Classically, a blackhole is decribed by it's mass, angular momentum and charge. We will ignore angular...

  

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Classically, a blackhole is decribed by it's mass, angular momentum and charge. We will ignore angular momentum and and charge for this homework. A blackhole is characterized by it's event horizon inside of it causally disconnected to the outside world. Therefore, once something goes inside the blackhole all the information about that is lost (classically). To circumvent the above information issue, in 1972 Bekenstein proposed an entropy formula for a blackhole, s- ks Gh where A is the area of the event horizon of the blackhole. This is a very unusal relationship because for nearly all of the entities in the nature entropy is proportional to it's volume not the area. One can calculate the area of the event horizon by, A = inr where rn is the horizon radius. Since, we stated that an ordinary blackhole is characterized only by it's mass, TA- 2GM/2. Using the entropy formula above and the first law of thermodynamics dE = Tds find the temperature associated with a blackhole with mass M. Remember E = Me and mass is only parameter you can vary. Calculate the temperature of a blackhole with a mass equal to the Sun's mass. 1.2 Heat Capacity Using the temperature you have found, calculate the constant-volume heat capacity, Cy = dE/dT. What is the sign of this quantity and how can you explain it? 1.3 Higher Dimensions We want to generalize blackhole entropy as a function of it's mass relation to higher dimensions. In a four- dimensional universe (3 space + 1 time dimensions) entropy is proportional to the second power of the horizon radius, S x r. In D-dimensions Sox rf, what is k in terms of D? In D-spacetime dimensions rA, x M, Then, if Sx M', write t in terms of D. For a thermodynamical system to be stable, 20 dE/V.N Show why the above relation holds. Then, using the previous results, discuss the stability of blackhole's in higher dimensions. Classically, a blackhole is decribed by it's mass, angular momentum and charge. We will ignore angular momentum and and charge for this homework. A blackhole is characterized by it's event horizon inside of it causally disconnected to the outside world. Therefore, once something goes inside the blackhole all the information about that is lost (classically). To circumvent the above information issue, in 1972 Bekenstein proposed an entropy formula for a blackhole, s- ks Gh where A is the area of the event horizon of the blackhole. This is a very unusal relationship because for nearly all of the entities in the nature entropy is proportional to it's volume not the area. One can calculate the area of the event horizon by, A = inr where rn is the horizon radius. Since, we stated that an ordinary blackhole is characterized only by it's mass, TA- 2GM/2. Using the entropy formula above and the first law of thermodynamics dE = Tds find the temperature associated with a blackhole with mass M. Remember E = Me and mass is only parameter you can vary. Calculate the temperature of a blackhole with a mass equal to the Sun's mass. 1.2 Heat Capacity Using the temperature you have found, calculate the constant-volume heat capacity, Cy = dE/dT. What is the sign of this quantity and how can you explain it? 1.3 Higher Dimensions We want to generalize blackhole entropy as a function of it's mass relation to higher dimensions. In a four- dimensional universe (3 space + 1 time dimensions) entropy is proportional to the second power of the horizon radius, S x r. In D-dimensions Sox rf, what is k in terms of D? In D-spacetime dimensions rA, x M, Then, if Sx M', write t in terms of D. For a thermodynamical system to be stable, 20 dE/V.N Show why the above relation holds. Then, using the previous results, discuss the stability of blackhole's in higher dimensions.

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We discuss the stability of charged static black holes in higher dimensional spacetimes with and wit... View the full answer