Hawking Radiation Power Equations for Black Holes

We derive the Hawking radiation power equations for black holes in asymptotically flat, asymptotically Anti-de Sitter (AdS) and asymptotically de Sitter (dS) black holes, This is done by using the greybody factor for these black holes. We observe that the radiation power equation for asymptotically flat black holes, corresponding to greybody factor at low frequency, depends on both the Hawking temperature and the horizon radius. However, for the greybody factors at asymptotic frequency, it only depends on the Hawking temperature. We also obtain the power equation for asymptotically AdS black holes both below and above the critical frequency. The radiation power equation for at asymptotic frequency is same for both Schwarzschild AdS and Reissner-Nordstr\"om AdS solutions and only depends on the Hawking temperature. We also discuss the power equation for asymptotically dS black holes at low frequency, for both even or odd dimensions.


I. INTRODUCTION
The evaporation of a black hole can be understood in terms of the black body factor and the greybody factor. The black body factor is calculated from the probability of a particle being created in the vicinity of a horizon, and the greybody factor is calculated from the probability that this particle penetrates the potential barrier and escapes to infinity. The analysis of the black hole greybody factors shows that Hawking emission from a highly rotating black hole is strongly spin dependent, with particles of highest spin (gravitons) dominating the energy spectrum [1].
Gravitational greybody factors are analytically computed for static, spherically symmetric black holes, including black holes with charge and in the presence of a cosmological constant [2]. In this context, the greybody factors for both asymptotically dS and AdS spacetimes can be obtained. There are many distinct models with exact black hole solutions in literature, which in turn implies the need for concrete calculations of the corresponding greybody factors (some recent developments can be found in [3][4][5][6][7][8][9][10][11][12][13][14]).
These greybody factors were studied long ago in [15,16]. one can easily follow the basic setup of the calculations. There might be some trouble in order to obtain exact results with such set-up. The scattering problem of black holes has common features of scattering in media with some index of refraction. In fact, the curvature of space-time itself involved in the scattering of black holes. [17]. The quantum Hawking evaporation of near-extremal Reissner-Nordström black holes is studied recently, where the effective curvature potential causes distortion in the familiar radiation spectrum of genuine (3 + 1)-dimensional perfect black-body emitters [18].
The Hawking radiations of black holes in asymptotically flat, AdS and dS spacetimes through greybody factors are subject of interesting investigations. With the help of radiation power expression, one can compute total energy emitted as Hawking radiation by multiplying the power with black hole evaporation time scale and this must be equal to the total mass of the black hole by virtue of energy conservation. This may provide an insight to the process of black hole evaporation.
It may be noted that we use a formalism that depends on the frequency, and can be consistently applied to the greybody factors which are functions of all frequencies [2]. So, we use a form of the greybody factor which is defined for all the frequencies. Here we note that the low energy expression for the greybody factor for scalar fields in the background of a higher-dimensional Schwarzschild black hole have been obtained derived in [19]. The low-energy greybody factor for a higher-dimensional dS black hole have also been studied [20]. The emission of Hawking radiation of higher-dimensional black holes in the bulk (greybody factors and radiation spectra) are studied in details for the emission of scalar modes, and the ratio of the missing energy over the visible one is calculated for different values of the number of extra dimensions [21]. The procedure of scalar wave absorption by a black hole is followed here. This is because of nature of scalar wave which spreads from infinity over whole spacetime and becomes reflected by the black hole potential barrier. The transmitted scalar wave, near the horizon, appears as the incoming radiation. In fact, for low frequency scattering, the greybody factor is found equal to the absorption probability of the black hole. The reason to consider the real frequencies at low energy is the opposite nature of scattering and absorption processes. Our work is mainly based on the results of [2], where it is emphasized that the main contribution to the greybody factor of black holes comes from the l = 0 mode in the low frequency limit.
With this motivation, in this paper, we calculate the Hawking radiation power equation for black holes in d + 1-dimensional asymptotically flat, AdS and dS spacetimes with the help of following equation 1 [22,23] where x ≡ ω T H , T H is the Hawking temperature and γ(ω) represents greybody factor. First, we consider the greybody factor at the low frequency limit for black holes in asymptotically flat spacetime and compute the respective Hawking radiation power equation which depends on both, the Hawking temperature and horizon radius R H under different power law. Then, we derive the Hawking radiation power equation corresponding to the greybody factor at asymptotic frequency for Schwarzschild solution and find that it depends on the Hawking temperature only and does not depend on horizon radius. Interestingly, we observe that the radiation power equations corresponding to greybody factors at asymptotic frequency are same for all spacetime dimensions as it does not depend on the dimension d. In fact, it depends on Hawking temperature only with power law. Further, we compute the Hawking radiation power equation along greybody factor at the low frequency limit for asymptotically AdS black holes. We consider here both cases for which the frequencies are much lower and higher than the critical frequency (at which the black hole absorbs all of the radiation which is sent towards it). The radiation power equations for asymptotically AdS black holes at asymptotic frequency are also derived for both the Schwarzschild and Reissner-Nordström solutions which are found same and depend on the Hawking temperature only. Corresponding to greybody factor at the low frequency, we demonstrate the Hawking radiation power equations for asymptotically dS black holes for both even and odd spacetimes. Here, we find that the radiation power equation in five-dimensional dS spacetimes depends on both the horizon radius and Hawking temperature. In case of even spacetime dimensions, we get simpler form of it, however it has an infinite sum series of Hurwitz Zeta function for the case of odd spacetime dimensions. It may be noted that as the greybody factors depend on all frequencies [2], our formalism can be applied for any given greybody factors. We have plotted diagrams also to understand the behavior of the radiation power equation with respect to both the Hawking temperature and horizon radius. This paper is organized as following. In section II, we compute the radiation power equation for asymptotically flat spacetimes. Specifically, we derive radiation power equation for the black holes with the help of greybody factors at both the low frequency and asymptotic frequency for Schwarzschild solution. In section III, we evaluate the Hawking radiation power equations for the asymptotically AdS black holes. Here, the radiation power equations corresponding to greybody factors at both low and asymptotic frequencies for Schwarzschild solution and Reissner-Nordström solution are discussed. Further, in section IV, we derive the radiation power equation for greybody factors at low frequency only for asymptotically dS black holes in even and odd spacetimes. In the last section, we draw conclusion with final remarks.

