Can First Year Undergrad Physics Concepts Solve the Evaporation of Black Holes?

[hasan_researc]

Homework Statement

"Given that a black hole emits black body radiation at a temperature (in degrees Kelvin) given by T = hc³/16pi²MGk, how long would it take a solar mass black hole to evaporate?

If small black holes were created during the big bang, what mass of black hole would now be evaporating? "

Homework Equations

This problem was set as part of my first year undergrad "Professional Skills Problem Solving" seminar. So I have to use first year undergrad physics concepts (or possibly more advanced tools) to solve this problem.

The Attempt at a Solution

No idea! That's why I'm here.

Thanks in advance for any help!

 

[betel]

You should try to express the power of the emitted radiation in terms of the Temperature.
Then you can try to connect this energy loss to the time evolution of the mass of the black hole.

 

[hasan_researc]

Energy (of the emitted radiation) is on the order of kT (is it not?).

So the (average) power is kT/t, where t is some time interval.

Therefore, using the formula in my previous post, Power = hc³/16pi2MGt.
Therefore, t = hc³/(16pi2MG*power).

But do I know what the power is?

 

[betel]

Simply putting some time interval is not quite correct.

There is a certain law relating power and temperature of radiation of a black body.

 

[hasan_researc]

Actually, I don't know what the formula is. I found Plankc's law on wikipedia, but that looks horribly complicated, so I omitted that.

Please help me out on this.

 

[betel]

Try Stefan-Boltzmann Law.

 

[hasan_researc]

Okay, so the power radiated = sigma*T⁴
= sigma * (hc³/16pi²MGk)⁴

Now, how shall we assume that the mass radiated per unit time is a constant, and thus eliminate the time t in the equation?

 

[vela]

That's actually the power per unit area. You need to find the surface area of the black hole to get the total power radiated. Then try setting up a differential equation to work in the time dependence.

 

[hasan_researc]

Thank you for your reply. It was informative.

But you're saying that I have to set up a differential equation to work out the time dependence. But the structure of the course (in which the problem was set) was such that no differential equation needed to be used to solve the problems, but rather the concepts of energy and energy flux and idealised models are all we would need to solve the problem.

So I am wondering if there is a simpler solution.

 

[betel]

You need the concepts of energy and energy flux to set the way to the solution.
But it will not work without a differential equation, which are just a mathematical tool.