Static spherical vacuum solutions in the bumblebee gravity model

The field equations are obtained by taking variations with respect to the metric and the bumblebee field in Eq. (2). Under the assumption that the bumblebee field does not couple to conventional matter, the field equations can be written as

	$\displaystyle G_{\mu\nu}=\kappa\left(T_{\rm m}\right)_{\mu\nu}+\kappa\left(T_{B}\right)_{\mu\nu},$
	$\displaystyle D^{\mu}B_{{\mu\nu}}-2B_{\nu}\frac{dV}{d(B^{\lambda}B_{\lambda})}+\frac{\xi}{\kappa}B^{\mu}R_{\mu\nu}=0,$		(3)

where $\left(T_{\rm m}\right)_{\mu\nu}$ is the energy-momentum tensor for conventional matter, and the contribution of the bumblebee field to the Einstein equations is

	$\displaystyle\left(T_{B}\right)_{\mu\nu}$	$\displaystyle=$	$\displaystyle\frac{\xi}{2\kappa}\Big{[}g_{\mu\nu}B^{\alpha}B^{\beta}R_{\alpha\beta}-2B_{\mu}B_{\lambda}R_{\nu}^{\phantom{\nu}\lambda}-2B_{\nu}B_{\lambda}R_{\mu}^{\phantom{\mu}\lambda}-\Box_{g}(B_{\mu}B_{\nu})$		(4)
			$\displaystyle-g_{{\mu\nu}}D_{\alpha}D_{\beta}(B^{\alpha}B^{\beta})+D_{\kappa}D_{\mu}\left(B^{\kappa}B_{\nu}\right)+D_{\kappa}D_{\nu}(B_{\mu}B^{\kappa})\Big{]}$
			$\displaystyle+B_{\mu\lambda}B_{\nu}^{\phantom{\nu}\lambda}-g_{\mu\nu}\left(\frac{1}{4}B^{\alpha\beta}B_{\alpha\beta}+V\right)+2B_{\mu}B_{\nu}\frac{dV}{d(B^{\lambda}B_{\lambda})}.$

The d’Alembertian in the curved spacetime is $\Box_{g}:=g^{\alpha\beta}D_{\alpha}D_{\beta}$. Note that the potential $V$ has been assumed to be a function of the scalar product $B^{\mu}B_{\mu}$.

We are interested in nontrivial background configurations of the bumblebee field that are compatible with vacuum, namely $\left(T_{\rm m}\right)_{\mu\nu}=0$. The potential $V$ is supposed to take extrema at these background configurations. Denoting $B_{\mu}=b_{\mu}$ as a background bumblebee field, we have

$\displaystyle\frac{dV}{d(B^{\lambda}B_{\lambda})}\Big{|}_{B_{\mu}=b_{\mu}}=0.$

(5)

In addition, the value of $V$ at $B_{\mu}=b_{\mu}$ is equivalent to a cosmological constant. We drop it in the current study of BH solutions. Therefore, the vacuum field equations are simplified to

	$\displaystyle G_{\mu\nu}=\kappa\left(T_{b}\right)_{\mu\nu},$
	$\displaystyle D^{\mu}b_{{\mu\nu}}+\frac{\xi}{\kappa}b^{\mu}R_{\mu\nu}=0,$		(6)

where $b_{\mu\nu}=D_{\mu}b_{\nu}-D_{\nu}b_{\mu}$, and

	$\displaystyle\left(T_{b}\right)_{\mu\nu}$	$\displaystyle=$	$\displaystyle\frac{\xi}{2\kappa}\Big{[}g_{\mu\nu}b^{\alpha}b^{\beta}R_{\alpha\beta}-2b_{\mu}b_{\lambda}R_{\nu}^{\phantom{\nu}\lambda}-2b_{\nu}b_{\lambda}R_{\mu}^{\phantom{\mu}\lambda}-\Box_{g}(b_{\mu}b_{\nu})$		(7)
			$\displaystyle-g_{{\mu\nu}}D_{\alpha}D_{\beta}(b^{\alpha}b^{\beta})+D_{\kappa}D_{\mu}\left(b^{\kappa}b_{\nu}\right)+D_{\kappa}D_{\nu}(b_{\mu}b^{\kappa})\Big{]}$
			$\displaystyle+b_{\mu\lambda}b_{\nu}^{\phantom{\nu}\lambda}-\frac{1}{4}g_{\mu\nu}b^{\alpha\beta}b_{\alpha\beta}.$

