Number of photons to make a black hole

[quasar_4]

Homework Statement

By setting the total energy of photons to Mc^2, estimate the maximum number of photons that could be used to make a black hole of mass M.

This is a thermal physics problem, in case anyone needs a context; we're discussing the entropy of a black hole.

Homework Equations

I'm given the entropy for a black hole. I also know the Schwarzschild radius.

The Attempt at a Solution

I'm just having trouble seeing where to start. Probably this is something very basic and I'm just missing it. Can someone just help me to start?

 

[Jhero]

Thank you for bringing up this interesting question. To estimate the maximum number of photons that could be used to make a black hole of mass M, we can use the formula for the entropy of a black hole, which is given by S = (A/4)G, where A is the area of the event horizon and G is the gravitational constant.

We also know that the Schwarzschild radius, which is the radius of the event horizon, is given by R = 2GM/c^2, where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.

By setting the total energy of photons to Mc^2, we can equate it to the energy needed to create a black hole of mass M, which is given by E = Mc^2.

We can then use the formula for the energy of a photon, E = hf, where h is the Planck constant and f is the frequency, to calculate the maximum number of photons needed.

By equating E = Mc^2 and E = hf, we get f = Mc^2/h.

Next, we can use the relationship between frequency and wavelength, c = fλ, to calculate the wavelength of the photons.

Substituting c = fλ into the equation f = Mc^2/h, we get λ = h/Mc, which gives us the maximum wavelength of the photons needed.

Finally, we can use the formula for the area of a sphere, A = 4πr^2, to calculate the area of the event horizon, which is equal to the entropy of the black hole.

Substituting the value of R = 2GM/c^2 into the equation A = 4πr^2, we get A = 16πG^2M^2/c^4.

We can then substitute this value into the formula for entropy, S = (A/4)G, to get S = 4πG^2M^2/c^4.

To find the maximum number of photons, we can divide the entropy by the maximum wavelength of the photons, which gives us N = S/λ = (4πG^2M^2/c^4)/(h/Mc) = 4πG^2M^3/hc^5.

Therefore, the maximum number of photons needed to make