Quantum Radiation Properties of General Nonstationary Black Hole
aa r X i v : . [ g r- q c ] F e b Quantum radiation properties of general non-stationary black hole T . Ibungochouba SinghDepartment of Mathematics, Manipur University,Canchipur, 795003 Manipur (India)Email: ibungochouba@rediffmail.com
Abstract
Using the generalized tortoise coordinate transformations the quantum radiationproperties of Klein-Gordon scalar particles, Maxwell’s electromagnetic field equa-tions and Dirac equations are investigated in general non-stationary black hole. Thelocations of the event horizon and the Hawking temperature depend on both timeand angles. A new extra coupling effect is observed in the thermal radiation spec-trum of Maxwell’s equations and Dirac equations which is absent in the thermalradiation spectrum of scalar particles. We also observe that the chemical potentialderived from scalar particles is equal to the highest energy of the negative energystate of the scalar particle in the non-thermal radiation in general non-stationaryblack hole. Applying generalized tortoise coordinate transformation a constant term ξ is produced in the expression of thermal radiation in general non-stationary blackhole. It indicates that generalized tortoise coordinate transformation is more accu-rate and reliable in the study of thermal radiation of black hole. Key-words : General non-stationary black hole; Thermal radiation; Generalizedtortoise coordinate transformation. Introduction
Hawking discovered the thermal radiation of black hole using the techniques of quan-tum field theory in curve space-time background and the derived radiation spectrum ispurely thermal in nature [1, 2]. An important aspects in the study of black hole is toreveal the thermal and non-thermal radiation of black hole. Refs. [3-5] have shown thatthe black hole has a non zero finite entropy and the entropy of black hole is proportionalto the horizon area. Refs. [6, 7] also proposed the Hawking radiation as quantum tun-neling process where the particles moves in the dynamical geometry. They recovereda leading correction to the emission rate arising from the loss mass of the black hole.Following their works, Zhang and Zhao [8-10] have extended the method to more generalcircumstances for rotating black hole and they have shown the spectrum is no longerprecisely thermal. Further, some information of the black hole can be obtained.Akhmedov, et al. [11] investigated the Hawking radiation as tunneling picture in theSchwarzschild black hole using the relativistic Hamilton-Jacobi method. The importanceof this investigations are (i) if Schwarzschild co-ordintes or any other coordinates relatedto them via a transformation of spatial coordinates is used, we will get twice originalHawking temperature and (ii) Any transformation involving time coordinates is utilizedin the black hole evaporation, it will give the original Hawking result. Following thismethod, many fruitful results have been obtained in [12-14]. This factor of two issue hasbeen resolved via the discovery of the temporal contribution to the tunneling amplitude1n the literatures [15-17]. The thermodynamics of black hole in lovelock gravity and inAdS space time have been investigated by Cai [18-20]. One of the important aspects inthis investigation is to calculate the Hawking temperature.Recently, Anghenben, et al. [21] investigated Hawking radiation as tunneling ofextremal and rotating black hole using the relativistic Hamilton-Jacobi method andWKB approximation without considering the particles back reaction. Since then manyother authors applied the Hamilton-Jacobi method to study the Hawking radiation formore general space time [22-25]. Choosing appropriate Gamma matrices, the Hawkingradiation as tunneling from Dirac particles was investigated by Ref. [26] for the generalstationary black hole. By inserting the wave function into the Dirac field equations,the action of the radiant particles is derived. This result is related to the Boltzmannfactor of emission at the Hawking radiation temperature in accordance with semiclassicalWKB approximation. Refs. [27, 28] introduced tortoise coordinate transformation tostudy the Hawking radiation of black hole in which gravitational field is independentof time. Using tortoise coordinate transformation, Klein-Gordon equation, Maxwell’selectromagnetic field equations and Dirac particles can be transformed into single formof wave equation near the event horizon. Separating the variables from the standardwave equation, ingoing wave and outgoing wave can be obtained. Extending the waveequation from outside the event horizon into the inside by rotating − π through the lowerhalf of the complex plane, the thermal radiation spectra can be derived. Generalizingthis method, many more works have been done [29-31].Refs. [32-33] have also investigated the Hawking radiation by calculating the vacuumexpectation values of the renormalized energy momentum tensor of spherically symmetricnon-stationary black hole. This results are consistent with the Refs. [27-28]. However allthese research works were confined to the quantum thermal radiation of black hole only.In addition to the quantum thermal radiation, the importance of quantum non-thermalradiation of black hole has been studied by different authors in different types of spacetime in the literatures [34-38].The main aim of this paper is to investigate thermal radiation from Klein-Gordonequation, Maxwell’s electromagnetic field equations, Dirac particles and also non-thermalradiation of Hamilton-Jacobi equation in general non-stationary black hole. It givesthe relationship between two kind of radiation in the case of scalar particles. A newextra coupling effect is derived from the thermal radiation spectrum of Maxwell’s fieldequations and Dirac particles. In section 2, we derive the location of horizon in generalnon-stationary black hole using null surface condition and generalized tortoise coordinatetransformation. In section 3, adjusting the parameter κ , the Klein-Gordon equation istransformed into a wave equation near the event horizon in general non-stationary blackhole. In section 4, we derive asymptotic behaviors of first order form of four equations andsecond order