II. ASYMPTOTICALLY FLAT SPACETIMES
In this section, we analyse the the radiation power equation for asymptotically flat black holes. First, we shall derive the radiation power equation for greybody factor at low frequency limit ω ≪ T H , ωR H ≪ 1 for Schwarzschild solution. Here, the low frequency limit means that the characteristic scales associated with the black hole is much smaller than the scalar wave wavelength. Then, we discuss the radiation power equation for greybody factor at asymptotic frequency for the case of Schwarzschild solution.

A. Power Equation for Greybody Factors at Low Frequency
In this subsection, we compute the radiation power equation for greybody factor at low frequency. Since the greybody factor in the low frequency limit for asymptotically flat black holes is given by [2] where R H is the horizon radius. Here we clarify that the above greybody factor is calculated by assuming that scalar wave, which spreads from infinity over whole spacetime, get reflected by the black hole potential barrier and the transmitted part of scalar wave, near the horizon, appears as the incoming radiation to black hole. We note that, for low frequency scattering, the greybody factor is found equal to the absorption probability of the black hole.
For a convenience, we change dimension d → (d + 1) 2 and, hence, the greybody factor for d + 1 dimensional black hole takes following form: For greybody factor in the low frequency limit as given in (2), the Hawking radiation power equation for asymptotically flat black holes is given by where we have utilized relation (1). After further simplification, this reduces to the following expression: In order to solve above equation, we recall Riemann Zeta function, which is given by, To match this Zeta function with the desired integral, we replace y = ω T H in above equation to get, Now, exploiting (3) and (5), the Hawking radiation power equation for asymptotically flat black holes is given by respect to Hawking temperature and horizon radius can been seen from plot (1).

