Differentiation woes with temperature/entropy relations.

[EricVT]

Alright, this is probably a really redundant question but for some reason it is giving me trouble. Let's say you are given the entropy of a black hole as:

$$S=\frac{8\pi^2GM^2k}{hc}$$ (thanks Stephen Hawking)

And you have the relation between temperature and entropy/energy

$$\frac{1}{T}= \frac{\partial S}{\partial U}$$ (U is energy, S is entropy)

Now if you want an expression for the temperature of a black hole in terms of it's mass and you are using U = Mc^2, then should you rewrite:

$$M^2 = \frac{U^2}{c^4}$$

$$S = \frac{8\pi^2GkU^2}{hc^5}$$

And then differentiate with respect to U to get:

$$\frac{1}{T} = \frac{16\pi^2GkU}{hc^5}$$

$$T = \frac{hc^5}{16\pi^2GkU}$$

First of all, as the ENERGY increases the TEMPERATURE decreases? Is this really possible here? I'm confused by this. Finishing the problem, though, and rewriting U = Mc^2 gives:

$$T = \frac{hc^3}{16\pi^2GkM}$$

Does this seem correct? I tried working the problem a different way by writing c in terms of U as well at the start, and differentiating that expression and got a completely different answer...one that is always negative no less. So with that approach you get constantly negative temperatures...I'm very confused by what result I should be looking for.

 

[learningphysics]

What you did here looks right to me.

 

[Hurkyl]

First of all, as the ENERGY increases the TEMPERATURE decreases?

(Heuristically speaking) Remember that as a black hole gains energy, it expands, which has a cooling effect. This apparently dominates other effects. This is why we expect large black holes to be very stable, whereas tiny black holes should evaporate away very quickly.

 

[EricVT]

Thanks then.

Cheers.