Tidal response and near-horizon boundary conditions for spinning exotic compact objects

Starting from the Boyer-Lindquist coordinate system $(t,r,\theta,\phi$), FIDOs in the MP are characterized by constant $r$ and $\theta$, but $\phi={\rm const}+\omega_{\phi}t$, with

$$\omega_{\phi}=\frac{2ar}{\Xi}\,.$$

(5)

and

$$\Xi=(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta\,.$$

(6)

Each FIDO carries an orthonormal tetrad of ¹¹1Note that MP uses different notations for the $\rho$, $\Sigma$.

	$\displaystyle\vec{e}_{\hat{r}}$	$\displaystyle=\sqrt{\frac{\Delta}{\Sigma}}\vec{\partial}_{r}\,,\;$	$\displaystyle\vec{e}_{\hat{\theta}}$	$\displaystyle=\frac{\vec{\partial}_{\theta}}{\sqrt{\Sigma}}\,,\;$		(7)
	$\displaystyle\vec{e}_{\hat{\phi}}$	$\displaystyle=\sqrt{\frac{\Sigma}{\Xi}}\frac{\vec{\partial}_{\phi}}{\sin\theta}\,,\;$	$\displaystyle\vec{e}_{\hat{0}}$	$\displaystyle=\frac{1}{\alpha}\left(\vec{\partial}_{t}+\omega_{\phi}\vec{\partial}_{\phi}\right)\,,$

with

$$\alpha=\sqrt{\frac{\Sigma\Delta}{\Xi}}\,.$$

(8)

Here $\vec{e}_{\hat{0}}$ is the four-velocity of the FIDO. The FIDOs have zero angular momentum (hence are also known as Zero Angular-Momentum Observers, or ZAMOs), since $\vec{e}_{\hat{0}}$ has zero inner product with $\vec{\partial}_{\phi}$. Here $\alpha$ is called the redshift factor, also known as the lapse function, since it relates the proper time of the FIDOs and the coordinate time $t$.

Near the horizon, we have $\alpha\rightarrow 0$; FIDO’s tetrads are related to the Kinnersly tetrad Kinnersley (1969) via

$\displaystyle\vec{l}\approx\sqrt{\frac{\Sigma}{\Delta}}(\vec{e}_{\hat{0}}+\vec{e}_{\hat{r}})\,,\vec{n}\approx\sqrt{\frac{\Delta}{\Sigma}}\frac{\vec{e}_{\hat{0}}-\vec{e}_{\hat{r}}}{2}\,,\vec{m}\approx\frac{e^{i\beta}(\vec{e}_{\hat{\theta}}+i\vec{e}_{\hat{\phi}})}{\sqrt{2}}\,,$

(9)

where

$$\beta=-\tan^{-1}\left(\frac{a\cos\theta}{r_{H}}\right)\,.$$

(10)

Let us now introduce the electric-type tidal tensor $\mathcal{E}$ as viewed by FIDOs, which can be formally defined as Zhang et al. (2012)

$$\mathcal{E}_{ij}={h_{i}}^{a}{h_{j}}^{c}C_{abcd}U^{b}U^{d}\,.$$

(11)

Here $U=\vec{e}_{\hat{0}}$ is the four-velocity of FIDOs as in Eq. (7), and ${h_{i}}^{a}={\delta_{i}}^{a}+U_{i}U^{a}$ is the projection operator onto the spatial hypersurface orthogonal to $U$. In particular, we look at the $mm$-component of the tidal tensor, as the gravitational-wave stretching and squeezing will be along these directions. Near the horizon, the tidal tensor component is then given by

$$\mathcal{E}_{mm}=C_{\hat{0}m\hat{0}m}\approx-\frac{\Delta}{4\Sigma}\psi_{0}-\frac{\Sigma}{\Delta}\psi_{4}^{*}\,.$$

(12)

For convenience, let us define a new variable ${}_{s}\Upsilon$ which is the solution to the Teukolsky equation with spin weight $s$. For $s=\pm 2$ we have

$\displaystyle{}_{-2}\Upsilon\equiv\rho^{-4}\psi_{4}\,,$

$\displaystyle{}_{+2}\Upsilon\equiv\psi_{0}$

$\displaystyle\,.$

(13)

We briefly review the Teukolsky formalism in Appendix. A. For perturbations that satisfy the linearized vacuum Einstein’s equation (in this case the Teukolsky equation), at $r_{*}\rightarrow-\infty$, in general we can decompose ${}_{s}\Upsilon$ using the spin-weighted spheroidal harmonics ${}_{s}S_{\ell m\omega}(\theta)$, and write