form of three equations from the Maxwell’s electromagnetic field equationsnear the event horizon using generalized tortoise coordinate transformation (GTCT).In section 5, the asymptotic behaviors of the first order form of four equations andsecond order form of two equations are deduced from the Dirac particles using the GTCT2ear the event horizon. In section 6, the second order form of Klein-Gordon equation,the three second order form of Maxwell’s equations and the two second order form ofDirac equations are transformed into single standard form of equation near the blackhole event horizon. By separating the wave equation the chemical potential and thethermal radiation spectra can be obtained. In section 7, the highest energy of non-thermal radiation is obtained from the relativistic Hamilton-Jacobi equation. In section8, we derive the expression of a new extra coupling effect which is absent from thethermal radiation spectrum of scalars particles. The relationship between thermal andnon-thermal radiation in general non-stationary black hole is also established. Someconclusions are given in the last section.2. General non-stationary black hole . The line element describing the mostgeneral non-stationary black hole is given by ds = g µν dx µ dx ν (1)where we assume the retarded Eddington-Finkelstein coordinates ( x = u, x = r, x = θ and x = φ ) and make the conventions that all indices of Greek letters µ, ν = 0 , , , j, k = 0 , ,
3. The event horizon in general non-stationary blackhole can be characterized by null hypersurface condition: F = F ( u, r, θ, φ ) = 0. The nullhypersurface condition gives the position of horizon of stationary or non-stationary blackhole [39] g µν ∂ µ F ∂ ν F = 0 · (2)The location and the temperature of the horizon in a non-stationary black hole may beobtained by applying tortoise coordinate transformation. By using tortoise coordinatetransformation of the form r ∗ = r + (2 κ ) − ln ( r − r h ), the Klein-Gordon equation, theMaxwell’s electromagnetic field equations and the Dirac particles can be combined intoa standard form of wave equation in a non-stationary or stationary space-time (where κ is the surface gravity). The location of horizon may be assumed as functions of retardedtime coordinate u = t − r ∗ and different angles θ, φ . The space time geometry outside theevent horizon is described by tortoise coordinate only and in this condition, r ∗ approachesto positive infinity when tending to infinite point and r ∗ tends to negative infinity atthe event horizon. It is also assumed that the geometry of space time in the generalnon-stationary black hole is symmetric about φ -axis. According to Refs. [40-46], thegeneralized tortoise coordinate transformation is defined as r ∗ = r + 12 κ ( u , θ , φ ) ln n r − r h ( u, θ, φ ) r h ( u, θ, φ ) o ,u ∗ = u − u , θ ∗ = θ − θ , φ ∗ = φ − φ , (3)where u , θ , φ are the parameters under the tortoise coordinate transformation. FromEq. (3), we get ∂∂r = ∂∂r ∗ + 12 κ ( r − r h ) ∂∂r ∗ , ∂x j = ∂∂x j ∗ − rr h,j κr h ( r − r h ) ∂∂r ∗ ,∂ ∂r = [2 κ ( r − r h ) + 1] [2 κ ( r − r h )] ∂ ∂r ∗ − κ ( r − r h ) ∂∂r ∗ ,∂ ∂r∂x j = [1 + 2 κ ( r − r h )]2 κ ( r − r h ) ∂ ∂r ∗ ∂x j ∗ + r h,j κ ( r − r h ) ∂∂r ∗ − r h rr h,j [1 + 2 κ ( r − r h )][2 r h κ ( r − r h )] ∂ ∂r ∗ ,∂ ∂x j ∂x k = ∂ ∂x j ∗ ∂x k ∗ − rr h,j κ ( r − r h ) ∂ ∂r ∗ ∂x k − rr h,k κ ( r − r h ) ∂ ∂r ∗ ∂x j + r h r h,j r h,k [2 κr h ( r − r h )] ∂ ∂r ∗ − r κ [( r − r h ) r h,jk + r h,j r h,k (2 r h − r )][ r h ( r − r h ] ∂∂r ∗ . (4)Using Eqs. (4) in (2), the horizon equation in general non-stationary black hole isobtained as g − g j r h,j + g jk r h,j r h,k = 0 , (5)where r h,j = ( ∂r h ∂u , ∂r h ∂θ , ∂r h ∂φ ). r h,u = ∂r h ∂u represents the evaporation rate in general non-stationary black hole near the event horizon. The event horizon is expanded graduallyif ∂r h ∂u > ∂r h ∂u <
0, the event horizon is contracted. Inaddition, r h,θ = ∂r h ∂θ and r h,φ = ∂r h ∂φ denote the rate of event horizon varying with anglesand also describe the rotation effect of non-stationary black hole. r h is the location ofevent horizon and depends on retarded time u and angular coordinates θ , φ and also κ ≡ κ ( u , θ , φ ) is an adjustable parameter that depends on retarded time and angularcoordinates.3. Klein-Gordon Equation . In this section, the asymptotically behaviour of min-imally electromagnetic coupling Klein-Gordon Equation near the black hole will be dis-cussed. The Klein-Gordon equation describes the explicit form of wave equation of thescalar particles with mass µ in curve space time which is given by1 √− g h ∂∂x a √− gg ab ∂∂x b i Φ − µ Φ = 0 . (6)Using generalized tortoise coordinate transformation to Eq. (6) and subsequently mul-tiplying by the factor 2 κr h ( r − r h ) / [ r h g { κ ( r − r h ) } − rg j r h,j ] to both sides ofEq. (6) and finally taking limit near the event horizon as r −→ r h ( u , θ , φ ), u −→ u , θ −→ θ and φ −→ φ , the second order form of wave equation is obtained as follows A ∂ Φ ∂r ∗ + 2 ∂ Φ ∂r ∗ ∂u ∗ + A ∂ Φ ∂r ∗ ∂θ ∗ + A ∂ Φ ∂r ∗ ∂φ ∗ + A ∂ Φ ∂r ∗ = 0 , (7)where A = g r h [1 + 2 κ ( r − r h )] − g j rr h r h,φ [1 + 2 κ ( r − r h )]2 κr h ( r − r h )[ r h g { κ ( r − r h } − rg j r h,j ] ,A = 2 g − g j r h,j ( g − g j r h,j ) , = 2 g − g j r h,j ( g − g j r h,j ) ,A = − ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − √− g ( g − g j r h,j ) [( √− g ,ν g ν + √− gg ν,ν ) − ( √− g ,ν g ν + √− gg ν,ν ) − ( √− g ,ν g ν + √− gg ν,ν ) − ( √− g ,ν g ν + √− gg ν,ν )] . (8)By adjusting parameter κ , the coefficient of ∂ Φ ∂r ∗ is assumed to be unity near the eventhorizon, we getlim r −→ r h κr h ( r − r h ) [ g r h { κ ( r − r h ) } − g j rr h r h,φ { κ ( r − r h ) } + r g jk r h,j r h,k ] = r h ( g − g j r h,j ) . (9)It is also observed that, in left hand side of Eq. (9), both numerator and denominatortend to zero near the event horizon r = r h . Hence Eq. (9) is an indeterminate form of0 /
0. Using L’ Hospital rule and using Eq. (5), the surface gravity is obtained from theKlein-Gordon scalar particles as follows κ = ∂g ∂r − ∂g j ∂r r h,j + ∂g jk ∂r r h,j r h,k g − g + (2 g j − g j ) r h,j ] + g jk r h,j r h,k − g j r h,j r h [ g − g + (2 g j − g j ) r h,j ] · (10)4. Maxwell’s electromagnetic field equations.
