Asymptotic flatness and nonflat solutions in the critical $2+1$ Hořava theory

Abstract

The Hořava theory in $2+1$ dimensions can be formulated at a critical point in the space of coupling constants where it has no local degrees of freedom. This suggests that this critical case could share many features with $2+1$ general relativity, in particular its large-distance effective action that is of second order in derivatives. To deepen on this relationship, we study the asymptotically flat solutions of the effective action. We take the general definition of asymptotic flatness from $2+1$ general relativity, where an asymptotically flat region with a nonfixed conical angle is approached. We show that a class of regular asymptotically flat solutions are totally flat. The class is characterized by having nonnegative energy (when the coupling constant of the Ricci scalar is positive). We present a detailed canonical analysis on the effective action showing that the dynamics of the theory forbids local degrees of freedom. Another similarity with $2+1$ general relativity is the absence of a Newtonian force. In contrast to these results, we find evidence against the similarity with $2+1$ general relativity: we find an exact nonflat solution of the same effective theory. This solution is out of the set of asymptotically flat solutions.

A Newtonian force

We may ask whether there is a place for a Newtonian force in this theory. The computations can be taken from Ref. [14], since the solution is the same for the critical $\lambda = 1/2$ and noncritical $\lambda \ne 1/2$ cases. Here we summarize the solution with the aim of showing explicitly the coupling between the critical Hořava gravity and the particle and how the Newtonian potential is lacked.

We couple the gravitational theory to a massive particle at rest. We assume that the dynamics of the particle is governed by an action of relativistic nature. Hence we assume that the action of the particle is proportional to the lenght of its worldline, embedded in a spacetime ambient. The ambient is taken from a solution of the Hořava theory. We choose the time coordinate t of the ambient foliation to parameterize the worldline of the particle. The mechanics of the particle is characterized by the embedding fields $q^0 = q^0(t)$ and $q^i = q^i(t)$, which define the position of the particle in the foliation. Thus, the combined system critical $2+1$ Hořava gravity–point particle is given by the action

$$\begin{aligned} S = \frac{1}{2\kappa } \int dt d^2x \sqrt{g} N \left( K_{ij}K^{ij} - \frac{1}{2} K^2 + \beta R + \alpha a_k a^k \right) - m \int dt \sqrt{ L } \,, \qquad \end{aligned}$$

(A.1)

where

$$\begin{aligned} L = (N^{2} - N_{k}N^{k}) \left( {\dot{q}}^0 \right) ^2 - 2 N_k {\dot{q}}^0 {\dot{q}}^k - g_{kl} {\dot{q}}^k {\dot{q}}^l \,, \end{aligned}$$

(A.2)

m is the mass of the particle and $\kappa$ is a coupling constant. L is the squared line element of the particle evaluated on the background of the ADM variables, and these variables are evaluated at the position of the particle in L. The field equations of the ADM variables are obtained by varying the action (A.1) with respect to them. We shown the resulting equations directly evaluated on the gauge $N_i = 0$,

$$\begin{aligned}&K^{ij}K_{ij} - \frac{1}{2} K^{2} + \beta R + \alpha a_i a^i - 2 \alpha \frac{\nabla ^{2} N}{N} = 2 \kappa m \frac{ ({\dot{q}}^0)^2 N }{ \sqrt{gL} } \delta ^{(d)} (x^k - q^k) \,, \qquad \end{aligned}$$

(A.3)

$$\begin{aligned}&G^{ijkl} \nabla _{j} K_{kl} = - \frac{ \kappa m }{ \sqrt{gL} } {\dot{q}}^0 {\dot{q}}^i \delta ^{(d)} (x^m - q^m) \,, \end{aligned}$$

(A.4)

$$\begin{aligned}&\frac{1}{\sqrt{g}} \frac{\partial }{\partial t} \left( \sqrt{g} G^{ijkl} K_{kl} \right) + 2 N (K^i{}_k K^{jk} - \frac{1}{2} KK^{ij}) - \frac{1}{2} N g^{ij} G^{klmn} K_{kl} K_{mn} \nonumber \\&\qquad - \beta \left( \nabla ^{ij} N - g^{ij} \nabla ^{2}N \right) + \frac{\alpha }{N} \left( \nabla ^i N \nabla ^j N - \frac{1}{2} g^{ij} \nabla _{k} N \nabla ^{k} N \right) \nonumber \\&\quad = \frac{ \kappa m}{ \sqrt{gL}} {\dot{q}}^{i} {\dot{q}}^{j} \delta ^{(d)} (x^m - q^m) \,. \end{aligned}$$

(A.5)

In the $N_i = 0$ gauge we have that $K_{ij} = {\dot{g}}_{ij}/2N$. The equations of motion corresponding to the variations of the coordinates of the particle are

$$\begin{aligned} {q}^i{}'' + \Gamma ^{i}_{kl} {q}^k{}' {q}^l{}' + \frac{1}{2} \partial ^i N^2 ( {q}^0{}' )^2= & {} 0 \,, \end{aligned}$$

(A.6)

$$\begin{aligned} \frac{2}{\sqrt{L}} \frac{d}{d t} N^{2} {q}^{0}{}' + \partial _0 N^2 ({q}^0{}')^2 - \partial _{0} g_{ij} {q}^{i}{}' {q}^{j}{}'= & {} 0 \,, \end{aligned}$$

(A.7)

where the prime means

$$\begin{aligned} {\psi }' \equiv \frac{1}{\sqrt{L}}\frac{\partial \psi }{\partial t} \,. \end{aligned}$$

(A.8)

In the equations of motion (A.3)–(A.7) we consider that the particle and its gravitational field (the N and $g_{ij}$ fields) are static. A suitable choice for the location of the particle is $q^0 = t$, $q^i = 0$, which means that the particle remains at the origin. Under these settings, Eq. (A.5) acquires exactly the same form displayed in (3.18) for the vacuum theory. Its trace gives again the condition of harmonicity on N (3.19). Imposing the asymptotic condition (2.5)–(2.6), which is appropiated for the Newtonian case, we have again that the only solution is $N = 1$ everywhere. Since a nonconstant $-N^2$ would be the analogue of the Newtonian potential, we have that there is no Newtonian force in this theory. Hence, the situation is the same as in $2+1$ general relativity. Indeed, the rest of the analysis for solving the remaining field equations is parallel to general relativity. Equation (A.4) is automatically solved. Particle’s equations of motion, given in Eqs. (A.6)–(A.7), are automatically solved considering that $N=1$. The remaining is the analogue of the time-time component of the Einstein equations, Eq. (A.3). It takes the form

$$\begin{aligned} \sqrt{g} R = \frac{2 \kappa m }{\beta } \delta ^{(2)}(x^i) \,. \end{aligned}$$

(A.9)

This equation was solved in [13]. Its solution, in polar coordinates, is

$$\begin{aligned} ds^{2} = r^{-\frac{ \kappa m}{\pi \beta }} ( dr^2 + r^2 d\theta ^2 ) \,. \end{aligned}$$

(A.10)

The generic geometry is a flat cone (other geometries are possible in the space of parameters [12, 13]).