	$\displaystyle{}_{-2}\Upsilon(t,r_{*},\theta,\phi)=$	$\displaystyle\sum_{\ell m}\int\frac{d\omega}{2\pi}e^{-i\omega t}\,{}_{-2}S_{\ell m\omega}(\theta)\,e^{im\phi}$		(14)
		$\displaystyle\times\left[Z^{\rm hole}_{\ell m\omega}\Delta^{2}e^{-ikr_{*}}+Z^{\rm refl}_{\ell m\omega}\,e^{ikr_{*}}\right]\,,$
	$\displaystyle{}_{+2}\Upsilon(t,r_{*},\theta,\phi)=$	$\displaystyle\sum_{\ell m}\int\frac{d\omega}{2\pi}e^{-i\omega t}\,{}_{+2}S_{\ell m\omega}(\theta)\,e^{im\phi}$		(15)
		$\displaystyle\times\left[Y^{\rm hole}_{\ell m\omega}\,\Delta^{-2}e^{-ikr_{*}}+Y^{\rm refl}_{\ell m\omega}\,e^{ikr_{*}}\right]\,,$

where $k\equiv\omega\,-\,m\,\Omega_{H}$. We use the shorthand $\sum_{\ell m}\equiv\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}$, in which $\ell$ is the multipolar index, and $m$ is the azimuthal quantum number. Note this $m$ here should not be confused with the label $m$ in the Kinnersly tetrad basis. Here $Z_{\ell m\omega}$ and $Y_{\ell m\omega}$ are amplitudes for the radial modes, with “hole” labeling the left-propagation modes into the compact object (in this paper, left means toward direction with decreasing $r$)  ²²2Of course here we refer to ECOs instead of black holes, the label “hole” is for matching the notations from Teukolsky and Press (1974). and “refl” labeling the right-propagation (reflected) modes (in this paper, “right” means toward direction with increasing $r$).

Note that for outgoing modes of either ${}_{+2}\Upsilon$ or ${}_{-2}\Upsilon$, we have

$$e^{-i\omega t}\,e^{ikr_{*}}\,e^{im\phi}=e^{-i\omega(t-r_{*})}\,e^{im(\phi-\Omega_{H}r_{*})}\,,$$

(16)

therefore the outgoing modes are functions of the retarded time $u=t-r_{*}$, and the position-dependent angular coordinate $\phi-\Omega_{H}r_{*}$. Similarly for ingoing modes, we have

$$e^{-i\omega t}\,e^{-ikr_{*}}\,e^{im\phi}=e^{-i\omega(t+r_{*})}\,e^{im(\phi+\Omega_{H}r_{*})}\,,$$

(17)

indicating that the ingoing modes are functions of the advanced time $v=t+r_{*}$, and another position-dependent angular coordinate $\phi+\Omega_{H}r_{*}$. We can then write down one schematic expression for either ${}_{+2}\Upsilon$ or ${}_{-2}\Upsilon$, by decomposing both of them into left- and right- propagation components:

	$\displaystyle{}_{+2}\Upsilon(t,r_{*},\theta,\phi)=$	$\displaystyle{}_{+2}\Upsilon^{R}(u,\theta,\varphi_{-})+\frac{1}{\Delta^{2}}{}_{+2}\Upsilon^{L}(v,\theta,\varphi_{+})\,,$		(18)
	$\displaystyle{}_{-2}\Upsilon(t,r_{*},\theta,\phi)=$	$\displaystyle{}_{-2}\Upsilon^{R}(u,\theta,\varphi_{-})+\Delta^{2}{}_{-2}\Upsilon^{L}(v,\theta,\varphi_{+})\,,$		(19)

where we have defined

$\displaystyle\varphi_{-}$

$\displaystyle=\phi-\Omega_{H}r_{*}\,,$

$\displaystyle\varphi_{+}$

$\displaystyle=\phi+\Omega_{H}r_{*}\,.$

(20)

Here both the $L$ and $R$ components are finite, and the $\Delta$ represents the divergence/convergence behaviors of the components. As we can see here, once we specify these $L$, and $R$ components on a constant $t$ slice, as functions of $(r_{*},\theta,\phi)$, we will be able to obtain their future, or past, values by inserting $t$.

