Entropy of Black hole

Introducing thermodynamics to Black holes.

Entropy is the term which when we read about it leads us to the path towards thermodynamics. In the early 1800s, researchers and scientists started studying heat, temperature, and the behaviour of gases, which later evolved into thermodynamics. According to thermodynamics and the famous three laws of thermodynamics, it says that:

• The zeroth law states that if two bodies are each in thermal equilibrium with a third body, then the first two bodies are also in thermal equilibrium with each other.
• The first law states that the total energy of an isolated system always remains constant. It can only transform from one state to another but never be destroyed.
• The second law states that the change in the entropy of the entire universe can never be negative.
• The third law states that the entropy of a system at absolute zero is a well-defined constant.

So, considering all this above, many controversies and paradoxes arise when we try to apply them to black holes. As black holes have mass, rotation, and temperature, it is obvious for them to have entropy, so as the second law states that (the total energy of an isolated system always remains constant, it can only transform from one state to another but never be destroyed). The energy of a black hole should always remain constant, but if you could throw an object (with a considerable amount of entropy) into a black hole, the entropy would simply go away. It would vanish nowhere. In other words, the entropy of the system would get smaller and smaller, which would violate the second law of thermodynamics. Considering another situation is that the classical black hole has a temperature of absolute zero. This means you could take a bucket full of hot water and throw it into a black hole, which would essentially be cooling an object to absolute zero. It is a violation of the third law of thermodynamics.

Bekenstein-Hawking entropy :

 Bekenstein-Hawking entropy, also known as black hole entropy, is the amount of entropy that a black hole must have in order to obey thermodynamic laws as interpreted by observers outside the black hole. A black hole can be formed in many ways . After it settles down, space and time outside are described by only M and J. The radiation it emits is essentially thermal. It can’t depend on the information inside without violating causality or locality. 

There are several ways to justify the entropy of a black hole.

• Considering the loss of signal with a body outside the black hole, when a body enters into a black hole, it is the same as the loss of information, and in ordinary physics, entropy is the measure of the loss of information. Hence, entropy can be defined for a black hole.
• A black hole is usually formed from the collapse of matter under its own gravity or radiation. Both the terms which relate to the formation of a black hole, i.e., matter and radiation, are associated with entropy. However, the black hole’s matter inside is unknown to the observer outside the black hole. Thus, a thermodynamic layout of the collapse from that observer's point of view cannot be based on the entropy of that matter or radiation (the key roles in the formation of a black hole) because these are unobservable. Associating entropy with the black hole provides a handle on thermodynamics.

 

Formulation for a concrete formula for entropy…

There is a need for a concrete formulation to describe the entropy of a black hole, but from the above discussion it is clear that only the observable parameters can be considered for the formulation of black holes. So the major observable parameters were mass, angular momentum, and electric charge. So, taking the area theorem into consideration, all these parameters come into the same combination as that which represents the surface area of a black hole. The area theorem states that the surface area of a black hole can’t decrease; it can only increase in black hole transformation. So, the final formulation provided as a solution to all these is

                                                   

Where,

             A represents the surface area of black hole.

             G Newton's gravity constant.

             h the Planck-Dirac constant (h/(2π)).

             c speed of light.

   For, Schwarzschild or spherically symmetric black hole the horizon's radius is 

So,                      

                                     A=16π(GM/c2)2

Hence, the considerable efforts to make all the parameters fit to make a defined formulation to find the entropy of black was possible, but still, much more research is going on to find more reasonable formulations in the present and future.