Axial gravitational quasinormal modes of a self-dual black hole in loop quantum gravity

The first direct detection of the gravitational wave (GW) in 2015 LIGOScientific:2016aoc marked an all-new era of physics and astronomy Cai:2017cbj ; Bian:2021ini . The Event Horizon Telescope has taken the first picture of a supermassive object at the center of galaxy M87 EventHorizonTelescope:2019dse ; EventHorizonTelescope:2019uob ; EventHorizonTelescope:2019jan ; EventHorizonTelescope:2019ths ; EventHorizonTelescope:2019pgp ; EventHorizonTelescope:2019ggy , and the picture of the central black hole in our Milky Way EventHorizonTelescope:2022wkp ; EventHorizonTelescope:2022apq ; EventHorizonTelescope:2022wok ; EventHorizonTelescope:2022exc ; EventHorizonTelescope:2022urf ; EventHorizonTelescope:2022xqj . Human beings can observe the universe with multi-messenger, both the gravitational wave and the electromagnetic wave. Until now, the LIGO–Virgo–KAGRA collaboration has finished three observing runs and detected 90 confident GW-burst events LIGOScientific:2018mvr ; LIGOScientific:2020ibl ; LIGOScientific:2021usb ; LIGOScientific:2021djp . GW-bursts, emitted from the merger of binary compact objects, bring information about gravitational theories and sources and provide us with a new approach to test general relativity in the strong gravitational field LIGOScientific:2016lio ; LIGOScientific:2019fpa ; LIGOScientific:2020tif ; LIGOScientific:2021sio . The whole gravitational wave waveform of a GW-burst event can be divided into three parts: inspiral, merger, and ringdown. And the ringdown part can be successfully described by the black hole perturbation theory Chandrasekhar:1985kt ; Maggiore:2018sht .

A black hole with perturbations is a dissipative system, and the eigenmodes of this system are named quasinormal modes (QNMs). The QNMs are the spectroscopy of a black hole, because the quasinormal frequencies depend only on the black hole’s parameters, while their amplitudes depend on the source exciting the oscillations Kokkotas:1999bd ; Nollert:1999ji ; Berti:2009kk ; Konoplya:2011qq . According to the behavior under space inversions, the gravitational perturbations of a spherically symmetric black hole can be divided into the odd (axial) parity part and the even (polar) parity part Chandrasekhar:1985kt . As the most successful theory for gravitational interaction, general relativity has passed many astrophysical tests Weinberg:1972kfs . In general relativity, Regge, Wheeler Regge:1957td , and Zerilli Zerilli:1970se first studied the odd parity and the even parity gravitational perturbations of the Schwarzschild black hole. Moncrief first studied both the odd parity and the even parity gravitational perturbations of the Reissner-Nordstrom black hole Moncrief:1974gw ; Moncrief:1974ng . And Teukolsky first studied the gravitational perturbations of the Kerr black hole Teukolsky:1972my . To get the quasinormal frequencies for the black hole perturbation problem, numerical methods are needed to solve the eigenvalue problem. With the development of the black hole perturbation theory, more and more numerical methods were proposed, such as the Wentzel-Kramers-Brillouin (WKB) approximations Mashhoon ; Schutz:1985km ; Iyer:1986np ; Konoplya:2003ii ; Matyjasek:2017psv ; Konoplya:2019hlu , the asymptotic iteration method Cho:2011sf , the monodromy technique Motl:2003cd , the series solution Horowitz:1999jd , the resonance method Berti:2009wx , and the Leaver’s continued fraction method Leaver:1985ax .

