# Key Mathematical Formulas in Black Hole Thermodynamics and Bulk/Boundary Duality (Holography) in Gravitating Systems

Below is a list of key equations commonly used in these areas. Keep in mind that this is a simplified overview, and a deeper understanding would require further study.

1. **Black Hole Thermodynamics**:
- *First Law*: `dM = TdS + ΦdQ`, where M is the mass, T is the temperature, S is the entropy, Φ is the electrostatic potential, and Q is the charge of the black hole.
- *Second Law*: `ΔS ≥ 0`, stating that the entropy of an isolated black hole can never decrease.
- *Temperature*: `T = (ħc³)/(8πGkBM)`, where ħ is the reduced Planck constant, c is the speed of light, G is the gravitational constant, kB is the Boltzmann constant, and M is the mass of the black hole.
- *Entropy*: `S = (kBA)/(4ħG)`, where A is the area of the black hole's event horizon.

2. **Holographic Principle and Bulk/Boundary Duality**:
- *Holographic Entropy*: `S = (Area)/(4G)`, relating the entropy of a gravitational system to the area of its boundary.
- *AdS/CFT correspondence*: This principle states that a quantum gravity theory in an Anti-de Sitter (AdS) space is dual to a conformal field theory (CFT) on the space's boundary.

3. **Covariant Mapping between Bulk and Boundary**:
- *Stress-Energy Tensor (SET)*: `T^(μν) = (ρ + p)u^(μ)u^(ν) + pg^(μν)`, where ρ is the energy density, p is the pressure, u^(μ) is the four-velocity, and g^(μν) is the metric tensor.
- *Fluid Conservation Equations*: `∇_(μ)T^(μν) = 0`, representing the conservation of energy-momentum in the fluid system.

4. **Entropy Production and Gravitational Waves**:
- *Entropy Production*: `σ = (kappa/T)∇^(μ)T_(μν)u^(ν)`, where kappa is the thermal conductivity, T is the temperature, and u^(ν) is the four-velocity.

These equations and principles provide a starting point for understanding the relationships between black hole thermodynamics, holography, and gravitational systems. However, a thorough understanding requires a deeper study of the subject matter, including the derivation of these equations and their applications.

@phdthesis{Simovic:2021uwd,
author = "Simovic, Fil",
title = "{Gravitational Thermodynamics: From Black Holes to Holography}",
school = "Waterloo U.",
year = "2021"
}

https://academic.oup.com/ptep/article/2020/3/033E03/5814072