Hawking radiation - quasi-normal modes correspondence and effective states for nonextremal Reissner-Nordstr\"{o}m black holes

It is known that the nonstrictly thermal character of the Hawking radiation spectrum harmonizes Hawking radiation with black hole (BH) quasi-normal modes (QNM). This paramount issue has been recently analyzed in the framework of both Schwarzschild BHs (SBH) and Kerr BHs (KBH). In this assignment, we generalize the analysis to the framework of nonextremal Reissner-Nordstr\"{o}m BHs (RNBH). Such a generalization is important because in both SBHs and KBHs an absorbed (or emitted) particle has only mass. Instead, in RNBHs the particle has charge as well as mass. In doing so, we expose that for the RNBH, QNMs can be naturally interpreted in terms of quantum levels for both particle emission and absorption. Conjointly, we generalize some concepts concerning the RNBH's"effective states".


Introduction
A RNBH of mass M is identical to a SBH of mass M except that a RNBH has the nonzero charge quantity Q. In this paper, we are interested in RNBHs with the nonextremal constraint M > Q [1]. The quantity Q is the physical mechanism for the RNBH's dual horizons from eq. (1) in [1] because the RNBH outer (event) horizon radius R +RNBH (M, Q) and the RNBH inner (Cauchy) horizon radius R −RNBH (M, Q) are clearly functions of both M and Q, not just M as in the well known case of the SBH horizon radius Energy conservation plays a fundamental role in BH radiance [2] because the emission or absorption of a Hawking quanta with mass m and energy-frequency ω causes a BH of mass M to undergo a transition between discrete energy spectrum levels [3]- [7], where for G = c = k B = = 1 4πǫ0 = 1 (Planck units). Given that emission and absorption are reverse processes for the quantized energy spectrum conservation [3]- [7], we consider this pair of transitions as being equal in magnitude but opposite in direction from the neutral radius perspective of r 0 = (r + + r − )/2.
It is known that the countable character of successive emissions of Hawking quanta which is a consequence of the nonstrictly thermal character of the Hawking radiation spectrum (see [3]- [7] and [22]- [26]), generates a natural correspondence between Hawking radiation and BH QNM [3]- [7]. Moreover, it has also been shown that QNMs can be naturally interpreted in terms of quantum levels, where the emission or absorption of a particle is interpreted as a transition between two distinct levels on the discrete energy spectrum [3]- [7]. The thermal spectrum correction is an imperative adjustment to the physical interpretation of BH QNMs because these results are important to realize the underlying unitary quantum gravity theory [3]- [7]. Hod's intriguing works [8,9] suggested that BH QNMs carry principle information regarding a BH's horizon area quantization. Hod's influential conjecture was later refined and clarified by Maggiore [10]. Moreover, it is also believed that QNMs delve into the micro structure of spacetime [11].
To make sense of the state space for the energy spectrum states and the underlying BH perturbation field states, an effective framework based on the nonstrictly thermal behavior of Hawking's framework began to emerge [3]- [7]. In the midst of this superceding BH effective framework [3]- [7], the BH effective state concept was originally introduced for KBHs in [6] and subsequently applied through Hawking's periodicity arguments [30,31], to the BH tunneling mechanism's nonstrictly black body spectrum [7]. The effective state is meaningful to BH physics and thermodynamics research because one needs additional features and knowledge to consider in future experiments and observations.
In this paper, our objective is to apply the nonstrictly thermal BH effective framework of [3]- [7] to nonextremal RNBHs. Thus, upon recalling that a RNBH of mass M is identical to a SBH of mass M except that a RNBH has the charge Q, we prepare for our BH QNM investigation by reviewing relevant portions of the SBH effective framework [3]- [7] for quantities related to SBH states and transitions in Section 2. Then in Section 3, we launch our RNBH QNM exploration by introducing a RNBH effective framework for quantities pertaining to RNBH states and transitions. Finally, we conclude with a brief comparison between the fundamental SBH and RNBH results of Section 4 followed by the recapitulation of Section 5.
2 Schwarzschild black hole framework: background and review

Schwarzschild black hole states and transitions
Here, we recall some quantities that characterize the SBH. First, consider a SBH of initial mass M , when the SBH emits or absorbs a quantum of energy-frequency ω (for particle mass m and SBH mass change ∆M , such that m = ω = ∆M ) to achieve a final mass of M − ω or M + ω, respectively, for the SBH mass-energy transition between states in state space. Thus, we follow [3]- [5], where the SBH initial and final horizon area are respectively, for the SBH area quanta number such that the SBH horizon area change for the corresponding mass change ∆M is because the transition's minus (−) and plus (+) signs depend on emission and absorption, respectively. Next, in [3]- [5], the Bekenstein-Hawking SBH initial and final entropy are respectively, where the corresponding SBH entropy change is Subsequently, the SBH initial and final total entropy are [3]-[5] respectively. Additionally, the SBH initial and final Hawking temperature are [3]- [5] T HSBH (M ) = 1 8πM respectively. Therefore, the quantum transition's SBH emission tunneling rate

