR Lorentz-deformation parameter ${\lambda }$ is at the conformal point, ${\lambda }=1/D$, or not. It is found that, for Cases A and C, the dynamical degrees of freedom are the same as in general relativity, while, for Case B, there is one additional phase-space degree of freedom, representing an extra (odd) scalar graviton mode. This would achieve the dynamical consistency of a restricted model at the fully non-linear level and be a positive result in resolving the long-standing debates about the extra graviton modes of the Hořava gravity.

The organization of the paper is as follows. In Sect. 2, we consider the set-up for the Hamiltonian formulation of the non-projectable Hořava gravity in arbitrary dimensions. In Sect. 3, we consider Dirac’s constraint analysis when ${\lambda }\ne 1/D$ for the IR Lorentz-deformation parameter ${\lambda }$ and study two cases, A and B, depending on whether the Hamiltonian constraint generates new (second-class) constraints or just fixes the associated Lagrange multipliers. In Sect. 4, we consider the third case, C, when ${\lambda }=1/D$, which is at a conformal point and generates new (second-class) constraints similar to Case A. We show that the dynamical degrees of freedom are the same as in GR for Cases A and C, while there is one additional phase-space degree of freedom for Case B which representing an extra (odd) scalar graviton mode. In Sect. 5, we study several exact solutions as some explicit examples of the new constraints. In Sect. 6, we conclude with remarks on several challenging theoretical problems, which include the path integral quantization and the Dirac bracket quantization.

2 The Hamiltonian formulation in arbitrary dimensions: set-up

In this section, we consider the Hamiltonian formulation of the non-projectable Hořava gravity in arbitrary dimensions. To this ends, we start by considering the ADM decomposition of the metric [18]

$$\begin{aligned} ds^2=-N^2 dt^2+g_{ij}\left( dx^i+N^i dt\right) \left( dx^j+N^j dt\right) \end{aligned}$$

(1)

with the arbitrary space-time dependent lapse and shift functions¹, $N, N^i$, and induced metric $g_{ij}~ (i,j=1,2, \ldots D)$ on a time-slicing hypersurface $\Sigma _t$. Then, the action on a $(D+1)$-dimensional manifold $\mathcal{M}$ with the boundary $\partial \mathcal{M}$ is given by

$$\begin{aligned} S= & {} \int _\mathcal{M} d t d^{D}{x} \sqrt{g} N \left\{ \frac{2}{\kappa ^2} \left( K_{ij}K^{ij} - \lambda K^2 \right) - \mathcal{V}[g_{ij}] \right\} \nonumber \\&+S_{\partial \mathcal{M}} \end{aligned}$$

(2)

with an appropriate boundary action² $S_{\partial \mathcal{M}}$ and the potential $\mathcal{V}[g_{ij}]$, which depends only on the metric $g_{ij}$ and its spatial derivatives [1, 2]³. Here,

$$\begin{aligned} K_{ij}=\frac{1}{2N} \left( \dot{g}_{ij}-\nabla _i N_j-\nabla _jN_i\right) \end{aligned}$$

(3)

is the extrinsic curvature (the overdot $(~\dot{}~)$ denotes the time derivative) of the hypersurface $\Sigma _t$ and $K \equiv g_{ij} K^{ij}$ denotes its trace.

It is well known that GR is not renormalizable in the conventional way. Its Lorentz invariant higher-curvature modifications have some improvements of UV behaviors but there are more dynamical degrees of freedom than those of GR generally, and the existence of negative kinetic energy modes in the additional degrees of freedom, called ghost modes, is unavoidable due to higher-time derivatives [24,25,26]. In order to avoid the possible problems of ghost degrees of freedom, we do not simply consider the higher-time-derivative terms, like $K^4,(K_{ij}K^{ij})^2$, etc., but only consider the second-order time-derivative terms, like $K_{ij}K^{ij}$ and $K^2$, in the kinetic part of the action (2). Whereas, in order to achieve the renormalizable theory with the improved UV behaviors, we consider the higher-spatial-derivative terms, like $R^2, R_{ij}R^{ij}$, etc., in the potential part $\mathcal{V}[g_{ij}]$ with the intrinsic Ricci curvature $R_{ij}$ and its trace $R\equiv g_{ij}R^{ij}$. In order that the theory be power-counting renormalizable, the potential part needs to contain “$2 \times D$” (spatial) derivatives at least, which is sometimes represented by the dynamical critical exponent, $z=D$ [2]. In order that this construction of a renormalizable action is not spoiled by the mixing of space and time (derivatives) in the general coordinate transformations, we need to further constrain the allowed coordinate transformations into the foliation-preserving diffeomorphisms $({ Diff}_\mathcal{F})$,

$$\begin{aligned} \delta x^i= & {} -\zeta ^i (t, \mathbf{x}), ~\delta t=-f(t), \nonumber \\ \delta g_{ij}= & {} {\partial }_i\zeta ^k g_{jk}+{\partial }_j \zeta ^k g_{ik}+\zeta ^k {\partial }_k g_{ij}+f \dot{g}_{ij},\nonumber \\ \delta N_i= & {} {\partial }_i \zeta ^j N_j+\zeta ^j {\partial }_j N_i+\dot{\zeta }^j g_{ij}+f \dot{N}_i+\dot{f} N_i,\nonumber \\ \delta N= & {} \zeta ^j {\partial }_j N+f \dot{N}+\dot{f} N. \end{aligned}$$

(4)

In ${ Diff}_\mathcal{F}$, each term in the kinetic part is invariant separately and ${\lambda }$ can be arbitrary⁴ [23]. If we consider ${\lambda }=1$ and $\mathcal{V}[g_{ij}] =-(2/{\kappa }^2) R-{\Lambda }$ as in GR, then there is an “accidental” symmetry enhancement which mixes each term in the action so that the full Diff is recovered [8]. So, there are two sources of the Lorentz violations, either from the higher-spatial-derivative (UV) terms in the potential part or from the deformation of kinetic part with an arbitrary ${\lambda }$ in IR, generally.

For the potential part, one may consider any function having 2D spatial derivatives for the power-counting renormalizability, but in this paper we consider only the function of Ricci curvature scalar, R, i.e., $\mathcal{V}[g_{ij}] \equiv \mathcal{V}(R)$, for simplicity⁵. Then, the first-order formulation of the action (2) is given by

$$\begin{aligned} S= & {} \int _{\mathcal{M}} d t d^{D}x \left\{ \pi ^{ij} \dot{g}_{ij}-N \mathcal{H}-N_i \mathcal{H}^i-{\partial }_i {\gamma }^i\right\} \end{aligned}$$

(5)

with appropriate boundary terms, ${\partial }_i {\gamma }^i$, the conjugate momenta,

$$\begin{aligned} \pi ^{ij} \equiv \frac{{\delta }S}{{\delta }\dot{g}_{ij}}= \frac{2 \sqrt{g}}{\kappa ^2} \left( K^{ij} - \lambda K g^{ij} \right) , \end{aligned}$$

(6)

and

$$\begin{aligned} \mathcal{H}\equiv & {} \frac{\kappa ^2}{2 \sqrt{g}} \left[ \pi ^{ij}\pi _{ij} - \left( \frac{\lambda }{D {\lambda }-1}\right) \pi ^2 \right] +\sqrt{g} \mathcal{V}, \end{aligned}$$

(7)

$$\begin{aligned} \mathcal{H}^i\equiv & {} -2 \nabla _j \pi ^{ij}, \end{aligned}$$

(8)

where $\pi \equiv g_{ij} \pi ^{ij}$. Here, we first consider the case ${\lambda }\ne 1/D$ so that $\mathcal{H}$ in (7) and the first-order action (5) are not singular.

The Poisson brackets for the canonical variables are given by

$$\begin{aligned} \{ g_{ij} (x), \pi ^{kl} (y) \}={\delta }_{ij}^{kl} {\delta }^D (x-y) \end{aligned}$$

(9)

with ${\delta }_{ij}^{kl}\equiv ({\delta }_i^k {\delta }_j^l+{\delta }_i^l {\delta }_j^k)/2$.

3 Constraint analysis à la Dirac (${\lambda }\ne 1/D$)

The primary constraints of the action (2) are given by

$$\begin{aligned} \Phi ^t \equiv \pi _N \approx 0,~\Phi ^i \equiv \pi ^i \approx 0, \end{aligned}$$

(10)

from the definition of conjugate momenta, $\pi _N \equiv {\delta }S/{\delta }\dot{N}$ and $\pi ^i \equiv {\delta }S/{\delta }\dot{N_i}$. Here, the weak equality ‘$\approx$’ means that the constraint equations are used only after calculating the Poisson brackets.

The preservation of the primary constraints, $\Phi ^\mu \equiv (\Phi ^t , \Phi ^i)$, i.e., $\dot{\Phi }^{\mu } =\{\Phi ^{\mu }, H_C \}\approx 0$, as being required by the consistency of the constraints, with the canonical Hamiltonian,

$$\begin{aligned} H_C= & {} \int _{{\Sigma }_t} d^{D}x \left\{ N \mathcal{H}+N_i \mathcal{H}^i\right\} +H_\mathcal{B} \end{aligned}$$

(11)

produces the secondary constraints,

$$\begin{aligned} \mathcal{H} \approx 0,~~ \mathcal{H}^i \approx 0. \end{aligned}$$

(12)

Here, $H_\mathcal{B}$ is an appropriate boundary Hamiltonian, $H_\mathcal{B}=\oint _\mathcal{B} d^{D}x~ \hat{n}_i {\gamma }^i$, for the intersection $\mathcal{B}$ of an arbitrary time-like boundary ($\hat{n}_i$ is its unit normal) with a time slice ${\Sigma }_t$ so that the total boundaries are ${\partial }\mathcal{M}= {\Sigma }_{t_f} \cup {\Sigma }_{t_i} \cup \mathcal{B}$. On the other hand, the dynamical equations for $g_{ij}$ and $\pi ^{kl}$ are obtained as follows, neglecting boundary contributions (see Appendix A for the details),

$$\begin{aligned} \dot{g}_{ij}= & {} \{ g_{ij}, H_C \} = \frac{{\delta }H_C}{{\delta }\pi ^{ij}} \nonumber \\= & {} \left( \frac{\kappa ^2}{2} \right) \frac{2 N}{ \sqrt{g}} \left( \pi _{ij}-\widetilde{{\lambda }} g_{ij} \pi \right) + \nabla _i N_j + \nabla _j N_i, \end{aligned}$$

(13)

$$\begin{aligned} \dot{\pi }^{ij}= & {} \{ \pi ^{ij}, H_C \}= -\frac{{\delta }H_C}{{\delta }g_{ij}} \nonumber \\= & {} \left( \frac{\kappa ^2}{2} \right) \frac{ N}{ \sqrt{g}} \left[ \frac{1}{2} g^{ij} \left( \pi _{mn} \pi ^{mn}-\widetilde{{\lambda }} \pi ^2 \right) \right. \nonumber \\&\left. -2 \left( \pi ^{im} \pi ^j_m-\widetilde{{\lambda }} \pi \pi ^{ij} \right) \right] \nonumber \\&-N \sqrt{g} \left[ \frac{ 1}{2} g^{ij} \mathcal{V}(R)-R^{ij} \mathcal{V}'(R) \right] \nonumber \\&-\sqrt{g} \left[ \nabla ^i \nabla ^j \left( N \mathcal{V}'(R) \right) -g^{ij} \nabla _m \nabla ^m \left( N \mathcal{V}'(R)\right) \right] \nonumber \\&+ \nabla _m \left( N^m \pi ^{ij} \right) -\left( \nabla _m N^i\right) \pi ^{jm}-\left( \nabla _m N^j\right) \pi ^{im}, \nonumber \\ \end{aligned}$$

(14)

where $\widetilde{{\lambda }}\equiv {\lambda }/(D {\lambda }-1)$ and $(~)' \equiv d( ~)/dR$.

With the primary constraints in (10), one can consider the extended Hamiltonian with the Lagrange multipliers $u_{\mu }$,

$$\begin{aligned} H_E= & {} H_C + \int _{{\Sigma }_t} d^{D}x~( u_{\mu } \Phi ^{\mu } ), \end{aligned}$$

(15)

from the arbitrariness in the equations of motion, due to the primary constraints.