Focusing on static spherical solutions, we use the metric ansatz

$\displaystyle ds^{2}=-e^{2\nu}dt^{2}+e^{2\mu}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right),$

(8)

and assume the background bumblebee field to be $b_{\alpha}=\left(b_{t},\,b_{r},\,0,\,0\right)$, where the unknowns, $\mu\,,\nu,\,b_{t}$ and $b_{r}$, are functions of $r$. Then the Einstein field equations are nonvanishing for the diagonal components as well as the $tr$-component, and the vector field equation has nonvanishing $t$-component and $r$-component. It turns out that the $tr$-component of the Einstein field equations is guaranteed by the $r$-component of the vector field equation, leaving 5 equations for the 4 unknowns, $\mu\,,\nu,\,b_{t}$ and $b_{r}$. So one of the equations depends on the others. These equations are displayed in Appendix A.

Inspired by the solution obtained in Ref. Casana et al. (2018), we checked

	$\displaystyle\nu=\frac{1}{2}\ln{\left(1-\frac{2M}{r}\right)},$
	$\displaystyle\mu=\mu_{0}-\frac{1}{2}\ln{\left(1-\frac{2M}{r}\right)},$
	$\displaystyle b_{t}=\lambda_{0}+\frac{\lambda_{1}}{r},$		(9)

against Eqs. (49) and (50), or equivalently Eq. (6) with the static spherical ansatz, to find that as long as

	$\displaystyle b_{r}^{2}$	$\displaystyle=$	$\displaystyle e^{2\mu_{0}}\Bigg{[}\frac{1}{\xi}\frac{\left(e^{2\mu_{0}}-1\right)r}{r-2M}-\frac{\kappa\lambda_{1}^{2}}{3\xi M(r-2M)}$		(10)
			$\displaystyle+\frac{\lambda_{1}^{2}(2r-M)+6\lambda_{0}\lambda_{1}Mr+6\lambda_{0}^{2}M^{2}r}{3M(r-2M)^{2}}\Bigg{]},$

Eq. (9) is a solution to the field equations. It is an analytical solution characterized by 4 integral constants, $\mu_{0},\,M,\,\lambda_{0}$ and $\lambda_{1}$. The 2-parameter solution obtained in Ref. Casana et al. (2018) simply corresponds to $\lambda_{0}=\lambda_{1}=0$. We also notice that the 3-parameter solution corresponding to $\mu_{0}=0$ has been previously obtained in Ref. Fan (2018).

The following two observations can be made based upon the solution given by Eqs. (9) and (10).

1. 1.

   The Schwarzschild metric corresponding to $\mu_{0}=0$ is no longer necessarily accompanied by a vanishing bumblebee field. Among the valid choices of $\lambda_{0}$ and $\lambda_{1}$ that give nonnegative $b_{r}^{2}$, an intriguing choice is $\lambda_{1}=-2M\lambda_{0}$, which simplifies Eq. (10) to

   $\displaystyle b_{r}^{2}$

   $\displaystyle=$

   $\displaystyle\frac{2}{3}\left(1-\frac{2\kappa}{\xi}\right)\frac{\lambda_{0}^{2}M}{r-2M},$

   (11)

   for $\mu_{0}=0$. As we require $b_{r}^{2}\geq 0$ outside the event horizon, this choice of $\lambda_{1}$ can only be made for $\xi\geq 2\kappa$. For the special case of $\xi=2\kappa$, the Schwarzschild metric is a solution to the field equations together with a nontrivial bumblebee field that has only a temporal component $b_{t}\propto 1-2M/r$.

2. 2.

   The Minkowski metric is recovered with $\mu_{0}=0$ and $M=0$. The limit of Eq. (10) exists if $\lambda_{1}\propto\sqrt{M}$ while $M\rightarrow 0$. That is to say, the Minkowski metric can be accompanied by a nontrivial bumblebee field that has a radial component

   $\displaystyle b_{r}^{2}=\frac{1}{3}\left(2-\frac{\kappa}{\xi}\right)\frac{C}{r},$

   (12)

   where $C$ being the ratio $\lambda_{1}^{2}/M$ has the dimension of length, and might represent the length scale of possible Lorentz violation in an underlying theory.

