How close can we approach the event horizon of the Kerr black hole from the detection of the gravitational quasinormal modes?

Using the WKB method, we show that the peak location ($r_{\rm peak}$) of the potential, which determines the quasinormal mode frequency of the Kerr black hole, obeys an accurate empirical relation as a function of the specific angular momentum $a$ and the gravitational mass $M$. If the quasinormal mode with $a/M \sim 1$ is observed by gravitational wave detectors, we can confirm the black-hole space-time around the event horizon, $r_{\rm peak}=r_+ +O(\sqrt{1-q})$ where $r_+$ is the event horizon radius. While if the quasinormal mode is different from that of general relativity, we are forced to seek the true theory of gravity and/or face to the existence of the naked singularity.


Introduction Coalescing binary black holes (BHs) form a BH in numerical relativity
simulations [1][2][3]. The BH radiates characteristic gravitational waves (GWs) with the quasinormal mode (QNM) frequencies which dominates in the final phase of the merger of two BHs. QNM will be detected by the second generation GW detectors such as Advanced LIGO (aLIGO) [4], Advanced Virgo (AdV) [5], and KAGRA [6,7]. QNM is also one of the targets for space based GW detectors such as eLISA [8] and DECIGO [9]. However to confirm the QNM GWs, we need a sufficiently high signal-to-noise ratio (see e.g., Refs. [10,11]).
To calculate GWs from the Kerr BH [12], we need to solve the Teukolsky equation [13]. The radial Teukolsky equation for gravitational perturbations in the Kerr space-time which is expressed as where T is the source and with where ∆ = r 2 − 2M r + a 2 with M and a being the mass and the spin parameter, respectively.
In this paper, we use the geometric unit system, where G = c = 1. Here, we consider the Kerr metric in the Boyer-Lindquist coordinates as where Σ = r 2 + a 2 cos 2 θ. The constants m and λ in the Teukolsky equation come from the spin-weighted spheroidal function Z aω ℓm (θ, φ). A prime is the derivative with respect to r. When we consider GWs emitted by a test particle falling into the Kerr BH, the source term T diverges as ∝ r 7/2 and the potential V is the long range one, which motivated Sasaki and Nakamura [14][15][16] to consider the change of the variables and the potential. Using two functions α(r) and β(r) for the moment, let us define various variables as Then, we have a new wave equation for X from the Teukolsky equation as where dr * /dr = (r 2 + a 2 )/∆. We define α and β by where with P = r 2 + a 2 gh Here, g and h are free functions under the restrictions to guarantee the convergent source term and the short range potential which is given by 2/8 for r * → −∞, and for r * → +∞. In Refs. [14,15], h and g are adopted as Defining a new variable Y by X = √ γ Y , we have where In our previous paper [17], it was shown that this V SN has double peaks for q = a/M > 0.8 to refuse the approach to determine the complex frequency of QNMs from the peak location r peak which is real-valued, of the absolute value of V SN . Our new g is defined by The above new g seems to violate the needed dependence for r → ∞. However Eq. (18) tells only the sufficient condition but not the necessary one. In reality, our new V NNT 1 is confirmed to be short-ranged, that is, V NNT is in promotion to 1/r 2 for r → ∞ and becomes the Regge-Wheeler potential [18] for a = 0 as V SN . After adopting the new potential V NNT , what we will do is to ask which part of the Kerr metric determines the QNMs for given 0.8 < q < 1. Conversely, if the QNM GWs are observed, which part of the Kerr BH we can say being confirmed, which is the main theme of this paper.
The essential technique to determine the QNMs by using the WKB method was proposed by Schutz and Will [19] for the Schwarzschild BH using the Regge-Wheeler potential V RW [18,20]. They approximated V RW near its peak radius at r 0 ≈ 3.28M as where r * 0 = r 0 + 2M ln(r 0 /2M − 1). The QNM frequencies are expressed as with n = 0, 1, 2, · · ·. As for the accuracy of fundamental n = 0 QNM frequency with ℓ = 2, the errors of the real (ω r = Re(ω)) and the imaginary (ω i = Im(ω)) parts are 7% and 0.7%, respectively, compared with the numerical results of Chandrasekhar and Detweiler [21]. This suggests that for the fundamental QNM of the a = 0 case, the space-time of a Schwarzschild BH around r ≈ 3.28M is confirmed through the detection of the QNM GWs. The word "around" has two meaning that the GW cannot be localized due to the equivalence principle and the imaginary part of the QNMs is determined by the curvature of the potential, which reflects the space-time structure of the Schwarzschild BH around r ≈ 3.28M . In our previous papers [17,22], we succeeded in doing similar procedures up to q = 0.98 for the Kerr BH with the Detweiler potential [23] (see also Ref. [24]). However, above q = 0.98, we could not derive the consistent QNMs in the method to use r peak of the absolute value of the potential.
In Fig. 1, we show that new Sasaki-Nakamura potential V NNT has only a single strong peak to allow us to identify r peak up to q → 1 successfully. Here, the QNM frequencies have been calculated accurately by the Leaver's method [25]. It is noted that when we consider an additional function a(M − a)/r 2 for Eq. (22), the peak location r peak changes only 0.4% and 0.06% for q = 0.98 and 0.9999, respectively, which show that the results do not depend on the choice of g significantly.
which is derived from 13 data between q = 0.98 and q = 0.9999 yielding the correlation coefficient of 0.999995 and the chance probability of 7.4 × 10 −29 . The red and green dotted curves present the event horizon radius, and the inner light ring radius [26], r lr /M = 2 + 2 cos 2 3 cos −1 (−q) The latter radius is evaluated in the equatorial (θ = π/2) plane. It is noted that there are various studies on the relation between the QNMs and the orbital frequency of the light ring orbit (see a useful lecture note [27]  The light blue dashed curve in Fig. 2 is obtained as follows. Here, we introduce a fitting curve to evaluate λ by using s A ℓm which is a constant defined in Eq. (25) of Ref. [28], as −2 A 22 = 0.545652 + (6.02497 + 1.38591 i)(− ln q) 1/2 , where λ is related to s A ℓm as λ = s A ℓm − 2amω + a 2 ω 2 . And also, the (n = 0) QNM frequency with (ℓ = 2, m = 2) is described by Ref. [29] as Then, using q = 1 − ǫ and expanding V NNT with respect to ǫ, we estimate the r peak analytically. Instead of finding the peak location of |V NNT |, we derive the location of dV NNT /dr * = 0. The location has a O( √ ǫ) term which is consistent with r fit because (− ln(1 − ǫ)) 1/2 = √ ǫ + O(ǫ 3/2 ), and the result is Here, we have ignored the tiny imaginary contribution of −0.020157 i. It should be noted that a different choice of g makes a difference of O((1 − q) 1/2 ) in the estimation of the peak location. We will discuss the detail in our future work.

