- #1
EricVT
- 166
- 6
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:
[tex] S=\frac{8\pi^2GM^2k}{hc} [/tex] (thanks Stephen Hawking)
And you have the relation between temperature and entropy/energy
[tex] \frac{1}{T}= \frac{\partial S}{\partial U} [/tex] (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:
[tex] M^2 = \frac{U^2}{c^4} [/tex]
[tex] S = \frac{8\pi^2GkU^2}{hc^5} [/tex]
And then differentiate with respect to U to get:
[tex] \frac{1}{T} = \frac{16\pi^2GkU}{hc^5} [/tex]
[tex] T = \frac{hc^5}{16\pi^2GkU} [/tex]
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:
[tex] T = \frac{hc^3}{16\pi^2GkM} [/tex]
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.
[tex] S=\frac{8\pi^2GM^2k}{hc} [/tex] (thanks Stephen Hawking)
And you have the relation between temperature and entropy/energy
[tex] \frac{1}{T}= \frac{\partial S}{\partial U} [/tex] (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:
[tex] M^2 = \frac{U^2}{c^4} [/tex]
[tex] S = \frac{8\pi^2GkU^2}{hc^5} [/tex]
And then differentiate with respect to U to get:
[tex] \frac{1}{T} = \frac{16\pi^2GkU}{hc^5} [/tex]
[tex] T = \frac{hc^5}{16\pi^2GkU} [/tex]
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:
[tex] T = \frac{hc^3}{16\pi^2GkM} [/tex]
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.