B. Power Equation for Greybody Factors at Asymptotic Frequency
In this subsection, we determine power equation for greybody factors at asymptotic frequency for Schwarzschild solution. In order to determine the radiation power equation, we first write the Schwarzschild greybody factor at asymptotic frequency by [2], The poles of above greybody factor precisely correspond to the asymptotic quasinormal frequencies. Now, using binomial theorem, we can write the greybody factor at asymptotic frequency With the help of expression (1) and greybody factor (8), the radiation power equation corresponding to the greybody factors at asymptotic frequency for Schwarzschild solution takes the following form: In order to simplify the above integral, we can use the Hurwitz (generalized Riemann) Zeta function 3 as the Hurwitz Zeta function has the following definition: In order to match the form of above Hurwitz Zeta function integral with the radiation power equation (9), we make the following identification: y = ω T H . With such identification, the expression of Hurwitz Zeta function results to Comparing expressions (9) and (10), one can easily derive the the radiation power equation corresponding to greybody factors at asymptotic frequency for Schwarzschild solution as where, coefficient C ) . Here, we have used Γ(n) = (n − 1)! for Γ(2) = 1! = 1. Remarkably, we find that the radiation power equations corresponding to greybody factors at asymptotic frequency in any arbitrary dimensions are same 3 Here we note that for this type of Zeta function there are two different forms exists 1) ζ(s, q) = ∞ a=0 (a + q) −s for q > 0 and 2) ζ(s, q) = ∞ a=0 [(a + q) 2 ] − s 2 for q < 0. We also note that both are identical for Re(q) > 0.
as it does not depend on the dimension d. Also, we notice that the radiation power equation corresponding to greybody factors at asymptotic frequency depends on Hawking temperature only with power law T 2 H and does not depend on horizon radius. However, for greybody factor at low frequency, it depends on both the Hawking temperature and horizon radius.

III. ASYMPTOTICALLY ADS SPACETIMES
In this section, we derive the radiation power equation of black hole in asymptotically AdS spacetimes corresponding to the greybody factors at both the low frequency and asymptotic frequency.

A. Power Equation for Greybody Factors at Low Frequency
The greybody factor for asymptotically AdS black holes, in the low frequency regimeω ≪ 1 is given by [2], whereω = ω k is a dimensionless variable for the frequency and quantity Now, we define  (12), the radiation power equation (1) takes the following form: Here we see that the first integral of above expression can be solved by using (5), as a result we With this simplification, the expression for radiation power (14) reads, here we have utilized z = αω d−1 . Now, taking αω d−1 = δ, the above expression can further be simplified as follows, Here, exploiting binomial expansion for (1+δ) −1 and (1+δ) −2 , we find that (1+δ) −1 −(1+δ) −2 = ∞ n=1 (−1) n+1 nδ n . With this result, the above expression for the radiation power equation reduces to, In order to solve above integral, we plug y = 1 T H δ α Finally, using (15) and (16), we get simplified expression for the radiation power equation corresponding to greybody factor in low frequency limit as where, C 2n ζ(nd − n + 2)Γ(nd − n + 2) . Here, the radiation power equation has an infinite sum series with terms depending on Hawking temperature and horizon radius differently. Also, we notice that, contrary to flat spacetime case, the radiation power equation depends on horizon radius with inverse power law.
Here, we note that there exists a critical frequencyω c for which there is no reflection of scalar wave occurs for black hole which means that the black hole absorbs all of the radiation sent towards it. Alternatively, for emission of scalar wave from the black hole, all of the emitted scalar wave at critical frequency should reach the asymptotic region.
In this condition, z(ω c ) = 1 and critical frequencyω c simplifies from (13) as [2] ω c = 2 Γ d−1 Here we stress that the critical frequency can be achieved for small AdS black holes only. Now, there are two cases possible in calculation of power radiation equation for AdS black hole. Firstly, if we consider frequencies much lower than the critical frequency (ω <<ω c ) and secondly, if we consider instead frequencies much higher than the critical frequency (ω >>ω c ).