Here we also note that, while the vacuum/homogeneous perturbation of space-time geometry are encoded in both $\psi_{0}$ and $\psi_{4}$ — either of them suffices to describe the perturbation field Starobinskil and Churilov (1974); Teukolsky and Press (1974) ³³3One may imagine a very rough Electromagnetic analogy: for a vacuum EM wave (without electro- or magneto-static fields), both $E$ and $B$ fields contain the full information of the wave, since one can use Maxwell equations to convert one to the other. Nevertheless, when it comes to interacting with charges and currents, $E$ and $B$ play very different roles, and sometimes it is important to evaluate both $E$ and $B$ fields. . Near the horizon, the numerical value of $\psi_{0}$ is dominated by left-propagating waves, while the numerical value of $\psi_{4}$ is dominated by right-propagating waves. According to Eq. (12), we then have

$$\mathcal{E}_{mm}\approx-\frac{1}{4\Sigma\Delta}{}_{+2}\Upsilon^{L}(v,\theta,\varphi_{+})-\frac{{\rho^{*}}^{4}\Sigma}{\Delta}\left[{{}_{-2}\Upsilon^{R}}(u,\theta,\varphi_{-})\right]^{*}\,.$$

(21)

Note that both terms diverge toward $r_{*}\rightarrow-\infty$ and at the same order. This divergence correctly reveals the fact that the FIDOs will observe gravitational waves with the same fractional metric perturbation, but because the frequency of the wave gets increased, the curvature perturbation will diverge as $\alpha^{-2}$.

Now, suppose we have a surface, $\mathcal{S}$ at a constant radius $r_{*}=b_{*}$ (or in the Boyer-Lindquist coordinates $r=b$), with $e^{\kappa b_{*}}\ll 1$. Here $\kappa=(r_{H}-r_{C})/2(r^{2}_{H}+a^{2})$ is the surface gravity of the Kerr black hole. To the right of the surface, for $r_{*}>b_{*}$, we have completely vacuum, and to the left of the surface, we have matter that are relatively at rest in the FIDO frame, we shall refer to this as the ECO region. The ECO is assumed to be extremely compact and $\mathcal{S}$ is close to the position, as viewed as part of its external Kerr spacetime.

For the moment, let us assume that linear perturbation theory holds throughout the external Kerr spacetime of the ECO. On $\mathcal{S}$ and to its right, $\mathcal{E}_{mm}$ will be the sum of two pieces,

$$\mathcal{E}_{mm}=\mathcal{E}^{\rm src}_{mm}+\mathcal{E}^{\rm resp}_{mm}\,,$$

(22)

with the first term

$$\mathcal{E}^{\rm src}_{mm}=-\frac{\Delta}{4\Sigma}{}_{+2}\Upsilon^{\rm src}(v,\theta,\varphi_{+})$$

(23)

a purely left-propagating wave that is sourced by processes away from the surface, e.g., an orbiting or a plunging a particle. The second term can be written as

$\displaystyle\mathcal{E}^{\rm resp}_{mm}=-\frac{{\rho^{*}}^{4}\Sigma}{\Delta}\left[{}_{-2}\Upsilon^{\rm refl}(u,\theta,\varphi_{-})\right]^{*}\,,$

(24)

as the ECO’s response to the incoming gravitational wave.

Now we are prepared to discuss the reflecting boundary condition of the Teukolsky equations in terms of the tidal response of the ECO. According to the linear response theory, we can assume the linear tidal response of the ECO is proportional to the total tidal fields near the surface of the ECO. That is, we may introduce a new parameter $\eta$, and write

$$\mathcal{E}^{\rm resp}_{mm}=\eta(b,\theta)\,\mathcal{E}_{mm}\,.$$

(25)

Here $\eta$ is analogous to the tidal love number. This leads to the following relation at $r_{*}=b_{*}$:

$\displaystyle\frac{\left[{}_{-2}\Upsilon^{\rm refl}(t-b_{*},\theta,\phi-\Omega_{H}b_{*})\right]^{*}}{{}_{+2}\Upsilon^{\rm src}(t+b_{*},\theta,\phi+\Omega_{H}b_{*})}=\frac{\eta}{1-\eta}\frac{e^{-4i\beta}}{4}\Delta^{2}\,.$

(26)

In particular, when $\eta\rightarrow\infty$, we will have the Dirichlet boundary condition,

$$\frac{\left[{}_{-2}\Upsilon^{\rm refl}(t-b_{*},\theta,\phi-\Omega_{H}b_{*})\right]^{*}}{{}_{+2}\Upsilon^{\rm src}(t+b_{*},\theta,\phi+\Omega_{H}b_{*})}=-\frac{e^{-4i\beta}}{4}\Delta^{2}\,.$$

(27)

This then provides us with a prescription for obtaining the boundary condition at $r_{*}=b_{*}$. Once we know the left-propagating $\psi_{0}^{\rm src}$, the reflected waves due to the tidal response are simply given by Eq. (26).

Let us now define a new parameter $\mathcal{R}(b,\theta)$ as

$$\mathcal{R}(b,\theta)\equiv-\frac{\eta}{1-\eta}\,.$$

(28)

This parameter has the physical meaning of being the reflectivity of the tidal fields on the ECO surface. This local response assumes that different angular elements of the ECO act independently, which is reasonable since on the ECO surface, and in the FIDO frame, the gravitational wavelength is blue shifted by $\alpha$, hence, much less than the radius of the ECO.

