Entropy of Black Hole - Hawking Radiation - TIT

Entropy of Black Hole & Hawking Radiation

Black Hole thermodynamics is a field of study where we relate the laws of thermodynamics with the existence of black-hole or rather black-hole event horizons. We can get the entropy of a black hole in terms of the event area of horizon and three fundamental constants of nature, by applying the laws of Black Hole mechanics. 

Laws of Black Hole Thermodynamics:

(I)                 The Zero’th Law: The horizon has constant surface gravity for a stationary black hole.

The temperature (T) is constant throughout a body in thermal equilibrium. It suggests that the surface gravity (k) is analogous to temperature ‘T’ constant for thermal equilibrium for a normal system is analogous to ‘k’ constant over the horizon of a stationary black hole.

(II)               The First Law: The change of energy of a stationary Black hole is related to change of area, angular momentum and electric charge by

                                           dE = (k/8π) dA +  Ω dω + Ф dQ

Where,  dE = Change in energy,
                k = Surface gravity,
                A = Horizon area,
                Ω = Angular velocity,
                ω = Angular momentum,
                Ф = Electrostatic potential,
                Q = Electric charge

The first law of thermodynamics is a statement of energy conservation, which contains on the right side of the equation, the term TdS.

(III)             The Second Law: The horizon area (assuming the weak energy condition), is a non-decreasing function of time:

                                                   dA/dt ≥ 0
                                         ⇒    dA ≥ dt

This states the Hawking’s area theorem, which states that black hole radiates. It causes both the black hole’s mass and area of its horizon to decrease over time.

The second law of thermodynamics states that change in entropy (e) in an isolated system will be grater than or equal to 0 [ e  0 ] for a spontaneous process.

The entropy of a black-hole horizon doesn’t violet the second law of thermodynamics. Generalizing the second law as the sum of black-hole entropy and outside entropy, shows that the second law of thermodynamics is not violated in a system including the universe beyond the horizon.

(IV)             The Third Law: It is not possible to form a black hole without surface gravity (k). It means, k=0 is not possible to achieve.

It is analogous to the third law of third law of thermodynamics, which states that the entropy of a system at absolute zero temperature is a well defined constant. Because a system at zero temperature exists in its ground state.

                          Δs will reach Zero ( at zero temperature)
                  Also, s will reach Zero (s is entropy)

Only the Extremal Black Holes have ‘0’ surface gravity.

Interpretation of the laws:

All the four laws of the Black Hole dynamics suggests that, one can identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants.
However, when quantum mechanical effects are taken into the calculations, one finds that black holes emit thermal radiation at a temperature,

                                                        TH = k/2π

This thermal electromagnetic radiation is called the Hawking Radiation.

Now, from the first law of thermodynamics, this determines the multiplicative constant of the Bekenstein-Hawking entropy, which is

                                                        SBH = A/4

If we sum up the entropy a measure of the disorder of a system in a formula,

                              S = (Akc3)/(4G ħ)

Where,  S = Entropy of Black Hole,
              A = Event area of the horizon
              c = Speed of light,
              G = Newton’s constant of Gravitation,
             ħ = h/2π,           h = Plank’s constant,
               k = Surface gravity

This formula expresses the entropy in terms of the area of the horizon and the three fundamental constants of nature.

The emission of this thermal radiation from the black hole is now called Hawking Radiation.


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