Schwarzschild black hole effective states and transitions
Here, we recall some effective quantities that characterize the SBH. Given that M is the mass state before and M ± ω is the mass state after the quantum transition, the SBH effective mass and SBH effective horizon are respectively identified in [3]- [5] as which are average quantities between the two states before and after the process [3]- [5]. Consequently, using eqs. (4) and (12) we define the SBH effective horizon area as which is the average of the SBH's initial and final horizon areas. Subsequently, utilizing eq. (7), the Bekenstein-Hawking SBH effective entropy is defined as and consequently employs eqs. (13) and (14) to define the SBH effective total entropy as Thus, employing eqs. (3) and (10), the SBH effective temperature is which is the inverse of the average value of the inverses of the initial and final Hawking temperatures. Consequently, eq. (16) lets one rewrite eq. (11) to define the SBH effective emission tunneling rate (in the Boltzmann-like form) such that eq. (14) defines the SBH effective entropy change as because the SBH effective horizon area change is and the SBH effective area quanta number is

Effective application of quasi-normal modes to the Schwarzschild black hole
Here, we recall how the SBH perturbation field QNM states can be applied to the SBH effective framework. The quasi-normal frequencies (QNF) are typically labeled as ω nl , where l is the angular momentum quantum number [3]- [5,10,12]. Thus, for each l, such that l ≥ 2 for gravitational perturbations, there is a countable sequence of QNMs labeled by the overtone number n, which is a natural number [3]- [5,10]. Now |ω n | is the damped harmonic oscillator's proper frequency that is defined as [3]- [5,10] Maggiore [10] articulated that the establishment |ω n | = ω n R is only correct for the very long-lived and lowly excited QNMs approximation |ω n | ≫ ω n I , whereas for a lot of BH QNMs, such as those that are highly excited, the opposite limit is correct [3]- [5,10]. Therefore, the ω parameter in eqs. (12) − (20) is substituted for the |ω n | parameter [3]- [5] because we wish to employ BH QNFs. When n is large the SBH QNFs become independent of l and thereby exhibit the nonstrictly thermal structure [3]- [5] Thus, when referring to highly excited QNMs one gets |ω n | ≈ p n [3]- [5], where the quantized levels differ from [10] because they are not equally spaced in exact form. Therefore, according to [3]- [5], we have which is solved to yield when we obey |ω n | < M because a BH cannot emit more energy than its total mass.

Reissner-Nordström black hole states and transitions
Here, we recall some quantities that characterize the RNBH. First, consider a RNBH of initial mass M and initial charge Q. Using eq. (1), we define the RNBH initial event horizon area as the Bekenstein-Hawking RNBH initial entropy as and the RNBH initial electrostatic potential as Consequently, eq. (17) of [2] identifies the RNBH initial Hawking temperature as Second, consider when the RNBH emits or absorbs a quantum of energy-frequency ω with charge q to achieve a final mass of M − ω or M + ω and a final charge of Q − q or Q + q, respectively, for the RNBH mass-energy transition between states in state space. For this, all we need to do is replace the RNBH's mass and charge parameters in eqs. (26) and (29). Thus, eq. (26) establishes the RNBH final event horizon area as and eq. (28) defines the RNBH final electrostatic potential as for usage in eq. (29) of [13], where it is proposed that the RNBH adiabatic invariant is because ∆Q = q. Hence, eq. (29) identifies the RNBH final Hawking temperature as Next, upon generalizing eq. (16) in [2] and the work [27], we define the RNBH tunneling rate as such that the RNBH event horizon area change is so we can define the RNBH event horizon area quanta number as

Reissner-Nordström black hole effective states and transitions
Here, we define some effective quantities that characterize the RNBH. The RNBH effective mass is equivalent to the SBH effective mass component of eq. (12), which is Next, we define the RNBH effective charge as which is the average of the RNBH's initial charge Q and final charge Q ± q. From this, eqs. (1), (39) and (40) are used to define the corresponding RNBH effective event horizon and RNBH effective cauchy horizon as q), (41) with respect to the energy conservation and pair production neutrality of eq. (39). Next, we employ eqs. (26), (39) and (41) to define the RNBH effective event horizon area as which is then used to define the RNBH effective entropy as Afterwards, we use eqs. (28) and (42) to define the RNBH effective electrostatic potential as such that the RNBH effective entropy change is defined as for the RNBH effective event horizon area change and the RNBH effective event horizon area quanta number