Then, after tedious computations, we obtain the following constraint algebra (see Appendix A for the details),

$$\begin{aligned} \{ \mathcal{H} (x),\mathcal{H} (y) \}= & {} C^i (x) \nabla ^x_i {\delta }^D (x-y) \nonumber \\&-C^i (y) \nabla ^y_i {\delta }^D (x-y), \end{aligned}$$

(16)

$$\begin{aligned} \{ \mathcal{H} (x),\mathcal{H}_i (y) \}= & {} -\mathcal{H} (y) \nabla ^y_i {\delta }^D (x-y), \end{aligned}$$

(17)

$$\begin{aligned} \{ \mathcal{H}_i (x),\mathcal{H}_j (y) \}= & {} \mathcal{H}_i (y) \nabla ^x_j {\delta }^D (x-y) \nonumber \\&+\mathcal{H}_j (x) \nabla ^x_i {\delta }^D (x-y), \end{aligned}$$

(18)

or, for the smeared constraints, $\langle \eta \mathcal{H} \rangle \equiv \int d^D x~ \eta \mathcal{H}$, etc., with the smearing functions $\eta$ and $\eta _i$,

$$\begin{aligned} \{ \langle \eta \mathcal{H} \rangle ,\langle \zeta \mathcal{H}\rangle \}= & {} \langle \left( \eta \nabla _i \zeta -\zeta \nabla _i \eta \right) C^i \rangle , \end{aligned}$$

(19)

$$\begin{aligned} \{ \langle \eta \mathcal{H} \rangle ,\langle \zeta ^i \mathcal{H}_i\rangle \}= & {} - \langle \zeta ^i \nabla _i \eta \mathcal{H} \rangle , \end{aligned}$$

(20)

$$\begin{aligned} \{ \langle \eta ^i \mathcal{H}_i \rangle ,\langle \zeta ^j \mathcal{H}_j \rangle \}= & {} \left\langle \left( \eta ^i \nabla _i \zeta ^j-\zeta ^i \nabla _i \eta ^j \right) \mathcal{H}_j \right\rangle , \end{aligned}$$

(21)

where⁶

$$\begin{aligned} C^i\equiv & {} \sqrt{g}{\mathcal{V}' }^2 (R) \nabla _j \left( \frac{K^{ij}}{ \mathcal{V}' (R) } \right) \nonumber \\= & {} \left( \frac{{\kappa }^2}{2} \right) 2 {\mathcal{V}' }^2 (R) \nabla _j \left( \frac{\pi ^{ij}- \widehat{{\lambda }} g^{ij} \pi }{ \mathcal{V}' (R) } \right) \nonumber \\= & {} \left( -\frac{{\kappa }^2}{2} \right) \left[ \left( \mathcal{H}^i + 2 \widehat{{\lambda }} \nabla ^i \pi \right) \mathcal{V}' (R) \right. \nonumber \\&\left. +2 \left( \pi ^{ij}-\widehat{{\lambda }} g^{ij} \pi \right) \nabla _j \mathcal{V}' (R) \right] \end{aligned}$$

(22)

with $\widehat{{\lambda }}\equiv ({\lambda }-1)/(D {\lambda }-1)$. If we consider $- \mathcal{V}(R) \equiv {\Lambda }+\xi R+ {\alpha }R^n$ as a typical example with an arbitrary power n, we can obtain

$$\begin{aligned} C^i= & {} \left( \frac{{\kappa }^2}{2} \right) \left[ \left( \mathcal{H}^i + 2 \widehat{{\lambda }} \nabla ^i \pi \right) \left( \xi +{\alpha }n R^{n-1} \right) \right. \nonumber \\&\left. +2 \left( \pi ^{ij}-\widehat{{\lambda }} g^{ij} \pi \right) {\alpha }n \nabla _j R^{n-1} \right] . \end{aligned}$$

(23)

Note that, when the higher-derivative contributions are absent, i.e., ${\alpha }=0$, (23) reduces to $C^i =({{\kappa }^2}/{2}) \xi \mathcal{H}^i \approx 0$ and the Hamiltonian constraint, $\mathcal{H} \approx 0$, becomes the first-class constraint as in GR, for either (i) ${\lambda }=1$, i.e., $\widehat{{\lambda }}=0$, or (ii) $\pi \approx 0$, representing the maximal slicing, for an arbitrary ${\lambda }$ [18]. However, when the higher-derivative terms are present, $C^i$ does not vanish generally, so that “ the Hamiltonian constraint, $\mathcal{H} \approx 0$, becomes the second-class constraint ” even for the maximal slicing, $\pi \approx 0$, due to the genuine dynamical degrees of freedom in $\pi ^{ij}$, other than the trace part, $\pi$.

On the other hand, we note that the momentum constraint, $\mathcal{H}_i\approx 0$, in (12) has the same canonical form as in GR with no higher-derivative corrections⁷ so that we also have the same constraint algebra for $\mathcal{H}_i$, (18) or (21), which representing the spatial Diff generator,

$$\begin{aligned} {\delta }_{\zeta } g_{ij}= & {} \{ g_{ij}, \langle \zeta ^k \mathcal{H}_k \rangle \} =\frac{{\delta }\langle \zeta ^k \mathcal{H}_k\rangle }{{\delta }\pi ^{ij}}\nonumber \\= & {} \nabla _i \zeta _j +\nabla _j \zeta _j, \end{aligned}$$

(24)

$$\begin{aligned} {\delta }_{\zeta } \pi ^{ij}= & {} \{ \pi ^{ij}, \langle \zeta ^k \mathcal{H}_k \rangle \}=-\frac{{\delta }\langle \zeta ^k \mathcal{H}_k\rangle }{{\delta }g_{ij}}\nonumber \\= & {} \nabla _m \left( \zeta ^m \pi ^{ij} \right) -\left( \nabla _m \zeta ^i\right) \pi ^{jm}-\left( \nabla _m \zeta ^j\right) \pi ^{im},\nonumber \\ \end{aligned}$$

(25)

as in GR. Moreover, the algebras (17) and (18) show that

$$\begin{aligned} {\delta }_{\zeta }\mathcal{H}= & {} \{ \mathcal{H}, \langle \zeta ^k \mathcal{H}_k \rangle \}= \nabla _k \left( \zeta ^k \mathcal{H} \right) , \end{aligned}$$

(26)

$$\begin{aligned} {\delta }_{\zeta } \mathcal{H}_i= & {} \{ \mathcal{H}_i, \langle \zeta ^k \mathcal{H}_k \rangle \}=\nabla _k \left( \zeta ^k \mathcal{H}_i \right) +\left( \nabla _i \zeta ^k\right) \mathcal{H}_k, \end{aligned}$$

(27)

which tells that $\mathcal{H}$ and $\mathcal{H}_i$ behave as, under the spatial Diff, the scalar and vector densities, respectively, as in GR.

Using the above constraint algebra, one can easily find that the preservation of the secondary constraints give

$$\begin{aligned} \dot{\mathcal{H}}= & {} \{\mathcal{H} , H_E \} \nonumber \\= & {} \frac{1}{N} \nabla _i (N^2 C^i) +\nabla _i (N^i \mathcal{H}) \approx \frac{1}{N} \nabla _i (N^2 \widetilde{C}^i) , \end{aligned}$$

(28)

$$\begin{aligned} \dot{\mathcal{H}_i}= & {} \{\mathcal{H}_i , H_E \} \nonumber \\= & {} \mathcal{H} \nabla _i N +\nabla _j (N^j \mathcal{H}_i)+\mathcal{H}_j \nabla _i N^j \approx 0, \end{aligned}$$

(29)

which produces the tertiary constraint,

$$\begin{aligned} \widetilde{\Omega } \equiv \nabla _i (N^2 \widetilde{C}^i) \approx 0, \end{aligned}$$

(30)

where

$$\begin{aligned} \widetilde{C}^i\equiv & {} C^i |_{\mathcal{H}_i \approx 0} =\left( \frac{{\kappa }^2}{2} \right) \left[ 2 \widehat{{\lambda }} \nabla ^i \pi \left( \xi +{\alpha }n R^{n-1} \right) \right. \nonumber \\&\left. +2 \left( \pi ^{ij}-\widehat{{\lambda }} g^{ij} \pi \right) {\alpha }n \nabla _j R^{n-1} \right] \end{aligned}$$

(31)

from the preservation of $\mathcal{H} \approx 0$ in (28), excluding the trivial case of $N=0$ for all space-time. Here we note that, in the above computations, there are no contributions from the multiplier terms in $H_E$ so that we need to consider further steps of preserving the constraints until the multipliers are determined.

Now, one more step of preserving the new tertiary constraint, $\widetilde{\Omega } \approx 0$, gives

$$\begin{aligned} \dot{\widetilde{\Omega }}= & {} \{\widetilde{\Omega }, H_E\} \nonumber \\= & {} \{\widetilde{\Omega }, H_C\}+2 \widetilde{\Omega } \left( \frac{u_t}{N} \right) +2 \widetilde{C}^i N^2 \nabla _i \left( \frac{u_t}{N} \right) \nonumber \\\approx & {} \{\widetilde{\Omega }, H_C\}+2 \widetilde{C}^i N^2 \nabla _i \left( \frac{u_t}{N} \right) \approx 0. \end{aligned}$$

(32)

Then, there are two different constraint systems, with different subsequent procedures, depending on whether $\widetilde{C}^i = 0$ or $\widetilde{C}^i \ne 0$.

A. Case $\widetilde{C}^i = 0$: In this case⁸, (32) does not determine the multiplier $u_t$ but reduces to

$$\begin{aligned} \dot{\widetilde{\Omega }}\approx & {} \{\widetilde{{\Omega }} , H_C \} \nonumber \\\approx & {} \nabla _i \left( N^2 \widetilde{\Theta }^i \right) \equiv \widetilde{\Xi } \approx 0 \end{aligned}$$

(33)

for preserving the tertiary constraint $\widetilde{{\Omega }}$ with

(34)

where $-\mathcal{V}'(R) \equiv \left( \xi +{\alpha }n R^{n-1} \right)$. Here, the higher-derivative contributions come, either from the constraint (30), $\widetilde{{\Omega }}$, i.e., $\widetilde{C}^i$ in (31), or from the Hamiltonian constraint $\mathcal{H}$ in (7) and (11). One can compute the explicit forms of the higher-derivative contributions with the help of (13) and (14) (see Appendix B for some more details; see also [15] for the case of $n=2$) but, due to its messy expression, we will not consider the explicit forms in the analysis below unless it is crucial.

