Revisiting linear stability of black hole odd-parity perturbations
in Einstein-Aether gravity

Under a redefinition of the metric accompanied with the Aether field in the form $\tilde{g}_{\mu\nu}=g_{\mu\nu}+Bu_{\mu}u_{\nu}$, where $B$ is a constant, the structure of the action (1) is preserved with a change of the coupling constants $\tilde{c}_{1,2,3,4}$ in the transformed frame [69]. This redefinition stretches the metric tensor in the Aether direction by a factor $1-B$. On choosing $B=1-c_{I}^{2}$ for the Minkowski metric $g_{\mu\nu}=\eta_{\mu\nu}$, where the subscript $I$ is either $S,V,T$ with the squared propagation speeds $c_{I}^{2}$ given by Eqs. (10)-(12), it is possible to transform to a metric frame $\tilde{g}_{\mu\nu}$ in which one of the speeds is equivalent to 1 [32, 70].

One can perform a more general disformal transformation [71] of the form

where the conformal factor $\Omega$ and the disformal factor $B$ are constants, and the Aether field $u_{\mu}$ satisfies the unit-vector constraint (4). The corresponding inverse metric and determinant are

where we are assuming that $1-B>0$. The Aether field has a different (but constant) norm with respect to $\bar{g}^{\mu\nu}$ as

Hence, it makes sense to define

The first covariant derivative of the Aether field with respect to $\bar{g}_{\mu\nu}$ is given by [72]

where

The Einstein-Hilbert action transforms as [72]

On using these relations, in the absence of matter fields, Einstein-Aether theory for the combination ($g_{\mu\nu}$, $u_{\mu}$) with the constant parameters ($G_{\ae}$, $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$) is equivalent to Einstein-Aether theory for the combination ($\bar{g}_{\mu\nu}$, $\bar{u}_{\mu}$) with a different set of parameters ($\bar{G}_{\ae}$, $\bar{c}_{1}$, $\bar{c}_{2}$, $\bar{c}_{3}$, $\bar{c}_{4}$), where [69]

More explicitly, the coefficients ${\bar{c}}_{i}$ are related to $c_{i}$, as

We consider the Minkowski background characterized by the metric tensor $g_{\mu\nu}=\eta_{\mu\nu}$ and perform the disformal transformation (21). Upon choosing

and using Eqs. (10)-(15) and Eqs. (29)-(32), the squared propagation speeds $\bar{c}_{I}^{2}$ in the frame ($\bar{g}_{\mu\nu}$, $\bar{u}_{\mu}$) yield

for each subscript $I=S,V,T$. Furthermore, the coefficients of the time kinetic terms yield

where²²2For the vector perturbation, the no-ghost condition changes from $q_{V}>0$ to $Q_{V}>0$ if one swaps the roles of the dynamical variable and its canonical momentum by a canonical transformation. See e.g., Appendix B of [73] or/and Section IV of [74] for a technique to perform canonical transformations at the level of the Lagrangian.

Here, we have assumed

in order to avoid the gradient instability of perturbations. Under this assumption, the inequality $1-B>0$ holds and thus the disformal transformation does not change the Lorentzian signature of the metric.

If the background Aether field has zero vorticity, which is the case for any spherically symmetric configurations, one can locally choose the time coordinate $\phi$ such that

where $\eta$ is a nonvanishing function. This choice of the time coordinate $\phi$ for the metric frame $g_{\mu\nu}$ is called the Aether-orthogonal frame. The unit-vector constraint (4) gives $\left.g^{\phi\phi}\right|_{\rm background}=-\eta^{-2}$ and $\left.u^{\phi}\right|_{\rm background}=\eta^{-1}$.

At each point of physical interest, one can choose a local Lorentz frame and then perform a local Lorentz transformation so that the metric and Aether field are of the form

At leading order in the geometrical optics approximation, i.e., for modes whose wavelengths are much shorter than the time and length scales of the background, the background in the vicinity of the point of interest can be approximated by Eq. (39). In particular, in the vicinity of the point of interest, one can decompose the perturbations into spin-$0$ (scalar, $I=S$), spin-$1$ (vector, $I=V$) and spin-$2$ (tensor, $I=T$) modes. Let us consider the perturbations $\delta\chi_{I}$ corresponding to the $I$-excitation ($I=S,V,T$). Then, by definition of $Q_{I}$ and $c_{I}^{2}$, the kinetic and gradient terms of $\delta\chi_{I}$ should locally have the following structure

where $C_{I}$ are positive definite coefficients, which should not be confused with $C_{1},C_{2},\cdots$ introduced later in Sec. IV, and $\cdots$ represents higher-order terms in the geometrical optics approximation. By undoing the local Lorentz transformation and going back to the original coordinate system before choosing the local Lorentz frame, the leading kinetic and gradient terms (40) can be written in a general coordinate system as

at leading order in the geometrical optics approximation, where $\cdots$ again represents higher-order terms in the geometrical optics approximation. We have assumed that the time and spatial scales involved in the coordinate transformation between the original coordinate system and the local Lorentz frame in the vicinity of the point of interest are sufficiently longer than the wavelengths of the modes of interest.