General solutions to Eqs. (49) and (50) are obtained numerically in two cases enumerated in Appendix A according to the two factors in the $r$-component of the vector equation: (i) $b_{r}=0$; (ii) $R_{rr}=0$. For the first case, the system of equations reduce to two coupled second-order ODEs, indicating a family of solutions determined by 4 parameters. For the second case, the system of equations reduce to three coupled second-order ODEs, indicating a family of solutions determined by 6 parameters. Both families of solutions turn out to have the asymptotic expansions

	$\displaystyle\nu=\sum\limits_{n=1}^{n=\infty}\frac{\nu_{n}}{r^{n}},$
	$\displaystyle\mu=\sum\limits_{n=0}^{n=\infty}\frac{\mu_{n}}{r^{n}},$
	$\displaystyle b_{t}=\sum\limits_{n=0}^{n=\infty}\frac{\lambda_{n}}{r^{n}}.$		(13)

After substituting them into the ODEs to find the recurrence relations for the expansion coefficients $\nu_{n},\,\mu_{n}$ and $\lambda_{n}$, these expansions can be utilized to assign the initial values to start the numerical integrations from large enough $r$. Note that a constant term in the expansion of $\nu$ is unnecessary as it amounts to a change of the time coordinate. Therefore, the number of parameters characterizing the solutions is actually three for the first family ($b_{r}=0$) and 5 for the second family ($R_{rr}=0$).

The recurrence relations for the first few coefficients are given in Appendix B. The conclusion is that for the case of $b_{r}=0$, the solutions must have $\mu_{0}=0$ and are fixed once the three free coefficients $\mu_{1},\,\lambda_{0}$ and $\lambda_{1}$ are specified, and that for the case of $R_{rr}=0$, there are 5 free coefficients, $\mu_{0},\,\mu_{1},\,\mu_{2},\,\lambda_{0}$ and $\lambda_{1}$, describing the solutions. We point out that the analytical solution given by Eqs. (9) and (10) belongs to the second case and is recovered when taking $\mu_{2}=\mu_{1}^{2}=M^{2}$.

Before further discussing the numerical solutions, let us relate the parameters $\mu_{1}$ and $\lambda_{1}$ to the mass and the bumblebee charge of the spacetime. With $\mu_{0}=0$, the Arnowitt-Deser-Misner (ADM) mass of the spacetime can be calculated and it is $\mu_{1}$ Arnowitt et al. (2008). However, the definition of the ADM mass does not apply to the solutions with $\mu_{0}\neq 0$, when the metric is not asymptotically flat. Therefore we adopt the Komar mass as a measure of the spacetime mass when $\mu_{0}\neq 0$. The Komar mass is defined by the conserved current associated with the time-translation Killing vector $K^{\mu}=(1,0,0,0)$. The conserved current is Poisson (2009)

$\displaystyle J_{M}^{\mu}:=K_{\nu}R^{\mu\nu}=D_{\nu}D^{\mu}K^{\nu},$

(14)

where the curvature identity $[D_{\mu},D_{\nu}]K^{\alpha}=R^{\alpha}_{\phantom{\alpha}\lambda{\mu\nu}}K^{\lambda}$ and the Killing equation $D_{\mu}K_{\nu}+D_{\nu}K_{\mu}=0$ have been used. The Komar mass then can be calculated as

	$\displaystyle M_{\rm K}$	$\displaystyle:=$	$\displaystyle-\frac{1}{4\pi}\int d^{3}x\sqrt{-g}\,J_{M}^{t}$		(15)
		$\displaystyle=$	$\displaystyle-\frac{1}{4\pi}\lim\limits_{r\rightarrow\infty}\iint d\theta d\phi\sqrt{-g}\,D^{t}K^{r}$
		$\displaystyle=$	$\displaystyle e^{-\mu_{0}}\mu_{1}.$

It is evident that the Komar mass reduces to the ADM mass when $\mu_{0}=0$. The bumblebee charge can be defined in a similar way. Instead of using a Killing vector, the current

$\displaystyle J_{Q}^{\mu}:=\frac{\xi}{\kappa}b_{\nu}R^{\mu\nu}=-D_{\nu}b^{\nu\mu}$