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To confirm our analysis, we present the QNM frequencies via the WKB approximation in Fig. 3. The errors in the real and imaginary parts of the QNM frequencies are plotted in Fig. 4. WKB ω r WKB ω i × 10 Fig. 3 The real and imaginary parts of the fundamental (n = 0) QNM frequencies with V NNT (ℓ = 2, m = 2) evaluated for various spin parameters q = a/M . The exact frequencies ω r and ω i are from Refs. [30,31], and the Leaver's method [25]. δ r δ i Fig. 4 Absolute value of relative errors for the real and imaginary part of the QNM frequencies with V NNT (ℓ = 2, m = 2), δ r = |(WKB ω r )/ω r − 1| and δ i = |(WKB ω i )/ω i − 1| between the exact value and that of the WKB approximation in Fig. 3. 3. Discussion There is the ergoregion in the Kerr BH where the timelike Killing vector turns out to be spacelike. The boundary of the ergoregion is given by which is called the ergosphere. Note here that r ergo (0) = r + and r ergo (π/2) = 2M . The physical origin of the Penrose mechanism [32] and the Blanford-Znajek mechanism [33] is the 6/8 eroregion, which enables the extraction of the rotational energy of the Kerr BH. There are many papers using these mechanisms while the confirmation of the existence of the ergoregion has not been done observationally. The detection of QNM GWs, for example, for q = 0.9999 can confirm its existence. Let us define the covering solid angle 4πC of the ergoregion for the given sphere of the radius r peak . Then, C is given by C = cos θ m with r peak /M = 1 + 1 − q 2 cos 2 θ m . Then if, for example, the QNM with q = 0.9999 is observed by GW detectors, we can confirm the space-time around r = 1.01445M covering 99.9996% of the ergoregion. Here, the horizon radius for q = 0.9999 is 1.01414M . Therefore, the spacetime at only 1.0003 times the event horizon can be confirmed. If the QNM is confirmed to be different from that of general relativity by the detection of the GWs, a very serious problem is raised since the Kerr BH is the unique solution of the stationary vacuum solution of the Einstein equation under the assumption of the cosmic censorship [34][35][36]. The Einstein equation and/or the cosmic censorship are wrong. In the former case, the true theory of gravity should be determined to be compatible with the data of QNMs. In the latter case, we face the existence of the naked singularity, which requires a new physics law possibly related to quantum gravity. As for the detection rate, the population Monte Carlo simulation by Kinugawa et al. [37][38][39] showed that for the Pop III binary BHs, 0.43% have the final q > 0.98 in their standard model, in which they adopted the various parameters and functions for the Pop I stars like the sun except for the initial mass function. Since the Pop III star which is the first star in our universe without metal that has atomic number larger than carbon has not been observed so that these parameters are highly unknown. This suggests that the percentage of BHs with q > 0.98 can be either larger or smaller than 0.43% . While the massive BHs from Pop I and Pop II BH binaries are also expected [40] although the spin parameter data are not available from their paper at present. This suggests that the second generation detectors might detect the QNM GWs of such BH with q > 0.98. The third generation detectors such as the Einstein Telescope (ET) [41] will increase the detection number ∼ 1000 larger. Another possibility is QNM GWs from very massive BHs of mass ∼ 10 4 M ⊙ and ∼ 10 7 M ⊙ for DECIGO [9] and eLISA [8], respectively.
In conclusion, the present and the future GW detectors with the frequency ∼ 10 −3 Hz to 100 Hz, would observe the very strong gravity almost at the event horizon radius to clarify the true theory of gravity.