Case I: whenω <<ω c
In this case, the greybody factor forω <<ω c is given as follows, Here we notice that the greybody factor is inversely proportional to the area of the black hole, whereas it is proportional to ω d−2 . Therefore, the frequencies much lower than the critical frequency is identical toω << kR H . This means that large AdS black holes (with kR H >> 1) are always in a frequency regime much lower than the critical frequency. For convenience (without loss of generality), we rewrite the above expression of the greybody factor for d → (d + 1) as, For this greybody factor, the power radiation equation (1) is given by here, subscript low − I stands for low frequency in region I (i.e.ω <<ω c ). In order to obtain explicit expression for the power radiation, we need to solve the integral, which can be done easily with the help of Zeta function (5). Hence, we get where the constant C (d+1) Here we observe that the radiation power corresponding to greybody factor for frequencies much lower than the critical frequency depends on both the Hawking temperature with power law ∼ T d+1 H and horizon radius with power law ∼ R d−1 H . In four-dimensional spacetimes, the radiation power equation reduces to P  Hawking radiation power versus horizon radius for d = 3 and k = T H = 1.

Case II: whenω >>ω c
In this case of frequencies much higher than the critical frequency, the greybody factor for d → (d + 1) is given by, Here, greybody factor is proportional to the area of the black hole, whereas it is instead inversely proportional to ω d−2 . Here we note that this frequency regime is possible for small AdS black holes with kR H << 1 only. Now, we follow the same procedure as discussed in case I and get the following expression for radiation power equation: where constant C (d+1) Here subscript low − II stands for low frequency in region II (i.e.ω >>ω c ). Remarkably, we observe that the radiation power depends both on Hawking temperature with power law ∼ T −d+3 and horizon radius with power law ∼ R d−1 H . This clearly means that for four-dimensional spacetimes the radiation power depends only on horizon radius and does not depend on Hawking temperature. However, for space dimensions d = 3, the radiation power depends on temperature with different nature. For d > 3, it depends on temperature with inverse power law and, for d < 3, it depends on temperature with direct power law.

B. Power Equation for Greybody Factors at Asymptotic Frequency
Remarkably, the greybody factor at asymptotic frequency for the Schwarzschild solution and the Reissner-Nordström solution is same and given by [2], For this greybody factor, the Hawking radiation power equation reads, The above integration can be performed with the help of (5). Thus, we find the value of radiation power equation in simplified form as where the coefficient C Here we conclude that the radiation power equation corresponding to greybody factor at asymptotic frequency depends on Hawking temperature only.

IV. ASYMPTOTICALLY DE SITTER SPACETIMES
In this section we shall find the Hawking radiation power equation corresponding to the greybody factor, at low frequencies, for black holes in asymptotically dS spacetimes. The greybody factor (after considering d → (d + 1)) in this case is given by [2] where function h(ω) for even (d + 1) ≥ (3 + 1) is expressed by however, for odd (d + 1) ≥ (4 + 1) we have, Here we see that h(ω) → 1 and h(ω) → 0 asω → 0, for even and odd spacetime dimension respectively. Now, we shall calculate the power radiation equation for the given greybody factor by considering specific cases. For example, we shall study (a) (d + 1) = 4 and (b) (d + 1) = 6 for the even spacetimes, and (c) (d + 1) = 5 and (d) (d + 1) = 7 for the odd spacetimes.
A. Even Spacetimes case I: (d+1)=4 In this case, the function h(ω) (25) leads to With this value of h(ω), the greybody factor, at low frequencies, for asymptotic dS black holes (24) reads,