(16)

is automatically conserved as $b_{\mu\nu}$ is antisymmetric. Therefore the bumblebee charge having the dimension of length can be defined as

	$\displaystyle Q$	$\displaystyle:=$	$\displaystyle-\frac{1}{4\pi}\sqrt{\frac{\kappa}{2}}\int d^{3}x\sqrt{-g}\,J_{Q}^{t}$		(17)
		$\displaystyle=$	$\displaystyle-\frac{1}{4\pi}\sqrt{\frac{\kappa}{2}}\lim\limits_{r\rightarrow\infty}\iint d\theta d\phi\sqrt{-g}\,b^{tr}$
		$\displaystyle=$	$\displaystyle\sqrt{\frac{\kappa}{2}}\,e^{-\mu_{0}}\lambda_{1},$

where the factor $\sqrt{\kappa/2}$ has been chosen so that the Reissner-Nordström metric in Eq. (54) can be recovered when $\xi=0$.

Back to the numerical results, as we integrate the system of ODEs from a large radius to $r=0$, there exist three types of solutions. The one that we are interested in has $g_{rr}=e^{2\mu}$ diverging at a certain radius $r=r_{h}$. The other two types that we will ignore are solutions of naked singularity and solutions with $g_{tt}=-e^{2\nu}$ diverging at a finite radius. Concentrating on the interesting solutions that have $g_{rr}$ diverging at a finite radius $r=r_{h}$, the BH solutions are selected with two extra conditions: (i) vanishing $g_{tt}$ at $r_{h}$ and (ii) finite curvature scalars at $r_{h}$. It turns out that the finiteness of the curvature scalars at $r_{h}$ is trivially satisfied by all the solutions with $g_{rr}$ diverging at $r_{h}$. But among these solutions, it turns out that not all of them have $g_{tt}=0$ at $r_{h}$. In fact, for the solutions in the first family ($b_{r}=0$), most of them have nonvanishing $g_{tt}$ at $r_{h}$.

To comprehend what happens near $r_{h}$, we seek expansions of the variables in terms of $\delta:=r-r_{h}$. It is after a careful investigation of the numerical solutions that we find the variables taking the following expansions near $r_{h}$,

	$\displaystyle g_{tt}=-e^{2\nu}=-\sum\limits_{n=0}^{\infty}N_{1n}\,\delta^{n}-\sqrt{\delta}\sum\limits_{n=0}^{\infty}N_{2n}\,\delta^{n},$
	$\displaystyle g_{rr}=e^{2\mu}=\frac{1}{\delta}\sum\limits_{n=0}^{\infty}M_{1n}\,\delta^{n}+\frac{1}{\sqrt{\delta}}\sum\limits_{n=0}^{\infty}M_{2n}\,\delta^{n},$
	$\displaystyle b_{t}=\sum\limits_{n=0}^{\infty}L_{1n}\,\delta^{n}+\sqrt{\delta}\sum\limits_{n=0}^{\infty}L_{2n}\,\delta^{n},$		(18)

where $\big{\{}N_{1n},\,N_{2n},\,M_{1n},\,M_{2n},L_{1n},\,L_{2n}\big{\}}$ is the set of expansion coefficients. The recurrence relations can be obtained by substituting the expansions into the ODEs and setting each order of $\delta$ to zero. The recurrence relations for the first few coefficients are given in Appendix C. The most important fact we learn is that the coefficient $N_{10}$ is closely related to the coefficients for the terms involving $\sqrt{\delta}$ in the following way. If $N_{10}=0$, then $N_{2n}=M_{2n}=L_{2n}=0$, and vice versa.

The solutions with $g_{rr}$ diverging at a finite radius $r_{h}$, namely those can be expanded in the form of Eq. (18), thus can be divided into two categories according to whether $N_{10}$ is zero. The BH solutions have $N_{10}=0$, while the solutions with $N_{10}\neq 0$, for their unexpected property of bouncing geodesics at $r_{h}$, which will be discussed in Sec. II.4, will be dubbed CHs in the remaining of the paper. Note that both the BH solutions and the CH solutions have finite curvature scalars at $r_{h}$ (see Appendix D).

