1.School of Physics and Electronic Engineering, Qilu Normal University, Jinan 250200, China 2.College of Physics and Space Science, China West Normal University, Nanchong 637002, China Received Date:2020-02-11 Accepted Date:2020-04-27 Available Online:2020-12-01 Abstract:According to a corrected dispersion relation proposed in the study on the string theory and quantum gravity theory, the Rarita-Schwinger equation was precisely modified, which resulted in the Rarita-Schwinger-Hamilton-Jacobi equation. Using this equation, the characteristics of arbitrary spin fermion quantum tunneling radiation from non-stationary Kerr-de Sitter black holes were determined. A number of accurately corrected physical quantities, such as surface gravity, chemical potential, tunneling probability, and Hawking temperature, which describe the properties of black holes, were derived. This research has enriched the research methods and enabled increased precision in black hole physics research.
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2.Correction of the dynamic equation for arbitrary spin fermions in non-stationary Kerr-de Sitter space-timeThe modified Lorentz dispersion relation proposed in string theory and quantum gravity theory is [36-38, 58, 59]
where L is a constant on the Planck scale. When $ \alpha = 2 $ is considered, the general fermions' dynamic equation—R-S equation [60, 61] can be extended to the Kerr-de Sitter curved space-time as [19-21]
where $ \Omega_{\mu} $ is the rotational connection in curved space-time. As quantum scale corrections, $ 0<\sigma\ll1 $, so $ \sigma\hbar\gamma^{t}D_{t}\gamma^{j}D_{j}\psi_{\alpha_{1}\cdots\alpha_{k}} $ is a small term. This matrix equation can only be solved in the specific curved space-time, so the fermions' wave function is set first as
where $ \mu = 0,1,2,3; j = 1,2,3 $. For a non-stationary black hole, we use the advanced Eddington coordinate v to represent the dynamic characteristics. To solve Eq. (6), define
We have maintained the correction item of $ \sigma $ in Eq. (19), so Eq. (19) is the precisely corrected R-S equation considering the corrected Lorentz dispersion relation. Obviously, it is actually also an accurately corrected H-J equation; we call Eq. (19) the Rarita-Schwinger-Hamilton-Jacobi equation (R-S-H-J equation). In the following, we will use this precisely corrected R-S-H-J equation to study the dynamic behavior of arbitrary spin fermions in the non-stationary Kerr-de Sitter space-time; thus, we will study the properties of black holes.
3.Accurate correction of arbitrary spin fermion tunneling from the non-stationary Kerr-de Sitter black holeIn the advanced Eddington-Finkelstein coordinate, the line element of the non-stationary Kerr-de Sitter black hole can be written as [62, 63]
M is the mass of the black hole, and a is the angular momentum per unit mass of the black hole. From Eq. (20), the non-zero components of the contravariant metric tensor of the black hole are
where f is a hypersurface. Since the Kerr-de Sitter space-time is axisymmetric, Eq. (23) is independent of $ \varphi $, so $ \frac{\partial f}{\partial\varphi} = 0 $, and the equation of the event horizon of the Kerr-de Sitter black hole can be expressed as
where r is the function of v and $ \theta $, $ r = r(v,\theta) $. In order to obtain the position of the event horizon of the black hole, we need to calculate the rate of change of f with respect to each of its components. Taking the partial derivative of Eq. (25), we obtain
Substituting the expressions of $ g^{00},g^{11},g^{22},g^{01} $ in Eq. (22) into Eq. (27), when $ r\rightarrow r_{H} $, the equation of the event horizon is obtained as follows
which represents the change of the position of the event horizon with time and angle, respectively. For the Kerr-de Sitter black hole, substituting Eq. (22) into Eq. (19) — the accurately corrected R-S-H-J equation, the specific expression of Eq. (19) is
Since the event horizon of the black hole varies with time, it requires using a generalized tortoise coordinate transformation to solve the tunneling probability of fermions from the event horizon of the black hole. The Kerr-de Sitter black hole is an axisymmetric black hole, so we perform the following transformation
which implies that the position of the event horizon varies with time and angle, respectively. In the Kerr-de Sitter space-time, the action of arbitrary spin fermions can be expressed as follows:
$ S = S(v_{\ast},r_{\ast},\theta_{\ast},\varphi). $
(34)
Although S cannot be separated, it is certain that
Eq. (30) is further simplified by writing the second-order small quantity as $ O(\sigma^{2}) $, which is not considered in the separation of variables; thus, we obtain
Substituting Eqs. (32), (33), (35), and (36) into Eq. (37), we obtain (for compactness, the symbol (v, θ) of the independent variables is omitted in the following.)