The singularity of general relativity is a good motivation to probe new physics. It is generally believed that a complete theory of quantum gravity has no singularity. Loop quantum gravity is exactly this case Rovelli:2004tv . In loop quantum gravity, spacetime is made up of some basic building blocks called spin networks. In the framework of loop quantum gravity, Modesto and Premont-Schwarz constructed the Reissner-Nordstrom-like self-dual black hole Modesto:2009ve ; Modesto:2008im . Many works investigated the phenomenological implications of this black hole Alesci:2011wn ; Barrau:2014yka ; Dasgupta:2012nk ; Sahu:2015dea ; Hossenfelder:2012tc ; Zhu:2020tcf ; Yan:2022fkr . The perturbations of the self-dual black hole also have been studied in some works, which can be divided into two categories by whether using the Arnowitt-Deser-Misner (ADM) mass of the black hole as one of the parameters fixed during calculation. Fixing the parameter $M/(1+P)^{2}$ instead of the ADM mass of the self-dual black hole, Chen and Wang studied the QNMs of a massless scalar field Chen:2011zzi , Santos $et~{}al.$ studied QNMs of a massive scalar field nonminimally coupled to gravity Santos:2021wsw , Cruz $et~{}al.$ studied axial Cruz:2015bcj and polar gravitational perturbations Cruz:2020emz . But it is worth pointing out that the effective potential in Ref. Cruz:2015bcj cannot be reduced to the Schwarzschild black hole case when setting all loop quantum gravity parameters equal to zero. Fixing the ADM mass of the self-dual black hole, Liu $et~{}al.$ studied QNMs of the massless scalar field and electromagnetic field Liu:2020ola , and Momennia studied the QNMs of a test scalar field Momennia:2022tug .

In this work, we focus on the axial gravitational perturbation of the self-dual black hole with fixed ADM mass, because the ADM mass is the physical mass of a black hole measured in astronomical observations. Following Ref. Modesto:2009ve , we assume that the self-dual black hole is described by Einstein’s gravity minimally coupled to an anisotropic fluid, and derive the master equation of the axial gravitational perturbation of the self-dual black hole. This method also was used to study the gravitational perturbations of nonsingular black holes in conformal gravity Chen:2019iuo and non-singular Schwarzschild black holes in loop quantum gravity Bouhmadi-Lopez:2020oia . Then, we calculate the corresponding quasinormal frequencies with the WKB approximation and the asymptotic iteration method. The influence of the quantum correction parameter $P$ on the QNMs is also studied. We find that the parameter $P$ has a positive effect on the absolute values of both the real part and the imaginary part of quasinormal frequencies, which is consistent with the conclusions for the QNMs of the scalar field and the electromagnetic field on the self-dual black hole with fixed ADM mass during calculating Liu:2020ola ; Momennia:2022tug . Assuming the perturbation is a Gaussian packet, we investigate the numerical evolution of an initial wave packet on the self-dual black hole. Besides, Cardoso, Lemos, and Yoshida found that, in the eikonal limit, quasinormal modes of a stationary, spherically symmetric, and asymptotically flat black hole in any dimension are determined by the parameters of the circular null geodesics Cardoso:2008bp . We obtain the relation between the quasinormal frequencies in the eikonal limit of the axial gravitational perturbation and the parameters of the circular null geodesics in the self-dual black hole, and numerically verify this relation. The numerical results show that the relation between the parameters of the circular null geodesics and quasinormal frequencies in the eikonal limit is right in the self-dual black hole in loop quantum gravity.

This paper is organized as follows. In Sec. II, we derive the master equation of the axial gravitational perturbation of the self-dual black hole. In Sec. III, we calculate the corresponding quasinormal frequencies with the WKB approximation method and the asymptotic iteration method. And we investigate the numerical evolution of an initial wave packet on the self-dual black hole spacetime. Then we obtain the relation between the parameters of the circular null geodesics and quasinormal frequencies in the eikonal limit in the self-dual black hole, and numerically verify this relation in Sec. IV. Finally, the conclusions and discussions of this work are given in Sec. V.