However, the response may also depend on the history of the exerted tidal perturbation, and to account for this we can write a more general boundary condition on the ECO surface as

$\displaystyle\!\!\left[{}_{-2}\Upsilon^{\rm refl}(t-b_{*},\theta,\phi-\Omega_{H}b_{*})\right]^{*}=-\frac{e^{-4i\beta}}{4}\int_{-\infty}^{t}dt^{\prime}\mathcal{R}(b,\theta;t-t^{\prime})\Delta^{2}{}_{+2}\Upsilon^{\rm src}(t^{\prime}+b_{*},\theta,\phi+\Omega_{H}b_{*}-\Omega_{H}(t-t^{\prime}))\,.$

(29)

Here the $-\Omega_{H}(t-t^{\prime})$ term has been inserted into the argument of ${}_{+2}\Upsilon^{\rm src}$ because the FIDO follows $\phi=\phi_{0}+\Omega_{H}t$ (see Fig. 1). This is the key equation of our reflection model.

We now have obtained the modified boundary condition (29) in terms of the Newman-Penrose quantities, and are ready to apply it to the Teukolsky formalism. The solutions to the $s=-2$ Teukolsky equation, ${}_{-2}\Upsilon$, admits the near-horizon decomposition as in Eq. (14). In this equation, $Z^{\rm hole}$ is the amplitude of the ingoing wave down to the ECO, which is contributed by the source, and $Z^{\rm refl}$ is the amplitude of the reflected wave due to the tidal response. For $s=+2$, the corresponding amplitudes are $Y^{\rm hole}$ and $Y^{\rm refl}$. We would like to derive a relation among the four amplitudes.

Near the ECO surface $\mathcal{S}$, ${}_{+2}\Upsilon^{\rm src}$ is given by

$\displaystyle{}_{+2}\Upsilon^{\rm src}(v,\theta,\varphi_{+})$

$\displaystyle=\sum_{\ell m}\int\frac{d\omega}{2\pi}e^{-i\omega v}Y^{\rm hole}_{\ell m\omega}\,\Delta^{-2}\,{}_{+2}S_{\ell m\omega}(\theta,\varphi_{+})\,,$

(30)

where we have kept only the dominant piece—the left-propagating mode, and $Y^{\rm hole}_{\ell m\omega}$ is the amplitude of that mode. The quantity ${}_{-2}\Upsilon^{\rm refl}$ is given by

$\displaystyle{}_{-2}\Upsilon^{\rm refl}(u,\theta,\varphi_{-})$

$\displaystyle=\sum_{\ell m}\int\frac{d\omega}{2\pi}e^{-i\omega u}Z^{\rm refl}_{\ell m\omega}\,{}_{-2}S_{\ell m\omega}(\theta,\varphi_{-})\,,$

(31)

Inserting the above two equations into Eq. (29), we obtain

		$\displaystyle\sum_{\ell}Z_{\ell m\omega}^{\rm refl}\,{}_{-2}S_{\ell m\omega}(\theta,\phi)$
	$\displaystyle=$	$\displaystyle\frac{1}{4}\sum_{\ell^{\prime}}e^{4i\beta-2ikb_{*}}(-1)^{m+1}\mathcal{R}^{*}_{-k^{*}}\,Y^{\rm hole^{*}}_{\ell^{\prime}\,\text{-}m\,\text{-}\omega^{*}}\,{}_{-2}S_{\ell^{\prime}m\omega^{*}}(\theta,\phi)\,,$		(32)

where $\mathcal{R}_{\omega}(b,\theta)$ is the Fourier transform of $\mathcal{R}(b,\theta;t-t^{\prime})$, and $k\equiv\omega-m\Omega_{H}$. During the derivation, we have used the fact that the spheroidal harmonic functions satisfy the relation

$${}_{-2}S^{*}_{\ell m\omega}(\theta,\phi)=(-1)^{m}{}_{+2}S_{\ell\,\text{-}m\,\text{-}\omega^{*}}(\theta,\phi)\,.$$

(33)

Assuming the normalization that

$$\int_{0}^{\pi}|{}_{-2}S_{\ell m\omega}(\theta)|^{2}\sin\theta d\theta=1\,,$$

(34)

from Eq. (II.4), we can write

$$Z_{\ell m\omega}^{\rm refl}=(-1)^{m+1}\frac{1}{4}e^{-2ikb_{*}}\sum_{\ell^{\prime}}\mathcal{M}_{\ell\ell^{\prime}m\omega}Y^{\rm hole^{*}}_{\ell^{\prime}\,\text{-}m\,\text{-}\omega^{*}}\,,$$

