Chapter 1: Introduction to Black Hole Entropy

The concept of black hole entropy, introduced by the Mexican-born Israeli-American physicist J. Bekenstein, suggests that black holes—regions of spacetime where gravity is so intense that light cannot escape—should possess a defined entropy. Bekenstein proposed that black hole entropy can be understood as follows:

"Black hole entropy represents the quantity of entropy that must be attributed to a black hole to ensure adherence to thermodynamic laws as interpreted by external observers."

— Jacob Bekenstein

Section 1.1: Understanding Black Hole Entropy

Bekenstein outlined three justifications for the existence of black hole entropy:

1. The matter and radiation that collapse to form a black hole are concealed from external observers, preventing them from describing the collapse thermodynamically based on the entropy of the collapsed matter. Associating entropy with the black hole itself provides an explanation.
2. To characterize a stationary black hole, only three parameters are necessary: mass (M), charge (Q), and angular momentum (J). Given various formation scenarios for a black hole with specific values for these parameters, multiple internal states must exist, indicating a relationship with thermodynamic entropy.
3. Black holes obscure information, offering only the parameters (M, Q, J) while blocking all incoming signals. This aligns with the principle that entropy reflects missing information, further implying that black holes possess entropy.

Section 1.2: The Formula for Black Hole Entropy

It is reasonable to assert that the entropy of a black hole relies solely on observable quantities M, Q, and J. The area theorem states that the area of a black hole's event horizon cannot diminish, akin to the behavior of thermodynamic entropy in closed systems. Consequently, it is logical to propose that a black hole's entropy is an increasing function of the event horizon's area.

The relationship can be expressed as follows:

Equation 1: The entropy of a black hole is proportional to the area (A) of its event horizon.

Chapter 2: Types of Black Holes

There are four primary types of black holes:

1. Schwarzschild Black Holes: Characterized by mass (M), with no charge (Q=0) or angular momentum (J=0).
2. Kerr Black Holes: Defined by mass (M) and angular momentum (J), with no charge (Q=0).
3. Kerr-Newman Black Holes: Generalized black holes with mass (M), charge (Q), and angular momentum (J).
4. Reissner-Nordström Black Holes: Distinguished by mass (M), charge (Q), and no angular momentum (J=0).

Section 2.1: The Schwarzschild Black Hole

The Schwarzschild black hole is stationary and exhibits spherical symmetry, defined solely by its mass (M). Its line element in spherical coordinates can be expressed mathematically.

Changing to Eddington–Finkelstein coordinates reveals that the point r=2M is not a real singularity, with the only true singularity located at r=0. The event horizon's radius is r=2M, leading to the corresponding surface area and entropy.

Section 2.2: The Kerr-Newman Black Hole

The Kerr-Newman black hole is the most generalized stationary black hole, characterized by its mass (M), charge (Q), and angular momentum (J). Unlike the Schwarzschild black hole, the Kerr-Newman horizon does not maintain a spherical shape.

In Boyer-Lindquist coordinates, the line element can be expressed mathematically, along with the equations governing the motion of test particles orbiting the black hole.

The first video titled "Entropy of a Black Hole, Thermodynamics & Physics" explores the foundational concepts of black hole entropy and its thermodynamic implications.

Section 2.3: Important Features of the Kerr-Newman Black Hole

The Kerr-Newman metric reveals critical surfaces, including inner and outer event horizons and ergospheres, essential for understanding black hole behavior.

The second video, "Black Holes obey the Laws of Thermodynamics. Here's how," delves into how black holes adhere to thermodynamic principles.

Section 2.4: The Laws of Black Hole Thermodynamics

The laws governing black hole thermodynamics encompass the first, second, third, and zeroth laws, drawing intriguing parallels to classical thermodynamics.

The First Law of Thermodynamics for black holes establishes a relationship akin to standard systems, incorporating energy, entropy, and mechanical properties.

In summary, black hole entropy remains a captivating topic, intertwining quantum mechanics, general relativity, and thermodynamics.

For further exploration of physics, mathematics, and data science, visit my GitHub and personal website at www.marcotavora.me.