The local structure (41) clearly states that the perturbations $\delta\chi_{I}$ in the Aether-orthogonal frame do not behave as ghosts so long as $Q_{I}>0$. Indeed, if we choose the time coordinate $\phi$ in the Aether-orthogonal frame, then the leading time kinetic term is

which is positive.

Rigorously speaking, we have only given a heuristic argument for the local structure (41) without a proof, which requires analysis similar to that in Ref. [75]. In the rest of the present paper, we shall show that the kinetic and gradient terms for odd-parity perturbations (including $I=V$ and $I=T$ modes) around spherically symmetric BHs indeed have the local structure (41). The same analysis for even-parity perturbations (including $I=S,V,T$ modes) will be left for a future work.

Let us consider the SSS background given by the line element

together with the Aether-field configuration

where $f,h,a,b$ are functions of $r$. In Eq. (43) the angular part contains two angles $\theta$ and $\varphi$, such that $\Omega_{pq}{\rm d}\vartheta^{p}{\rm d}\vartheta^{q}={\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2}$.

The unit-vector constraint (4) gives the following relation

where $\epsilon=\pm 1$. The existence of the Aether-field profile (45) with $b\neq 0$ requires that $(a^{2}f-1)h>0$.

A metric (Killing) horizon is defined at the radius $r=r_{\rm g}$ at which the time translation Killing vector $\zeta^{\mu}=\delta^{\mu}_{t}$ is null, i.e., $\zeta^{\mu}\zeta_{\mu}=0$. This translates to the condition

So long as $h$ also vanishes on the metric horizon, the product $a^{2}fh$ is at least a positive constant at $r=r_{\rm g}$ to satisfy the condition $(a^{2}f-1)h>0$. This means that, as $r\to r_{\rm g}^{+}$, the temporal Aether-field component diverges as $a^{2}\propto(fh)^{-1}$ for the coordinate system (43).

The leading-order kinetic and gradient terms of perturbations $\delta\chi_{I}$ (where $I=S,V,T$) for the effective metric $\bar{g}^{I}_{\mu\nu}=\Omega_{I}^{2}[g_{\mu\nu}+(1-c_{I}^{2})u_{\mu}u_{\nu}]$ (under the geometric optics approximation) are

where

Therefore, the coefficient $Q_{I}^{\rm apparent}$ of the kinetic term with respect to the Killing time $t$ may become negative, despite the fact that the kinetic term with respect to the time coordinate in the Aether-orthogonal frame is always positive as in Eq. (42). Also, the apparent angular sound speed squared $(c_{I,\Omega}^{\rm apparent})^{2}$ would have nontrivial position dependence, although the propagation speed squared $c_{I}^{2}$ relative to the Aether-orthogonal frame is a constant given by the theory.

These behaviors are due to the deviation of the Killing time slicing from the Aether-orthogonal frame. The sound cones are not only narrowed or widened but also tilted relative to the Killing time slicing³³3See Ref. [76] for similar behaviors of sound cones for open string modes in the context of inhomogeneous tachyon condensation in string theory.. On the other hand, the sound cones are not tilted relative to the Aether-orthogonal frame.

Near the metric horizon $r=r_{\rm g}$, we have the following expansions

where $f_{1}$ and $\alpha_{0}$ are constants [62]. Note that $\alpha_{0}$ is the value of $\alpha$ at $r=r_{\rm g}$, where $\alpha$ is defined later in Eq. (56). In this regime, the quantities (48) and (49) can be estimated as⁴⁴4For $I=V$ and $c_{13}=0$, the result is to be compared with (5.32) and (5.33) of Ref. [62] up to an arbitrary positive overall factor for $q_{V}^{\rm apparent}$.

Therefore, if $c_{I}^{2}>1$, the apparent no-ghost condition is violated near the horizon. Moreover, the apparent angular sound speed squared vanishes at the horizon. Let us stress again that the kinetic term with respect to the time coordinate in the Aether-orthogonal frame is always positive for $Q_{I}>0$ and that the propagation speed squared $c_{I}^{2}$ relative to the Aether-orthogonal frame is constant given by the theory. In Sec. IV, we will address the linear stability of BHs in Einstein-Aether theory by choosing the Aether-orthogonal frame as a timelike coordinate.

The SSS background is described by the line element (43) with the Aether-field profile (44). Alternatively, we can choose the following Eddington-Finkelstein coordinate [48]

which is related to the coordinate (43) as

The transformation to the coordinate (53) is valid in the regime $f/h>0$, i.e., for the same signs of $f$ and $h$. On this background, we take the Aether-field configuration

The $r$-dependent functions $\alpha$ and $\beta$ are related to $a$ and $b$ in Eq. (44), as

From Eq. (45), we obtain the following relations

The sign of $b$ depends on that of $(1-f\alpha^{2})/(f\alpha)$. Then, the Aether field $u^{\mu}$ has nonvanishing components

Even though $a$ is divergent on the metric horizon ($r=r_{\rm g}$), the quantity $\alpha$ can take a finite value $\alpha_{0}$ due to the relation $fa=-1/(2\alpha_{0})$ at $r=r_{\rm g}$, see Eq. (50). On using the metric components $g_{vv}=-f$, $g_{vr}=g_{rv}=B$, and $g_{rr}=0$, the nonvanishing components of $u_{\mu}$ are $u_{v}=f\alpha-B\beta$ and $u_{r}=-B\alpha$, so that

A vector field $s_{\mu}$ that is orthogonal to $u^{\mu}$ obeys the relation $s_{\mu}u^{\mu}=0$. This has the following nonvanishing components

Moreover, it satisfies the relation $s_{\mu}s^{\mu}=1$.