Then, one more time-evolution of the new constraint, $\widetilde{\Xi }\approx 0$, will read,

$$\begin{aligned} \dot{\widetilde{\Xi }}= & {} \{\widetilde{\Xi }, H_E \} \nonumber \\= & {} \{\widetilde{\Xi }, H_C \}+\nabla _i \left( N^2 ~ \widetilde{\Theta }^i|_{N \rightarrow u_t}+2N u_t \widetilde{\Theta }^i \right) \approx 0. \end{aligned}$$

(35)

After a long computation, we obtain (see also Appendix B for the details)⁹

(36)

Here, it is important to note that the multipliers, $u_t,\nabla ^i u_t$, etc., have generally non-vanishing coefficients in the left-hand side so that (36) may provide the equation for determining the multiplier $u_t$. However, the similar equation for $u_i$ does not exist and $u_i$ is still undetermined but this is just a reflection of the first-class nature of the constraint, $\pi ^i\approx 0$, in (10). This would now complete the Dirac’s procedure for finding the complete set of constraints, though we would not try to solve for the explicit solution of the multiplier $u_t$. Then, the full set of constraints are given by $\chi _A \equiv ( \pi _N, \mathcal{H}, \widetilde{{\Omega }}, \widetilde{\Xi } )\approx 0, {\Gamma }_B \equiv (\pi ^i, \mathcal{H}_i ) \approx 0$. Here, the constraints $\chi _A \approx 0$ are the second-class constraints with the constraint algebra,

$$\begin{aligned} \{ \pi _N (x), \mathcal{H} (y) \}= & {} 0 , \nonumber \\ \{ \pi _N (x), \widetilde{{\Omega }} (y) \}= & {} -2 \nabla ^y_i \left( N \widetilde{C}^i(y) {\delta }^D(x-y)\right) \approx 0, \nonumber \\ \{ \pi _N (x), \widetilde{\Xi } (y) \}= & {} \Delta (x-y), \nonumber \\ \{ \mathcal{H} (x), \mathcal{H} (y) \}= & {} C^i (x) \nabla ^x_i {\delta }^D (x-y) \nonumber \\&-C^i (y) \nabla ^y_i {\delta }^D (x-y) \approx 0 , \nonumber \\ \{ \mathcal{H} (x), \widetilde{\Omega } (y) \}\approx & {} \{ \pi _N (x), \widetilde{\Xi } (y) \} ,\text { etc.}, \end{aligned}$$

(37)

whose determinant, $\text{ det } \{\chi _A, \chi _B \}$, is generally non-vanishing,

$$\begin{aligned}&\text{ det } \{\chi _A (x), \chi _B (y) \} \nonumber \\&\quad \approx \left( \text{ det } \{ \mathcal{H} (x), \widetilde{{\Omega }} (y) \} \right) ^2 \left( \text{ det } \{\widetilde{{\Omega }} (x), \mathcal{H} (y) \} \right) ^2 \nonumber \\&\quad \approx \left( \text{ det } [ \Delta (x-y) \Delta (y-x) ] \right) ^2, \end{aligned}$$

(38)

where

(39)

with

(40)

On the other hand, the constraints, ${\Gamma }_A \equiv (\pi ^i, \mathcal{H}_i ) \approx 0$, are the first-class constraints with the vanishing determinant, $\text{ det } (\{{\Gamma }_A, {\Gamma }_B \} )=0$. Then, the resulting number of dynamical degrees of freedom in the “configuration” space is given by

$$\begin{aligned} s= & {} \frac{1}{2} \left( P-2 N_1-N_2\right) \nonumber \\= & {} \frac{1}{2} \left[ (D+1)(D+2)-2 \times 2 D -``4\hbox {''}\right] \nonumber \\= & {} \frac{1}{2} (D+1)(D-2), \end{aligned}$$

(41)

where $P=(D+1)(D+2)$ is the number of canonical variables in the “phase” space $(N, \pi _N, N_i, \pi ^i,g_{ij}, \pi _{ij})$, $N_1=2D$ is the number of the first-class constraints $(\pi ^i, \mathcal{H}_i)\approx 0$, and $N_2=``4\hbox {''}$ is the number of the second-class constraints, $(\pi _N, \mathcal{H}, \widetilde{{\Omega }}, \widetilde{\Xi } )\approx 0$. Note that, for Case A, the dynamical degrees of freedom are the same as that of GR (in arbitrary dimensions) though the constraint structure is different:¹⁰ Actually, in GR, i.e., $\widehat{{\lambda }}=0,~ {\alpha }=0$ or the ${\lambda }$-deformed GR (${\lambda }R$ model) with the condition, $\pi = 0$, we have $N_1=2 (D+1), N_2=0$ so that the 2 first-class constraints, $(\pi _N, \mathcal{H})\approx 0$, in GR or ${\lambda }$-deformed GR, transform into the 4 second-class constraints, $( \pi _N, \mathcal{H}, \widetilde{{\Omega }}, \widetilde{\Xi } )\approx 0$, in the Case A of full Hořava gravity, with maintaining the same dynamical degrees of freedom s. This completes the previous linear analysis in [10, 33], but now at the “fully non-linear” level. (cf. [14, 27]).

B. Case $\widetilde{C}^i \ne 0$: This is the more generic case where the conjugate momenta $\pi ^{ij}$ and the (scalar) curvature R are arbitrary, with the generic higher-derivative potential, $\mathcal{V}(R)$. In this case, (32) does not yield new constraints but determines the multiplier $u^t$ generally¹¹ so that the Dirac’s procedure may be completed, without further iterations. Then, in contrast to Case A, there are the second-class constraints, $\widetilde{\chi }_A \equiv ( \pi _N, \mathcal{H}, {\Omega })\approx 0$, whose determinant $\text{ det }(\{\widetilde{\chi }_A, \widetilde{\chi }_B \})$ is non-vanishing, generally,

$$\begin{aligned}&\text{ det } \{\widetilde{\chi }_A (x), \widetilde{\chi }_B (y) \}\nonumber \\&\quad =- \text{ det } \{ \pi _N (x), \widetilde{{\Omega }} (y) \} ~\nonumber \\&\qquad \times \text{ det } \{ \mathcal{H} (x), \mathcal{H} (y) \} ~\text{ det } \{ \widetilde{{\Omega }} (x),\pi _N (y)\} \nonumber \\&\quad \approx 4 ~ \text{ det } \left( N \widetilde{C}^j (y) \nabla ^y_j {\delta }^D(x-y)\right) \nonumber \\&\qquad \times \text{ det } \left( N \widetilde{C}^k (x) \nabla ^x_k {\delta }^D (x-y)\right) \text{ det } \left( C^i (x) \nabla ^x_i {\delta }^D (x-y) \right) -(x \leftrightarrow y),\nonumber \\ \end{aligned}$$

(42)

whereas the first-class constraints, ${\Gamma }_A \equiv (\pi ^i, \mathcal{H}_i )$, are the same as in Case A. Hence, the resulting number of dynamical degrees of freedom is

$$\begin{aligned} s= & {} \frac{1}{2} \left[ (D+1)(D+2)-2 \times 2 D -\text {``}3\text {''}\right] \nonumber \\= & {} \frac{1}{2} (D+1)(D-2) +\frac{1}{2}, \end{aligned}$$

(43)

with $N_1=2 D$ and $N_2=\text {``}3\text {''}$, which shows one extra degree of freedom in phase space, in addition to the usual $(D+1)(D-2)$ graviton (transverse traceless) modes in arbitrary $(D+1)$-dimensions.¹² In particular, in $(2+1)$-dimensions, the extra mode is the only dynamical degree of freedom. This result supports the previous case-by-case results [15, 27] but in a more generic set-up with arbitrary dimensions and cosmological constant.

The usual increase of dynamical degrees of freedom with higher-time derivatives is the result of the competition between the increased canonical momenta for the higher-time derivative fields and their increased, associated new constraints [25, 26]. However, for Hořava gravity, there are no increased canonical momenta but exist only the increased second-class constraints: For Case A, the increased second-class constraints are enough to preserve the dynamical degrees of freedom of GR, whereas for Case B, they are not enough and one extra degree of freedom persists in the phase-space.

4 Constraint analysis at the conformal point, ${\lambda }=1/D$: Case C

The kinetic part of Hořava action (2) can be written formally as,

$$\begin{aligned} S_K \equiv \left( \frac{2}{{\kappa }^2} \right) \int dt d^D x \sqrt{g} N \left( K_{ij} G^{ijkl} K_{kl} \right) , \end{aligned}$$

(44)

where $G^{ijkl} \equiv {\delta }^{ijkl}-{\lambda }g^{ij} g^{kl}$ is the (generalized) DeWitt metric [1, 2, 23]¹³. In the previous sections, we have considered the case, ${\lambda }\ne 1/D$, so that the DeWitt metric is not degenerated. Then the first-order action (5) can be obtained by considering the Legendre transformation of $S_K$,

$$\begin{aligned} S_K = \left( \frac{{\kappa }^2}{2} \right) \int dt d^D x \frac{N}{2 \sqrt{g}} \left( \pi ^{ij} \mathcal{G}_{ijkl}\pi ^{kl} \right) , \end{aligned}$$

(45)

with the conjugate momenta, $\pi ^{ij}=({2 \sqrt{g}}/{\kappa ^2}) {G}^{ijkl}K_{kl}$, and the inverse DeWitt metric, $\mathcal{G}_{ijkl}={\delta }_{ijkl}-{\lambda }g_{ij} g_{kl} /(D {\lambda }-1)$, satisfying $G^{ijmn}\mathcal{G}_{mnkl}=\mathcal{G}_{klmn} G^{mnij}={\delta }^{ij}_{kl}$ [1, 2].

On the other hand, for ${\lambda }=1/D$, the DeWitt metric is degenerated and we need to project out the non-degenerate parts only when considering the appropriate inverse of the DeWitt metric. Actually, using the fact that $\widehat{G}^{ijkl}\equiv G^{ijkl} |_{{\lambda }=1/D}$ has a null eigenvector $g_{ij}$,

$$\begin{aligned} \widehat{G}^{ijkl} g_{ij}=0, \end{aligned}$$

(46)

it is easy to see that its inverse, $\widehat{\mathcal{G}}_{ijkl}$, is given by

$$\begin{aligned} \widehat{\mathcal{G}}_{ijkl}= & {} {\delta }_{ijkl}-\frac{1}{D} g_{ij}g_{kl}, \nonumber \\ \widehat{\mathcal{G}}_{ijkl} g^{ij}= & {} 0,~\widehat{\mathcal{G}}_{ijmn} \widehat{G}^{mnkl}={\widetilde{{\delta }}_{ij}}^{kl} \end{aligned}$$

(47)

with the (projected) Kronecker-delta, ${\widehat{{\delta }}_{ij}}^{kl}={\delta }_{ij}^{kl}-g_{ij} g^{kl}/D$, satisfying ${\widehat{{\delta }}_{ij}}^{kl}g^{ij}={\widehat{{\delta }}_{ij}}^{kl}g_{kl}=0$ [10]. (See also [25, 26] for the corresponding analyses in Lorentz invariant higher-curvature gravities.)

In the Hamiltonian formulation, the existence of a null eigenvector in $g_{ij}$ is reflected in the primary constraint,

$$\begin{aligned} \chi \equiv {\widehat{\pi }^i}_i \equiv g_{ij} \widehat{\pi }^{ij} \approx 0 \end{aligned}$$

(48)

for the momenta,

$$\begin{aligned} \widehat{\pi }^{ij} \equiv \frac{{\delta }S}{{\delta }\dot{g}_{ij}} =\frac{2 \sqrt{g}}{\kappa ^2} \widehat{G}^{ijkl}K_{kl}, \end{aligned}$$

(49)

in addition to the usual primary constraints, $\Phi _{\mu }$, in (10).

Then, one can find the canonical Hamiltonian, up to boundary terms,

$$\begin{aligned} \widehat{H}_C= & {} \int _{{\Sigma }_t} d^{D}x \left\{ N \widehat{\mathcal{H}}+N_i {\widehat{\mathcal{H}}}^i\right\} \end{aligned}$$

(50)

with

$$\begin{aligned} \widehat{\mathcal{H}}\equiv & {} \frac{\kappa ^2}{2 \sqrt{g}} \widehat{\pi }^{ij}\widehat{\pi }_{ij} +\sqrt{g} \mathcal{V}, \end{aligned}$$

(51)

$$\begin{aligned} \widehat{\mathcal{H}}^i\equiv & {} -2 \nabla _j \widehat{\pi }^{ij}. \end{aligned}$$

(52)

Now, the preservation of the additional primary constraint (48),

$$\begin{aligned} \dot{\chi }= & {} \{ \chi , \widehat{H}_C\} \nonumber \\\equiv & {} \Psi \approx 0 \end{aligned}$$

(53)

produces a new secondary constraint,

$$\begin{aligned} \Psi\equiv & {} \left( \frac{{\kappa }^2}{2} \right) \left( \frac{D}{2 } \right) \frac{N}{\sqrt{g}} \widehat{\pi }_{mn} \widehat{\pi }^{mn} \nonumber \\&-N \sqrt{g} \left[ \left( \frac{D}{2}\right) \mathcal{V}(R)- R \mathcal{V}' (R) \right] \nonumber \\&+\sqrt{g}(D-1) \nabla ^2 \left( N \mathcal{V}'(R) \right) \nonumber \\= & {} \frac{D}{2} N \widehat{\mathcal{H}}-N \sqrt{g} \left[ D \mathcal{V}(R)-R \mathcal{V}' (R) \right] \nonumber \\&+\sqrt{g}(D-1) \nabla ^2 \left( N \mathcal{V}'(R) \right) \approx 0 \end{aligned}$$