The line element of the spherically symmetric self-dual black hole in loop quantum gravity is Modesto:2009ve

$$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+h(r)\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right),$$

(2.1)

where the functions $f(r)$, $g(r)$, and $h(r)$ have the following forms

	$\displaystyle f(r)$	$\displaystyle=$	$\displaystyle\frac{(r-r_{+})(r-r_{-})}{r^{4}+a^{2}_{0}}(r+r_{0})^{2},$		(2.2)
	$\displaystyle g(r)$	$\displaystyle=$	$\displaystyle\frac{(r-r_{+})(r-r_{-})}{r^{4}+a^{2}_{0}}\frac{r^{4}}{(r+r_{0})^{2}},$		(2.3)
	$\displaystyle h(r)$	$\displaystyle=$	$\displaystyle r^{2}+\frac{a^{2}_{0}}{r^{2}},$		(2.4)

where $a_{0}\simeq 5l^{2}_{P}/8\pi$ ($l_{p}$ is the Planck length) is related to the minimum area gap of loop quantum gravity, $r_{+}=2M/(1+P)^{2}$ is the outer (event) horizon, with $P$ a function of the polymeric parameter $\delta_{b}$ related to the geometric quantum effect of loop quantum gravity. $r_{-}=2MP^{2}/(1+P)^{2}$ is the inner (Cauchy) horizon, $r_{0}=\sqrt{r_{+}r_{-}}$, and $M$ is the ADM mass of the black hole. The deviation of the self-dual black hole from the Schwarzschild black hole is described by two quantum correction parameters $P$ and $a_{0}$. The constraints on the parameter $P$ have been obtained from various astrophysical observations Sahu:2015dea ; Zhu:2020tcf ; Yan:2022fkr , and the max one is $P<0.0675$ Zhu:2020tcf . Expanding Eqs. (2.2) and (2.3) in the power of $1/r$, one can see that the maximal correction from the parameter $P$ is at the order of $(MP)/r$, while the maximal correction from $a_{0}$ is at the order of $a_{0}^{2}/r^{4}$ Zhu:2020tcf . In this work, we focus on the physics of QNMs outside the event horizon. And the radius of the event horizon of a typical Schwarzschild black hole with the mass of the sun is of about $3$ km, then $P\sim\mathcal{O}(10^{-2})$ and $a_{0}^{2}/r^{4}\sim\mathcal{O}(10^{-67})$. So the effect of $a_{0}$ on astrophysical observation can be safely neglected, and we only care about the quantum correction from the parameter $P$.

To study the perturbations of a spherically symmetric black hole, one can first focus on axisymmetric modes of perturbations Chandrasekhar:1985kt . We consider a perturbed spacetime which is described by a non-stationary and axisymmetric metric as

$\displaystyle ds^{2}=-e^{2\nu}\left(dx^{0}\right)^{2}+e^{2\psi}\left(dx^{1}-\sigma dx^{0}-q_{2}dx^{2}-q_{3}dx^{3}\right)^{2}+e^{2\mu_{2}}\left(dx^{2}\right)^{2}+e^{2\mu_{3}}\left(dx^{3}\right)^{2},$

(2.5)

where $\nu$, $\psi$, $\mu_{2}$, $\mu_{3}$, $\sigma$, $q_{2}$, and $q_{3}$ depend on time coordinate $t$ $(t=x^{0})$, radial coordinate $r$ $(r=x^{2})$, and polar angle coordinate $\theta(\theta=x^{3})$. And a tetrad basis $e_{(a)}^{\mu}$ corresponding to the metric (2.5) is

	$\displaystyle e^{\mu}_{(0)}$	$\displaystyle=$	$\displaystyle(e^{-\nu},~{}~{}\sigma e^{-\nu},~{}~{}0,~{}~{}0),$
	$\displaystyle e^{\mu}_{(1)}$	$\displaystyle=$	$\displaystyle(0,~{}~{}e^{-\psi},~{}~{}0,~{}~{}0),$
	$\displaystyle e^{\mu}_{(2)}$	$\displaystyle=$	$\displaystyle(0,~{}~{}q_{2}e^{-\mu_{2}},~{}~{}e^{-\mu_{2}},~{}~{}0),$
	$\displaystyle e^{\mu}_{(3)}$	$\displaystyle=$	$\displaystyle(0,~{}~{}q_{3}e^{-\mu_{3}},~{}~{}0,~{}~{}e^{-\mu_{3}}).$		(2.6)

