黑洞热学常用公式

metric

$$d{s}^2=-f(r)d{t}^2+\frac{1}{f(r)}d{r}^2+Ad{\theta }^2+Bd{\varphi }^2+OtherParts$$

event horizon

$$fromf(r)=0,getr=r_h$$

surface gravity

$$\kappa =\frac{1}{2r_h}$$

temperture

$$T=\frac{\kappa }{2\pi }=\frac{1}{4\pi r_h}$$

$$\beta =\frac{1}{T}$$

GUP result

the $ϵ$:

$$2\sqrt{\lambda }={\int }_{r_h}^{r_h+ϵ}\frac{dr}{\sqrt{f}}$$

$$I(x)=\frac{f^{\prime }(r_h){ϵ}^2{r}_h^2}{2x^2(x^2+\frac{{\beta }^2{f}^{\prime }(r_h)ϵ}{4\lambda }{)}^2}$$

where $x=\frac{\beta \omega }{2}$

Entropy

$$S_{AH}=\frac{{\beta }_{AH}^3}{12\pi {\lambda }^3}{\int }_0^{\mathrm{\infty }}dx\frac{x^4}{{\sinh }^{2}x}I(x{)}_{AH}$$