Figures 1 and 2 show how $r_{h}$ depends on the asymptotic parameters $\lambda_{0}$ and $\lambda_{1}$. We take the asymptotic parameter $\mu_{1}$ as the length unit in the numerical calculations. For the first family of solutions ($b_{r}=0$), a solution is obtained once the values of the coupling constant $\xi$ and the asymptotic parameters $\lambda_{0}$ and $\lambda_{1}$ are specified properly. Sufficient amount of solutions obtained by varying $\lambda_{0}$ and $\lambda_{1}$ then generate the contour plots of $r_{h}$ on the $\lambda_{0}$-$\lambda_{1}$ plane as shown in Fig. 1. For the second family of solutions ($R_{rr}=0$), with $\mu_{1}$ being the unit, a solution is obtained when the asymptotic parameters $\mu_{0},\,\mu_{2},\,\lambda_{0},\,\lambda_{1}$, as well as the coupling constant $\xi$ are specified properly. To generate Fig. 2, we have taken $\mu_{0}=0$ and two representative values for $\mu_{2}$. Solutions with nonzero but small $\mu_{0}$ turn out to produce similar shapes of the contour plots, while solutions with $\mu_{0}$ deviating from zero significantly are uninteresting to us as $\mu_{0}$ has been constrained to be smaller than $10^{-12}$ using the advance of perihelia of the Solar-System planets in Ref. Casana et al. (2018).

Figures 3 and 4 show how $N_{10}$ depends on $\lambda_{0}$ and $\lambda_{1}$, using the same solutions producing Figs. 1 and 2. The values of the asymptotic parameters $\lambda_{0}$ and $\lambda_{1}$ for BH solutions and for CH solutions are conveniently revealed in Figs. 3 and 4. For the first family of solutions, the requirement of $N_{10}=0$ turns out to force $L_{10}$ to vanish, which acts as an extra constraint on the asymptotic parameters $\lambda_{0}$ and $\lambda_{1}$. Therefore, the BH solutions in the first family only exist along a single line on the $\lambda_{0}$-$\lambda_{1}$ plane for a given $\xi$, leaving the majority of the valid $\lambda_{0}$ and $\lambda_{1}$ to generate the CH solutions characterized by $N_{10}\neq 0$. For the second family of solutions, the requirement of $N_{10}=0$ forces no extra constraint on $\lambda_{0}$ and $\lambda_{1}$, so the BH solutions exist on the two-dimensional plane of $\lambda_{0}$ and $\lambda_{1}$. In fact, as shown in Fig. 4, there is no CH solution for any values of $\lambda_{0}$ and $\lambda_{1}$ as long as one takes $\mu_{2}\leq\mu_{1}^{2}$.

Comparing the BH solutions in the first family ($b_{r}=0$) with the charged BH solution in GR, i.e., the Reissner-Nordström metric, is worthwhile. The asymptotic expansions of the metric functions $\mu$ and $\nu$ given by the recurrence relations in Eq. (52) include the Reissner-Nordström metric as a special case for $\xi=0$, when the bumblebee model recovers the Einstein-Maxwell theory. This suggests that the bumblebee BH solutions in the first family, characterized by the two parameters, $M=\mu_{1}$ and $Q=\sqrt{\kappa/2}\,\lambda_{1}$, can be viewed as a generalization of the Reissner-Nordström metric. A plot of $r_{h}$ versus $Q$ in unit of $\mu_{1}$ as shown in Fig. 5 demonstrates how the family of solutions change with the coupling constant $\xi$. We would like to direct the readers’ attention to two special values of $\xi$, namely $\xi=0$ and $\xi=2\kappa$. When $\xi<0$, the maximal charge is less than $M$, and when $\xi>0$, the maximal charge is greater than $M$. When $\xi<2\kappa$, the maximal radius of the event horizon is $2M$ (Schwarzschild metric), and when $\xi>2\kappa$, the maximal radius of the event horizon is greater than $2M$. We notice that when $\xi>2\kappa$ or $\xi<0$ there exist two black-hole solutions with the same $r_{h}$ but different $Q$. The two black holes are certainly observationally distinguishable as the difference in $Q$ makes their metric functions different. In fact, when we calculate the shadow of the black holes in Sec. III.2, we see that they have different shadow radii (see Fig. 10). A perhaps more interesting but puzzling observation is that when $\xi=\kappa$, $r_{h}$ has two values given $Q$ near its maximum. When this happens, we suspect that the black hole with the smaller $r_{h}$ might be unstable. Further study on quasi-normal modes of the black holes found here will shed light on this point.