In this regard, one can project any vector or tensor field onto the tetrad frame by

$\displaystyle A_{(a)}=e^{\mu}_{(a)}A_{\mu},~{}~{}~{}~{}B_{(a)(b)}=e^{\mu}_{(a)}e^{\nu}_{(b)}B_{\mu\nu}.$

(2.7)

For a static and spherically symmetric spacetime, $\sigma$, $q_{2}$, and $q_{3}$ are zero. Then, comparing the metric (2.5) with (2.1), one can get

$\displaystyle e^{2\nu}=f(r),~{}~{}~{}~{}e^{-2\mu_{2}}=g(r),~{}~{}~{}~{}e^{2\mu_{3}}=h(r),~{}~{}~{}e^{2\psi}=h(r)\sin^{2}\theta.$

(2.8)

For a self-dual black hole, one can simulate the quantum corrections with an effective anisotropic matter fluid, and write the field equation as the Einstein equation form $G_{\mu\nu}=8\pi T_{\mu\nu}$, where $T_{\mu\nu}$ is the effective energy-momentum tensor Modesto:2009ve . In the tetrad frame, the field equation can be rewritten as

$\displaystyle R_{(a)(b)}-\frac{1}{2}\eta_{(a)(b)}R=8\pi T_{(a)(b)}.$

(2.9)

And it has been proven that the axial components of the perturbed energy-momentum tensor defined by an anisotropic fluid are zero in the tetrad formalism Chen:2019iuo . Therefore, the master equation of the axial gravitational perturbation of the self-dual black hole can be derived from the axial components of $R_{(a)(b)}=0$. The $(1,3)$ and $(1,2)$ components of $R_{(a)(b)}|_{\text{axial}}=0$ are

	$\displaystyle\left[he^{\nu-\mu_{2}}\left(q_{2,3}-q_{3,2}\right)\right]_{,2}$	$\displaystyle=$	$\displaystyle\left[he^{\mu_{2}-\nu}\left(\sigma_{,3}-q_{3,0}\right)\right]_{,0},$		(2.10)
	$\displaystyle\left[he^{\nu-\mu_{2}}\left(q_{3,2}-q_{2,3}\right)\sin^{3}\theta\right]_{,3}$	$\displaystyle=$	$\displaystyle\left[h^{2}e^{-\nu-\mu_{2}}\left(\sigma_{,2}-q_{2,0}\right)\sin^{3}\theta\right]_{,0},$		(2.11)

respectively, where $F_{,i}\equiv\frac{\partial F}{\partial x^{i}}$. Then, one can define

$\displaystyle Q=he^{\nu-\mu_{2}}\left(q_{2,3}-q_{3,2}\right)\sin^{3}\theta,$

(2.12)

and rewrite Eqs. (2.10) and (2.11) as

	$\displaystyle e^{\nu-\mu_{2}}\frac{Q_{,2}}{h\sin^{3}\theta}$	$\displaystyle=$	$\displaystyle\left(\sigma_{,3}-q_{3,0}\right)_{,0},$		(2.13)
	$\displaystyle e^{\nu+\mu_{2}}\frac{Q_{,3}}{h^{2}\sin^{3}\theta}$	$\displaystyle=$	$\displaystyle-\left(\sigma_{,2}-q_{2,0}\right)_{,0}.$		(2.14)

By differentiating Eqs. (2.13) and (2.14) and eliminating $\sigma$, one can obtain

$\displaystyle\frac{1}{\sin^{3}\theta}\left(\frac{e^{\nu-\mu_{2}}}{h}Q_{,2}\right)_{,2}+\frac{e^{\nu+\mu_{2}}}{h^{2}}\left(\frac{Q_{,3}}{\sin^{3}\theta}\right)_{,3}=\frac{Q_{,00}}{he^{\nu-\mu_{2}}\sin^{3}\theta}.$

(2.15)

