On Monday, April 8, 2019 at 4:03:19 PM UTC-5, John Clark wrote:
>
>
> On Mon, Apr 8, 2019 at 3:29 PM Mason Green <mason...@hotmail.com
> <javascript:>> wrote:
>
>
>>
>> *> Here’s another idea I just came up with, that doesn’t harness dark
>> energy itself so much as the Hawking radiation of the de Sitter horizon.A
>> civilization could build a sphere around a cold black hole (I.e., a
>> rotating or charged black hole whose Hawking temperature is lower than that
>> of the cosmological horizon; such a black hole would need to be very close
>> to extremal).The sphere would catch the Hawking radiation from the
>> cosmological horizon, and then feed some of it into the black hole in such
>> a way as to further decrease its temperature (by pushing it closer to
>> extremality). The sphere could use the rest of the energy for its own
>> needs. The black hole and the sphere would keep growing over time.*
>>
>
>
> That could work, for a while. As long as you have a temperature difference
> you can run a heat engine. The Cosmic Microwave Background Radiation is at
> 2.7K so if you used a solar mass Black Hole as a heat sink you could
> extract some work out of it because it has a temperature of only 00000006k.
> But it wouldn't be much as the efficiency would be very low; and even
> that pitiful trickle of work wouldn't last forever because over time the
> Cosmic Microwave Background Radiation will get colder and, as it starts to
> evaporate and gets smaller and smaller, the Black Hole will get hotter and
> hotter until it explodes and disappears.
>
> John K Clark
>
The CMB is not the Gibbon-Hawking radiation from the cosmological horizon.
The CMB is what happened when the plasma of the semi-early universe
coalesced into atom and the radiation was released. This happened about
380k years into the evolution of the cosmos. The Gibbon-Hawking result of
radiation due to the cosmological horizon.
To start I consider the elementary case is with the accelerated frame. An
accelerated observer, g = acceleration, is on a hyperbolic path that
asymptotes to a split horizon. The two horizons occur at some arbitrarily
chosen origin. A quantum flutuation as a loop at this origin with a radius
r has null geodesics connecting it to the accelerated frame. Assume this
accelerated observer approaches within r = c^2/g of the origin. Then the
observer has causal contact with the loop throughout this Rindler wedge.
The loop parameterized by a time or length d = 2πr = 2πct and in Euclidean
time, since this is an off shell state and an instanton, the unitary
operator e^{-iHt/ħ} → e^{-2πħωrħ}. Here the Hamiltonian is assumed to give
energy ħω. Substitute in r = c^2/g we have a Boltzmann term e^{-2πħωc/g}.
An identification of this with e^{-E/kT} leads easily to the temperature
T = ħg/2πkc.
So temperature is the same as acceleration! This is a quick an dirty
derivation of Unruh radiation, which I will admit glosses over some points.
Some work and the identification of the acceleration with a black hole
gives Bekenstein-Hawking temperature for a black hole.
The Gibbon-Hawking temperature can be found if we let g = c^2/(horizon
distance) and for the cosmological constant Λ = 10^{-56}m^{-2} and R =
sqrt{3/Λ} ~ 10^{28}m then g ~ 10^{-12}m/s^2 The temperature
T = ħc^2 sqrt{Λ/3}/2πkc ~ 10^{-30}K.
That is an absurdly cold temperature. In order to use that as an energy
source you would need to have a cold bath that is even colder. That is not
really possible. Another way to see it is the wavelength of most of this
radiation is on the order of the cosmological horizon scale. You would need
a detector on that scale to detect a boson.
LC
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