To write the explicit form of Maxwell’s electromagnetic field equations in Newman-Penrose formalism [47], the following complex null tetrad vectors ℓ , n , m and ¯ m arechosen at each point in four dimensional space, where ℓ and n are a pair of real nulltetrad vectors and m and ¯ m are a pair of complex null tetrad vectors. They are requiredto satisfy the following conditions ℓ ν ℓ ν = n ν n ν = m ν m ν = ¯ m ν ¯ m ν = 0 ,ℓ ν n ν = − m ν ¯ m ν = 1 ,ℓ ν m ν = ℓ ν ¯ m ν = n ν m ν = n ν ¯ m ν = 0 ,g µν = ℓ µ n ν + ℓ ν n µ − m µ ¯ m ν − m ν ¯ m µ ,g µν = ℓ µ n ν + ℓ ν n µ − m µ ¯ m ν − m ν ¯ m µ (11)and corresponding directional derivatives are given by D = ℓ ν ∂∂x µ , ∆ = n ν ∂∂x µ ,δ = m ν ∂∂x µ , ¯ δ = ¯ m ν ∂∂x µ . (12)The dynamical behavior of spin-1 particles in curve space time is given by four coupledMaxwell’s electromagnetic field equations expressed in Newman-Penrose formalism [48]5s follows Dφ − ¯ δφ − ( π − α ) φ − ρφ + κφ = 0 ,δφ − ∇ φ + νφ − µφ − ( τ − β ) φ = 0 ,δφ − ∇ φ − ( µ − γ ) φ − τ φ + σφ = 0 ,Dφ − ¯ δφ + λφ − πφ − ( ρ − ε ) φ = 0 , (13)where φ , φ and φ are the four components of Maxwell’s spinor in the Newman-Penrose formalism. ǫ, ρ, π, α, β, µ, τ and γ are spin coefficients introduced by Newmanand Penrose, they are given by ρ = ℓ µ ; ν m µ ¯ m ν ,π = − n µ ; ν ¯ m µ ℓ ν ,τ = ℓ µ ; ν m µ n ν ,α = 12 ( ℓ µ ; ν n µ ¯ m ν − m µ ; ν ¯ m µ ¯ m ν ) ,β = 12 ( ℓ µ ; ν n µ m ν − m µ ; ν ¯ m µ m ν ) ,γ = 12 ( ℓ µ ; ν n µ n ν − m µ ; ν ¯ m µ n ν ) ,ǫ = 12 ( ℓ µ ; ν n µ ℓ ν − m µ ; ν ¯ m µ ℓ ν ) , (14)where ¯ α , ¯ β , ¯ γ , ¯ τ , ¯ ǫ , ¯ π , ¯ µ and ¯ ρ are complex conjugates of α , β , γ , τ , ǫ , π , µ and ρ .From Eqs. (13), the three second order form of Maxwell’s equations for ( φ , φ , φ )components are given by D ∇ φ − δ ¯ δφ + ( µ − γ ) Dφ − ( π − α ) δφ − ρ ∇ φ + 2 τ ¯ δφ (15)+( m ν ℓ µ,ν − ¯ m µν ℓ ν ) ∂φ ∂x µ + κ ∇ φ − σ ¯ δφ = 0 . ∇ Dφ − ¯ δδφ − ρ ∇ φ + ( µ − γ ) Dφ + 2 τ ¯ δφ − ( π − α ) δφ (16)+( ¯ m ν n µ,ν − n ν ¯ m µ,ν ) ∂φ ∂x µ + κ ∇ φ − σ ¯ δφ = 0 . ¯ δδφ − ∇ Dφ − ( τ − β )¯ δφ + 2 πδφ + ( ρ − ǫ ) ∇ φ − µDφ (17)+( ¯ m µν n ν − n µ,ν ¯ m ν ) ∂φ ∂x µ − λ ∇ φ + ν ¯ δφ = 0 . Refs. [49-50] have shown that Eqs. (13) cannot be decoupled except only for the station-ary black hole space time. For studying the thermal radiation in general non-stationaryblack hole, the asymptotic behavior of the first order and second order form of Eq.(13) near the event horizon r = r h will be considered. Then, after taking the limit as r −→ r h ( u , θ , φ ), u −→ u , θ −→ θ and φ −→ φ , the first order form of Maxwell’sequations near the event horizon are as follows:( ℓ − ℓ j r h,j ) ∂φ ∂r ∗ − ( ¯ m − ¯ m j r h,j ) ∂φ ∂r ∗ = 0 , m − m j r h,j ) ∂φ ∂r ∗ − ( n − n j r h,j ) ∂φ ∂r ∗ = 0 , ( m − m k r h,k ) ∂φ ∂r ∗ − ( n − n j r h,k ) ∂φ ∂r ∗ = 0 , ( ℓ − ℓ k r h,k ) ∂φ ∂r ∗ − ( ¯ m − ¯ m k r h,k ) ∂φ ∂r ∗ = 0 . (18)We assume that the three derivatives ∂φ /∂r ∗ , ∂φ /∂r ∗ and ∂φ /∂r ∗ in Eqs. (18) arenonzero. Then non-trivial solutions for φ , φ and φ can be obtained if the determinantof their coefficients is zero, which will give the horizon equation like the null surfacecondition (5). The importance of Eqs. (18) is to eliminate the crossing terms involvedin the second order form of Maxwell’s equations near the event horizon.Utilizing the generalized coordinate transformation (3) to Eqs. (15), (16) and (17) andsubsequently multiplying by the factor 2 κr h ( r − r h ) / [ r h g { κ ( r − r h ) } − rg j r h,j ] toboth sides of three second order equations for the coefficients ∂ φ ∂r ∗ ∂u ∗ , ∂ φ ∂r ∗ ∂u ∗ and ∂ φ ∂r ∗ ∂u ∗ tobe 2 and finally taking the limit of r −→ r h ( u , θ , φ ), u −→ u , θ −→ θ and φ −→ φ ,then the three second order form of Maxwell’s electromagnetic field equations near theevent horizon can be expressed as follows I ∂ φ ∂r ∗ + 2 ∂ φ ∂r ∗ ∂u ∗ + B ∂ φ ∂r ∗ ∂θ ∗ + B ∂ φ ∂r ∗ ∂φ ∗ + ( B + 2 iB ) ∂φ ∂r ∗ = 0 , (19)where I = g r h { κ ( r − r h ) } − g j rr h r h,φ { κ ( r − r h ) } κr h ( r − r h )[ r h g { κ ( r − r h } − rg j r h,j ] ,B = 2 ( g − g j r h,j )( g − g j r h,j ) ,B = 2 ( g − g j r h,j )( g − g j r h,j ) ,B = − ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − ,B = − i ( g − g j r h,j ) [ ℓ ν ( n ,ν − n j,ν r h,j ) − m ν ( ℓ ,ν − ℓ j,ν r h,j ) − ρ ( n − n j r h,j )+( µ − γ )( ℓ ,ν − ℓ j,ν r h,j ) − ( π − α )( m − m j r h,j ) + 2 τ ( ¯ m − ¯ m j r h,j )+ ( ¯ m − ¯ m j r h,j )( ℓ − ℓ j r h,j ) { m ν ( ℓ ,ν − ℓ j,ν r h,j ) − ℓ ν ( ¯ m ,ν − ¯ m j r h,j ) }−{ σ ( ¯ m − ¯ m j r h,j ) − κ ( n − n j r h,j ) } ( n − n j r h,j )( ¯ m − ¯ m j r h,j )( m − m j r h,j )( ℓ − ℓ j r h,j ) ] (20)and I ∂ φ ∂r ∗ + 2 ∂ φ ∂r ∗ ∂u ∗ + B ∂ φ ∂r ∗ ∂θ ∗ + B ∂ φ ∂r ∗ ∂φ ∗ + ( B + 2 i ˜ B ) ∂φ ∂r ∗ = 0 , (21)where˜ B = − i ( g − g j r h,j ) [ − ρ ( n − n j r h,j ) + ( µ − γ )( ℓ − ℓ j r h,j ) + 2 τ ( ¯ m − ¯ m j r h,j ) − ( π − α )( m − m j r h,j ) + n ν ( ℓ ,ν − ℓ j,ν r h,j ) − ¯ m ν ( m ,ν − m j,ν r h,j )7 ( ℓ − ℓ j r h,j )( ¯ m − ¯ m j r h,j ) { ¯ m ν ( n ,ν − n j,ν r h,j ) − n ν ( ¯ m ,ν − ¯ m j,ν r h,j ) } + {− σ ( ¯ m − ¯ m j r h,j ) + κ ( n − n j r h,j ) } ( n − n j r h,j )( m − m j r h,j ) ] , (22)and I ∂ φ ∂r ∗ + 2 ∂ φ ∂r ∗ ∂u ∗ + B ∂ φ ∂r ∗ ∂θ ∗ + B ∂ φ ∂r ∗ ∂φ ∗ + ( B + 2 i ¯ B ) ∂φ ∂r ∗ = 0 , (23)where¯ B = − i ( g − g j r h,j ) [ n ν ( ℓ ,ν − ℓ j,ν r h,j ) − ¯ m ν ( m ,ν − m j,ν r h,j ) − π ( m ,ν − m j,ν r h,j ) + ( τ − β )( ¯ m ,ν − ¯ m j,ν ) − ρ − ǫ )( n ,ν − n j,ν r h,j )+2 µ ( ℓ ,ν − ℓ j,ν r h,j ) + ( ℓ − ℓ j r h,j )( ¯ m − ¯ m j r h,j ) ( m − m j r h,j )( n − n j r h,j ) { λ ( n ,ν − n j,ν r h,j ) − ν ( ¯ m ,ν − ¯ m j,ν r h,j ) } − { n ν ( ¯ m − ¯ m j r h,j ) − ¯ m ν ( n ,ν − n j,ν r h,j ) } ( m − m j r h,j )( n − n j r h,j ) ] . (24)We assume the value of I approaches to unity near the event horizon, then we getlim r −→ r h g r h { κ ( r − r h ) } − g j rr h r h,φ { κ ( r − r h ) } κr h ( r − r h )[ r h g { κ ( r − r h ) } − rg j r h,j ] = 1 , (25)which is an indeterminate form of 0 / κ = ∂g ∂r − ∂g j ∂r r h,j + ∂g jk ∂r r h,j r h,k g − g + (2 g j − g j ) r h,j ] + g jk r h,j r h,k − g j r h,j r h [ g − g + (2 g j − g j ) r h,j ] · (26)which is equal to the surface gravity derived from Klein-Gordon scalar particle given byEq. (10).5. Dirac equations.
