Massive Charged Fermion Emissions from Kerr – Newman – Vaidya Black Holes

In this article, we consider charged fermion particle radiation around Kerr – Newman – Vaidya black holes. Using the Hamilton – Jacobi method we derived the emission probability and the temperature of the Hawking radiations. We obtained that the temperature is equal to that due to scalar particles times a factor that contains some characteristics of fermions.


Introduction
In 1974, Hawking proved that black holes could radiate due to quantum mechanics processes.This result showed that a black hole is not a one-way object, thus there is a probability that objects inside a black hole could escape by radiation [1].The works from Bekenstein [2][3][4] gave derivations of the entropy formulae of black holes.Then, in 1975, Hawking completed the calculation with quantum view.These calculations resulted in Hawking's temperature and entropy: with  is the surface gravity and  is the surface area of the black hole.Another key point on black hole is black hole has no hair theorem, which explains that black hole only can be characterized by its mass, charge, and spin [5], causing the temperature formulation mainly formed by the parameters from this theorem.Hawking radiation refers to the occurrence of particle pairs being generated near the event horizon due to quantum fluctuations.The emitted particles from the event horizon are detected as Hawking radiation, while the corresponding antiparticles fall back into the interior of the black hole.This process occurs when the total energy of the particle and antiparticle is zero as explained in [6][7][8][9][10].This pair particle creation is similar with a classical radiation that occurred by particle tunnelling process.For this case we assume that there are two entangled particles.When each particle passes the event horizon, each particles changes their energy sign, but still in the same value.The process can be likened to the creation of a pair within or beyond the horizon, with an overall energy of zero.We use complex path analysis using the Hamilton -Jacobi method provided by Srinivasan and Padmanabhan [14] to analyze the Hawking radiation.Quantum correction for Hamilton -Jacobi was added by Banerjee and Majhi [15], but quantum corrections in term of minimal Planck length for black hole physics in general are given in reference [16][17].Several works in [18][19][20][21][22] provide tunnelling for several types of particle tunnelling from various types of black holes.Other works provide semiclassical methods for Hawking radiation on Vaidya type black holes [23][24][25][26], which are non-stationary black holes that their mass and charge depend on time and radius.
In this paper, we analyse the Hawking radiation on Kerr -Newman -Vaidya (KNV) black hole from massive-charged fermion particles.The Dirac equations that govern these particles' emission are solved by the Hamilton -Jacobi method.Firstly, a brief introduction on the Kerr -Newman -Vaidya metric is given.This metric has non-stationary components that imply the radiation from the black hole as derived by Ng.Ibohal [12][13] based on Vaidya metric [11].In Section 3, Dirac equations on the KNV background for the fermions are derived until they are ready to be solved by Hamilton -Jacobi method in Section 4. The result obtained by this method is the temperature formulation that depend on the characteristics of massive-charged fermions.

Kerr -Newman -Vaidya Metric
Given Kerr -Newman -Vaidya metric in the following equation [13], (2) with  is the nonstationary term, Δ * = ( 2 − 2( + ) +  2 +  2 ), and  2 =  2 +  2 cos 2 .The potential vector is given in the following term, From metric (2), the event horizons are located at, where thesign indicates the radius of the inner event horizon, while the + sign indicates the outer event horizon.Inner (outer) event horizon is in the null surface area that is located at the infinite redshift surface obtained from   = 0.This infinite red-shift surface is also known as an ergosphere within the non-stationary limit, which is expressed in the following equation, (5) The Kerr -Newman -Vaidya black hole angular velocity formulation is given in the following equation [13],