We end the general discussion on the solutions by tabulating several representative numerical solutions for further reference. Table 1 lists 4 BH solutions and 4 CH solutions in the first family. Table 2 lists 8 BH solutions and 3 CH solutions in the second family. To have a brief picture about how different from the Schwarzschild metric these solutions are, the metric functions, as well as the scalar $b^{2}:=g^{\mu\nu}b_{\mu}b_{\nu}$ and the Ricci scalar $R$, are plotted in Fig. 6, for several of the representative solutions.

The CH solutions share the same form of the asymptotic expansions with the BH solutions, so they mimic the BH solutions at large radius. But near the event horizon, the nonvanishing $g_{tt}$ causes a distinctive feature for them. For CHs, geodesics bounce back on the event horizon, which is the reason for us to name them “compact hill”. The very terminology “event horizon” actually seems improper as this feature of the CH solutions is radically different from the event horizon of BHs. However, in the sense that the null hypersurface $r=r_{h}$ of the CHs prevents geodesics outside of it from going in while the event horizon of BHs prevents geodesics inside from going out, we still refer $r=r_{h}$ as the event horizon of the CHs.

Now to show that geodesics bounce back on the event horizon of the CHs, one first finds that under the metric in Eq. (8) the radial components of the four-velocity and the four-acceleration can be written as

	$\displaystyle\left(\frac{dr}{d\lambda}\right)^{2}=e^{-2\mu}\left(\epsilon^{2}\,e^{-2\nu}-\frac{l^{2}}{r^{2}}+k\right),$
	$\displaystyle\frac{d^{2}r}{d\lambda^{2}}=e^{-2\mu}\left[-\epsilon^{2}\,e^{-2\nu}(\mu^{\prime}+\nu^{\prime})+\frac{l^{2}}{r^{2}}\left(\mu^{\prime}+\frac{1}{r}\right)-k\mu^{\prime}\right],$		(19)

where the prime denotes the derivative with respect to $r$, and the integral constants $\epsilon$ and $l$ are

	$\displaystyle\epsilon$	$\displaystyle=$	$\displaystyle e^{2\nu}\frac{dt}{d\lambda},$
	$\displaystyle l$	$\displaystyle=$	$\displaystyle r^{2}\frac{d\phi}{d\lambda},$		(20)

while the integral constant $k$ takes $-1,\,0$, and $1$, for timelike, lightlike, and spacelike geodesics as $\lambda$ is the suitable affine parameter in each case. With $g_{tt}=-e^{2\nu}$ being nonzero on the event horizon, we immediately see $dr/d\lambda\rightarrow 0$ on the event horizon. Moreover, using the expansions in Eq. (18) we find near the event horizon

	$\displaystyle\left(\frac{dr}{d\lambda}\right)^{2}\rightarrow\frac{r-r_{h}}{M_{10}}\left(\frac{\epsilon^{2}}{N_{10}}-\frac{l^{2}}{r_{h}^{2}}+k\right),$
	$\displaystyle\frac{d^{2}r}{d\lambda^{2}}\rightarrow\frac{1}{2M_{10}}\left(\frac{\epsilon^{2}}{N_{10}}-\frac{l^{2}}{r_{h}^{2}}+k\right).$		(21)

Because $M_{10}$ is positive, the combination in the parentheses must be nonnegative for $\left(dr/d\lambda\right)^{2}\geq 0$. Therefore near the event horizon the radial acceleration $d^{2}r/d\lambda^{2}$ is nonnegative. Together with the fact that the radial velocity $dr/d\lambda$ approaches zero on the event horizon, we conclude that geodesics bounce back there.

The gravitational force of a CH exerting on a test mass then can be thought as repulsive near the event horizon while attractive far away. As far as we know there is completely no evidence for this kind of gravitational interaction in experiments and astrophysical observations. Anyway, the effect is academically interesting. The simplest scenario is that a test mass is released at rest and then oscillates between the initial position and the event horizon. More generally, bound orbits exist near the event horizon. The repulsive nature near the event horizon ensures that even the lightlike circular orbit at the radius determined by $\nu^{\prime}=1/r$ is stable. Figure 7 shows example orbits bouncing back on the event horizon of a CH.