When $ r\rightarrow r_{H} $, the limit of the coefficient of $ \left\{\frac{\partial S}{\partial r_{\ast}}\right\}^{2} $ at the event horizon should be equal to one, that is
In Eq. (42), the limit of the denominator is zero when $ r\rightarrow r_{H} $; thus, the limit of the numerator should also be zero when $ r\rightarrow r_{H} $. Using L'hopital's rule, we can work out $ \kappa $ as
For the Schwarzschild space-time, it can be proved that $ \tilde{m} = m $. $ \kappa $ is the precisely corrected surface gravity at the event horizon. Taking $ \lim_{r\rightarrow r_{H}}\frac{C_{0}}{B_{0}} = \omega_{0} $, we obtain
$ \omega_{0} $ is the exact corrected chemical potential. Combined with Eqs. (42) and (45), Eq. (41) can be written as follows when $ r\rightarrow r_{H} $
In Eq. (51), $ T_{H} $ is the precisely corrected Hawking temperature at the event horizon of the black hole. This is a new form of the Hawking temperature expression for the Kerr-de Sitter black holes. From the demonstration in this section, we note that using the modified R-S-H-J equation based on the corrected Lorentz dispersion relation, we have obtained a series of precisely corrected physical quantities of the Kerr-de Sitter black holes, including the surface gravity, chemical potential, tunneling probability of arbitrary spin fermions, and Hawking temperature. The correction is indicated by the parameter $ \sigma $. From Eqs. (43), (45), (50), and (51), it is observed that for a non-stationary Kerr-de Sitter black hole, the surface gravity, chemical potential, tunneling probability, and Hawking temperature change with time as the event horizon surface changes. The chemical potential also varies with angle $ \theta $.
4.Discussion and conclusionIn this paper, based on the modified Lorentz dispersion relation on the quantum scale and with the condition $ \alpha = 2 $ selected, we derived the modified R-S-H-J equation by properly selecting the transformation matrix and using the semi-classical method. By using the derived R-S-H-J equation, we solved the dynamic Kerr-de Sitter black hole using the tortoise coordinate transformation, and obtained the accurately corrected surface gravity, chemical potential, tunneling probability, and Hawking temperature. Although the correction term $ \sigma $ is a small quantity, it is still worth further studying. Another important physical quantity in the thermodynamics of black holes is the black hole entropy. According to the first law of thermodynamics, the entropy $ S^{s} $ of a black hole can be expressed as
$ {\rm d}M = T{\rm d}S^{s}+V{\rm d}J+U{\rm d}Q. $
(52)
For the Kerr-de Sitter black hole,
$ {\rm d}S^{s} = \frac{{\rm d}M-V{\rm d}J}{T}. $
(53)
The exactly corrected entropy at the event horizon $ r = r_{H} $ of the black hole can be expressed as
In Eq. (53) and Eq. (54), the relational formula $\dfrac{{\rm d}M-V{\rm d}J}{T_{0}} = {\rm d}S_{0}^{s}$ represents the black hole entropy before the correction, and $ T_{0} $ represents the Hawking temperature before the correction, which is
When not considering the correction and returning to the Schwarzschild space-time, combining Eqs. (21) and (28), Eqs. (51) and (56) revert to the form of $ T = \frac{1}{8\pi M} $, which also proves the correctness of the obtained results. Equations (43), (45), (50), and (51) are all new expressions with precise corrections. The Lorentz dispersion relation is a theory worth studying in the field of high energies, and is a relation that needs to be considered in both the theories of strong gravitational fields and gravitational waves. It is worth pointing out that in the corrected Lorentz dispersion relation, we considered $ \alpha = 2 $ , and considering other values is also worth pursuing; this will be addressed in our future work.