Quasinormal modes and second order thermodynamic phase transition for Reissner-Nordstr\"om black hole

The relation between the quasinormal modes (QNMs) and the second order thermodynamic phase transition (SOTPT) for the Reissner-Nordstr\"om (RN) black hole is studied. It is shown that the quasinormal frequencies of the RN black hole start to get a spiral-like shape in the complex $\omega$ plane and both the real and imaginary parts become the oscillatory functions of the charge if the real part of the quasinormal frequencies arrives at its maximum at the second order phase transition point of Davies for given overtone number and angular quantum number. That is to say, we can find out the SOTPT point from the QNMs of the RN black hole. The fact shows that the quasinormal frequencies carry the thermodynamical information of the RN black hole.

The QNMs of a black hole are defined as proper solutions of the perturbation equations belonging to certain complex characteristic frequencies which satisfy the boundary conditions appropriate for purely ingoing waves at the event horizon and purely outgoing waves at infinity [1]. They are entirely fixed by the structure of the background spacetime and irrelevant of the initial perturbations [1,2]. Thus, it is generally believed that QNMs carry the footprint to directly identify the existence of a black hole. Meanwhile, the study of QNMs may lead to a deeper understanding of the thermodynamical properties of black holes in loop quantum gravity [3] [4], as well as the QNMs of anti-de Sitter black holes have a direct interpretation in terms of the dual conformal field theory [5,6,7].
On the other hand, one of important characteristics of a black hole is its thermodynamical properties. It is well known that the heat capacity of the Schwarzschild black hole is always negative and so the black hole is thermodynamical unstable. But for the RN black hole, the heat capacity is negative in some parameter region and positive in other region. Davies pointed out that the phase transition appears in black hole thermodynamics and the SOTPT takes place at the point where the heat capacity diverges [8,9,10].
Because the QNMs of a black hole are entirely fixed by the structure of the background spacetime and the SOTPT is only related to the parameters of the black hole, an interesting question is whether there are some relations between them. The aim of this paper is to study this question for the RN black hole, and we find that the quasinormal frequencies do carry the thermodynamical information of the black hole.
Assuming that the azimuthal and time dependence of the fields will be the form e −i(ωt−mϕ) and using the Newman-Penrose formalism [11], we can obtain the separated equations for massless scalar, Dirac, and Rarita-Schwinger (RS) perturbations around a RN black hole [12,13,14] [ [∆D † 1+s D 0 + 2(2s + 1)iωr − λ]R s = 0, where ∆ = r 2 − 2Mr + Q 2 , M and Q represent the mass and charge of the black hole, and λ is the angular separation constant [15,16,17] where l and j are the quantum number characterizing the angular distribution for the boson and fermion perturbations respectively. Introducing an usual tortoise coordinate dr * = r 2 /∆dr and resolving the equation in the form R s = ∆ −s/2 Ψ s /r, we can rewrite the radial wave equations in Eqs. (1) and (2) as with The boundary conditions on wave function Ψ s at the horizon and infinity can be expressed where κ ± = (r + − r − )/(2r 2 ± ) is the surface gravity on the horizons r ± . A solution to Eq. (4) that has the desired behavior at the boundary can be written as If we take r + + r − = 1, the sequence of the expansion coefficients {a m : m = 1, 2, ....} is determined by a three-term recurrence relation staring with a 0 = 1: The recurrence coefficient α m , β m and γ m are given in terms of m and the black hole parameters by α m = m 2 + (C 0 + 1)m + C 0 , and the intermediate constants C m are defined by where B = (r 2 The series in (7) converges and the r = +∞ boundary condition (6) is satisfied if, for a given s and λ, the frequency ω is a root of the continued fraction equation This leads to a simple method to find quasinormal frequencies of the RN black hole -defining a function which returns the value of the continued fraction for an initial guess at the frequency, and then use a root finding routine to find the zeros of this function in the complex ω plane.
The frequency for which happens is a quasinormal frequency [18,19].
The Figs. 1-3 describe the QNMs for the scalar (s=0), Dirac (s=-1/2), and RS fields (s=-3/2) obtained by the continued fraction method respectively. In Fig The critical value of the overtone number n c increases as the angular quantum number l (or j) increases for a given perturbation, and it also increases as the absolute value of spins |s|  increases for the same level of the angular quantum number (here the least level is defined as l = 0, j = 1/2, and j = 3/2 for the scalar, Dirac, and RS fields respectively).
On the other hand, we know that the heat capacity C Q of the RN black hole is given by [8,9] where T is the temperature and S is the entropy of the black hole. It is obvious that the singular    4 shows that the heat capacity is negative in the region Q < Q sp and positive in the region Q > Q sp . Davies [8,9,10] showed that the SPHC is just the SOTPT point of the black hole thermodynamics.
To find the relation between the QNMs and the SOTPT point, we present critical point Q cp , which is the value of the charge at which the real part of the first oscillatory quasinormal frequencies arrives at its maximum, with the different critical overtone number n c for the scalar, Dirac and RS fields in Table I  Besides, for the Schwarzschild black hole there is no SPHC because its heat capacity is always negative and there is no critical point because its QNMs never get a spiral-like.
To compare them farther more, we take K = dω I dω R as the slope of the QNMs. For Q < Q cp ,  τ R presented here is similar to that of the temperature of the black hole when it takes the same quality of heat. If we take a slope of the time scale as K τ = dτ I /dτ R , the trajectory of the slope takes the same form as that of C Q versus Q because K τ = −K(ω R /ω I ) 2 . Above discussions show us that the critical point of the QNMs may be associated with the SOTPT point.
In summary, we study the relation between the QNMs and the SOTPT for the RN black hole and find the following results: if the real part of the quasinormal frequencies arrives at its maximum at the SOTPT point for given overtone number and angular quantum number, the QNMs will start to get a spiral-like shape in the complex ω plane, and both the real and imaginary parts will become the oscillatory functions of the charge. The QNMs will (not) take a spiral-like shape if the angular quantum number or the overtone number larger (less) than this given value. If a black hole does not possess the SOTPT point, its QNMs never take a spiral-like in the complex ω plane. Besides the fact that the critical point Q cp of the QNMs is in good agreement with the SOTPT point Q sp , the transition from negative to positive value of the slope K (K τ ) of the QNMs (time scale) occurs via a infinite discontinuity, and the trajectories of K and K τ take the same form as that of the heat capacity. These facts