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Collector
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Re: RIP

March 15th, 2018, 11:20 pm

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 for emissivity somewhere...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 \(\frac{1}{2}\)   :mrgreen: . 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!!!. 
 
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ppauper
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Re: RIP

April 4th, 2018, 10:50 pm

mathematician Philip J. Davis who wrote large chunks of Abramowitz & Stegun, including the section on Gamma functions, died March 14, 2018 aged 95
 
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Cuchulainn
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Re: RIP

April 4th, 2018, 11:03 pm

ppauper wrote:
mathematician Philip J. Davis who wrote large chunks of Abramowitz & Stegun, including the section on Gamma functions, died March 14, 2018 aged 95

RIP.
A top numerical analyst. His "Interpolation and Approximation" is a classic.
95! A good innings.

//
I see that Cathleen Morawitz Synge also passed last year
https://en.wikipedia.org/wiki/Cathleen_Synge_Morawetz
 
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ppauper
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Re: RIP

April 4th, 2018, 11:12 pm

RIP  Cathleen Morawitz Synge
 
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Cuchulainn
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Re: RIP

April 4th, 2018, 11:19 pm

One began to hear it said that World War I was the chemists’ war, World War II was the physicists’ war, World War III (may it never come) will be the mathematicians’ war. 

Philip J. Davis
 
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katastrofa
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Re: RIP

April 14th, 2018, 3:20 am

Daphne Sheldrick, 86, Kenyan-British environmentalist and elephant conservationist
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ppauper
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Re: RIP

April 14th, 2018, 5:22 am

katastrofa wrote:
Daphne Sheldrick, 86, Kenyan-British environmentalist and elephant conservationist
Image


RIP
 
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tw
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Re: RIP

April 14th, 2018, 8:50 am

six horses at Cheltenham festival

Hopefully Aintree is more benign today but the soft running doesn't bode well.
 
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ppauper
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Re: RIP

April 14th, 2018, 10:01 am

tw wrote:
six horses at Cheltenham festival

Hopefully Aintree is more benign today but the soft running doesn't bode well.

RIP horsies
I'm looking forward to reading about this years Ladies Day
tagoma wrote:
 
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tw
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Joined: May 10th, 2002, 3:30 pm

Re: RIP

April 14th, 2018, 10:15 am

Well that is the UK for you; all buttoned up repression, demureness and inhibition. 

ppauper wrote:
tw wrote:
six horses at Cheltenham festival

Hopefully Aintree is more benign today but the soft running doesn't bode well.

RIP horsies
I'm looking forward to reading about this years Ladies Day
tagoma wrote:

 
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katastrofa
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Re: RIP

April 16th, 2018, 2:07 am

Milos Forman :-(
 
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ppauper
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Re: RIP

April 16th, 2018, 5:27 am

RIP
 
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rmax
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Re: RIP

April 16th, 2018, 1:51 pm

RIP
 
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Cuchulainn
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Re: RIP

April 16th, 2018, 2:23 pm

RIP
"I can't take it no more. I gotta get outta here. "
 
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Cuchulainn
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Re: RIP

April 17th, 2018, 11:44 am

RIP Big Tom McBride

https://www.rte.ie/entertainment/2018/0 ... ride-dies/

Oh we're going out the same way we came in 
Don't matter who you know or where you've been 
Makes no difference who you are Skid Row Joe or superstar 
You're going out the same way you came in
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