Considering the ansatz Chandrasekhar:1985kt

$\displaystyle Q(r,\theta)=Q(r)Y(\theta)$

(2.16)

with $Y(\theta)$ the Gegenbauer function satisfying

$\displaystyle\frac{d}{d\theta}\left(\frac{1}{\sin^{3}\theta}\frac{dY}{d\theta}\right)=-\mu^{2}\frac{Y}{\sin^{3}\theta},$

(2.17)

where $\mu^{2}=(l-1)(l+2)$, one can rewrite Eq. (2.15) as

$\displaystyle\left(\frac{e^{\nu-\mu_{2}}}{h}Q_{,r}\right)_{,r}+\left(\frac{\omega^{2}}{he^{\nu-\mu_{2}}}-\frac{e^{\nu+\mu_{2}}\mu^{2}}{h^{2}}\right)Q=0.$

(2.18)

Note that here we have used the Fourier transformation $\partial t\rightarrow-i\omega$. Then, one can define

$\displaystyle\Psi(r)=\frac{Q(r)}{\sqrt{h(r)}}.$

(2.19)

With this, we can obtain the Schrödinger-like master equation of the axial gravitational perturbation for the self-dual black hole

$\displaystyle\frac{\partial^{2}\Psi}{\partial r^{2}_{\ast}}+\left[\omega^{2}-V(r)\right]\Psi=0,$

(2.20)

where

$\displaystyle V(r)=\frac{f(r)(l-1)(l+2)}{h(r)}-\sqrt{f(r)g(r)h(r)}\frac{d}{dr}\left(\frac{\sqrt{f(r)g(r)}}{h(r)}\frac{d\sqrt{h(r)}}{dr}\right)$

(2.21)

is the effective potential, and $r_{\ast}$ is the tortoise coordinate defined by

	$\displaystyle r_{\ast}$	$\displaystyle=$	$\displaystyle\int\frac{dr}{\sqrt{f(r)g(r)}}$		(2.22)
		$\displaystyle=$	$\displaystyle r-\frac{a_{0}^{2}}{r_{+}r_{-}}\left(\frac{1}{r}-\frac{r_{+}+r_{-}}{r_{+}r_{-}}\ln(r)\right)+\frac{1}{(r_{+}-r_{-})}\left(\frac{a^{2}_{0}+r^{4}_{+}}{r^{2}_{+}}\ln(r-r_{+})-\frac{a^{2}_{0}+r^{4}_{-}}{r^{2}_{-}}\ln(r-r_{-})\right).$

It is worthwhile to mention that $r_{\ast}$ running from $-\infty$ to $+\infty$ matches $r$ from the event horizon to spatial infinity. With different values of the parameter $P$, the plots for the effective potential (2.21) in the tortoise coordinate (2.22) are shown in Fig. 1. It can be seen that, the height of the effective potential increases with the parameter $P$.

Assuming a stationary, spherically symmetric, and asymptotically flat line element, Cardoso $et~{}al.$ showed that, in the eikonal limit $l\rightarrow\infty$, the QNMs of a black hole in any dimensions are Cardoso:2008bp

$\displaystyle\omega_{\text{QNM}}=\Omega_{c}l-i(n+1/2)|\lambda_{c}|,$

(3.6)

where the subscript $c$ means that the quantity is evaluated at the radius $r=r_{c}$ of a circular null geodesic, $\Omega_{c}$ and $\lambda_{c}$ are the coordinate angular velocity and the Lyapunov exponent of the circular null geodesics, respectively. It is an interesting relation between quasinormal frequencies and the parameters of the circular null geodesics, but it may be not valid in a specific black hole Konoplya:2017wot . In this section, based on Eq. (3.6), we shall derive the explicit relation between the QNMs, in the eikonal limit, and the parameters of the circular null geodesics of the self-dual black hole. Then we numerically verify this relation.