The four couples Dirac equations [51] expressed in Newman-Penrose formalism are given by( D + ǫ − ρ ) f + (¯ δ + π − α ) f − iµ g √ , ( ∇ + µ − γ ) f + ( δ + β − τ ) F − iµ g √ , ( ∇ + ¯ µ − ¯ γ ) g − (¯ δ + ¯ β − ¯ τ ) g − iµ f √ , ( D + ¯ ǫ − ¯ ρ ) g − ( δ + ¯ π − ¯ α ) g − iµ f √ , (27)where D , ∇ , δ and ¯ δ are the directional derivatives given by Eqs. (12) and ǫ, ρ, π, α, β, µ, τ and γ are spin coefficients and also µ is the mass of the Dirac particles. f , f , g and g are the four components of Dirac spinor in the Newman-Penrose formalism. Eqs. (27)8an be decoupled only for the stationary black hole space time. From Eqs. (27) thesecond order form of Dirac equations for the components ( f , f ) are given by − ∇ + ¯ µ − ¯ γ ) × [( D + ǫ − ρ ) f + (¯ δ + φ − α ) f ] + 2(¯ δ + ¯ β − ¯ τ ) × [( ∇ + µ − γ ) f (28)+( δ + β − τ ) f ] − µ f = 0 . − D + ¯ ǫ − ¯ ρ ) × [( ∇ + µ − γ ) f + ( δ + β − τ ) f ] + 2( δ + ¯ π − ¯ α ) × [( D + ǫ − ρ ) f (29)+(¯ δ + π − α ) f ] − µ f = 0 . Applying generalized tortoise coordinate transformation to Eqs. (27), after taking limit r −→ r h ( u , θ , φ ), u −→ u , θ −→ θ and φ −→ φ , the first order form of Diracequations near the event horizon are given by ∂f ∂r ∗ = ¯ m − ¯ m j r h,j ℓ j r h,j − ℓ ∂f ∂r ∗ ,∂f ∂r ∗ = m − m j r h,j n j r h,j − n ∂f ∂r ∗ ,∂g ∂r ∗ = m − m k r h,k ℓ − ℓ k r h,k ∂g ∂r ∗ ,∂g ∂r ∗ = ¯ m − ¯ m k r h,k n − n k r h,k ∂g ∂r ∗ . (30)In order to study Hawking thermal radiation from Dirac particles, we should consider theasymptotic behaviour of Eqs. (27) near the black hole horizon. The non-trivial solutionsfor f , f , g and g can be obtained if the four derivatives ∂f /∂r ∗ , ∂f /∂r ∗ , ∂g /∂r ∗ and ∂g /∂r ∗ in Eqs. (30) are nonzero. Eqs. (30) may be used to eliminate the crossingterms appeared in the second order form of Dirac equation near the horizon.Applying generalized tortoise coordinate transformation to the Eqs. (28) and (29), viasome arrangement, multiplying by the factor 2 κr h ( r − r h ) / [ r h g { κ ( r − r h }− rg j r h,j ]to both sides of the two second order form of Dirac equations for the coefficient ∂ f ∂r ∗ ∂u ∗ and ∂ f ∂r ∗ ∂u ∗ to be 2 and taking the limit of r −→ r h ( u , θ , φ ), u −→ u , θ −→ θ and φ −→ φ , the two second order form of Dirac equations can be written as follows Q ∂ f ∂r ∗ + 2 ∂ f ∂r ∗ ∂u ∗ + C ∂ f ∂r ∗ ∂θ ∗ + C ∂ f ∂r ∗ ∂φ ∗ + ( C + 2 iC ) ∂f ∂r ∗ = 0 , (31)where Q = g r h { κ ( r − r h ) } − g j rr h r h,φ { κ ( r − r h ) } κr h ( r − r h )[ r h g { κ ( r − r h ) } − rg j r h,j ] ,C = 2 ( g − g j r h,j )( g − g j r h,j ) ,C = 2 ( g − g j r h,j )( g − g j r h,j ) ,C = − ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − , = − i ( g − g j r h,j ) [ − ( β − τ + ¯ β − ¯ τ )( ¯ m − ¯ m j r h,j ) + ( ǫ − ρ )( n − n j r h,j )+(¯ µ − ¯ γ )( ℓ − ℓ j r h,j ) − ( ℓ − ℓ j r h,j )( ¯ m − ¯ m j r h,j ) {− ¯ m ν ( n ,ν − n j,ν )+ n ν ( ¯ m − ¯ m j,ν ) } − (¯ µ − ¯ γ − µ + γ )( ℓ − ℓ j r h,j ) − ( π − α − ¯ β + ¯ τ ) × ( ℓ − ℓ j r h,j ) ( ¯ m − ¯ m j r h,j ) ] (32)and Q ∂ f ∂r ∗ + 2 ∂ f ∂r ∗ ∂u ∗ + C ∂ f ∂r ∗ ∂θ ∗ + C ∂ f ∂r ∗ ∂φ ∗ + ( C + 2 i ¯ C ) ∂f ∂r ∗ = 0 , (33)where¯ C = − ig − g j r h,j [( µ − γ )( ℓ − ℓ j r h,j ) − ( π − α )( m − m j r h,j )+ ℓ ν ( n ,ν − n j,ν ) + (¯ ǫ − ¯ ρ )( n − n j r h,j ) + ℓ ν ( n ,ν − n j,ν ) − m ν ( ¯ m − ¯ m j,ν ) − (¯ ǫ − ¯ ρ − ǫ + ρ )( n ,ν − n j,ν ) − ℓ ν ( m ,ν − m j,ν )( ¯ m − ¯ m jh,j ) ( n − n j r h,j )+ { m ν ( ℓ ,ν − ℓ j,ν )( m − m j r h,j ) − ( β − τ − ¯ π + ¯ α ) ( ℓ − ℓ j r h,j )( m − m j r h,j ) } ( n − n j r h,j )] , (34)when Q approaches to unity, we obtainlim r −→ r h g r h { κ ( r − r h ) } − g j rr h r h,φ { κ ( r − r h ) } κr h ( r − r h )[ r h g { κ ( r − r h ) } − rg j r h,j ] = 1 . (35)The above Eq. (35) is a 0 / κ = ∂g ∂r − ∂g j ∂r r h,j + ∂g jk ∂r r h,j r h,k g − g + (2 g j − g j ) r h,j ] + g jk r h,j r h,k − g j r h,j r h [ g − g + (2 g j − g j ) r h,j ] · (36)which is the same as Eqs. (10) and (26), the surface gravities derived from Klein-GordonScalar particles, and Maxwell’s electromagnetic field equations.6. Thermal radiation spectrum.
