Black hole entropy with minimal length in tunneling formalism

Abstract

Here we study the effects of the Generalized Uncertainty Principle in the tunneling formalism for Hawking radiation to evaluate the quantum-corrected Hawking temperature and entropy for a Schwarzschild black hole. We compare our results with the existing results given by other candidate theories of quantum gravity. In the entropy-area relation we found some new correction terms and in the leading order we found a term which varies as $\sim \sqrt{Area}$. We also get the well known logarithmic correction in the sub-leading order. We discuss the significance of this new quantum corrected leading order term.

Appendix

Let us consider a static line element in $(1+1)$ dimension

$$\begin{aligned} ds^2 = f(r)dt^2 - \frac{1}{f(r)}dr^2 \end{aligned}$$

(36)

where $f(r)$ is an arbitrary function of $r$. We also consider that at $r=r_0$ there is a horizon and $f(r)\vert _{r=r_0}=0$. If we take a massive minimally coupled scalar field $\phi$ in this background then the field $\phi$ satisfies the Klein-Gordon equation

$$\begin{aligned} \left[ \Box + \frac{m^2}{\hbar ^2}\right] \phi = 0. \end{aligned}$$

(37)

Evaluating $\Box$ operator in the background 36 we can write

$$\begin{aligned} \frac{1}{f(r)}\frac{\partial ^2 \phi }{\partial t^2} - \frac{\partial }{\partial r} \left\{ f(r)\frac{\partial \phi }{\partial r}\right\} =-\frac{m^2}{\hbar ^2}\phi . \end{aligned}$$

(38)

We can rewrite this equation as

$$\begin{aligned} \left[ \frac{1}{f(r)}\left( \frac{\partial S}{\partial t}\right) ^2 \!-\! f(r)\left( \frac{\partial S}{\partial r}\right) ^2 \!-\!m^2 \right] \!-\! i\hbar \left[ \frac{1}{f(r)}\frac{\partial ^2 S}{\partial t^2} \!-\! f(r) \frac{\partial ^2 S}{\partial r^2} \!-\! \frac{df(r)}{dr}\frac{\partial S}{\partial r}\right] = 0\nonumber \\ \end{aligned}$$

(39)

where we considered the ansatz $\phi (r,t) = e^{\frac{i}{\hbar } S(r,t)}$ for the semi-classical wave function of the Klein-Gordon equation. If we expand $S(r,t)$ in powers of ($\frac{\hbar }{i}$) then $S_0(r,t)$, the leading order in the expansion, satisfies the Hamilton-Jacobi equation

$$\begin{aligned} \frac{1}{f(r)} \left( \frac{\partial S_0}{\partial t}\right) ^2 - f(r)\left( \frac{\partial S_0}{\partial r}\right) ^2 - m^2 = 0. \end{aligned}$$

(40)

The solution of this equation is

$$\begin{aligned} S_0(r,t) = -Et \pm \int \frac{dr}{f(r)}\sqrt{E^2 - m^2 f(r)}. \end{aligned}$$

(41)

In the massless case the solution is

$$\begin{aligned} S_0(r,t)\vert _{m=0} = f_1 (t-r_t) + f_2 (t+r_t) \end{aligned}$$

(42)

where $r_t=\int \frac{dr}{f(r)}$ is the tortoise coordinate and $f_1$ and $f_2$ are arbitrary functions. The choice of $f_{1,2} = -Et\pm Er_t$ gives Eq. 41 in the massless case. Here $E$ is identified with energy. So we can see that the semiclassical ansatz is exact for the massless scalar field but one can also show that the final results remain same for the massive scalar field [15].