(35)

with

	$\displaystyle\mathcal{M}_{\ell\ell^{\prime}m\omega}$	$\displaystyle=\int_{0}^{\pi}\mathcal{R}^{*}_{-\omega^{*}+m\Omega_{H}}(b,\theta)\,e^{4i\beta(\theta)}\times$		(36)
		$\displaystyle\times{}_{-2}S_{\ell^{\prime}m\omega}(\theta){}_{-2}S^{*}_{\ell m\omega}(\theta)\sin\theta\,d\theta\,.$

In general the reflection will mix between modes with different $\ell$, but not different $m$. Note that the mixing not only arises from the $\theta$ dependence of $\mathcal{R}(\theta,b)$, but also from the $\theta$ dependence of $\beta$. This mixing vanishes for the Schwarzschild case. For our calculation, it will be good to discard the phase term $e^{4i\beta}$ and make the assumption that $\mathcal{R}$ is independent of the angle $\theta$. But we should keep in mind that these assumptions only work well in the Schwarzschild limit $a\rightarrow 0$.

In the simplified scenario where mode mixing is ignored, we can write

$$Z^{\rm refl}_{\ell m\omega}\approx\left(-1\right)^{m+1}\frac{1}{4}\,e^{-2ikb_{*}}\,\mathcal{R}^{*}_{-\omega^{*}+m\Omega_{H}}\,Y^{\rm hole^{*}}_{\ell\,\text{-}m\,\text{-}\omega^{*}}\,,$$

(37)

In this way, the $\omega$ frequency component of the $\psi_{4}$ amplitude of each $(l,m)$ mode is related to the $-\omega^{*}$ frequency component of $\psi_{0}$ of the $(l,-m)$ mode. Here in a Fourier analysis, $\omega$ is always real, but we have kept $\omega^{*}$ so that our notation will directly apply to quasi-normal modes, where frequency can be complex.

Let us now study the propagation of waves near the horizon, and explore how emergent gravity might influence the boundary condition there.

Inside the ECO boundary $\mathcal{S}$, we can consider propagation of metric perturbations in the near-horizon FIDO coordinate system. According to MP, the unperturbed metric takes the simple form Thorne et al. (1986):

$$ds^{2}=-\alpha^{2}d\bar{t\vphantom{\theta}}^{2}+\frac{d\alpha^{2}}{g_{H}^{2}}+\Sigma_{H}d\bar{\theta}^{2}+\frac{4r_{H}^{2}}{\Sigma_{H}}\sin^{2}\bar{\theta}d\bar{\phi}^{2}\,,$$

(38)

where

$\displaystyle g_{H}=\frac{r_{H}-1}{2r_{H}}\,,$

$\displaystyle\Sigma_{H}=r^{2}_{H}+a^{2}\cos^{2}\bar{\theta}\,.$

(39)

This metric, only valid for $\alpha\ll 1$, is a Rindler-like spacetime with spherical symmetry, with horizon located at $\alpha=0$. According to the membrane paradigm Suen et al. (1988), the new radial coordinates $(\alpha,\,\bar{\theta},\,\bar{\phi})$ are defined as

	$\displaystyle\bar{t\vphantom{\theta}}$	$\displaystyle=t\,,$		(40)
	$\displaystyle\alpha$	$\displaystyle=\left(2g_{H}-2a\Omega_{H}g_{H}\sin^{2}\theta\right)^{\frac{1}{2}}\left(r-r_{H}\right)^{\frac{1}{2}}\,,$		(41)
	$\displaystyle\bar{\theta}$	$\displaystyle=\theta-\frac{{\Sigma_{H}}_{,\theta}}{4g_{H}^{2}\Sigma_{H}^{2}}\alpha^{2}\,,$		(42)
	$\displaystyle\bar{\phi}$	$\displaystyle=\phi-\Omega_{H}t\,.$		(43)

The Kinnersley tetrad, near the horizon, can then be expressed in terms of the Rindler coordinates as

	$\displaystyle\vec{l}$	$\displaystyle=\frac{2r_{H}}{\Delta}(\vec{\partial}_{\bar{t\vphantom{\theta}}}+g_{H}\alpha\vec{\partial}_{\alpha})\,,$		(44)
	$\displaystyle\vec{n}$	$\displaystyle=\frac{r_{H}}{\Sigma}(\vec{\partial}_{\bar{t\vphantom{\theta}}}-g_{H}\alpha\vec{\partial}_{\alpha})\,,$		(45)
	$\displaystyle\vec{m}$	$\displaystyle=\frac{-\rho^{*}}{\sqrt{2}}\Big{[}ia\sin\bar{\theta}\vec{\partial}_{\bar{t\vphantom{\theta}}}-\frac{a\Omega_{H}\sin\bar{\theta}\cos\bar{\theta}}{1-a\Omega_{H}\sin^{2}\bar{\theta}}\alpha\vec{\partial}_{\alpha}$
		$\displaystyle\qquad\quad+\vec{\partial}_{\bar{\theta}}+\left(\frac{i}{\sin\bar{\theta}}-ia\Omega_{H}\sin\bar{\theta}\right)\vec{\partial}_{\bar{\phi}}\Big{]}\,,$		(46)

where we have used the near-horizon approximations and discarded all $\mathcal{O}(\alpha^{2})$ corrections.