To investigate the thermal radiation spectrumin general non-stationary black hole, we combine the second order form of Klein-Gordonequation (7), the three second order form Maxwell’s equations (19), (21), (23) and thetwo second order form of Dirac equations (31), (33) as single form of wave equation nearthe event horizon r = r h as follows ∂ Ψ ∂r ∗ + 2 ∂ Ψ ∂r ∗ ∂u ∗ + L ∂ Ψ ∂r ∗ ∂θ ∗ + L ∂ Ψ ∂r ∗ ∂φ ∗ + ( L + 2 iL ) ∂ Ψ ∂r ∗ = 0 , (37)where L = 2 ( g − g j r h,j )( g − g j r h,j ) ,L = 2 ( g − g j r h,j )( g − g j r h,j ) . (38)10he Eq. (37) may be assumed as standard form of wave equation in general non-stationary black hole near the horizon r = r h . It includes Klein-Gordon equation,Maxwell’s electromagnetic field equations and Dirac equations with different co-efficientof constant terms.For example, when (Ψ = Φ) for the Klein-Gordon equation, the Eq. (37) gives thefollowing constants L = − ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − √− g ( g − g j r h,j ) [( √− g ,ν g ν + √− gg ν,ν ) − ( √− g ,ν g ν + √− gg ν,ν ) − ( √− g ,ν g ν + √− gg ν,ν ) − ( √− g ,ν g ν + √− gg ν,ν )] ,L = 0 . (39)For Maxwell’s electromagnetic equations (Ψ = φ ), the constant terms are L = − ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − ,L = − i ( g − g j r h,j ) [ ℓ ν ( n ,ν − n j,ν r h,j ) − m ν ( ℓ ,ν − ℓ j,ν r h,j ) − ρ ( n − n j r h,j )+( µ − γ )( ℓ ,ν − ℓ j,ν r h,j ) − ( π − α )( m − m j r h,j ) + 2 τ ( ¯ m − ¯ m j r h,j )+ ( ¯ m − ¯ m j r h,j )( ℓ − ℓ j r h,j ) { m ν ( ℓ ,ν − ℓ j,ν r h,j ) − ℓ ν ( ¯ m ,ν − ¯ m j r h,j ) −{ σ ( ¯ m − ¯ m j r h,j ) − κ ( n − n j r h,j ) } ( n − n j r h,j )( ¯ m − ¯ m j r h,j )( m − m j r h,j )( ℓ − ℓ j r h,j ) ] . (40)For (Ψ = φ ) L = − ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − ,L = − i ( g − g j r h,j ) [ − ρ ( n − n j r h,j ) + ( µ − γ )( ℓ − ℓ j r h,j ) + 2 τ ( ¯ m − ¯ m j r h,j ) − ( π − α )( m − m j r h,j ) + n ν ( ℓ ,ν − ℓ j,ν r h,j ) − ¯ m ν ( m ,ν − m j,ν r h,j )+ ( ℓ − ℓ j r h,j )( ¯ m − ¯ m j r h,j ) { ¯ m ν ( n ,ν − n j,ν r h,j ) − n ν ( ¯ m ,ν − ¯ m j,ν r h,j ) } + {− σ ( ¯ m − ¯ m j r h,j ) + κ ( n − n j r h,j ) } ( n − n j r h,j )( m − m j r h,j ) ] . (41)Lastly for (Ψ = φ ), we get L = − ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − ,L = − i ( g − g j r h,j ) [ n ν ( ℓ ,ν − ℓ j,ν r h,j ) − ¯ m ν ( m ,ν − m j,ν r h,j ) − π ( m ,ν − m j,ν r h,j ) + ( τ − β )( ¯ m ,ν − ¯ m j r h,j ) − ρ − ǫ )( n ,ν − n j,ν r h,j )+2 µ ( ℓ ,ν − ℓ j,ν r h,j ) + ( ℓ − ℓ j r h,j )( ¯ m − ¯ m j r h,j ) ( m − m j r h,j )( n − n j r h,j ) { λ ( n ,ν − n j,ν r h,j ) − ν ( ¯ m ,ν − ¯ m j,ν r h,j ) } − { n ν ( ¯ m − ¯ m j r h,j )11 ¯ m ν ( n ,ν − n j,ν r h,j ) } ( m − m j r h,j )( n − n j r h,j ) ] . (42)Similarly for Dirac particles when (Ψ = f ), Eq. (37) gives the following constant terms L = − ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − ,L = − i ( g − g j r h,j ) [ − ( β − τ + ¯ β − ¯ τ )( ¯ m − ¯ m j r h,j ) + ( ǫ − ρ )( n − n j r h,j )+(¯ µ − ¯ γ )( ℓ − ℓ j r h,j ) − ( ℓ − ℓ j r h,j )( ¯ m − ¯ m j r h,j ) {− ¯ m ν ( n ,ν − n j,ν )+ n ν ( ¯ m − ¯ m j,ν ) } − (¯ µ − ¯ γ − µ + γ )( ℓ − ℓ j r h,j ) − ( π − α − ¯ β + ¯ τ ) × ( ℓ − ℓ j r h,j ) ( ¯ m − ¯ m j r h,j ) ] , (43)for (Ψ = f ), we obtain L = − [ ∂g ∂r − ∂g j ∂r r h,j + g jk r h,jk ( g − g j r h,j ) ] + g jk r h,j r h,k r h ( g − g j r h,j ) ( r h − ,L = − i ( g − g j r h,j ) [( µ − γ )( ℓ − ℓ j r h,j ) − ( π − α )( m − m j r h,j )+ ℓ ν ( n ,ν − n j,ν ) + (¯ ǫ − ¯ ρ )( n − n j r h,j ) + ℓ ν ( n ,ν − n j,ν ) − m ν ( ¯ m − ¯ m j,ν ) − (¯ ǫ − ¯ ρ − ǫ + ρ )( n ,ν − n j,ν ) − ℓ ν ( m ,ν − m j,ν )( ¯ m − ¯ m jh,j ) ( n − n j r h,j )+ { m ν ( ℓ ,ν − ℓ j,ν )( m − m j r h,j ) − ( β − τ − ¯ π + ¯ α ) ( ℓ − ℓ j r h,j )( m − m j r h,j ) } ( n − n j r h,j )] . (44)Eq. (37) may be assumed as second order partial differential equation near the eventhorizon in general non-stationary black hole since all the coefficients L , L , L and L are constant terms when r −→ r h ( u , θ , φ ), u −→ u , θ −→ θ and φ −→ φ .Using Refs. [29, 30, 36, 52], the variables in Eq. (37) may be separated for theanalysis of the field equations asΨ = R ( r ∗ , u ∗ )Θ( u ∗ , θ ∗ , φ ∗ ) e iωu ∗ + iK θ + iK φ , (45)where Θ( u ∗ , θ ∗ , φ ∗ ) is an arbitrary real function and ω is the energy of the particles whichdepend on tortoise coordinate transformation; K θ , K φ are components of generalizedmomentum of scalar particles. And we use K θ = ∂S∂θ ∗ , K φ = ∂S∂φ ∗ , where S is Hamiltonianfunction of scalar particles. Using Eq. (45) into Eq. (37) and after separating thevariables, the radial and angular parts are given by ∂ P∂r ∗ − (2 iω − L iK θ − L iK φ − L − i L − α ) ∂P∂r ∗ = 0 , ∂T∂u ∗ − ( ζ ( u ∗ ) − α ) T = 0 , (46)where α and ζ ( u ∗ ) are a constant and function of retarded time u ∗ in variable of separa-tion respectively, where ζ ( u ∗ ) = 2 ∂ Θ ∂u ∗ Θ + C ∂ Θ ∂θ ∗ Θ − C ∂ Θ ∂φ ∗ Θ and R ( r ∗ , u ∗ ) = P ( r ∗ ) T ( u ∗ ).12fter separation of variables, the radial components of two independent solutions aredefined byΨ inω ∼ e iω + iK θ + iK φ , Ψ outω ∼ e iω + iK θ + iK φ e i ( ω − KθL − KφL − L ) r ∗ e − ( α + L ) r ∗ , r > r h , (47)where Ψ inω represents an incoming wave which is analytic at r = r h ; Ψ outω denotes anoutgoing wave having singularity at the event horizon. Refs. [27-28] indicates that Ψ outω can continue analytically from outside of the event horizon r = r h into inside by rotating − π through lower half of the complex plane as˜Ψ outω ∼ e iω + iK θ + iK φ e πκ [ ω − KθL − KφL − L ] e iπ κ ( α + L ) × (cid:16) r h − rr h (cid:17) iκ ( ω − KθL − KφL − L ) (cid:16) r h − rr h (cid:17) − κ ( α + L ) , r < r h . (48)From Eqs. (47) and (48), one can obtain the relative scattering probability near theevent horizon r = r h (cid:12)(cid:12)(cid:12) Ψ outω ( r > r h )˜Ψ outω ( r < r h ) (cid:12)(cid:12)(cid:12) = e − πκ ( ω − ω ) , (49)where ω = K θ L K φ L L . (50)Following Damour and Ruffini [27] and extended by Sannan [28], the thermal radiationspectrum of Maxwell’s electromagnetic field equations (Dirac particles or scalar particles)from general non-stationary black holes is given by N ω = 1 e ω − ω κBT ± , (51)where κ B is Boltzmann constant and upper positive symbol stands for the Fermi-Diracdistribution and the lower negative symbol corresponds to the Bose-Einstein statistics.Eq. (51) shows that the black hole radiates like a black body. The Hawking temperatureis given by T ( u , θ , φ ) = 12 π h ∂g ∂r − ∂g j ∂r r h,j + ∂g jk ∂r r h,j r h,k { g − g + (2 g j − g j ) r h,j } + g jk r h,j r h,k − g j r h,j r h { g − g + (2 g j − g j ) r h,j } i , (52)and the chemical potential by ω = K θ g − g j r h,j g − g j r h,j + K φ g − g j r h,j g − g j r h,j + L . (53)Integrating the thermal radiation spectra (51) or distribution function over all ω ’s thecombined form of Hawking flux for Klein-Gordon scalar particles, Maxwell’s electromag-netic field equations and Dirac equations can be obtained as followsFlux = 1 π Z ∞ ωdωe πκ ( ω − ω ) ± . (54)13his is an exact result for the energy flux in general non-stationary black hole. If ω = 0in Eq. (54), the Hawking flux for fermions is given by(Flux) | fermions = 1 π Z ∞ ωdωe πωκ + 1 = κ π , (55)and the Hawking flux for boson is defined by(Flux) | boson = 1 π Z ∞ ωdωe πωκ − κ π . (56)This results are consistent with already obtained in the literature [53, 54]. From Eq.(52), we observe that T is a function of retarded time and different angles. Hence, itis a distribution of temperature of the thermal radiation near the event horizon r = r h due to the Klein-Gordon scalar field, the Maxwell’s electromagnetic field equations andthe Dirac equations in general non-stationary black hole. It has been shown that theconstant coefficient L appears in the expression of chemical potential and may representa particular energy term for Maxwell’s electromagnetic field and Dirac particles which isabsent in the thermal radiation spectrum of other scalar particles.7. Non-thermal radiation.
The relativistic Hamilton-Jacobi equation for the classical action of a particle of mass µ in a curve space time is given by [55] g ab (cid:16) ∂ Φ ∂x a (cid:17)(cid:16) ∂ Φ ∂x b (cid:17) − µ = 0 , (57)where Φ = Φ( u, r, θ, φ ) is the Hamiltonian principal function. Using Eq. (3) into Eq.(57), we obtain as follows G κ ( r − r h ) r h ( ∂S∂r ∗ ) − D ( ∂S∂r ∗ ) + 2 r h κ ( r − r h ) Y = 0 , (58)where G = g r h [1 + 2 κ ( r − r h )] + r g jk r h,j r h,k − rr h [1 + 2 κ ( r − r h )] r h,j g j ,D = rg jk ( ∂S∂x j ∗ ) − r h ( ∂S∂x j ∗ )[1 + 2 κ ( r − r h )] ,Y = g jk ( ∂S∂x j ∗ )( ∂S∂x k ∗ ) + µ . (59)Multiplying by the factor 1 / [ r h g { κ ( r − r h } − rg j r h,j ] to both sides of Eq. (58)and assuming the resulting coefficient of ( ∂S∂r ∗ ) tends to unity near the event horizon,then we getlim r −→ r h g r h { κ ( r − r h ) } − g j rr h r h,φ { κ ( r − r h ) } κr h ( r − r h )[ r h g { κ ( r − r h ) } − rg j r h,j ] = 1 . (60)Eq. (60) is an indeterminate form of 0 / κ , the surface gravity obtained from relativisticHamilton-Jacobi equation is given by κ = ∂g ∂r − ∂g j ∂r r h,j + ∂g jk ∂r r h,j r h,k g − g + (2 g j − g j ) r h,j ] + g jk r h,j r h,k − g j r h,j r h [ g − g + (2 g j − g j ) r h,j ] · (61)14hich is equal to the surface gravity obtained from Klein-Gordon equation, Maxwell’sequations and Dirac equations as given in Eqs. (10), (26) and (36). Using the generalizedtortoise coordinate transformation in Eq. (57) and defining ∂ Φ ∂u ∗ = − ω, ∂ Φ ∂θ ∗ = K θ , ∂ Φ ∂φ ∗ = K φ , for real ∂ Φ ∂r ∗ the distribution of energy levels of the particles is given by ω ≥ ω + and ω ≤ ω − . (62)Near the black hole event horizon there exist seas of positive and negative energy statesand a forbidden energy gap. Penrose [56] proposed that a particle entering close to thesurface of the event horizon decays into two particles - one particle having positive energyescapes to infinity and the other particle having negative energy gets absorbed by theblack hole. A quantum analogue as spontaneous pair creation was proposed by Zel’dovich[57] in the Kerr black hole. The energy states must satisfy the condition ω − ≤ ω ≤ ω + atthe forbidden region. The maximum value of the negative energy state after overlappingof the positive and negative energy states at the surface of the event horizon is ω h = K θ g − g j r h,j g − g j r h,j + K φ g − g j r h,j g − g j r h,j . (63)The width of the forbidden energy approaches to zero near the event horizon. Thisindicates that there exists a crossing of the positive and the negative energy levels near theevent horizon [38]. When ω h > µ , the particle can escape to infinity from the black holeevent horizon. The Starobinskii-Unruh process (spontaneous radiation) must occur in theregion near the black hole event horizon [57-61]. It indicates that the incident negativeenergy particle will become emerging positive energy particle via quantum tunnelingeffect. From this result, there is radiation near the event horizon. This type of radiationis independent of the temperature of the black hole and the type of quantum effect is anon-thermal. Application of this theory . The line element describing general non-stationarysymmetrical