(54)

from (A18), in addition to the usual (reduced) Hamiltonian and momentum constraints in (12),

$$\begin{aligned} \widehat{\mathcal{H}} \approx 0, ~\widehat{\mathcal{H}}^i \approx 0. \end{aligned}$$

(55)

The extended Hamiltonian is then given by

$$\begin{aligned} \widehat{H}_E= & {} \widehat{H}_C + \int _{{\Sigma }_t} d^{D}x ( u_{\mu } \Phi ^{\mu } + v \chi ) \end{aligned}$$

(56)

with a new Lagrange multiplier v. The constraint algebra, (16)–(18), are reduced to

$$\begin{aligned} \{ \widehat{\mathcal{H}} (x),\widehat{\mathcal{H}} (y) \}= & {} \widehat{C}^i (x) \nabla ^x_i {\delta }^D (x-y)\nonumber \\&-\widehat{C}^i (y) \nabla ^y_i {\delta }^D (x-y),\qquad \end{aligned}$$

(57)

$$\begin{aligned} \{ \widehat{\mathcal{H}} (x),\widehat{\mathcal{H}}_i (y) \}= & {} -\widehat{\mathcal{H}} (y) \nabla ^y_i {\delta }^D (x-y), \end{aligned}$$

(58)

$$\begin{aligned} \{ \widehat{\mathcal{H}}_i (x),\widehat{\mathcal{H}}_j (y) \}= & {} \widehat{\mathcal{H}}_i (y) \nabla ^x_j {\delta }^D (x-y) \nonumber \\&+\widehat{\mathcal{H}}_j (x) \nabla ^x_i {\delta }^D (x-y), \end{aligned}$$

(59)

where

$$\begin{aligned} \widehat{C}^i= & {} \left( -\frac{{\kappa }^2}{2} \right) \left[ \left( \widehat{\mathcal{H}}^i+2 \nabla ^i \widehat{\pi } \right) \mathcal{V}' (R)\right. \nonumber \\&\left. +2 \left( \widehat{\pi }^{ij}-g^{ij} \widehat{\pi } \right) \nabla _j \mathcal{V}' (R) \right] . \end{aligned}$$

(60)

Using the above reduced constraint algebra, one can find that

$$\begin{aligned} \dot{\widehat{\mathcal{H}}}= & {} \{\widehat{\mathcal{H}} , \widehat{H}_E \} \nonumber \\= & {} \frac{1}{N} \nabla _i (N^2 \widehat{C}^i) +\nabla _i (N^i \widehat{\mathcal{H}}) +\{\widehat{\mathcal{H}}, \langle v \pi \rangle \}, \end{aligned}$$

(61)

$$\begin{aligned} \dot{\widehat{\mathcal{H}}_i}= & {} \{\widehat{\mathcal{H}}_i , \widehat{H}_E \} \nonumber \\= & {} \widehat{\mathcal{H}} \nabla _i N +\nabla _j (N^j \widehat{\mathcal{H}}_i)+\widehat{\mathcal{H}}_j \nabla _i N^j +\widehat{\pi } \nabla _i v \approx 0, \end{aligned}$$

(62)

where

$$\begin{aligned} \{\widehat{\mathcal{H}}, \langle v \pi \rangle \}= & {} \left[ -\frac{v}{N} \Psi +\sqrt{g}(D-1)\left( \frac{v}{N} \nabla ^2 \left( N \mathcal {V}^\prime (R) \right) \right. \right. \nonumber \\&\left. \left. -\mathcal {V}^\prime (R) \nabla ^2 v \right) \right] . \end{aligned}$$

(63)

For the potential, $-\mathcal{V}(R)\equiv {\Lambda }+\xi R +{\alpha }R^n$, (63) becomes

$$\begin{aligned} \{\widehat{\mathcal{H}}, \langle v \pi \rangle \}= & {} -\left[ \frac{v}{N} \Psi +\sqrt{g}(D-1)\left( -\frac{v}{N} \nabla ^2 \Bigg (N \Bigg (\xi \right. \right. \nonumber \\&\left. \left. \left. +{\alpha }n R^{n-1}\Bigg ) \right) + \left( \xi +{\alpha }n R^{n-1}\right) \nabla ^2 v \right) \right] . \end{aligned}$$

(64)

Since we are considering the non-trivial case of $-\mathcal{V}' (R) \equiv \xi +{\alpha }n R^{n-1}\ne 0$, preserving the Hamiltonian constraint, $\widehat{\mathcal{H}}\approx 0$, i.e., $\dot{\widehat{\mathcal{H}}} \approx 0$ in (61), does not produce new constraints but determines the Lagrange multiplier v: For the ${\lambda }$-deformed GR (${\alpha }=0$), where $\widehat{C}^i = 0$, (61) and (64) determine $v=N$.

On the other hand, for the preservation of the secondary constraint, $\widetilde{\Psi }\equiv \Psi _{\widehat{\mathcal{H}}\approx 0}$,

$$\begin{aligned} \widetilde{\Psi }\equiv & {} N \sqrt{g} \left[ D {\Lambda }+(D-1) \xi R +(D-n) {\alpha }R^n \right] \nonumber \\&+ {\sqrt{g}(D-1) \nabla ^2 \left( N \mathcal {V}^\prime \right) }, \end{aligned}$$

(65)

one can find that

$$\begin{aligned} \dot{\widetilde{\Psi }}= & {} \{\widetilde{\Psi } , \widehat{H}_E \} \nonumber \\= & {} \{\widetilde{\Psi } , \widehat{H}_C \}+\{\widetilde{\Psi }, \langle u_t \pi _N\rangle \} +\{\widetilde{\Psi }, \langle v \widehat{\pi } \rangle \} , \end{aligned}$$

(66)

where¹⁴

$$\begin{aligned} \{\widetilde{\Psi }, \langle u_t \pi _N \rangle \}= & {} \frac{u_t}{N} \widetilde{\Psi } +\sqrt{g}(D-1)\left( \nabla ^2 \left( u_t \mathcal {V}^\prime \right) \right. \nonumber \\&\left. -\frac{u_t}{N} \nabla ^2 \left( N\mathcal {V}^\prime \right) \right) , \end{aligned}$$

(67)

$$\begin{aligned} \{\widetilde{\Psi }, \langle v \pi \rangle \}= & {} v N \sqrt{g}\left[ \frac{D^2}{2} {\Lambda }+\frac{(D-1)(D-2)}{2} \xi R \right. \nonumber \\&\left. +\frac{(D-2 )(D-2 n)}{2} {\alpha }R^n\right] \nonumber \\&+\sqrt{g}(D-1) \left[ \frac{(D-2)}{2} \nabla _i \left( v \nabla ^i \left( N \mathcal {V}^\prime \right) \right) \right. \nonumber \\&\left. -N \nabla ^2 v \Big ( (D-1) \xi \right. \nonumber \\&\left. \left. + (D-n) {\alpha }n R^{n-1} \right) \right] \nonumber \\&+{\alpha }n(n-1) (D-1) \sqrt{g}\nabla ^2 \left[ R^{n-2} N \Big ( R v \right. \nonumber \\&\left. \left. +(D-1) \nabla ^2 v \right) \right] . \end{aligned}$$

(68)

Now, (61) and (66) determine the multipliers v and $u_t$, respectively, so that one can finish the Dirac’s procedure without generating further constraints. Then, the complete set of constraints are $\widehat{\chi }_A \equiv ( \pi _N, \widehat{\mathcal{H}}, \pi , \widetilde{\Psi } )\approx 0$ and ${\Gamma }_A \equiv (\pi ^i, \mathcal{H}_i )\approx 0$. Here, the constraints, $\widehat{\chi }_A \approx 0$, are the second-class constraints with the algebra,

$$\begin{aligned} \{ \pi _N (x), \widehat{\mathcal{H}} (y) \}= & {} 0 ,~~\{ \pi _N (x), \chi (y) \} = 0, \nonumber \\ \{ \pi _N (x), \widetilde{\Psi } (y) \}= & {} \widehat{\Delta } (x-y), \nonumber \\ \{ \widehat{\mathcal{H}} (x), \widehat{\mathcal{H}} (y) \}= & {} \widehat{C}^i (x) \nabla ^x_i {\delta }^D (x-y) \nonumber \\&-\widehat{C}^i (y) \nabla ^y_i {\delta }^D (x-y) , \nonumber \\ \{ \widehat{\mathcal{H}} (x), \pi (y) \}\approx & {} \{ \pi _N (x), \widetilde{\Psi } (y) \} ,\text { etc.}, \end{aligned}$$

(69)

whose determinant, $\text{ det }(\{\widehat{\chi _A}, \widehat{\chi _B} \})$, is generally non-vanishing,

$$\begin{aligned} \text{ det } \{\widehat{\chi _A}, \widehat{\chi _B} \}\approx & {} \left( \text{ det } \{ \widehat{\mathcal{H}} (x), \pi (y) \} \right) ^2 \left( \text{ det } \{\pi (x), \widehat{\mathcal{H}} (y) \} \right) ^2 \nonumber \\\approx & {} \left( \text{ det } \left[ \widehat{\Delta } (x-y) \widehat{\Delta } (y-x) \right] \right) ^2, \end{aligned}$$

(70)

where

$$\begin{aligned}&\widehat{\Delta } (x-y) \nonumber \\&\quad \equiv -\sqrt{g}\left[ D {\Lambda }+(D-1) \xi R +(D-n) {\alpha }R^n \right] {\delta }^D(x-y) \nonumber \\&\qquad +\sqrt{g}(D-1) \nabla ^2_{y} \left[ ~{\delta }^D(x-y) \left( \xi +{\alpha }n R^{n-1} \right) (y) \right] .\nonumber \\ \end{aligned}$$

(71)

On the other hand, the constraints, ${\Gamma }_A \equiv (\pi ^i, \mathcal{H}_i )\approx 0$, are the first-class constraints, as in Cases A and B. So, the resulting number of dynamical degrees of freedom is the same as in Case A,

$$\begin{aligned} s= & {} \frac{1}{2} \left( P-2 N_1-N_2\right) \nonumber \\= & {} \frac{1}{2} (D+1)(D-2), \end{aligned}$$

(72)

which is the same as in GR. This provides the fully non-perturbative proof of the previous perturbative analysis (see also [35] for an earlier work), which does not show the extra degrees of freedom at the linear level [10]. There may exist some similarities with Case A due to the same physical degrees of freedom. Actually, one can consider the maximal slicing condition, i.e., $\pi =0$, for $\lambda$-deformed GR (with an arbitrary $\lambda$), as an example satisfying the condition, $\widetilde{C}^i =0$ for Case A. But, an importance difference is that Case C does not depend whether $\widehat{C}^i=0$, i.e., commuting Hamiltonian constraint $\widehat{\mathcal{H}}$, or $\widehat{C}^i \ne 0$, i.e., non-commuting Hamiltonian constraint $\widehat{\mathcal{H}}$: If we consider the maximal slicing, $\pi =0$, for Case B with an arbitrary ${\lambda }\ne 1/D$, we have basically the same results as Case C ! This implies that the constraint structure of Case C and so its number of dynamical degrees of freedom do not depend on spatially-higher-derivative terms in the potential, which are important for distinguishing Cases A and B. In other words, Case C does not depend on the UV conformal symmetry for the Cotton square term, $C^{ij}C_{ij}$, in the Hořava’s original potential [1, 2] and this is essentially due to the kinematic origin of the constraint, $\chi =\widehat{\pi }\approx 0$, in (48).¹⁵

5 Examples

In this section, we consider several exact solutions for our typical potential, $-\mathcal{V}(R) \equiv \xi R + {\alpha }R^n+{\Lambda }$, as some explicit examples of the constraint analysis.

