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Homework Help: Black hole question (temperature)

  1. Sep 25, 2012 #1
    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
  2. jcsd
  3. Sep 25, 2012 #2
    No one has any ideas?
  4. Sep 25, 2012 #3
    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.
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