Cuchulainn wrote:Collector wrote:RIP on \(\pi\)-day

One of several equations where Stephen Hawking used \(\pi\), Hawking temperature

\(T=\frac{\hbar c^3}{8\pi GM k_B} \)

and time slows down at the Black hole surface area \(16\pi N^2l_p^2\)

I'm with Wittgenstein on this one; this formula is a description, not an explanation.

Are black holes also black bodies? I am missing an entry foremissivitysomewhere...The Stefan-Boltzman law AFAIR is [$]\varepsilon\sigma T^4, [$] where [$] 0 < \varepsilon < 1[$].

Even humble soot - the closest thing to a black body - has a value of 0.97. (I used to simulate (black and white) TV CRTs using FEM at a Dutch company so I know a little bit and even phosphor was not a perfect absorbers).

Does anyone have a step-by-step derivation?

Not my area, but do perfect absorbers exist far away (or in people's heads) but not on planet Earth? Just wondering.Hawking used π

When does it appear?

"Does anyone have a step-by-step derivation?"

It is in his famous 1974 paper, on page 30 Hawking says"

"one would expect if the black hole was a body with temperature of \((\kappa/2\pi)(\hbar/2k)\)..."

And yes he says \(\kappa\) is the surface gravity of the Black hole. If we for simplicity assume Newton surface gravity at the Karl radius (better known as the Schwarzschild radius) we have the following Newtonian Schwarzschild gravitational acceleration field (Newton where only radius is replaced by Schwarzschild radius \(r_s\) :

\(g=\frac{GM}{r_s^2}\)

(after sleeping on it) I am quite sure we must divide this by c, then replace the k and we get

\begin{eqnarray*}

T&=& \frac{\frac{\frac{GM}{r_s^2}}{c}}{2\pi}\frac{\hbar}{2k} \\

T&=&\frac{ \frac{\frac{GM} { \left( \frac{2GM}{c^2}\right)^2 }}{c}} {2\pi}\frac{\hbar}{2k} \\

T&=&\frac{\frac{\frac{GM}{\frac{4G^2M^2}{c^4}}}{c}}{2\pi}\frac{\hbar}{2k} \\

T&=&\frac{c^3}{8\pi GM}\frac{\hbar}{2k}

\end{eqnarray*}

off by a factor of \(\frac{1}{2}\) . The formula \(T=\frac{\hbar c^3}{8\pi GM k_B} \) is based on different method for surface gravity taking into account the bending of space-time, the Schwarzschild metric solution of GR that is known to be different in strong gravitational fields than Newton.

I guess similar to also GR bending of light twice as Newton.

Anyway I am on Cuch on Wittgenstein on this one; this formula is a description, not an explanation.

Ohh yes Cuch, I now understand you for years have wondered about the temperature of Einstein in the Black Hole Hedge Fund ? It was a 10 solar masses Black Hole, so looks like the Hawking temperature is approx \(6.17 \times 10^{-9}\) kelvin (I assume he moved his space station down to the surface of the bh). Ohh that explain why so few Black Hole hedge funds, I always wondered. Their assets have been frozen so to say!!!.