Black hole question (temperature)

[linda300]

hey guys,

i'm working on this question to approximate the entropy of a black hole,

the approximation is that for the maximum entropy to be obtained you need a maximum number of particles to create the black hole, the particles must have low energies - large wavelength photons, but the maximum wavelength of the photons are twice the radius of the black hole.

so E = hc/λ=hc/2R

R = 2GM/c^2 (black hole of mass M)

The total energy of the black hole will be E_t = M c^2 and need

E_t = N hc/2R =M c^2

so from here you can find N to be N= 2MRc/h

and using the approximation that S = N k_b you can find the entropy of the black hole

S = (2 M R c k_b)/h

this entropy differs from the literature entropy by a factor of pi which i assume is due to the nature of the approximations,

my problem is, i want to find the temperature of the black hole using the entropy S i calculated

S = Q/T, but what is Q?

I have found online that Q=Mc^2, but I don't understand why.

using the literature expression for S i tried to derive the literature expression of the temperature by T = Q/S using Q=Mc^2 but the result differs by a factor of 2.

here is the working (literature case)

S=k_b (4 pi G M^2)/(hbar c)

T= Q/S = Mc^2 (hbar c)/ (4 pi G M^2 k_b)
=c^3 hbar/ (4 pi G M k_b)
but the literature value of T is
c^3 hbar/ (8 pi G M k_b)

which means that Q should = (1/2)Mc^2, but i don't understand why

what is the correct Q i should be using?

thank you in advance

 

[linda300]

No one has any ideas?

 

[marty1]

I'll be honest and tell you I don't really know, but what I think based on something I read once is that the energy (temperature) that escapes from a black hole (the only temperature you could measure) are the particle pairs that are created at the event horizon. It is 1/2 because only one of the pair escapes and gets far enough away to decay relasing the energy (e=mc^2) ? That really is a wild guess.