5.1 Non-rotating black holes in arbitrary dimensions

For non-rotating, spherically symmetric black holes with $N_i \equiv 0$, the extrinsic curvature, $K_{ij}$, and the conjugate momentum, $\pi _{ij}$, vanish so that the Hamiltonian constraint becomes simply $\mathcal{H}=\sqrt{g} \mathcal{V}(R) \approx 0$. The general solution is ‘$R=constant$’, whose constant value depends on the theory parameters, $\xi , {\alpha }, {\Lambda }$, and n. If we consider, as an explicit example, $D=3,~n=2$, i.e., $z=2$ case in $3+1$ dimensions [36],¹⁶ then the solution is given by

$$\begin{aligned} ds^2=-N^2(r) dt^2+\frac{dr^2}{f(r)}+r^2 d \theta ^2 +r^2 sin^2 \theta d \phi ^2 \end{aligned}$$

(73)

with

$$\begin{aligned} N^2= & {} f=1-\frac{2 M}{r} +\left( \frac{\xi -\sqrt{\xi ^2 -4 {\alpha }{\Lambda }}}{12 {\alpha }} \right) r^2, \end{aligned}$$

(74)

$$\begin{aligned} R= & {} \frac{-\xi +\sqrt{\xi ^2 -4 {\alpha }{\Lambda }}}{2 {\alpha }} . \end{aligned}$$

(75)

In this case, even though there are higher-derivative contributions, we have $\widetilde{C}^i =0$, trivially, due to $\pi _{ij}=0$ in (31). This would be a trivial example though its result is generally valid for arbitrary power n and dimension D. If we consider the more general, spherically symmetric solutions with $N_r \ne 0$ (cf. [37]), it would be a more non-trivial example with the extrinsic curvature and conjugate momenta.

5.2 Black string solutions in $3+1$ dimensions

For stationary black strings in $3+1$ dimensions, the general ansatz is given by

$$\begin{aligned} ds^2= & {} (-N^2+N_r N^r+N_\phi N^\phi ) dt^2 \nonumber \\&+2 (N_r dr+N_\phi d \phi ) dt+\frac{dr^2}{f}+r^2 d \phi ^2 +g dz^2, \end{aligned}$$

(76)

where all the metric functions, $N, N_i, f$, and g depend on the radial coordinate r. For $z=2$ black string solution which satisfies the vanishing Cotton tensor, $C_{ij}=0$, for simplicity, a simple solution with $g=constant, N_{\phi }=0$, and ${\lambda }=1$, which is called BTZ-type black string, is given by (with $g \equiv 1$)

$$\begin{aligned} N^2= & {} f=\eta r^2 -m, \end{aligned}$$

(77)

$$\begin{aligned} N_r^2= & {} f^{-1} \left[ {\delta }+\frac{{\kappa }^4 \mu ^2}{64} \left( \eta ^2-2 ({\Lambda }_W +{\omega })-3 {\Lambda }_W^2\right) \right] , \end{aligned}$$

(78)

where $\eta , m$, ${\delta }$ are integration constants [38] and the other parameters ${\kappa }, \mu , {\Lambda }_W, {\omega }$ are introduced by the usual parametrization,

$$\begin{aligned} \xi\equiv & {} \frac{{\kappa }^4 \mu ^2 ( {\Lambda }_W +{\omega })}{8 (1-3 {\lambda })}, \nonumber \\ ~ {\alpha }\equiv & {} \frac{{\kappa }^2 \mu ^2 (1-4 {\lambda })}{32 (1-3 {\lambda })}, ~ \nonumber \\ {\Lambda }\equiv & {} -\frac{2 {\kappa }^2 \mu ^2 {\Lambda }_W^2}{8 (1-3 {\lambda })}. \end{aligned}$$

(79)

In this case, even though the extrinsic curvatures and conjugate momenta are non-vanishing, it has a constant curvature, $R \sim f'/r=2 \eta =constant$. Since this is the solution for ${\lambda }=1$, one can easily find that $\widetilde{C}^i=0$ in (31) is trivially satisfied¹⁷: The first term in (31) vanishes due to ${\lambda }=1$ and the second term vanishes due to $R=constant$. If we consider the more general solutions with ${\lambda }\ne 1$ or the ${\lambda }=1$ solution without the condition, $C_{ij}=0$, it would be more non-trivial examples with the non-constant extrinsic curvatures and conjugate momenta.

5.3 Rotating black holes in $3+1$ dimensions

The exact solutions for rotating black holes in $3+1$ dimensions has not been found yet. However, for slowly rotating black holes, one can consider the ansatz [39, 40],

$$\begin{aligned} ds^2= & {} -N^2(r) dt^2+\frac{dr^2}{f(r)}+r^2 d \theta ^2 +r^2 sin^2 \theta d \phi ^2 \nonumber \\&+2 a g(r) sin^2 \theta dt d \phi . \end{aligned}$$

(80)

At the linear order in the rotation parameter a, the solution of g(r) for $n=2$ is given by

$$\begin{aligned} g(r)={\sigma }r^2 +\frac{{\gamma }}{r} \end{aligned}$$

(81)

with the integration parameters, ${\sigma }, {\gamma }$, and the same solution of $N^2=f(r)$ as in (74). In this case, the extrinsic curvature has a non-vanishing component, $K_{r \phi }=\mathcal{O}(a)$ but $K=0, \pi =0$, at the linear order in ‘a’. Since the curvature scalar is constant as in (75), we will have the case $\widetilde{C}^i =0$ again and this result is valid for arbitrary power n and dimensions D.¹⁸ If we consider higher orders in the rotation parameter a so that K or $\pi$ is non-vanishing or R is non-constant, it would be a more non-trivial example.

5.4 Rotating black holes in $2+1$ dimensions

For stationary black holes in $2+1$ dimensions, the most general ansatz is given by

$$\begin{aligned} ds^2= & {} (-N^2+N_r N^r+N_\phi N^\phi ) dt^2+2 (N_r dr+N_\phi d \phi ) dt\nonumber \\&+\frac{dr^2}{f}+r^2 d \phi ^2, \end{aligned}$$

(82)

where all the metric functions, $N, N_i, f$, depend on the radial coordinate r, as in (76), due to the symmetry property in $2+1$ dimensions. The general solution for the metric (82) is not known yet. For the simple case, however, of $N_r=0$, one can find that the there is non-vanishing conjugate momentum, $\pi ^{\phi r}=(f/2N) (N^{\phi })'$, whose trace is vanishing trivially, $\pi =0$, with the solution (for the $n=2$ case) [41, 42]

$$\begin{aligned} f= & {} -\mathcal{M} +\frac{b r^2}{2} \left[ 1- \sqrt{a+ \frac{c}{r^4}}+\sqrt{\frac{c}{r^4}} ln\left( \sqrt{\frac{c}{ar^4}} \right. \right. \nonumber \\&\left. \left. +\sqrt{1+\frac{c}{ar^4}} \right) \right] , \nonumber \\ N^2/f= & {} 1/\left( 1+\frac{c}{ar^4}\right) , \nonumber \\ N^{\phi }= & {} -\frac{\mathcal{J}}{2}\sqrt{\frac{a}{c}}~ ln \left[ \sqrt{\frac{c}{a r^4}}+\sqrt{1+\frac{c}{a r^4}} \right] , \end{aligned}$$

(83)

where

$$\begin{aligned} a=1-\frac{4 {\alpha }\Lambda }{\xi ^2},~b=\frac{\xi }{2 {\alpha }},~ c=\frac{2 {\alpha }\mathcal{J}^2}{\xi ^2}. \end{aligned}$$

(84)

In this case, even though $K=0,\pi =0$ trivially, the curvature scalar is non-constant,

$$\begin{aligned} R=-\frac{f'}{r}=-b \left( 1-\sqrt{a+ \frac{c}{r^4}} \right) , \end{aligned}$$

(85)

so that we have a non-vanishing component of $\widetilde{C}^i$,

$$\begin{aligned} \widetilde{C}^{\phi } \sim {\alpha }\pi ^{\phi r} \partial _r R \ne 0. \end{aligned}$$

(86)

But, it easy to find that the constraint $\widetilde{{\Omega }}\approx 0$ is satisfied again

$$\begin{aligned} \widetilde{{\Omega }} \sim {\partial }_{\phi }\left( N^2 \widetilde{C}^{\phi } \right) =0, \end{aligned}$$

(87)

due to the spherical symmetry, i.e., no $\phi$-dependence, in the solution. It is important to note that the non-vanishing $\widetilde{C}^i$ is the genuine result of higher-derivative terms (${\alpha }\ne 0$) with a rotation ($\pi ^{\phi r} \ne 0$). In other words, if we turn off the higher-derivative term, i.e., ${\alpha }=0$, similarly to BTZ black hole in GR, the system is reduced to the case with the vanishing $\widetilde{C}^i \sim \xi \widehat{{\lambda }} \nabla ^i \pi =0$. Of course, this does not correspond to Case B but Case C, due to the fact of $\pi =0$ for the solution (83), as can be seen by checking the constraint, $\widetilde{\Psi } \approx 0$, in (65). However, if we consider the $N_r \ne 0$ case, one obtains $\pi \ne 0$ generally so that it may correspond to a “genuine” case of Case B. This example would show the importance of higher-derivative terms for Case B, where the extra scalar graviton mode is involved.

6 Concluding remarks

We study the full constraint analysis of the non-projectable Hořava gravity whose potential, $\mathcal{V}(R)$, is an arbitrary function of the (intrinsic) Ricci scalar R in arbitrary dimensions but without the extension terms which depend on the proper acceleration $a_i$, for simplicity. We find that there are generally three distinct cases in this theory, depending on

• (i) whether the Hamiltonian constraint, $\mathcal{H}\approx 0$, generates new (second-class) constraints (Cases A, C) or just fixes the associated Lagrange multipliers (Case B), or

• (ii) whether the IR Lorentz-deformation parameter ${\lambda }$ is at the conformal point, ${\lambda }=1/D$ (Case C), or not (Cases A, B).

We find that, for Cases A and C, the dynamical degrees of freedom of Hořava gravity are the same as in GR, while for Case B, there is one extra phase-space degree of freedom, representing an extra (odd) scalar graviton mode. This would achieve the dynamical consistency of a restricted model at the fully non-linear level and be positive in resolving the long-standing debates about the extra graviton modes of the Hořava gravity. Several further remarks about other challenging problems are in order.

1. We have obtained the new “extended” constraint algebra for the Hamiltonian and momentum constraints, (16)–(18) ((57)–(59) for Case C) or (19)–(21), for the non-projectable Hořava gravity whose potential is an arbitrary function of the Ricci scalar R. The structure of the newly obtained “extended” constraints algebra seems to be generic to Hořava gravity itself, analogous to that of general higher-curvature gravities [25, 26], where $C^i$ becomes the momentum constraint $\mathcal{H}^i$ with higher-curvature corrections. An important difference is that the momentum constraint $\mathcal{H}^i$ in Hořava is the same form as in GR with no higher-derivative corrections and satisfies the same algebra as in GR [23] but the full algebra with the Hamiltonian constraint, $\mathcal{H}$, is not closed, whereas the momentum constraints, $\mathcal{H}^i \equiv C^i$, in generic higher-curvature gravities satisfy exactly the same closed algebra as in GR, “as has been argued generically in [43]”, even with the generic higher-curvature terms which include Riemann tensors also [25, 26, 44]. We suspect that a similar general argument in Hořava gravity exists also so that the extended constraint algebra reflects the generic space-time structure of our Lorentz violating gravities with Ricci and Riemann tensors, $R_{ij}, R_{ijkl}$, etc., i.e., with the potential, $\mathcal{V}(R,R_{ij}, R_{ijkl}, {\ldots })$,¹⁹ and its general proof would be a challenging problem.

2. Our constraint analysis shows the dynamical degrees of freedom in a restricted model of the Hořava gravity at the fully non-linear level. Comparison to the previous linear perturbation analyses [8, 10, 33], which do not show the extra degrees of freedom, implies that the extra modes would be the genuine consequence of non-linear effect (around the homogenous background) with Lorentz-violating higher-derivative terms [27]: For a direct proof in the $n=2$ constraint algebra, see [15]. On the other hand, it has been also argued that the extra mode may appear when considering (even linear) perturbations around the spatially-inhomogeneous and time-dependent background [5], even with the IR Lorentz-violating terms only, as anticipated from the usual connection between non-linear perturbations for a homogenous background and linear perturbations for a corresponding inhomogeneous background.²⁰ This may be in contrast to the Birkhoff’s theorem analysis in spherically symmetric, vacuum configuration which does not show time-dependent solutions, representing the “extra gravitational” modes [27]. However, this might be due to its high symmetry, i.e., lower inhomogeneity, and there might exist still some possibilities for higher inhomogeneities: This might be consistent with a related analysis in [15] which does not show the extra mode only up to the first order of inhomogeneity, $L^{-1}$. So, a direct proof of the argument of [5] in the generic constraint algebra would be an interesting open problem. The role of non-linear, UV scalar graviton in Big Bang cosmology and gravitational wave physics for compact objects with strong gravities, like black holes and neutron stars, would be also a challenging problem.

