# Dec 6, 2013 more on black hole evaporation lifetime. Did Hawking underestimate it?

We compute the total power P by multiplying the far-away redshifted energy density by the area A of the horizon. This may be a conceptual error in Hawking's original estimate of black hole evaporation time. The clock hovering at L distant from A is running much faster than our far away clock where we are. A is an infinite redshift surface, but including L makes it finite but large. Also it takes infinite far away clock time for objects to reach A etc.

Therefore, we can argue that the Wikipedia calculation is wrong. That is, for Hawking's surface gravity case, replace

P ~ A (energy density) ~ A^-1

by

P ~ g00(L)^1/2A (energy density)
~  [1 + z(L)]^-1A (energy density)
~ (L/A^1/2)^1/2A^-1

Therefore, even in Hawking's case,

P ~ dM/dt ~ L^1/2/A^5/4

Therefore,

dM/dt ~ L^1/2/M^5/2

tHawking ~ M^7/2 /L^1/2  not M^3

remember there is no actual evidence for M^3.

Next our new case

P' ~ AT'^4 ~ A/L^2A ~ L^-2 ~ mp^-2

dM'/dt ~ mp^-2

P' ~ [1 + z(L)]^-1AT'^4
~ (L^1/2/A^1/4)A/L^2A
~ 1/L^1/2A^1/4

dM'/dt ~ 1/L^1/2M^1/2

t' ~ L^1/2M^3/2

to be continued

Putting in some numbers
From Wiki

Stefan–Boltzmann–Schwarzschild–Hawking black hole radiation power law derivation:
For a solar mass black hole
Putting in the gravity time dilation factor L^1/2/A^1/4

L ~ 10^-35 meters

L^1/2 ~ (1/3) 10^-17

A^1/2 ~ 10^3 meters

A^1/4 ~ 3 x10

L^1/2/A^1/4 ~ 10^-17/3x3s10 ~ 10^-19

so

P ~ 10^-28 x 10^-19 ~ 10^-47 watts

P' ~ [1 + z(L)]^-1AT'^4 ~ (L^1/2/A^1/4)A/L^2A ~ 1/L^1/2A^1/4

http://en.wikipedia.org/wiki/Stefan–Boltzmann_constant

P' ~ 6 x 10^-8T'^4

Our T' = (A^1/2/L)^1/2T ~  10^23(M/mp)^1/2(1/M) ~ 10^23/(mpM)^1/2  deg K

Therefore, energy density is

6 x 10^-8 x 10^92/mp^2M^2

Multiply by the area A and the gravity time dilation factor L^1/2/A^1/4

So that's effective area    L^1/2A^3/4

Total power is then

P' ~ 10^85 L^1/2A^3/4/mp^2M^2  Watts

for a solar mass scale black hole that's roughly

P' ~ 10^85 (1/3) 10^-17 (10^6^)3/4 10^10 x 10^-60  Watts

P' ~ 10^23 Watts - very roughly in gravity wave black body radiation ~

peak wavelength ~ 10^-16 meters ~ 10^24 Hz

to be continued - next order of biz evaporation lifetime

The 10^23 Watts is only the initial output - that increases as the black hole evaporates

Putting in some numbers
From Wiki

In our new theory this is I think

t'ev = c^2(mpM)^3/2 /3Kev

(mpM)^3/2 = xM^3

x = (mpM)^3/2/M = (mp/M)^3/2

t'ev = (mp/M)^3/2 tev ~  (mp/M)^3/2 10^-16[M/kg]^3

For a ~ solar mass black hole

(10^ -35)3/2 10^67 years ~ 10^-52 10^67 ~ 10^15 years

On Dec 5, 2013, at 7:55 PM, JACK SARFATTI <This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:

From the beginning:

First Hawking

L = Schwarzschild radial coordinate distance to horizon classical 2D surface g00 = 0.

Newton's surface gravity ~ A^-1/2

A = area-entropy of g00 = 0

What they do in Wikipedia above comes down to this

Redshifted Unruh temperature a long distant from the black hole is

THawking ~ A^-1/2

Stefan-Boltzmann law

energy density ~ THawking^4 ~ A^-2

Total redshifted power

P ~ A (energy density) ~ A^-1

A ~ M^2

P ~ dM/dt

OK now my new prediction following the same argument as above

The redshifted thickness gravity Unruh temperature is

T' ~ (LA^1/2)^-1/2

If we take

Lp ~ mp = Planck mass

T' ~ (mpM)^-1/2

P' ~ AT'^4 ~ A/L^2A ~ L^-2 ~ mp^-2

dM'/dt ~ mp^-2

t' ~ mp^2M << t ~ M^3
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