Dirac Equation for Dirac Particle Emission
The Dirac equation in a curved spacetime for massive-charged fermions is written in the following equation, with  is the charge of particles and ∇  is covariant derivative, with and Γ   is Christoffel symbols and   are gamma matrices that satisfy the following anti-commutative relationship, {  ,   } = 2  .(11)   is the curved spacetime metric and  the identity matrix.For a flat spacetime, gamma matrices in the following terms, with   are Pauli matrices, 10th with  is an advanced standard time in Finkelstein -Eddington coordinates (, , , ).Furthermore, some variables in (15) written in the following equations, The corresponding  5 matrix is There are two cases of particles in the Dirac field, namely particles with spin-ups and particles with spin-downs which are represented in the following two ansatz wave functions: Fermion particles have associated actions  ↑ and  ↓ which are functions of the coordinates , r, , and .In the WKB approximation ((ℏ)), the derivatives and components Ω  from equation (8) for , , , and  can be ignored to the lowest order.In the scenario where there is no rotation, a statistical reasoning was employed to support the idea that an equilibrium of zero angular momentum is upheld during fermion emission.This is because an equal number of particles with outward radial spin (spin up) would be emitted as particles with inward radial spin (spin down).This rationale remains applicable even when rotation is considered.The statistical arrangement of spins in the fermion emission spectrum should not impact the angular momentum of the black hole [20].Furthermore, only the spin-up case is considered.An ansatz action  ↑ was used to evaluate the Dirac equations solutions as follows,  ↑ (, , , ) = − +   (, ) + Θ() + , (19) with  and  are the energy and magnetic quantum number (can be defined as the angular momentum of particles) from emitted particles.The use of this ansatz solution was inspired by the calculations in [20].Then, we simplify for the near horizon case and making use the ansatz solution to obtain these following equations,  20) and (21) would describe Hawking radiations from massless,  = 0, and massive,  ≠ 0, fermion particle emissions.For massless case, equations ( 20) and ( 21) are decoupled, just like equations ( 22) and ( 23), while for massive case equations ( 20) and ( 21) are coupled equations.Equations ( 22) and (23)  ) .( 24) Equation (24) show that the relation between Θ action and θ coordinate.The solutions of ( 20) and ( 21) for region around event horizon would be obtained by Hamilton -Jacobi method explained in the next sections.

Hamilton -Jacobi Method for Dirac Particles Emissions.
In this section, Dirac equation around event horizons would be evaluated by using Hamilton -Jacobi method.For any region around event horizons, the functions , , ,  could be approximated to, Equations ( 22)-( 23) do not contain   so that they would not contribute to radiation.The non-stationary term of (, ) corresponds to the change of mass with time.The change of (, ) with respect of time is ignorable as compared to that with respect of radial coordinate.Hence, the above equations simplify into Multiplying (28) by  and (29) by  and then add them up, we have giving the / ratio At a region around event horizons, (32) Therefore, the / is equal to 0 or −∞ on the event horizon, which implies that  → 0, corresponding to massive ( ≠ 0) righthand particles or  → 0, corresponding to massive ( ≠ 0) lefthand particles.
For the case of  = 0, we have Contour integral was used to evaluate the imaginary part of (33) and (34).As it is imaginary, it fully contributes to black hole's radiation probability. + corresponds to outgoing while  − corresponds to ingoing particle radiation actions.The tunnelling probabilities for Dirac particles are, Prob(out) describes outgoing particle probabilities, while Prob(in) describes ingoing particle probabilities.Since incoming particles are almost completely absorbed by the black hole, and considering normalized states, the probability of particles from inside to outside is equivalent to, The Hawking temperature   is obtained by equalizing the above expression with exp[−/    ].Accordingly, .
(38) The temperature formulation (35) is equal to the temperature formulation for massive scalar particle from the same black hole [26], where (, , , ) = 2    ′ .The magnetic quantum number  shows that the particles have spin-½.Setting  =  = 0 would cause (, , ) to go to unity and the temperature (38) reduces to (39).Hence, the formulation would be the same as massless scalar particle radiation.
Calculations in this section specifically for the emission of spin-up particles.Calculations for spin-down particles are similar, and the same formulation of probability and temperature can be obtained.Spindown and spin-up particles contribute equally to the radiation.

Conclusion
We calculated radiation probability of massive-charged fermion emissions from a Kerr -Newman -Vaidya black hole.This result yielded temperature of this black hole.Equation (38) has a similar form with massless scalar particle emissions as showed in reference [26], but with a factor (, , ) containing some characteristic of fermions.This factor including the energy, mass, charge, and spin of the particles.If the charge and spin parameters were neglected, the radiation formulation would be the same as that of massless scalar particle emissions with the same physical properties in [26].