For convenience, we introduce a new radial coordinate $x$, which is related to the lapse function via

$$\alpha=e^{g_{H}x}\,.$$

(47)

The regime $x\rightarrow-\infty$ is the horizon, where $\alpha\rightarrow 0$. In fact, $(t,x)$ is exactly the Cartesian coordinate of the Minkowski space in which this Rindler space is embedded. Now we consider metric perturbations of the trace-free form

	$\displaystyle h_{\bar{\theta}\bar{\theta}}(t,x,\bar{\theta},\bar{\phi})\$	$\displaystyle=\Sigma_{H}H_{+}(t,x,\bar{\theta},\bar{\phi})\,,$		(48)
	$\displaystyle h_{\bar{\theta}\bar{\phi}}(t,x,\bar{\theta},\bar{\phi})$	$\displaystyle=2r_{H}\sin\bar{\theta}H_{\times}(t,x,\bar{\theta},\bar{\phi})\,,$		(49)
	$\displaystyle h_{\bar{\phi}\bar{\phi}}(t,x,\bar{\theta},\bar{\phi})$	$\displaystyle=-4r^{2}_{H}\sin\bar{\theta}^{2}H_{+}(t,x,\bar{\theta},\bar{\phi})/\Sigma_{H}\,.$		(50)

Note that $H_{+,\times}$ are metric perturbations in the angular directions, measured in orthonormal bases. We first find that the Einstein’s equations reduce to

$$(-\partial_{t}^{2}+\partial_{x}^{2})H_{p}=0\,,\quad p=+,\times.$$

(51)

Again, to obtain the equations above we have only kept the leading terms in $\alpha$-series. The absence of $\bar{\theta}$ and $\bar{\phi}$ derivatives in this equation supports the argument that tidal response of the ECO is local to each angular element on its surface, as we have made in Sec. II.3.

We can further decompose $H_{p}(t,x)$ to the left- and right-propagating piece as

$$H_{p}(t,x,\bar{\theta},\bar{\phi})=H_{p}^{L}(t+x,\bar{\theta},\bar{\phi})+H_{p}^{R}(t-x,\bar{\theta},\bar{\phi})\,,$$

(52)

Using the Rindler approximations, we then find that the Weyl quantities $\psi_{0}$ and $\psi_{4}$ near horizon can be written as

	$\displaystyle\psi_{0}(t,x,\bar{\theta},\bar{\phi})$	$\displaystyle=\frac{8r_{H}^{2}e^{2i\beta}}{\Delta^{2}}(\partial_{t}^{2}-g_{H}\partial_{x})\mathcal{H}^{L}(t,x,\bar{\theta},\bar{\phi})\,,$		(53)
	$\displaystyle\Big{[}\rho^{-4}\psi_{4}(t,x,\bar{\theta},\bar{\phi})\Big{]}^{*}$	$\displaystyle=2r_{H}^{2}e^{-2i\beta}(\partial_{t}^{2}-g_{H}\partial_{x})\mathcal{H}^{R}(t,x,\bar{\theta},\bar{\phi})\,,$		(54)

where we have defined

$\displaystyle\mathcal{H}^{L}$

$\displaystyle=H^{L}_{+}+iH^{L}_{\times}\,,$

$\displaystyle\mathcal{H}^{R}$

$\displaystyle=H^{R}_{+}+iH^{R}_{\times}\,.$

(55)

Note that in this approximation, we only extract the leading behavior of $\psi_{0}$ and $\psi_{4}$ near the horizon, namely a left-going wave $\sim(r-r_{H})^{-2}$ for $\psi_{0}$, and a right-going wave $\sim(r-r_{H})^{0}$ for $\psi_{4}$. Here, we are considering wave propagation and reflection independently for each $(\bar{\theta},\bar{\phi})$. Eq. (53) and Eq. (54) are consistent with our reflection model given in Eq. (29). For instance, in the case of total reflection, we have $\mathcal{R}=-1$, and all left-propagating modes $H^{L}_{p}$ become right-propagating modes $H^{R}_{p}$.

