The Core

Black Hole Evaporation

Just now we have seen that a black hole "shines" with Hawking radiation. The escaping member of a virtual particle pair carried away energy from the black hole, and the black hole loses mass as a result. Eventually the black hole loses all its energy, or equivalently mass, and evaporates. Let's derive a formula to calculate the lifetime of a black hole. Jump Calculation

The power of Hawking radiation is just the same as the luminosity of a black hole:

On the other hand, the amount of energy carried away by Hawking radiation is the same as the energy loss by the black hole. Therefore, the power P of Hawking radiation is the rate of loss of total energy E of a black hole, that is:

P = -dE/dt

From Einstein's mass-energy equivalence relation,

(Speed of light is a constant!)

Equating P in equation (1) and (2) gives:

Em... The left-hand side of the equation looks pretty clumsy eh? Let's get rid of the chains of constants by a big constant K:

We now have a nicely written simple equation:

Finally, we are going to make use of a powerful math weapon... Yes, it's integration! As a black hole slowly evaporates, its mass drops from (its initial mass) to zero. The time required for evaporation starts from zero to (total time for evaporation). Integrating both-hand sides of equation (3),

The lifetime of a black hole is:

Cheers! We have derived the formula for calculating the lifetime of a black hole! The formula tells us that the lifetime of a black hole is proportional to the cube of its mass. That means a massive black hole takes proportionally much longer time to evaporate, and the process of evaporation accelerates as the black hole slowly loses its mass. This is known as the "runaway" effect. Moreover, take a look at the temperature formula of a black hole:

h: the Planck's constant;
c: the speed of light;
k: the Boltzmann constant;
G: the gravitational constant;
M: mass of a black hole

We see that as the black hole loses its mass, its temperature increases. When a black hole get very very small, its temperature may become so high that it may burn up and cause an explosion! Let's do some more calculations.

Some example calculations

What will be the lifetime of a black hole having the mass of our sun?

Wooow! The lifetime of such a black hole is even longer than that of our universe!

What about calculating the lifetime of the black hole in "The Hole Man"?

Well... the lifetime of this black hole is comparatively "shorter". If this black hole was formed not long after the beginning of our universe, we might be able to see its evaporation.

"Black holes aren't black - After Hawking they shine!"
Presented by Angie, Matthias and Thorsten
Team C007571,ThinkQuest Internet Challenge 2000.
Last modified: 2000-08-02.