black hole in retarded time coordinates ( u, r, θ, φ ) is defined by ds = g du + 2 g dudr + 2 g dudθ + g dudφ + g drdφ + g dθ + 2 g dθdφ + g dφ · (64)Using Eqs. (4), (5) and (64), the expression of temperature in general non-stationaryaxial symmetrical black hole is T = 14 πr h h r h ( ∂g ∂r − ∂g j ∂r r h,j + ∂g jk ∂r r h,j r h,k ) + 2( g jk r h,j r h,k − g j r h,j ) { g − g + (2 g j − g j ) r h,j } i (65)and the expression of chemical potential is defined by ω = K θ g − g j r h,j g − g j r h,j + K φ g − g j r h,j g − g j r h,j · (66)From Eqs. (63) and (66), we observe that the chemical potential derived from scalarparticle is equal to the highest energy of the negative-energy state.15n particular, the line element of Kerr black hole in retarded time coordinate is givenby [62] ds = (1 − M rR − ) du + 2 dudr + 4 arM R − sin θdudφ − a sin θdrdφ − R dθ − { ( r + a ) − ∆ a sin θ } R − θ dφ , (67)where ∆ = r − M r + a and R = r + a cos θ . The Kerr black hole has singularityat ∆ = 0. The roots r h = M + √ M − a and r h = M − √ M − a represent externalevent horizon and internal Cauchy horizon respectively. From Eqs. (65) and (67), theHawking temperature of non-stationary Kerr black hole is obtained as T = 12 π r h (1 + 2 r h,u ) − r h M ( u ) − { ∆ h + r h,u ( r h + a ) + ar h,φ } r h { ( r h + a + a sin θ )(1 + 2 r h,u ) + Z } (68)where Z = 2 r h,θ + r h,φ (4 ar h,u + r h,φ sin θ + 3 a ) and the chemical potential is ω = K θ r h,θ [( r h + a ) + a sin θ r h,u + ar h,φ ]+ K φ ( r h,u + a sin θ + r h,φ )sin θ [( r h + a ) + a sin θ r h,u + ar h,φ ] · (69)This indicates that the chemical potential derived from scalar particles is equal to highestenergy of the negative-energy state. This was mentioned at the beginning and has beenshown by direct calculation to a special case.8. Discussion . The total interaction energy of scalar particles of Klein-Gordon,Maxwell’s electromagnetic field equations and Dirac particles in general non-stationaryblack hole is given by ω = K θ g − g j r h,j g − g j r h,j + K φ g − g j r h,j g − g j r h,j + L (70)where L = 0 (71)for Ψ = Φ (Klein-Gordon scalar particle), L = − i ( g − g j r h,j ) [ ℓ ν ( n ,ν − n j,ν r h,j ) − m ν ( ℓ ,ν − ℓ j,ν r h,j ) − ρ ( n − n j r h,j )+( µ − γ )( ℓ ,ν − ℓ j,ν r h,j ) − ( π − α )( m − m j r h,j ) + 2 τ ( ¯ m − ¯ m j r h,j )+ ( ¯ m − ¯ m j r h,j )( ℓ − ℓ j r h,j ) { m ν ( ℓ ,ν − ℓ j,ν r h,j ) − ℓ ν ( ¯ m ,ν − ¯ m j r h,j ) }−{ σ ( ¯ m − ¯ m j r h,j ) − κ ( n − n j r h,j ) } ( n − n j r h,j )( ¯ m − ¯ m j r h,j )( m − m j r h,j )( ℓ − ℓ j r h,j ) ] (72)for Ψ = φ (Maxwell’s electromagnetic field), L = D = − i ( g − g j r h,j ) [ − ρ ( n − n j r h,j ) + ( µ − γ )( ℓ − ℓ j r h,j ) + 2 τ ( ¯ m − ¯ m j r h,j )16 ( π − α )( m − m j r h,j ) + n ν ( ℓ ,ν − ℓ j,ν r h,j ) − ¯ m ν ( m ,ν − m jν r h,j )+ ( ℓ − ℓ j r h,j )( ¯ m − ¯ m j r h,j ) { ¯ m ν ( n ,ν − n j,ν r h,j ) − n ν ( ¯ m ,ν − ¯ m j,ν r h,j ) } + {− σ ( ¯ m − ¯ m j r h,j ) + κ ( n − n j r h,j ) } ( n − n j r h,j )( m − m j r h,j ) ] (73)for Ψ = φ and L = D = − i ( g − g j r h,j ) [ n ν ( ℓ ,ν − ℓ j,ν r h,j ) − ¯ m ν ( m ,ν − m j,ν r h,j ) − π ( m ,ν − m j,ν r h,j ) + ( τ − β )( ¯ m ,ν − ρ − ǫ )( n ,ν − n j,ν r h,j )+2 µ ( ℓ ,ν − ℓ j,ν r h,j ) + ( ℓ − ℓ j r h,j )( ¯ m − ¯ m j r h,j ) ( m − m j r h,j )( n − n j r h,j ) { λ ( n ,ν − n j,ν r h,j ) − ν ( ¯ m ,ν − ¯ m j,ν r h,j ) } − { n ν ( ¯ m − ¯ m j r h,j ) − ¯ m ν ( n ,ν − n j,ν r h,j ) } ( m − m j r h,j )( n − n j r h,j ) ] (74)for Ψ = φ . L = − i ( g − g j r h,j ) [ ℓ ν ( n ,ν − n j,ν r h,j ) − m ν ( ℓ ,ν − ℓ j,ν r h,j ) − ρ ( n − n j r h,j )+( µ − γ )( ℓ ,ν − ℓ j,ν r h,j ) − ( π − α )( m − m j r h,j ) + 2 τ ( ¯ m − ¯ m j r h,j )+ ( ¯ m − ¯ m j r h,j )( ℓ − ℓ j r h,j ) { m ν ( ℓ ,ν − ℓ j,ν r h,j ) − ℓ ν ( ¯ m ,ν − ¯ m j r h,j ) }−{ σ ( ¯ m − ¯ m j r h,j ) − κ ( n − n j r h,j ) } ( n − n j r h,j )( ¯ m − ¯ m j r h,j )( m − m j r h,j )( ℓ − ℓ j r h,j ) ] (75)for Ψ = f ( Dirac Particles) L = − i ( g − g j r h,j ) [( µ − γ )( ℓ − ℓ j r h,j ) − ( π − α )( m − m j r h,j )+ ℓ ν ( n ,ν − n j,ν ) + (¯ ǫ − ¯ ρ )( n − n j r h,j ) + ℓ ν ( n ,ν − n j,ν ) − m ν ( ¯ m − ¯ m j,ν ) − (¯ ǫ − ¯ ρ − ǫ + ρ )( n ,ν − n j,ν ) − ℓ ν ( m ,ν − m j,ν )( ¯ m − ¯ m jh,j ) ( n − n j r h,j )+ { m ν ( ℓ ,ν − ℓ j,ν )( m − m j r h,j ) − ( β − τ − ¯ π + ¯ α ) ( ℓ − ℓ j r h,j )( m − m j r h,j ) } ( n − n j r h,j )] (76)for Ψ = f This chemical potential ω is composed of two parts. The sum of the first two termsi.e, 2 K θ g − g j r h,j g − g j r h,j + 2 K φ g − g j r h,j g − g j r h,j is the rotational energy arising from the coupling be-tween different components of generalized momentum of Maxwell’s electromagnetic fieldor Dirac particles and different rotations of black hole. The second term L gives a newextra spin-rotation coupling and spin acceleration coupling effect. The physical origin ofextra coupling effect comes from the interaction between intrinsic spin of particles andgeneralized momentum of evaporating black hole but it has no classical correspondence[29, 52]. The value of L will vanish for the stationary black hole and scalar particle.When L = 0, Eqs. (63) and (70) show the chemical potential derived from the thermalradiation spectrum of Maxwell’s electromagnetic field or Dirac particle is equal to high-est energy of the negative energy state of scalar particles in general non-stationary black17ole. This brings out a clear relationship between the two kinds of radiation processesof black holes.On the other hand, the quantitative causal relation with no-loss-no-gain characterwould be satisfied by a lot of general physical process [63, 64]. Using the no-loss-no-gain homeomorphic map transformation satisfying causal relation, Ref. [65] derived theexact strain tensor formulas in Weitzenbock manifold. Ref. [66] investigated the cosmicquantum birth by studying the Wheeler-DeWitt equation which satisfies quantitativecausal relation. In fact, some effect of change (cause) of some quantities in (2) mustresult in the relative changes (cause) in (2) so as to keep no-loss-no-gain in the righthand side of (2), that is zero. Simarly Eqs.