3. In the literature, there have been claims of inconsistencies of the Hořava gravity, in its original form. It seems that some originate from the limited analysis, like linear approximations [7], or incomplete constraint analysis [4, 11]. Here, we consider the second case,²¹ especially about the claim in [11], which seems to be the most rigorous criticism against the (non-projectable) Hořava gravity. The basic claim in [11] is that “$N=0$ is the only possible solution” for the constraint (4.1), $\nabla _i (N^2 \nabla ^i \pi )\approx 0$, which corresponds to our constraint (30), $\widetilde{{\Omega }}\equiv \nabla _i (N^2 \widetilde{C}^i)\approx 0$, for the asymptotically flat ($\Lambda =0$) and ${\lambda }$-deformed GR, otherwise N blows up at infinity, generically.²² But, the elaborate analysis in [11] is just another proof of an adequacy of the condition, $\pi =0$, in that example, which is the only remaining solution for the constraint (4.1) in [11] or (30) in this paper, as has been argued also in [14, 15]. Actually, contrary to the argument in [11], the perturbations from $\pi =0$ are not arbitrary but restricted by another constraint (33), $\widetilde{\Xi } \approx 0$ in this paper (or (20) in [14]), when ${\lambda }\ne 1$. Moreover, when the metric is not asymptotically flat ($\Lambda \ne 0$), the argument of [14] (and possibly of [11] or [5] also) would not be valid generally, as can be seen in the general, $(2+1)$-dimensional solution, (82) with $N_r \ne 0$, which shows the non-vanishing $\pi$ and N.

4. With the full set of constraints, we can now consider Feynman’s path-integral for the S-matrix elements [45, 46], whose Hamiltonian expression for Case A, for example, is given by

(88)

where we have integrated out for the primary constraints $\pi _N, \pi ^i \approx 0$ with the Faddeev–Popov’s anti-commuting fields, $c_j, \bar{c}_i, b, \bar{b}$, the gauge-fixing conditions, $\gamma _j=0$, for the first-class constraint, $\mathcal{H}_i\approx 0$, and the determinant factor for the second-class constraints, $\left( \text{ det } \{ \chi _A, \chi _B \} \right) ^{1/2} \approx \text{ det } \left( \{ \mathcal{H}, \widetilde{{\Omega }} \} \{\widetilde{{\Omega }}, \mathcal{H} \} \right)$. One can obtain similarly the path integral for Cases B and C also. It would be desirable to study the renormalizability for the non-projectable Hořava gravity, based on the above S-matrix elements, beyond the recent proof of (perturbative) renormalizability for the projectable cases [47,48,49].²³

5. In the canonical quantization with the second-class constraints, we need to compute the Dirac brackets [17],²⁴ whose definition for Case A, for example, is given by, for any variable P, Q,

$$\begin{aligned}&\{ P, Q\}^* \equiv \{P, Q\}\nonumber \\&\quad - \int d^D z \int d^D w ~\{P, \chi _A (z) \} C^{-1}_{AB}(z,w) \{\chi _B (w), Q \},\nonumber \\ \end{aligned}$$

(89)

where $C^{-1}_{AB}$ is defined as $\int d^D z~ C^{-1}_{AB} (x,z) C_{BC} (z,y)=\int d^D z ~C_{CB} (y,z) C^{-1}_{BA} (z,x)={\delta }_{AC} {\delta }^D (x-y)$ for the Poisson brackets of the second-class constraints $C_{AB} \equiv \{ \chi _A, \chi _B \}$, and given by

$$\begin{aligned} C^{-1}_{AB}(x,z)=\left( \begin{array}{cccc} 0 &{}-\Delta ^{-1}(z-x) &{}0 &{} 0 \\ \Delta ^{-1}(x-z) &{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} -\Delta ^{-1}(z-x)\\ 0&{} 0&{} \Delta ^{-1}(x-z) &{}0 \end{array} \right) \end{aligned}$$

$\Delta ^{-1}$, defined by $\int d^D z~ \Delta ^{-1} (x-z) \Delta (z-y)=\int d^D z ~\Delta (y-z) \Delta ^{-1} (z-x)={\delta }^D (x-y)$. The bracket satisfies $\{ \chi _A, Q \}^* =0$ for any variable Q so that the second-class constraints, $\chi _A \approx 0$, can be imposed consistently in the Hamiltonian dynamics, i.e., $\{ \mathcal{H}, \mathcal{H} \}^* =0, \{ \mathcal{H}, \mathcal{H}_i \}^* =0,\text { etc.}$,²⁵ which corresponds to the delta-function insertions for the constraints, $\chi _A \approx 0$, in the path integral, (88). If we consider the gauge-fixing conditions as in the path integral formalism, we can obtain further corrections to the Dirac brackets. Then the Dirac’s quantization rule is given by $[ \widehat{P}, \widehat{Q} ] \equiv (i/\hbar ) \{ P, Q \}^*$ for the quantum operators, $\widehat{P}$ and $\widehat{Q}$, corresponding to classical variables P and Q, respectively, with the“appropriate” operator orderings. One can consider also Cases B and C similarly, but it would be more involved for the former case.

Note added: After finishing this paper, a related paper [53] appeared which is overlapping with ours for $D=2$ case. But due to the (full) $a_i$ extensions in [53], it shows a different constraint structure, as noted in our footnote No. 9.

Appendices

Appendix A: Computing $\{ \mathcal{H}, \mathcal{H} \}, \{ \mathcal{H}, \mathcal{H}_i \}$, and $\{ \mathcal{H}_i, \mathcal{H}_j \}$

In this Appendix, we compute the constraint algebra, $\{ \mathcal{H}, \mathcal{H} \}, \{ \mathcal{H}, \mathcal{H}_i \}$, and $\{ \mathcal{H}_i, \mathcal{H}_j \}$ in (16)–(18). To this ends, it is useful to consider the variations of the smeared constraints, $\langle \eta \mathcal{H} \rangle \equiv \int d^D x~ \eta \mathcal{H}\approx 0$, etc., with the smearing functions, $\eta$, $\eta _i$ (neglecting the boundary terms),

$$\begin{aligned} {\delta }\langle \eta \mathcal{H} \rangle= & {} \langle A^{kl} {\delta }g_{kl} +B_{kl} {\delta }\pi ^{kl}\rangle , \end{aligned}$$

(A1)

$$\begin{aligned} {\delta }\langle \eta ^i \mathcal{H}_i \rangle= & {} \langle C^{kl} {\delta }g_{kl} +D_{kl} {\delta }\pi ^{kl}\rangle . \end{aligned}$$

(A2)

After long computations, one can obtain the coefficients, A, B, C, D as follows:

$$\begin{aligned} A^{kl}\equiv & {} \frac{{\delta }\langle \eta \mathcal{H} \rangle }{{\delta }g_{kl}} =\left( \frac{\kappa ^2}{2} \right) \frac{\eta }{ \sqrt{g}} \left[ -\frac{1}{2} g^{kl} \left( \pi _{mn} \pi ^{mn}-\widetilde{{\lambda }} \pi ^2 \right) \right. \nonumber \\&\left. +2 \left( \pi ^{km} \pi ^l_m-\widetilde{{\lambda }} \pi \pi ^{kl} \right) \right] \nonumber \\&+\eta \sqrt{g} \left[ \frac{ 1}{2} g^{kl} \mathcal{V}(R)-R^{kl} \mathcal{V}'(R) \right] \nonumber \\&+\sqrt{g} \left[ \nabla ^k \nabla ^l \left( \eta \mathcal{V}'(R) \right) -g^{kl} \nabla _m \nabla ^m \left( \eta \mathcal{V}'(R)\right) \right] , \nonumber \\ \end{aligned}$$

(A3)

$$\begin{aligned} B_{kl}\equiv & {} \frac{{\delta }\langle \eta \mathcal{H} \rangle }{{\delta }\pi ^{kl}} =\left( \frac{\kappa ^2}{2} \right) \frac{2 \eta }{ \sqrt{g}} \left( \pi _{kl}-\widetilde{{\lambda }} g_{kl} \pi \right) , \end{aligned}$$

(A4)

$$\begin{aligned} C^{kl}\equiv & {} \frac{{\delta }\langle \eta ^i \mathcal{H}_i \rangle }{{\delta }g_{kl}} = -\nabla _m \left( \eta ^m \pi ^{kl} \right) \nonumber \\&+\left( \nabla _m \eta ^k\right) \pi ^{lm}+\left( \nabla _m \eta ^l\right) \pi ^{km}, \end{aligned}$$

(A5)

$$\begin{aligned} D_{kl}\equiv & {} \frac{{\delta }\langle \eta ^i \mathcal{H}_i \rangle }{{\delta }\pi ^{kl}} =\nabla _k \eta _l + \nabla _l \eta _k. \end{aligned}$$

(A6)

As a byproduct, one can also obtain a useful formula,

$$\begin{aligned} \{ R(x), \pi ^{kl} (y) \}&=\frac{{\delta }R (x)}{{\delta }g_{kl} (y)}=-R^{kl}(x) {\delta }^D (x-y)\nonumber \\&\quad +\nabla ^k_x \nabla ^l_x {\delta }^D (x-y)-g^{kl} \nabla ^2_x {\delta }^D (x-y). \end{aligned}$$

(A7)

Then, after some manipulations, one can find the Poisson bracket algebras for the smeared constraints as follows:

$$\begin{aligned}&\{ \langle \eta \mathcal{H} \rangle ,\langle \zeta \mathcal{H} \rangle \} \nonumber \\&\quad =\int d^D z \left[ \frac{{\delta }\langle \eta \mathcal{H} \rangle }{{\delta }g_{kl} (z)} \frac{{\delta }\langle \zeta \mathcal{H} \rangle }{{\delta }\pi ^{kl} (z)}- \frac{{\delta }\langle \eta \mathcal{H} \rangle }{{\delta }\pi ^{kl}(z)} \frac{{\delta }\langle \zeta \mathcal{H} \rangle }{{\delta }g_{kl} (z)} \right] \nonumber \\&\quad =\langle \left( \eta \nabla _i \zeta -\zeta \nabla _i \eta \right) C^i \rangle , \end{aligned}$$

(A8)

$$\begin{aligned}&\{ \langle \eta \mathcal{H} \rangle ,\langle \zeta ^i \mathcal{H}_i\rangle \} \nonumber \\&\quad =\int d^D z \left[ \frac{{\delta }\langle \eta \mathcal{H} \rangle }{{\delta }g_{kl} (z)} \frac{{\delta }\langle \zeta ^i \mathcal{H}_i \rangle }{{\delta }\pi ^{kl} (z)}- \frac{{\delta }\langle \eta \mathcal{H} \rangle }{{\delta }\pi ^{kl}(z)} \frac{{\delta }\langle \zeta ^i \mathcal{H}_i \rangle }{{\delta }g_{kl} (z)} \right] \nonumber \\&\quad =-\langle ~ \zeta ^i \nabla _i \eta \mathcal{H} \rangle , \end{aligned}$$

(A9)

$$\begin{aligned}&\{ \langle \zeta ^i \mathcal{H}_i \rangle ,\langle \zeta ^j \mathcal{H}_j \rangle \} \nonumber \\&\quad =\int d^D z \left[ \frac{{\delta }\langle \zeta ^i \mathcal{H}_i \rangle }{{\delta }g_{kl} (z)} \frac{{\delta }\langle \zeta ^j \mathcal{H}_j \rangle }{{\delta }\pi ^{kl} (z)}- \frac{{\delta }\langle \zeta ^i \mathcal{H}_i \rangle }{{\delta }\pi ^{kl}(z)} \frac{{\delta }\langle \zeta ^j \mathcal{H}_j \rangle }{{\delta }g_{kl} (z)} \right] \nonumber \\&\quad =\left\langle \left( \zeta ^i \nabla _i \zeta ^j-\zeta ^i \nabla _i \zeta ^j \right) \mathcal{H}_j \right\rangle , \end{aligned}$$

