Quantum Radiation Properties of Dirac Particles in General Nonstationary Black Holes

Quantum radiation properties of Dirac particles in general nonstationary black holes in the general case is investigated by both using the method of generalized tortoise coordinate transformation and considering the asymptotic behaviors of both the first and second order forms of Dirac equations near the event horizon. It is generally shown that the temperature and shape of event horizon of this kind of black holes depend on both the time and different angles. Further, we give a general expression of the new extra coupling effect in thermal radiation spectrum of Dirac particles which is missing in that of scalar particles. Also, we reveal a relationship that is ignored before between thermal radiation and non-thermal radiation in the case of scalar particles, which is that the chemical potential in thermal radiation spectrum is equal to the highest energy of the negative energy state of scalar particles in non-thermal radiation for general nonstationary black holes.

11618. [19][20][21]   To investigate the thermal radiation of spin-1/2 particles, we to need deal with the behavior of the second order Dirac equations near the event horizon. It is consistent to consider the asymptotic behavior of both the first and second order Dirac equations at the same time since the four-component Dirac spinors should satisfy both of them. By substituting Eq. (7) and Eq. (8) into Eq. (9) and Eq. (10), one can obtain the second order form of Dirac equations for ( 1 F , 2 F ) components as follows Introducing the generalized tortoise coordinate transformation [5-9, 12-14, 16-18, 30] Here the numerator of I is 11 1 which is the "surface gravity" of event horizon.
By the same reason, 3 C * is also an indeterminate form of where Following Damour, Ruffini [5] and Sannan [6], we can get the thermal radiation spectrum of Dirac particles (or scalar particles) from general nonstationary black holes and the Hawking temperature where B k is Boltzmann constant, "  " correspond to fermion and boson, respectively. The temperature We know from Eq. (29) that T depends on the time and different angles, so it is a distribution of temperatures. It is also interesting to find that the coefficient 1 C appears in the chemical potential, which may represent a particular energy term for Dirac particles. The physical meaning of 1 C will be discussed in section 4.

Research on non-thermal radiation of general nonstationary black holes
Now we use the methods and conclusions of references [22][23][24][25][26][27][28] where example, the chemical potential at zero temperature is equivalent to the Fermi energy of a fermions system, which is a kind of boundary for energy levels), and the thermal radiation and non-thermal radiation have the forenamed relationship that the chemical potential (as its ordinary meaning) in thermal radiation spectrum is equal to the highest energy of negative energy state of scalar particles in non-thermal radiation.
On the other hand, because a lot of general physical processes should satisfy quantitative causal relation with no-loss-no-gain character [32][33][34], e.g., Ref. [35] uses the no-loss-no-gain homeomorphic map transformation satisfying the quantitative causal relation to gain exact strain tensor formulas in Weitzenböck manifold. In fact, some changes ( cause ) of some quantities in (7) must result in the relative some changes ( result ) of the other quantities in (7) so that (7)'s right side keep no-loss-no-gain, i.e., zero, namely, (7) also satisfies the quantitative causal relation. And (2), (8)(9)(10)(11), (31), (37) also satisfy the quantitative causal relation in the same way. Hence the researches in this letter are consistent. Also, the researches of this letter provide an alternative and convenient way to obtain the Hawking temperature.

Summary and Conclusion
This letter carefully investigates quantum radiation of Dirac particles in general nonstationary black holes, generally shows that the temperature and shape of event horizon of black holes depend on both the time and different angles in general condition, and further obtains a general expression of the new extra coupling effect arising from the interaction between the intrinsic spin of Dirac particles and the generalized momentum of the general nonstationary black hole. Finally, this letter shows that the new extra coupling effect is absent in the thermal radiation spectrum in this case, and generally reveals a relationship that is ignored before between thermal radiation and non-thermal radiation of black holes in detail, which is that the chemical potential in thermal radiation spectrum is equal to the highest energy of the negative energy state in non-thermal radiation for general nonstationary black holes.