(6), (13), (15-17), (27-29) and (57) mustsatisfy the quantitative causal relation in the same way. Hence the findings in this paperare consistent.Refs. [29-31] proposed the tortoise coordinate transformation which is applicable tothe black hole event horizon r ∗ = r + 12ˆ κ ( u , θ , φ ) ln n r − r h ( u, θ, φ ) o ,u ∗ = u − u , θ ∗ = θ − θ , φ ∗ = φ − φ , (77)it gives the surface gravity at the event horizon asˆ κ ( u , θ , φ ) = ∂g ∂r − ∂g j ∂r r h,j + ∂g jk ∂r r h,j r h,k g − g + (2 g j − g j ) r h,j ] , (78)which is different from one given by Eqs. (10), (26) and (36) corresponding to the gen-eralized tortoise coordinate transformation (3). Using the different tortoise coordinatetransformations (3) and (77), it has been observed that the surface gravities of the generalnon-stationary black hole for Klein-Gordon scalar particles, Maxwell’s electromagneticfield equations, Dirac equations and relativistic Hamilton-Jacobi equation near the blackhole event horizon are the same as shown in (10), (26), (36) and (61) under the transfor-mation (3) but they are different for different tortoise coordinate transformations. FromEqs. (61) and (78), the surface gravity can be written as κ ( u , θ , φ ) = ˆ κ ( u , θ , φ ) + ξ ( u , θ , φ ) , (79)where κ ( u , θ , φ ) and ˆ κ ( u , θ , φ ) are the surface gravities obtained from tortoise co-ordinate transformations (3) and (77). Similarly, Eq. (52) can be expressed as T ( u , θ , φ ) = ˆ T ( u , θ , φ ) + 12 π ξ ( u , θ , φ ) (80)where ˆ T ( u , θ , φ ) is the thermal radiation temperature under tortoise coordinate trans-formation (77) and ξ ( u , θ , φ ) denotes the difference of thermal radiation under thedifferent tortoise coordinate transformations and it is given by ξ ( u , θ , φ ) = g jk r h,j r h,k − g j r h,j r h [ g − g + (2 g j − g j ) r h,j ] . (81)18ccording to Ref. [43, 46], the correction rate under the different tortoise coordinatetransformations is given byΥ( u , θ , φ ) = 2( g jk r h,j r h,k − g j r h,j ) r h ( ∂g ∂r − ∂g j ∂r r h,j + ∂g jk ∂r r h,j r h,k ) . (82)If ξ ( u , θ , φ ) approaches to zero, the two surface gravities are equal due to the tortoisecoordinate transformations (3) and (77). This indicates that different tortoise coordi-nate transformations correspond to different Hawking radiation temperatures in a non-stationary rotating black hole space time [45]. Eq. (80) implies that when ˆ T ( u , θ , φ )become zero, the temperature T ( u , θ , φ ) will not be vanished due to the presence ofextra term ξ ( u , θ , φ ). From this research paper, we conclude that the generalized tor-toise coordinate transformation (3) provides an alternative and convenient way to obtainHawking radiation and is also generalization and development of the works of [36, 52]about quantum radiation for general non-stationary black holes.9. Conclusions . In this paper the thermal radiation and chemical potentialis investigated using the generalized tortoise coordinate transformation in general non-stationary black hole. The locations of the horizon and the thermal radiation near theblack hole event horizon are determined. It has been found that they are functionsof retarded time co-ordinate u and different angles θ, φ . By adjusting properly thevalue of κ , the second order form of Klein-Gordon equation, the three second orderform of Maxwell’s electromagnetic field equations and the two second order form ofDirac equations are transformed into a standard form of wave equation near the eventhorizon r = r h . Separating the variables of the wave equation and applying the wellknown Damour-Ruffini-Sannan method, we determine accurately the thermal radiationand chemical potential at the event horizon r = r h in general non-stationary black hole.It is also observed that the constant term L appeared in the expression of chemicalpotential which gives the interaction between the intrinsic spin of the particles and thegeneralized momentum in general non-stationary black hole but it is found to be absent inthe chemical potential derived from Klein-Gordon scalar particles. This paper reveals thechemical potential obtained from the thermal radiation spectrum of Klein-Gordon scalarparticles is equal to the highest energy of the negative energy states of the non-thermalradiation in general non-stationary black hole near the event horizon. Besides, underthe generalized coordinate transformation a constant term ξ ( u , θ , φ ) is appeared inthe expression of surface gravity and the thermal radiation of black hole near the eventhorizon. As ξ ( u , θ , φ ) tends to zero, our results are consistent with already resultsobtained by Hua and Huang [36, 52]. In conclusion, the generalized tortoise coordinatetransformation is found to be more reliable and accurate in the study of thermal radiationspectrum in general non-stationary black hole near the event horizon r = r h . Conflict of Interests
The author declares that there is no conflict of interests regarding the publication ofthis paper.
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