# Derivation of Equation 26 in Bardeen's Four Laws of Black Hole Thermodynamics

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• thatboi
In summary, the conversation discusses a struggle with understanding equation 26 in Bardeen's paper on The Four Laws of Black Hole Thermodynamics, specifically how the Lie derivative statement is used to make two terms disappear. There is also mention of a possible typo in the comment about the Lie derivative.
thatboi
Hi all,
I am currently reading Bardeen's Paper on The Four Laws of Black Hole Thermodynamics: https://projecteuclid.org/journals/...ws-of-black-hole-mechanics/cmp/1103858973.pdf
and am struggling with the derivation of equation 26. Specifically, I do not see how he uses the Lie derivative statement just above the equation to make the 2 terms disappears in the second equality because one of the terms involves the differential of n as opposed to l.

Hm, that's indeed quite a headache. The little comment about the Lie derivative looks like it contains a typo (!), ought to be ##(L_{l} \delta l)^a = l^b \nabla_b (\delta l)^a - (\delta l)^b \nabla_b l^a = 0##, no?

robphy and vanhees71

## 1. What is Equation 26 in Bardeen's Four Laws of Black Hole Thermodynamics?

Equation 26 in Bardeen's Four Laws of Black Hole Thermodynamics is a mathematical expression that relates the change in entropy of a black hole to its mass and area. It is a key equation in understanding the thermodynamic properties of black holes.

## 2. How was Equation 26 derived?

Equation 26 was derived by applying the laws of thermodynamics to the horizon of a black hole. This involves considering the black hole as a thermodynamic system and using the first and second laws of thermodynamics to derive the equation.

## 3. What does Equation 26 tell us about black holes?

Equation 26 tells us that the entropy of a black hole is proportional to its horizon area and inversely proportional to its mass. This means that as a black hole gains mass, its entropy also increases, and as a black hole loses mass, its entropy decreases.

## 4. Why is Equation 26 important in black hole physics?

Equation 26 is important because it provides a link between the thermodynamic properties of black holes and their geometric properties. It also helps us understand the behavior of black holes in terms of entropy and energy, and has implications for the study of black hole thermodynamics.

## 5. Are there any limitations to the use of Equation 26?

Yes, there are limitations to the use of Equation 26. It is based on classical thermodynamics and does not take into account quantum effects. It also assumes that black holes are in equilibrium, which may not always be the case. Additionally, it is only applicable to non-rotating black holes.

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