(A10)

where

$$\begin{aligned} C^i= & {} \left( -\frac{{\kappa }^2}{2} \right) \left[ \left( \mathcal{H}^i + 2 \widehat{{\lambda }} \nabla ^i \pi \right) \mathcal{V}' (R) \right. \nonumber \\&\left. +2 \left( \pi ^{ij}-\widehat{{\lambda }} g^{ij} \pi \right) \nabla _j \mathcal{V}' (R) \right] . \end{aligned}$$

(A11)

Now, one can easily check that the Poisson algebra for the local constraints are given by (16)–(18):

$$\begin{aligned} \{ \mathcal{H} (x),\mathcal{H} (y) \}= & {} C^i (x) \nabla ^x_i {\delta }^D (x-y)\nonumber \\&-C^i (y) \nabla ^y_i {\delta }^D (x-y), \end{aligned}$$

(A12)

$$\begin{aligned} \{ \mathcal{H} (x),\mathcal{H}_i (y) \}= & {} -\mathcal{H} (y) \nabla ^y_i {\delta }^D (x-y), \end{aligned}$$

(A13)

$$\begin{aligned} \{ \mathcal{H}_i (x),\mathcal{H}_j (y) \}= & {} \mathcal{H}_i (y) \nabla ^x_j {\delta }^D (x-y) \nonumber \\&+\mathcal{H}_j (x) \nabla ^x_i {\delta }^D (x-y). \end{aligned}$$

(A14)

Moreover, from (A1)–(A6), one can easily obtain the dynamical equations of motion (13)–(14) as follows,

$$\begin{aligned} \dot{g}_{ij}&=\{ g_{ij}, \langle N \mathcal{H}+N_i \mathcal{H}^i \rangle \} = \frac{{\delta }\langle N \mathcal{H} \rangle }{{\delta }\pi ^{ij}} +\frac{{\delta }\langle N_i \mathcal{H}^i \rangle }{{\delta }\pi ^{ij}} \nonumber \\&=\left( \frac{\kappa ^2}{2} \right) \frac{2 N}{ \sqrt{g}} \left( \pi _{ij}-\widetilde{{\lambda }} g_{ij} \pi \right) + \nabla _i N_j + \nabla _j N_i, \end{aligned}$$

(A15)

$$\begin{aligned} \dot{\pi }^{ij}&=\{ \pi ^{ij}, \langle N \mathcal{H}+N_i \mathcal{H}^i \rangle \}= -\frac{{\delta }\langle N \mathcal{H} \rangle }{{\delta }g_{ij}} -\frac{{\delta }\langle N_i \mathcal{H}^i \rangle }{{\delta }g_{ij}} \nonumber \\&=\left( \frac{\kappa ^2}{2} \right) \frac{ N}{ \sqrt{g}} \left[ \frac{1}{2} g^{ij} \left( \pi _{mn} \pi ^{mn}-\widetilde{{\lambda }} \pi ^2 \right) \right. \nonumber \\&\quad \left. -2 \left( \pi ^{im} \pi ^j_m-\widetilde{{\lambda }} \pi \pi ^{ij} \right) \right] \nonumber \\&\quad -N \sqrt{g} \left[ \frac{ 1}{2} g^{ij} \mathcal{V}(R)-R^{ij} \mathcal{V}'(R) \right] \nonumber \\&\quad -\sqrt{g} \left[ \nabla ^i \nabla ^j \left( N \mathcal{V}'(R) \right) -g^{ij} \nabla _m \nabla ^m \left( N \mathcal{V}'(R)\right) \right] \nonumber \\&\quad + \nabla _m \left( N^m \pi ^{ij} \right) -\left( \nabla _m N^i\right) \pi ^{jm}-\left( \nabla _m N^j\right) \pi ^{im}. \end{aligned}$$

(A16)

Combining (A15) and (A16), one can also obtain the dynamical equation for $\sqrt{g}$ and the momentum trace $\pi$ as follows,

$$\begin{aligned} {{\sqrt{\dot{g}}}}= & {} \left( \frac{\kappa ^2}{2} \right) \frac{N \pi }{1-D {\lambda }} +2 \nabla _i N^i, \end{aligned}$$

(A17)

$$\begin{aligned} \dot{\pi }= & {} \left( \frac{\kappa ^2}{2} \right) \left( \frac{D}{2}\right) \frac{ N }{ \sqrt{g}} \left( \pi _{mn} \pi ^{mn}-\widetilde{{\lambda }} \pi ^2 \right) + \nabla _m \left( N^m \pi \right) \nonumber \\&-N \sqrt{g} \left[ \left( \frac{ D}{2} \right) \mathcal{V}(R)-R \mathcal{V}'(R) \right] \nonumber \\&+\sqrt{g}(D-1) \nabla ^2 \left( N \mathcal{V}'(R) \right) \nonumber \\= & {} \frac{D}{2} N \mathcal{H} -N \sqrt{g} \left[ D \mathcal{V}(R)-R \mathcal{V}'(R) \right] \nonumber \\&+\sqrt{g}(D-1) \nabla ^2 \left( N \mathcal{V}'(R) \right) + \nabla _m \left( N^m \pi \right) . \end{aligned}$$

(A18)

Appendix B: More details of computations in Sect. 3 A

In this Appendix, we present some more details of the computations in Sect. 3 A.

First, we consider the variations of the smeared constraints, $\langle \eta \widetilde{\Omega } \rangle \equiv \int d^D x~ \eta \widetilde{\Omega }$ and $\langle \zeta \widetilde{\Xi } \rangle \equiv \int d^D x~ \zeta \widetilde{\Xi }$ with the smearing functions $\eta$ and $\zeta$,

$$\begin{aligned} {\delta }\langle \eta \widetilde{\Omega } \rangle= & {} \langle E^{mn} {\delta }g_{mn} +F_{mn} {\delta }\pi ^{mn}\rangle , \end{aligned}$$

(B1)

$$\begin{aligned} {\delta }\langle \zeta \widetilde{\Xi } \rangle= & {} \langle G^{mn} {\delta }g_{mn} +H_{mn} {\delta }\pi ^{mn}\rangle . \end{aligned}$$

(B2)

After quite tedious computations, we obtain the coefficients, E, F, G, H as follows²⁶:

$$\begin{aligned} E^{mn}\equiv & {} \frac{{\delta }\langle \eta \widetilde{\Omega } \rangle }{{\delta }g_{mn}} =-\left( \frac{\kappa ^2}{2}\right) 2 \widehat{\lambda } \mathcal{V}'(R) \left[ \pi ^{mn}\nabla _{i}(N^{2}\nabla ^{i}\eta ) \right. \nonumber \\&\left. -\frac{1}{2}g^{mn}\nabla _{i}(N^{2}\pi \nabla ^{i}\eta ) +N^{2}\nabla ^{(m}\pi \nabla ^{n)}\eta \right] \nonumber \\&- \left( \frac{{\kappa }^2}{2} \right) 2 \widehat{{\lambda }} \left\langle \eta \nabla _i \left[ N^2 \nabla ^i \pi \frac{{\delta }\mathcal{V}' (R)}{{\delta }g_{mn}} \right. \right. \nonumber \\&\left. + N^2 \left( \pi ^{ij}-\widehat{{\lambda }} g^{ij} \pi \right) \frac{{\delta }\nabla _j \mathcal{V}' (R) }{{\delta }g_{mn}} \right. \nonumber \\&\left. \left. -N^2 \left( {{\delta }^{ij}}_{mn} \pi +g^{ij} \pi ^{mn} \right) \nabla _j \mathcal{V}' (R) \right] \right\rangle , \end{aligned}$$

(B3)

$$\begin{aligned} F_{mn}\equiv & {} \frac{{\delta }\langle \eta \widetilde{\Omega } \rangle }{{\delta }\pi ^{mn}} =-\left( \frac{\kappa ^2}{2}\right) 2 \widehat{\lambda } \mathcal{V}'(R) \nabla _{i}\left[ g_{mn}N^{2}\nabla ^{i}\eta \right] \nonumber \\&- \left( \frac{{\kappa }^2}{2} \right) 2 \Big \langle \eta \nabla _i \left[ N^2 \left( {{\delta }^{ij}}_{mn} -\widehat{{\lambda }}g^{ij}g_{mn} \right) \nabla _j \mathcal{V}' (R) \right] \Big \rangle , \nonumber \\ \end{aligned}$$

(B4)

$$\begin{aligned} G^{mn}\equiv & {} \frac{{\delta }\langle \zeta \widetilde{\Xi } \rangle }{{\delta }g_{mn}} =\left( \frac{\kappa ^{2}}{2}\right) 2(D-1)\xi ^{2}\widehat{\lambda }\sqrt{g}\Bigg \{ \frac{1}{2}g^{mn}N^{2} \nonumber \\&\times \nabla _{i}\zeta (\nabla ^{i}\nabla ^{j}\nabla _{j}N-N\nabla ^{i}R-R\nabla ^{i}N)\nonumber \\&+N^{2}\Big [\nabla _{i}(NR^{mn})\nabla ^{i}\zeta -\nabla ^{(m}\nabla _{i}\nabla ^{i}N\,\nabla ^{n)}\zeta \nonumber \\&+\nabla ^{(m}(NR)\nabla ^{n)}\zeta -\nabla _{i}\nabla ^{m}\nabla ^{n}N\nabla ^{i}\zeta \Big ]\nonumber \\&+\frac{\Lambda D}{2\xi (D-1)}\Big [2N^2\nabla ^{(m}N\nabla ^{n )} \zeta -g^{mn}N^{2}\nabla ^{i}N\nabla _{i}\zeta \Big ] \nonumber \\&+\nabla _{i}\left[ N^{2}(X^{imnkl}{}_{l}\nabla _{j}\nabla _{k}N\,\nabla ^{j}\zeta \right. \nonumber \\&\left. +2 X^{imnkjl}\nabla _{l}\nabla _{k}N\,\nabla _{j}\zeta -NR^{mn}\nabla ^{i}\zeta )\right] \nonumber \\&+\nabla _{k}\nabla _{i}\left[ N^{2}(X^{kmnl}{}_{l}{}^{i} -X^{kmnil}{}_{l})\nabla _{j}N\nabla ^{j}\zeta \right. \nonumber \\&\left. -N^{2}X^{kmnjl}{}_{l}\nabla _{j}N\nabla ^{i}\zeta \right] \nonumber \\&+\nabla _{k}\nabla _{j}\nabla _{i}\Big [N^{3}\nabla ^{i}\zeta (X^{kmnjl}{}_{l}-X^{kmnl}{}_{l}{}^{j})\Big ]\Bigg \}\nonumber \\&+\left( \frac{\kappa ^2}{2}\right) ^2 \frac{\xi \widehat{\lambda }}{(\lambda D-1)\sqrt{g}}\Bigg \{(2\lambda +1)\left[ 2\pi ^{mn}\pi \right. \nonumber \\&\left. \times \nabla _{i}(N^{3}\nabla ^{i}\zeta )-g^{mn}\nabla _{i}(N^{3}\pi ^{2}\nabla ^{i}\zeta ) \right. \nonumber \\&\left. +g^{mn}N^{3}\pi \nabla ^{i}\pi \nabla _{i}\zeta +2N^{3}\pi \nabla ^{(m}\pi \nabla ^{n)}\zeta \right] \nonumber \\&+N^{2}\left[ g^{mn}\pi ^{2}\nabla _{i}N\nabla ^{i}\zeta +2\pi ^{2}\nabla ^{(m}N\nabla ^{n)}\zeta \right. \nonumber \\&\left. -4\pi ^{mn}\pi \nabla _{i}N\nabla ^{i}\zeta \right] +4(\lambda D-1) \nonumber \\&\times \left[ \left( \frac{1}{2}g^{mn}\pi ^{ij}\pi -\pi ^{mn}\pi ^{ij}\right) \nabla _{j}(N^{3}\nabla _{i}\zeta )\right] \Bigg \} \nonumber \\&+\text{(higher-derivative } \text{ contributions) }, \end{aligned}$$

