P: 989

Quote by wikipedia
Current predictions for the behavior of a black hole with a mass less than Planck mass are inconsistent and incomplete.

One problem with calculating a constant rate of absorption, is the fact that a quantum black hole's horizon radius decreases with increasing mass. Therefore the horizon radius becomes a function of its mass:
[tex]r_h(E_b) = \frac{\hbar c}{E_b}[/tex]
This means that the cross section and reaction rate decreases as the quantum black hole mass increases.
However, I can determine what the upper limit of my equation is:
[tex]E_b = E_e = m_e c^2[/tex]
[tex]r_h = \frac{\hbar}{m_e c}[/tex]
[tex]t_e = \frac{4m_p}{3} \sqrt{\frac{(m_e c)^3 r_e^7}{2 \hbar^5}}[/tex]
[tex]t_e = 6.874 \cdot 10^{131} \; \text{s}[/tex]  2.180*10^124 years
An exponentially decreasing reaction rate should be even longer than this.
An equation demonstrating an exponentially decreasing cross section and reaction rate would probably show that the time required to absorb the Earth is infinite.
Is this infinite?, close enough...
