Bekenstein Entropy Formula

Black holes! Who does not want to talk about them? So today, in the 20th article of our Month of Equation series, we will touch upon a formula that explains the thermodynamics of these mysterious objects.Named after Jacob Bekenstein and Stephen Hawking, The Bekenstein-Hawking formula describes the thermodynamic entropy of a black hole with a given mass.

Bekenstein Entropy Formula says that the entropy of a black hole is proportional to the area of its event horizon.

{S_{BH} = \frac{k c^{3} A}{4 G \hslash }}

  • BH in subscript refers to black hole.
  • S : entropy
  • k : Boltzmann's constant
  • c : speed of light
  • G : Gravitational constant
  • {\hslash = \frac{h}{2 \pi}}, where h is the Planck's constant
  • A : area of Event Horizon. For a Schwarzschild black hole, {A = \frac{16 \pi G^{2} M^{2}}{c^{4}}}

IMPORTANT TERMS:

First of all let's try to understand the meaning of basic terms associated with this formula, the Entropy and Event horizon.

Entropy: Coined by Rudolf Clausius in 1850, Entropy refers to the measure of a system’s thermal energy per unit temperature that is unavailable for doing useful work. Since, ordered motion of system gives the obtainable work. So, the amount of entropy is also taken as the measure of the disorder or randomness of a system.

Event horizon: A possible solution of Einstein's general theory of Relativity, black holes are region of space-time, which exhibits such strong gravitational acceleration so strong that not even the electromagnetic radiation such as light—can escape from it. According to The theory of general relativity, a sufficiently compact mass can deform space-time to form a black hole. The boundary of the region from which no escape is possible is the event horizon.

Bekenstein Entropy Formula
An Artist's Illustration of a Black Hole

WHY TO ASSOCIATE ENTROPY WITH A BLACK HOLE?

According to the Second law of thermodynamics, the total entropy of our universe either remains constant or increases. However, Bekenstein realized that the accepted description of a black hole (at that time) violated the second law of thermodynamics. Let's understand how! Suppose, we throw a hot gas with some entropy into a black hole. Once it crosses the event horizon, the entropy would disappear. We can no longer have an account of the random properties of the gas once the black hole has absorbed it. So, In order to handle the thermodynamics of the total system, it become necessary to associate entropy with the black hole.

This paved way towards the generalized second law which considers the sum of black-hole entropy and outside entropy. Thus, if the increase in entropy of black hole is more than the decrease in entropy of the gas or other objects being swallowed, no violation of laws is seen. Moreover, In ordinary physics entropy is also referred as the measure of missing information. Since, a black hole can also be said to hide information.Hence it makes sense to attribute entropy to a black hole.

Also Read: The three types of Black Holes in the universe.

FORMULATION OF THE BEKENSTEIN ENTROPY FORMULA:

Now, it came the turn to associated properties with the observable properties of the black hole like mass, electric charge and angular momentum. It turned out that these three parameters entered only in the same combination which represented the surface area of the black hole. Moreover, according to Hawking's "area theorem", the event horizon area of a black hole cannot decrease. It increases in most transformations of the black hole, just like entropy. Thus led Jacob Bekenstein to conjecture that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. In 1974, Hawking confirmed Bekenstein's conjecture and fixed the constant of proportionality. This gave the Final Black hole entropy formula.

Also Read: All the 30 Articles of Basics of Astrophysics series

WHY IS IT IMPORTANT?

Associating entropy with the Black hole helps to handle the thermodynamics. Moreover, From the entropy, it is possible to use the usual thermodynamic relations to calculate the temperature of black holes.

Image result for black hole entropy formula
Bekenstein Entropy Formula

AUTHOR'S MESSAGE:

Black holes are the solutions to Einstein's theory of general Relativity.These are also one of the hottest topic in pop science as well. But, instead of getting misguided by the pop science facts associated with them, one must try to understand the physics behind them. The equation above gives a method to study the thermodynamics of these giant bodies. I hope that this article was helpful in introducing you to a section of physics concerned with Black holes.

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