The Lagrangian for a photon in the equatorial plane ($\theta=\pi/2$) in the self-dual black hole is

$\displaystyle\mathscr{L}=\frac{1}{2}\left[-f(r)\dot{t}^{2}+\frac{1}{g(r)}\dot{r}^{2}+h(r)\dot{\varphi}^{2}\right],$

(3.7)

and the generalized momentum from this Lagrangian is

	$\displaystyle p_{t}$	$\displaystyle=$	$\displaystyle-f(r)\dot{t}=-E,$		(3.8)
	$\displaystyle p_{\varphi}$	$\displaystyle=$	$\displaystyle h(r)\dot{\varphi}=L,$		(3.9)
	$\displaystyle p_{r}$	$\displaystyle=$	$\displaystyle\frac{\dot{r}}{g(r)},$		(3.10)

where $E$ is the energy, $L$ is the angular momentum, and the dot denotes differentiation to an affine parameter along the geodesics of the photon. Because the Lagrangian (3.7) is independent of both $t$ and $\varphi$, $E$ and $L$ are conserved. From Eqs. (3.8) and (3.9), one can get

$\displaystyle\dot{\varphi}=\frac{L}{h(r)},~{}~{}\dot{t}=\frac{E}{f(r)}.$

(3.11)

The Hamiltonian for the photon is

$\displaystyle\mathscr{H}$

$\displaystyle=$

$\displaystyle\frac{1}{2}\left[-E\dot{t}+L\dot{\varphi}+\frac{1}{g(r)}\dot{r}^{2}\right]=0.$

(3.12)

With Eqs. (3.11) and (3.12), one can define the effective potential as

$\displaystyle V_{r}\equiv\dot{r}^{2}=g(r)\left[\frac{E^{2}}{f(r)}-\frac{L^{2}}{h(r)}\right].$

(3.13)

For the circular ($r=r_{c}$) null geodesics on the equatorial plane, the conditions $V_{r}=V_{r}^{\prime}=0$ lead to

$\displaystyle\frac{E}{L}=\pm\sqrt{\frac{f_{c}}{h_{c}}},~{}~{}f_{c}h_{c}^{\prime}=f_{c}^{\prime}h_{c}.$

(3.14)

In this case

$\displaystyle V_{r}^{\prime\prime}=\frac{L^{2}g_{c}}{f_{c}h_{c}^{2}}\left(f_{c}h_{c}^{\prime\prime}-f_{c}^{\prime\prime}h_{c}\right),$

(3.15)

and the coordinate angular velocity is

$\displaystyle\Omega_{c}=\frac{\dot{\varphi}}{\dot{t}}=\left(\frac{f_{c}}{h_{c}}\right)^{1/2}.$

(3.16)

The principal Lyapunov exponent is a quantity that characterizes the rate of separation of infinitesimally close geodesics. Using the expression of the principal Lyapunov exponent Cardoso:2008bp

$\displaystyle\lambda=\sqrt{\frac{V_{r}^{\prime\prime}}{2\dot{t}^{2}}},$

(3.17)

we get

$\displaystyle\lambda_{c}=\sqrt{\frac{g_{c}}{2h_{c}}\left(f_{c}h_{c}^{\prime\prime}-f_{c}^{\prime\prime}h_{c}\right)}$

(3.18)

for the circular null geodesics of the self-dual black hole. Taking Eqs. (3.16) and (3.18) into (3.6), we derive the relation between the QNMs, in the eikonal limit, and the parameters of the circular null geodesics of the self-dual black hole

$\displaystyle\omega_{\text{QNM}}=l\left(\frac{f_{c}}{h_{c}}\right)^{1/2}-i(n+1/2)\sqrt{\frac{g_{c}}{2h_{c}}\left(f_{c}h_{c}^{\prime\prime}-f_{c}^{\prime\prime}h_{c}\right)}.$

(3.19)

Setting $l=100$ and ${n=0,1,2,3,4}$, we calculate the quasinormal frequencies in the eiknoal limit by the WKB approximation method and the relation (3.19). We list the numerical results in Tab. 5. Considering the error of calculation, one can find that the quasinormal frequencies obtained by the WKB approximation method and the relation (3.19) agree well with each other in Tab. 5. It means that the general relation (3.6), between quasinormal frequencies and the parameters of the circular null geodesics, is right for the self-dual black hole in loop quantum gravity.