B. Even Spacetimes case II: (d+1)=6
In case (d + 1) = 6, the expression for h(ω) given in (25) has the following form: With this value of h(ω), the greybody factor, at low frequencies, for asymptotically dS black holes (24) is given by Once the expression for the greybody factor is known, it is matter of calculation to obtain the radiation power equation which utilizes relation (1). For the greybody factor (30), the expression for the radiation power equation is given by In order to solve the integrals in above expression, we utilize the Zeta function (5). After doing so, the expression for the radiation power equation simplified to powers of Hawking radiation. In order to see behavior of the radiation power equation with respect to horizon radius and Hawking temperature, we plot Fig. (4). The expression for h(ω) given in (26) for Odd Spacetimes (d + 1) = 5 takes following value: Now, in order to simplify above expression we utilize following definition: coth(x) = 1+e −2x 1−e −2x . With this definition, we have With the help of binomial expansion, we can write 1 − e −π ω k −1 = ∞ m=1 e (1−m)π ω k . As a result, the expression (34) reduces to coth πω . By inserting this value of coth πω 2k into (33), the function h ω k takes the following expression: Plugging this value of h ω k in (24), the greybody factor for five-dimensional asymptotic dS black holes at low frequencies reduces to Now, it is matter of calculation to evaluate the radiation power equation for a given greybody factor. So, utilizing relations (1) and (35), we write the radiation power equation for balck hole in five-dimensional dS spacetimes as In order to simplify the integrals, we utilize the Hurwitz Zeta function (10) and by doing so, we get the following explicit expression for the radiation power equation: Here, it is evident that although the radiation power equation in five-dimensional dS spacetimes depends on both the horizon radius and Hawking temperature but has an infinite sum series of Hurwitz Zeta function as well.
Next, we shall derive the radiation power equation for the (d + 1) = 7.
D. Odd Spacetimes case II: (d+1)=7 The expression for function h(ω) (26) for Odd Spacetimes (d + 1) = 7 is given by: Now, plugging the value of coth πω 2k calculated in the above last subsection, the above function h(ω) reduces to the following form: With this value of h ω k (39), the greybody factor for seven-dimensional asymptotically dS black holes at low frequencies (24) has following value: Furthermore, we have evaluated the radiation power equations for black hole in asymptotically AdS spacetimes. Here, also we have considered the greybody factors in different regime of frequency in order to calculate power equations. For the low energy regime, we note that the critical frequency exists only for small AdS black holes at which there are no reflection of radiation for black hole and/or no emission of radiation from the black hole. Here, we have discussed the cases: 1) if one considers frequencies much lower than the critical frequency and 2), if one considers instead frequencies much higher than the critical frequency. For former case, we have observed that the radiation power depends on both the Hawking temperature with power law and horizon radius. However, for later case, remarkably, we observed that though it depends on both Hawking temperature and horizon radius but with different power law. In fact, for four-dimensional spacetimes the radiation power depends on horizon radius only but not on the Hawking temperature. The radiation power for greybody factors at asymptotic frequency is independent of horizon radius and dimensionality of spacetime, however, depends on Hawking temperature only.
For asymptotically dS spacetime, the Hawking radiation power corresponding to greybody factor at low frequency highly depends on dimensionality of spacetime. We have computed this for even and odd spacetimes. In case of even dimensions, we have obtained simpler form of the radiation power equation which depends on both the horizon radius and Hawking temperature.
However, the radiation power equation in odd dS spacetimes although depends on both the horizon radius and Hawking temperature but has an infinite sum series of Hurwitz Zeta function.
It is known that by multiplying the radiation power expression of black hole with black hole evaporation time, one can estimate the total energy emitted as Hawking radiation which must be equal to the total mass of the black hole by virtue of energy conservation. This might give an insight to understand the process of black hole evaporation. Such analysis a subject of future work.