(B5)

$$\begin{aligned} H_{mn}\equiv & {} \frac{{\delta }\langle \zeta \widetilde{\Xi } \rangle }{{\delta }\pi ^{mn}} =\left( \frac{\kappa ^{2}}{2}\right) ^{2}\frac{ 4\xi }{\sqrt{g}}\widehat{\lambda }\, \nonumber \\&\quad \Big \{N^{3}\nabla _{(m}\pi \nabla _{n)}\zeta -g_{mn}\nabla _{i}(N^{3}\pi ^{ij}\nabla _{j}\zeta ) \nonumber \\&\quad +\frac{N^{2}g_{mn}}{2(\lambda D-1)}\left[ (2\lambda +1)N\pi \nabla _{i}\nabla ^{i}\zeta \right. \nonumber \\&\left. \quad +(6\lambda +1)\pi \nabla _{i}N\nabla ^{i}\zeta \right] \Big \} \nonumber \\&\quad +\text{(higher-derivative } \text{ contributions) }, \end{aligned}$$

(B6)

where $f_{(ij)} \equiv (f_{ij}+f_{ji})/2$, $-\mathcal{V}'(R) \equiv \left( \xi +{\alpha }n R^{n-1} \right)$, and

$$\begin{aligned} -\frac{{\delta }\mathcal{V}' (R) (y)}{{\delta }g_{mn}(x)}= & {} {\alpha }n (n-1) R^{n-2} \frac{{\delta }R (y)}{{\delta }g_{mn}(x)} \nonumber \\= & {} -{\alpha }n (n-1) R^{n-2} \left( R^{mn} {\delta }^D (x-y) \right. \nonumber \\&\left. +A^{mnpq}{}_{kl}g^{kl}\nabla _p \nabla _q {\delta }^D (x-y) \right) \nonumber \\= & {} -{\alpha }n (n-1) R^{n-2} \left[ R^{mn} {\delta }^D (x-y) \right. \nonumber \\&\left. {-(\nabla ^{n}\nabla ^{m}-g^{mn}\nabla ^2)}{\delta }^D (x-y) \right] , \end{aligned}$$

(B7)

$$\begin{aligned} A^{mnpq}{}_{kl}\equiv & {} g^{pi} X^{qmn}{}_{ikl}-g^{ij} X^{qmn}{}_{ij(k}\delta ^{p}{}_{ l)}, \nonumber \\ X^{mij}{}_{nkl}\equiv & {} \frac{1}{2}\left[ g^{m}{}_{k}g^{(i}{}_{l}g^{j)}{}_{n} +g^{m}{}_{l}g^{(i}{}_{n}g^{j)}{}_{k} \right. \nonumber \\&\left. -g^{m}{}_{n}g^{(i}{}_{k}g^{j)}{}_{l} \right] . \end{aligned}$$

(B8)

Then, after some manipulations one can find the Poisson bracket algebras as follows:

$$\begin{aligned} \{ \langle \eta \widetilde{\Omega }\rangle ,\langle \zeta \widetilde{\Omega }\rangle \}= & {} \left( \frac{\kappa ^2}{2}\right) \widehat{\lambda } (D-2) \int d^D z \left[ \nabla _{i}\eta \nabla _{j}({N^2\nabla ^{j}\zeta }) \right. \nonumber \\&\left. -\nabla _{i}\zeta \nabla _{j}({N^2\nabla ^{j}\eta })\right] \mathcal{V}'(R) N^2 \widetilde{C}^i \nonumber \\&+\text{(higher-derivative } \text{ contributions) }, \nonumber \\ \{ \langle \eta \widetilde{\Omega }\rangle ,\langle \zeta \mathcal{H}\rangle \}\approx & {} -\int d^D z ~(\nabla _{i}\eta ) \zeta N ~{\widetilde{\Theta }}^i , \nonumber \\ \{ \langle \eta \widetilde{\Omega }\rangle ,\langle \zeta ^k \mathcal{H}_k\rangle \}= & {} -\int d^D z ~\zeta ^k \left[ (\nabla _{k}\eta ) ~\widetilde{{\Omega }}-(\nabla _k N^2) ~\widetilde{ C}^i \nabla _i \eta \right] , \nonumber \\ \{ \langle \eta \widetilde{\Xi }\rangle ,\langle \zeta \mathcal{H}\rangle \}\approx & {} -\int d^{D} z~ (\nabla _{i}\eta ) \zeta N ~ \widetilde{\Sigma }^i \nonumber \\&+\text{(higher-derivative } \text{ contributions) }, \nonumber \\ \{ \langle \eta \widetilde{\Xi }\rangle ,\langle \zeta ^k \mathcal{H}_k\rangle \}= & {} -\int d^D z ~ \zeta ^k \left[ (\nabla _{k}\eta ) \widetilde{\Xi }-(\nabla _k N^2)~\widetilde{\Theta }^i \nabla _i \eta \right. \nonumber \\&\left. - \nabla _k N \Big \langle N^2 \left( \frac{{\delta }\widetilde{\Theta }^i }{{\delta }N(z)} \right) \nabla _i \eta \Big \rangle \right] ,\nonumber \\&+\text{(higher-derivative } \text{ contributions) } \end{aligned}$$

(B9)

where,

$$\begin{aligned} \widetilde{\Sigma }^i= & {} \left( \frac{\kappa ^2}{2}\right) 2 (D-1) N \left[ \frac{(2\lambda +1) }{(\lambda D -1)} ~\pi ~\widetilde{\Theta }^{i} - 2\pi ^{i}{}_{k}~\widetilde{\Theta }^{k}\right] \nonumber \\&+\left( \frac{\kappa ^2}{2}\right) ^2 2\xi ^2 (D-1) \widehat{\lambda } \left[ \nabla ^{i} \mathcal {U}+\frac{ 2 \mathcal {V}^{i}}{(\lambda D-1)} \right] \nonumber \\&+\left( \frac{\kappa ^2}{2}\right) ^{3} \frac{8\xi \widehat{\lambda }}{\sqrt{g}} \mathcal {W}^{i} , \end{aligned}$$

(B10)

$$\begin{aligned} \mathcal {U}\equiv & {} \left( 2N\nabla _{j}\nabla _{k}N +\nabla _{k}N\nabla _{j}N \right) \left[ 2 \pi ^{jk}-\frac{(2\lambda -1)}{(\lambda D-1)}\pi \, g^{jk} \right] \nonumber \\&+ 2 N^{2}\left[ \frac{\lambda }{(\lambda D-1)}\pi R -\pi ^{jk}R_{jk} \right] ,\nonumber \\ \mathcal {V}^{i}\equiv & {} \pi \left( -N^{2} \nabla ^{i}R -\nabla ^{i}N \nabla _{j}\nabla ^{j}N+N \nabla ^{i}\nabla ^{j}\nabla _{j}N \right) \nonumber \\&+\nabla ^{i}\pi \left[ \frac{ \Lambda (D+2)}{2\xi (D-1)}N^{2} \right. \nonumber \\&\left. -\frac{(2\lambda -D-1)}{2(D-1)}\left( N^{2}R-N \nabla _{j}\nabla ^{j}N \right) \right] \nonumber \\&+ \frac{(\lambda D-1)}{(D-1)}\nabla _{j}\pi \left( N^{2}R^{ij}-N \nabla ^{j}\nabla ^{i}N \right) \nonumber \\&-(\lambda -1)\nabla ^{i}(N^{2}\nabla _{j}\nabla ^{j}\pi )-\frac{(6\lambda -5)}{2}\nabla ^{i}(\nabla ^{j}\pi N \nabla _{j}N),\nonumber \\ \mathcal {W}^{i}\equiv & {} - \frac{ (2 \lambda +1 ) ( D-2)}{2 ( \lambda D -1)^2}\pi ^{3} N\nabla ^{i}N \nonumber \\&- \frac{( 2 \lambda +1 ) (D (2 \lambda +1) -3 {\lambda }-2 )}{2 ( \lambda D-1)^2} N^{2}\pi ^2\nabla ^{i}\pi \nonumber \\&+\frac{(D-2)}{(\lambda D -1)}\pi ^{ij}\pi ^{2} N\nabla _{j}N-(2D -3)N^{2}\pi ^{ik}\pi _{jk} \nabla ^{j}\pi \nonumber \\&+ \frac{( 4D (2 \lambda +1) -12 {\lambda }-7)}{2 (\lambda D -1)}N^{2}\pi ^{ij}\pi \nabla _{j}\pi . \end{aligned}$$

(B11)

The Poisson algebra for the local constraints are given by

(B12)

Here, we have used

$$\begin{aligned} {}\{ \widetilde{\Omega }(x),\pi _N (y) \}= & {} \nabla _{i}^x \left[ 2 N \widetilde{C}^i (x) {\delta }^D (x-y) \right] , \end{aligned}$$

(B13)

$$\begin{aligned} {} \{ \widetilde{\Xi }(x),\pi _N (y) \}= & {} \nabla _{i}^x \left[ 2 N \widetilde{\Theta }^i (x) {\delta }^D (x-y) \right. \nonumber \\&\left. + N^2 (x) \left( \frac{{\delta }{\Theta }^i (x)}{{\delta }N(y)} \right) \right] , \end{aligned}$$

(B14)

where

$$\begin{aligned}&\left( \frac{{\delta }\widetilde{\Theta }^i (x)}{{\delta }N(y)} \right) \nonumber \\&\quad =- \left( \frac{\kappa ^{2}}{2}\right) ^2 \frac{2 \widehat{\lambda }}{(\lambda D-1) \sqrt{g}} \mathcal{V}' \left\{ \left[ (2\lambda +1 ) g^{ij}\pi \right. \right. \nonumber \\&\qquad \left. \left. -2(\lambda D-1)\pi ^{ij} \right] \nabla _{j}\pi (x) {\delta }^D (x-y) \right. \nonumber \\&\qquad \left. +\pi ^2 (x) \nabla ^i_x {\delta }^D (x-y) \right\} -\left( \frac{\kappa ^{2}}{2}\right) 2 \widehat{\lambda } (D-1)\nonumber \\&\qquad \times \sqrt{g} \mathcal{V}'\nabla ^{i}_x \left[ \left( \xi R+ \frac{{\Lambda }D}{D-1} + \frac{D-n}{D-1} \alpha R^{n} \right) {\delta }^D (x-y) \right. \nonumber \\&\left. \qquad +\nabla ^2 \left( \mathcal {V}^{\prime } {\delta }^D (x-y) \right) \right] \nonumber \\&\qquad - \left( \frac{{\kappa }^2}{2} \right) \left[ 2 \widehat{{\lambda }} \nabla _i \pi \{\mathcal{V}'(R)(x), \mathcal{H} (y)\} \right. \nonumber \\&\left. \qquad + \{2 (\pi ^{ij}-\widehat{{\lambda }} g^{ij} \pi ) \nabla _j \mathcal{V}'(R)(x), \mathcal{H} (y) \}\right] . \end{aligned}$$

(B15)

From (B9) or (B12), one can now compute

$$\begin{aligned} \{\widetilde{{\Omega }} , H_C \}&= \{\widetilde{{\Omega }} , \langle N \mathcal{H}+N_i \mathcal{H}^i \rangle \} \nonumber \\&\approx \nabla _i \left[ N^2 \widetilde{\Theta }^i -N_k (\nabla _k N^2) \widetilde{C}^i \right] +\nabla _k (N^k \widetilde{{\Omega }}), \end{aligned}$$

(B16)

$$\begin{aligned} \{\widetilde{\Xi }(x), H_C \}&=\{\widetilde{\Phi } (x), \langle N \mathcal{H}+N_i \mathcal{H}^i \rangle \} \nonumber \\&\approx \nabla _i \left[ N^2 ~\widetilde{\Sigma }^i -N_k (\nabla _k N^2) \widetilde{\Theta }^i \right] +\nabla _k (N^k \widetilde{\Xi }) \nonumber \\&\quad -\nabla _i^x \left[ N^2(x) \left\langle \left( \frac{{\delta }\widetilde{\Theta }^i (x)}{{\delta }N} \right) N^k \nabla _k N \right\rangle \right] \nonumber \\&\quad +\text{(higher-derivative } \text{ contributions) } , \end{aligned}$$

(B17)

which reduce to (33) and (36), respectively, for the case $\widetilde{C}^i = 0$.