Classical trajectories bounce back on the event horizon, suggesting that GWs echo, which is a genuine effect that can be searched for in GW observations Westerweck et al. (2018); Testa and Pani (2018). It turns out that the potentials for waves are finite on the event horizon so waves are not perfectly reflected. What marvelous is that the potentials become complex inside the event horizon, and it is well known in quantum mechanics that complex potentials cause absorption of waves (e.g. see Refs. Avishai and Knoll (1976); Vibók and Balint‐Kurti (1992)). Before we approximately demonstrate the reflection on the event horizon of a CH using a scalar wave, let us admit that not only the potentials for waves but also the Ricci scalar becomes complex inside the event horizon of CHs. The origination of their imaginary parts can be seen from Eq. (18), where $\sqrt{\delta}$ becomes imaginary when $r<r_{h}$. Though it first appears to us that the CH solutions are unphysical due to the complex Ricci scalar, a second thought about the absorption property related to the imaginary parts of the potentials strikes us the idea that a spacetime region with a complex Ricci scalar might be where information is lost. The idea needs to be further explored, but that is beyond the scope of the present work. In the remaining of this section, we only try to use a brute but manageable approximation to illustrate the reflection of a scalar wave on the event horizon of a CH.

Starting with the Klein-Gordon equation

$\displaystyle 0=\Box_{g}\Phi=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(g^{\mu\nu}\sqrt{-g}\,\partial_{\nu}\Phi\right),$

(22)

and using the spherical metric in Eq. (8) as well as the spherical wave ansatz for the scalar field $\Phi=e^{i\sigma t}\,\Psi(r)$, one obtains an equation for $\Psi(r)$

$\displaystyle\Psi^{\prime\prime}+\left(\nu^{\prime}-\mu^{\prime}+\frac{2}{r}\right)\Psi^{\prime}+\sigma^{2}e^{2(\mu-\nu)}\Psi=0.$

(23)

The changes of variables

	$\displaystyle z$	$\displaystyle=$	$\displaystyle r\Psi,$
	$\displaystyle\frac{dr_{*}}{dr}$	$\displaystyle=$	$\displaystyle e^{\mu-\nu}$		(24)

put Eq. (23) into the standard form Berti et al. (2009)

$\displaystyle\frac{d^{2}z}{dr_{*}^{2}}+\left(\sigma^{2}-V_{s}\right)z=0,$

(25)

with the potential for the scalar wave being

$\displaystyle V_{s}=e^{2(\nu-\mu)}\frac{\nu^{\prime}-\mu^{\prime}}{r}.$

(26)

Fully solving the scattering problem of Eq. (25) is beyond the scope of the present work. We shall focus on the behavior of the wave near the event horizon, where the potential $V$ and the derivative $dr_{*}/dr$ have the expansions

	$\displaystyle V_{s}=\frac{1}{2r_{h}}\frac{N_{10}}{M_{10}}\left[1+\frac{3}{2}\left(\frac{N_{20}}{N_{10}}-\frac{M_{20}}{M_{10}}\right)\sqrt{\delta}+...\right],$
	$\displaystyle\frac{dr_{*}}{dr}=\sqrt{\frac{M_{10}}{N_{10}}}\frac{1}{\sqrt{\delta}}\left[1-\frac{1}{2}\left(\frac{N_{20}}{N_{10}}-\frac{M_{20}}{M_{10}}\right)\sqrt{\delta}+...\right].$		(27)

Besides being complex inside the event horizon for the potential $V_{s}$, Eq. (27) is different from the BH expressions because of two other features. First, for BHs $V_{s}$ is zero on the event horizon while Eq. (27) is not. Then, the tortoise radius $r_{*}$ goes to negative infinity on the event horizon of BHs, but now that $dr_{*}/dr$ is only divergent as $1/\sqrt{\delta}$ for the CHs we find $r_{*}$ proportional to $\sqrt{\delta}$ plus an integral constant.

