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Posted by Bruce on May 23, 2003 04:54:08 UTC

that there is Hawking radiation at the DeSitter horizon is the same thing as saying virtual particles pop in and out of the vacuum. The issue is whether any of it becomes detectable with existing technology. Around the event horizon of a black hole a virtual pair can be separated by tidal effects [before they can annihilate]. In effect this is borrowing energy from the black hole via its gravitational field. Three outcomes are possible for a pair which gained enough energy to become real [detectable]. 1) The entire pair can fall into the black hole resulting in the energy of the hole remaining the same. 2) 1/2 the pair can escape to infinity carrying away black hole energy equal to the energy 1/2 pair gained from the tidal separation while 1/2 pair falls into the hole, 3) or the entire pair can escape to infinity carrying away all the energy gained by the tidal separation. Outcomes 2 and 3 result in a loss of mass for the black hole. The issue is what has to happen for Hawking radiation to gain enough energy to be detectable? Thorne says the tidal separation between the virtual pairs has to be ~ 1/4 the circumference of the event horizon. The Hawking temperature can be derived from the Unruh temperature

The Unruh result is based on a quantum explanation. Which is virtual particles pop in and out of the vacuum. So the Unruh temperature becomes

T_Unruh = h*g_conv/4(pi)^2*k_Boltzmann*c [eq.1]

In geometric units

g_shell = g_conventional/c^2 = (M/r^2)/(1-2M/r)^-1/2

g_conv = (M*c^2/r^2)(1-2M/r)^-1/2 [eq.2]

Substituting [eq.2] into [eq.1] and simplifying results in

T_Unruh = [h*c*M/4(pi)^2*k_Boltz*r^2](1-2M/r)^-1/2

To transform this result to Hawkings result you let r->2M [the event horizon] eliminating (1-2M/r)^-1/2 by factoring in (1-2M/r)^1/2 to
cover the redshift expected for the remote observer at boundary condition.

T_Hawking=[h*c*M/4(pi)^2*k_Boltz*(2M)^2](1-2M/r)^-1/2(1-2M/r)^1/2

T_Hawking = h*c/16(pi)^2*k_Boltz*M

Its hard for me to believe that the possibility exists for virtual radiation to become real [detectable] at a DeSitter horizon which doesn't include a black hole.







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