Let us now evaluate the Riemann tensor components in an orthonormal basis whose vectors point along the $(t,x,\bar{\theta},\bar{\phi})$ coordinate axes. The results are

	$\displaystyle R_{\hat{\bar{t\vphantom{\theta}}}\hat{\bar{\theta}}\hat{\bar{t\vphantom{\theta}}}\hat{\bar{\theta}}}=-R_{\hat{\bar{t\vphantom{\theta}}}\hat{\bar{\phi}}\hat{\bar{t\vphantom{\theta}}}\hat{\bar{\phi}}}$	$\displaystyle=-\frac{e^{-2g_{H}x}}{2}\left[-\partial_{t}^{2}+g_{H}\partial_{x}\right]H_{+}\,,$		(56)
	$\displaystyle R_{\hat{\bar{t\vphantom{\theta}}}\hat{\bar{\theta}}\hat{\bar{t\vphantom{\theta}}}\hat{\bar{\phi}}}$	$\displaystyle=-\frac{e^{-2g_{H}x}}{2}\left[-\partial_{t}^{2}+g_{H}\partial_{x}\right]H_{\times}\,.$		(57)

This also confirms the reflection model that we have obtained from the previous section.

We also point out that near the horizon, $x$ and $r_{*}$ differ by a additive constant for each $(\bar{\theta},\bar{\phi})$. Let’s work out the $\bar{\theta}$ dependence of the asymptotic shift between $x$ and $r_{*}$. More specifically, near the horizon, the tortoise coordinate $r_{*}$ is approximately given by

$$r_{*}\approx\frac{1}{2g_{H}}\ln[2g_{H}(r-r_{H})]+\mathcal{I}\,,$$

(58)

with a constant

$$\mathcal{I}=r_{H}+\ln\left(\frac{r_{H}}{2}\right)-\frac{1}{2r_{H}g_{H}}\ln\left(8r_{H}g^{2}_{H}\right)\,.$$

(59)

Here we have neglected $\mathcal{O}(r-r_{H})$ terms. Note that $r_{*}$ is independent of $\bar{\theta}$. We define the difference between the two radial coordinates as

$$x-r_{*}\equiv\delta(\bar{\theta})-\mathcal{I}\,,$$

(60)

where

$$\delta(\bar{\theta})=\frac{1}{2g_{H}}\ln(1-a\Omega_{H}\sin^{2}\bar{\theta})\,.$$

(61)

This may influence the mode mixing of reflected waves from an ECO whose surface has a constant redshift. In Fig. 3, we illustrate contant-$x$ contours in the $(r_{*},\cos\theta)$ plane.

Finally let us derive the Teukolsky reflectivity $\mathcal{R}$ using the Rindler approximation. Supposing for $x$ in certain regions we can write the wave solution as

$\displaystyle\mathcal{H}(t,x,\bar{\theta},\bar{\phi})=\mathcal{H}^{L}(t,x,\bar{\theta},\bar{\phi})+\mathcal{H}^{R}(t,x,\bar{\theta},\bar{\phi})\,,$

(62)

with

	$\displaystyle\mathcal{H}^{L}(t,x,\bar{\theta},\bar{\phi})$	$\displaystyle=\sum_{m}\int\frac{dk}{2\pi}\Theta_{k}(\bar{\theta})e^{-ikx}e^{-ikt}e^{im\bar{\phi}}\,,$		(63)
	$\displaystyle\mathcal{H}^{R}(t,x,\bar{\theta},\bar{\phi})$	$\displaystyle=\sum_{m}\int\frac{dk}{2\pi}\Theta_{k}(\bar{\theta})\zeta_{k}e^{ikx}e^{-ikt}e^{im\bar{\phi}}\,.$		(64)

Here $\Theta_{k}(\bar{\theta})$ gives the $k$-dependent angular distribution. $\zeta_{k}\equiv\zeta(k)$ is the reflection coefficient that converts left-propagating to right-propagating gravitational waves. Thus $\psi_{0}$ and $\psi_{4}$ are respectively given by

	$\displaystyle\psi$	${}_{0}(t,x,\bar{\theta},\bar{\phi})=\frac{8r^{2}_{H}e^{2i\beta(\bar{\theta})}}{\Delta^{2}}\sum_{m}\int\frac{dk}{2\pi}\Theta_{k}(\bar{\theta})(-k^{2}+ig_{H}k)e^{-ikx}e^{-ikt}e^{im\bar{\phi}}\,,$		(65)
	$\displaystyle(\rho^{-4}\psi$	${}_{4})^{*}(t,x,\bar{\theta},\bar{\phi})=2r_{H}^{2}e^{-2i\beta(\bar{\theta})}\sum_{m}\int\frac{dk}{2\pi}\Theta_{k}(\bar{\theta})(-k^{2}-ig_{H}k)\zeta_{k}e^{ikx}e^{-ikt}e^{im\bar{\phi}}\,.$		(66)