Now, we introduce the two coordinates $\phi$ and $\psi$, as

which mean that $\phi$ is constant on a hypersurface orthogonal to $u_{\mu}$ and that $\psi$ is constant on a hypersurface orthogonal to $s_{\mu}$. Note that $s_{v}$ goes to 0 as $r\to\infty$ and hence Eq. (62) is valid for a finite distance $r$. Since we already know the linear stability conditions in the asymptotically flat regime [18, 21], it is sufficient to focus on the stability in the region with finite $r$. If we introduce the coordinate $\tilde{\psi}$ as ${\rm d}\tilde{\psi}=-s_{v}{\rm d}v-s_{r}{\rm d}r$ instead of $\psi$ to avoid the divergence of $s_{r}/s_{v}$ at spatial infinity, then $\tilde{\psi}$ is not integrable due to the $r$ dependence in $s_{v}$. In this sense we choose the coordinate system $(\phi,\psi)$, which satisfies the integrability condition.

Since $\partial_{\mu}\phi=\delta^{v}_{\mu}-2\alpha^{2}/(1+f\alpha^{2})\sqrt{f/h}\,\delta^{r}_{u}$ from Eq. (61), the background Aether field can be expressed in the form

and hence $\eta=-(1+f\alpha^{2})/(2\alpha)=fa$ in Eq. (38). Thus, the coordinate system $(\phi,\psi)$ corresponds to the Aether-orthogonal frame in which $\phi$ is the time measured by observers comoving with the Aether field.

Substituting the first of Eq. (54) into Eqs. (61) and (62), the relation between the Aether-orthogonal frame and the coordinate (43) is given by [38]

Solving these equations for ${\rm d}t$ and ${\rm d}r$ and substituting them into Eq. (43), the line element is expressed as

Since $g_{\phi\phi}$ is always non-positive, $\phi$ is a timelike coordinate. Similarly, $\psi$ is a spacelike coordinate due to the positivity of $g_{\psi\psi}$. Note that a universal horizon is defined as the radius $r_{\rm UH}$ at which the time translation Killing vector $\zeta^{\mu}$ is orthogonal to $u_{\mu}$, i.e., $\zeta^{\mu}u_{\mu}=0$. This corresponds to the radius satisfying

at which the metric component $g_{\phi\phi}$ in Eq. (66) is vanishing. It is clear that Eq. (67) has solutions only when $f<0$, that is, the universal horizon always exists inside the metric horizon. For more details, see Ref. [38].

From Eqs. (64) and (65), we have

Then, for a given function ${\cal F}$ of $t$ and $r$, the $t$ and $r$ derivatives of ${\cal F}$ are given, respectively, by

where ${\cal F}_{,\phi}:=\partial{\cal F}/\partial\phi$ and ${\cal F}_{,\psi}:=\partial{\cal F}/\partial\psi$. The relations (70) and (71) will be used in Sec. IV.

On the SSS background (43) with $\Omega_{pq}{\rm d}\vartheta^{p}{\rm d}\vartheta^{q}={\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2}$, metric perturbations $h_{\mu\nu}$ can be separated into odd-parity (axial) and even-parity (polar) sectors [52, 53]. We express $h_{\mu\nu}$ in terms of the spherical harmonics $Y_{lm}(\theta,\varphi)$. We will focus on axial perturbations with the parity $(-1)^{l+1}$. We choose the gauge in which the components $h_{ab}$ vanish, where the subscripts $a$ and $b$ denote either $\theta$ or $\varphi$. Then, the nonvanishing components of odd-parity metric perturbations are given by

where $Q_{lm}$ and $W_{lm}$ are functions of $t$ and $r$. The tensor $E_{ab}$ is antisymmetric with nonvanishing components $E_{\theta\varphi}=-E_{\varphi\theta}=\sin\theta$.

In the odd-parity sector, the Aether field has the following components

where $\delta u_{lm}$ is a function of $t$ and $r$.

In the following, we will set $m=0$ without loss of generality. We also omit the subscripts $l$ and $m$ from the perturbations $Q_{lm}$, $W_{lm}$, and $\delta u_{lm}$. Expanding the action (1) up to quadratic order and integrating it with respect to $\theta$ and $\varphi$, we obtain the second-order action containing the fields $Q$, $W$, $\delta u$ and their $t,r$ derivatives. The dynamical field associated with the gravitational (tensor) perturbation is given by [62]