Effective application of quasi-normal modes to the Reissner-Nordström black hole
Here, we explain how the RNBH perturbation field QNM states can be applied to the RNBH effective framework. Similarly to SBH QNFs, the RNBH QNFs become independent of l for large n [14]. Thus, for large n, we have two families of the QNM: and (52) Now the approximation of eqs. (51) and (52) are only relevant under the assumption that the BH radiation spectrum is strictly thermal [3]- [5] because they both use the Hawking temperature T +H RN BH in eq. (29). Hence, to operate in compliance of [3]- [5] and thereby account for the thermal spectrum deviation of eq. (35), we opt to select the eq. (52) case and upgrade it by effectively replacing its T HRNBH in eq. (29) with the T +E RN BH in eq. (46). Therefore, the corrected expression for the RNBH QNFs of eq. (52) that encodes the nonstrictly thermal behavior of the radiation spectrum is defined as From eqs. (39), (41), and (46) we define the effective quantities associated to the QNMs as and respectively, for the quantum overtone number n in eq. (53). Hence, eq. (53) lets us rewrite the SBH case of eq. (23) to present the RNBH case Thus, we recall that if |ω n | ≈ p n , then we are referring to highly excited QNMs [3]- [5] . Therefore, the SBH case of eq. (24) becomes the RNBH case Hence, upon considering eqs. (40) and (54), one can rewrite eq. (58) as where the solution of eq. (59) in terms of |ω n | will be the answer of |ω n |. Therefore, given a quantum transition between the levels n and n − 1, we define |△ω n,n−1 | ≡ |ω n − ω n−1 | where eqs. (41)-(45) are rewritten as respectively.

A brief comparison
Here, we will show that the SBH results of Section 2 are in fundamental agreement with the RNBH results of Section 3 for small Q, where we recall that the RNBH of mass M is identical to a SBH of mass M except that a RNBH has the nonzero charge quantity Q.
First, for small Q, the SBH's T ESBH (M, ω) of eq. (16) is related to the RNBH's T +E RN BH (M, ω, Q, q) of eq. (46) as Second, for small Q, the SBH's A ESBH (M, ω) of eq. (13) complies with the RNBH's A +E RN BH (M, ω, Q, q) of eq. (42) as Third, for small Q, the SBH's S ESBH (M, ω) of eq. (14) corresponds with the RNBH's S +E RN BH (M, ω, Q, q) of eq. (43) as Fourth, for small Q, the SBH's QNF |ω n | of eq. (24) is consistent with the RNBH's QNF |ω n | of eqs. (58) and (59) as which can be applied to eqs. (69)-(70) by replacing the ω parameter with the pertinent |ω n |. Hence, eqs. (69)-(72) indicate that in general, the SBH results of Section 2 are fundamentally consistent with the RNBH results of Section 3 for small Q. Moreover, in eq. (72) for large n, the result is consistent with the SBH because ln 2 is negligible, but for small n there is an argument between scientists regarding ln 2 and ln 3 because these refer to the two distinct QNM families of eqs. (51) and (52).
Here, we provide the physical answer of eq. (72) for the case of emission by using the fact that Q is small, so the term which includes Q 2 is also very small and therefore negligible: Thus, by setting (ω 0 ) n ≡ |ω n | we obtain (ln 2) 2 + 4π 2 (n − 1 2 ) 2 (74) for an emission involving quantum levels n and n − 1, which becomes for large n.

Conclusion remarks
We began our paper by summarizing some basic similarities and differences between SBHs and RNBHs in terms of charge and horizon radii. Moreover, we briefly explored the Parikh-Wilczek statement that explains how energy conservation and pair production [2,21] are fundamentally related to such BHs. For a BH's discrete energy spectrum, the emission or absorption of a particle yields a transition between two distinct levels, where particle emission and absorption are reverse processes [3]- [7]. For this, we touched on the important issue that the nonstrictly thermal character of Hawking's radiation spectrum generates a natural correspondence between Hawking's radiation and BH QNMs, because these structures exemplify features of the BH's energy spectrum [3]- [5], which has been recently generalized to the emerging concept of a BH's effective state [6,7]. Next, we prepared for our nonextremal RNBH QNM investigation by first reviewing relevant portions of the SBH effective framework [3]- [5] in Section 2. Here, we listed the noneffective and effective quantities for SBH states and transitions, with direct application to the QNM characterization and framework of [3]- [5]. Subsequently, in Section 3, we identified some existing noneffective quantities and introduced new effective quantities for RNBH states and transitions so we could apply the BH framework of [3]- [5] to implement a RNBH framework. These results are crucial because the effective quantities in [3]- [5] have been achieved for the stable four dimensional RNBH solution in Einstein's general relativity-now effective frameworks exist for the SBH, KBH, and (nonextremal) RNBH solutions.
Ultimately, the RNBH effective quantities permitted us to utilize both the KBH's effective state concept [6,7] and the BH QNMs [3]- [5] to construct a foundation for the RNBH's effective state in this developing BH effective framework. The RNBH effective state concept is meaningful because, as scientists who wish to demystify the BH paradigm, we need additional features and knowledge to consider in future experiments and observations.
Finally, we stress that the nonstrictly thermal behavior of the Hawking radiation spectrum has been recently used to construct two very intriguing proposals to solve the BH information loss paradox. The first one received the First Award in the 2013 Gravity Research Foundation Essay Competition [26]. The latter won the Community Rating at the 2013 FQXi Essay Contest -It from Bit or Bit from It [28]. We are working to extend this second approach to the RNBH framework [29].