Now that we have obtained $\psi_{0}$ and $\psi_{4}$ using the Rindler approximations. We would like to relate $\zeta$ to the Teukolsky reflectivity $\mathcal{R}$. To accomplish this, recall that in Sec. II we have obtained a reflection relation (29) between $\psi_{0}$ and $\psi_{4}$ on the ECO surface. Since the relation is written in the Boyer-Lindquist coordinates, we first perform the coordinate transformations on $\psi_{0}$ and $\psi_{4}$ according to $x=r_{*}+\delta(\theta)-\mathcal{I}$, $\bar{\theta}=\theta$, and $\bar{\phi}=\phi-\Omega_{H}t$. During the coordinate transformation, we have used the near-horizon approximations and discarded all $\mathcal{O}(\alpha^{2})$ terms. The results are given by

	$\displaystyle\psi$	${}_{0}(v,\theta,\varphi_{+})=\frac{8r^{2}_{H}e^{2i\beta(\theta)}}{\Delta^{2}}\sum_{m}\int\frac{dk}{2\pi}\Theta_{k}(\theta)(-k^{2}+ig_{H}k)e^{-i(k+m\Omega_{H})v}e^{-ik\delta(\theta)}e^{ik\mathcal{I}}e^{im\varphi_{+}}\,,$		(67)
	$\displaystyle(\rho^{-4}\psi$	${}_{4})^{*}(u,\theta,\varphi_{-})=2r_{H}^{2}e^{-2i\beta(\theta)}\sum_{m}\int\frac{dk}{2\pi}\Theta_{k}(\theta)(-k^{2}-ig_{H}k)\zeta_{k}e^{-i(k+m\Omega_{H})u}e^{ik\delta(\theta)}e^{-ik\mathcal{I}}e^{im\varphi_{-}}\,.$		(68)

Using the reflection model (29), we obtain that

$\displaystyle\mathcal{R}_{k}=\zeta_{k}\left(\frac{-k-ig_{H}}{-k+ig_{H}}\right)\exp\left[2ikb_{*}+2ik\delta(\theta)-2ik\mathcal{I}\right]\,.$

(69)

Thus once we know $\zeta$, the Teukolsky reflectivity $\mathcal{R}$ can be readily obtained. Here we point out that the phase factor $e^{2ikb_{*}}$ here will cancel the $e^{-2ikb_{*}}$ factors in Sec. II.4. This is because in the previous section we chose $b_{*}$ as the location for the “surface of the ECO”, while in this section, the ECO is embedded into the $x$ coordinate system, therefore we no longer need to introduce a reference location $b_{*}$ as “surface of the ECO”. The information of ECO location will now be incorporated into $\zeta_{k}$.

Before the end of this subsection, let us look at the factor $\mathcal{M}_{\ell\ell^{\prime}m\omega}$ in Eq. (36), and see how the mode mixing show up in the reflected waves. We can pull out the angular dependence of this factor by defining

$\displaystyle\mathcal{M}_{\ell\ell^{\prime}m\omega}=\left(\frac{-k-ig_{H}}{-k+ig_{H}}\right)e^{2ikb_{*}-2ik\mathcal{I}}\hat{\mathcal{M}}_{\ell\ell^{\prime}m\omega}\,,$

(70)

where

$\displaystyle\hat{\mathcal{M}}_{\ell\ell^{\prime}m\omega}=\int_{0}^{\pi}\zeta^{*}_{-\omega^{*}+m\Omega_{H}}e^{i\Phi_{m\omega}(\theta)}{}_{-2}S_{\ell^{\prime}m\omega}(\theta){}_{-2}S^{*}_{\ell m\omega}(\theta)\sin\theta\,d\theta\,,$

(71)

with

$$\Phi_{m\omega}(\theta)=2(\omega-m\Omega_{H})\delta(\theta)+4\beta(\theta)\,.$$

(72)

As an example, we simply let $\zeta=1$, thus $\hat{\mathcal{M}}_{\ell\ell^{\prime}m\omega}$ directly shows the mixing of modes due to the phase. We plot the absolute value of $\hat{\mathcal{M}}_{\ell\ell^{\prime}m\omega}$ for $\ell=2$, $m=2$ and for various spin and $\ell^{\prime}$ in Fig. 4. For $a=0$, we have $\hat{\mathcal{M}}_{\ell\ell^{\prime}m\omega}=1$, indicating no mode mixing. As we raise the spin, modes get more mixed and the reflected waves attain more contributions from $\ell^{\prime}>2$ modes. This quantitatively shows the mixing of different $\ell$-modes is a significant feature for reflection of waves near horizon.