Taking the distinctive features of Eq. (27) into consideration, we reckon that the step potential

$\displaystyle V_{\rm step}=\begin{cases}V_{s1},&{\rm Re}(r_{*})\geq 0\\ V_{s2}+i\Gamma_{2},&{\rm Re}(r_{*})<0\end{cases}$

(28)

might be used to approximate the effective potential $V_{s}$ near the event horizon. In Eq. (28), the real constant $V_{s1}$ and the complex constant $V_{s2}+i\Gamma_{2}$ can be thought as certain average values of $V_{s}$ right outside and inside the event horizon. The tortoise radius, which also becomes complex inside the event horizon, has been set to zero at the event horizon. With $V_{s}$ brutely approximated by $V_{\rm step}$, the scattering problem of Eq. (25) is analytically solvable. Assuming no waves coming out of the center at $r=0$, the plane wave ansatz

$\displaystyle z=\begin{cases}C_{+}e^{ik_{1}r_{*}}+C_{-}e^{-ik_{1}r_{*}},&{\rm Re}(r_{*})\geq 0\\ D_{+}e^{ik_{2}r_{*}},&{\rm Re}(r_{*})<0\end{cases}$

(29)

satisfies the equation when

	$\displaystyle k_{1}$	$\displaystyle=$	$\displaystyle\sqrt{\sigma^{2}-V_{s1}},$
	$\displaystyle k_{2}$	$\displaystyle=$	$\displaystyle\sqrt{\sigma^{2}-V_{s2}-i\Gamma_{2}}.$		(30)

The continuity of $z$ and $dz/dr_{*}$ at $r_{*}=0$ fixes the ratio between the amplitudes of the reflected and the incident waves, which is

$\displaystyle\frac{C_{-}}{C_{+}}=\frac{k_{1}-k_{2}}{k_{1}+k_{2}}.$

(31)

The reflection rate is then

$\displaystyle{\cal{R}}=\bigg{|}\frac{C_{-}}{C_{+}}\bigg{|}^{2}=\frac{k_{1}^{2}+|k_{2}|^{2}-2k_{1}{\rm Re}(k_{2})}{k_{1}^{2}+|k_{2}|^{2}+2k_{1}{\rm Re}(k_{2})},$

(32)

where we have assumed the energy of the incident wave $\sigma^{2}$ to be greater than $V_{s1}$ so that $k_{1}$ is real. If we define $\theta_{2}\in[-\pi,\pi)$ by

	$\displaystyle\cos\theta_{2}=\frac{\sigma^{2}-V_{s2}}{\sqrt{\left(\sigma^{2}-V_{s2}\right)^{2}+\Gamma_{2}^{2}}},$
	$\displaystyle\sin\theta_{2}=\frac{-\Gamma_{2}}{\sqrt{\left(\sigma^{2}-V_{s2}\right)^{2}+\Gamma_{2}^{2}}},$		(33)

then $k_{2}$ can be explicitly written as

$\displaystyle k_{2}=\left[\left(\sigma^{2}-V_{s2}\right)^{2}+\Gamma_{2}^{2}\right]^{\frac{1}{4}}e^{\frac{i\theta_{2}}{2}}.$

(34)

A brief understanding on how $\Gamma_{2}$ affects the reflection rate can be achieved by assuming a small $\Gamma_{2}$ so that Eq. (32), up to the leading order of $\Gamma_{2}$, reads

$\displaystyle{\cal{R}}\approx\left(\frac{k_{1}-\tilde{k}_{2}}{k_{1}+\tilde{k}_{2}}\right)^{2}+\frac{k_{1}\left(3\tilde{k}_{2}^{2}-k_{1}^{2}\right)}{2\,\tilde{k}_{2}^{3}(k_{1}+\tilde{k}_{2})^{4}}\Gamma_{2}^{2},$

(35)

where $\tilde{k}_{2}=\sqrt{\sigma^{2}-V_{s2}}$. The first term independent of $\Gamma_{2}$ is the usual reflection rate due to a real step potential. The second term proportional to the square of $\Gamma_{2}$ represents the contribution from the imaginary part of the potential. The contribution is positive when $3\tilde{k}_{2}^{2}-k_{1}^{2}>0$, namely

$\displaystyle\sigma^{2}>\frac{3V_{s2}-V_{s1}}{2}.$

(36)

Because we have required $\sigma^{2}>V_{s1}$, Eq. (36) is always satisfied if $V_{s1}\geq V_{s2}$. If $V_{s1}<V_{s2}$, then the imaginary potential contributes negatively to the reflection rate when the frequency of the incident wave is lower than the value in Eq. (36).