# Thermodynamics of Horizons from a Dual Quantum System

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}where A, M and T are the horizon area, mass and Hawking temperature of a black hole and E′, S′ and T′ are the energy, entropy and temperature of the corresponding dual quantum system [8]. After working out the standard thermodynamics of the dual system, they apply the inverse transformations to get standard horizon thermodynamics as well as the logarithmic corrections to the original Bekenstein-Hawking entropy formula. This approach seem to provide a description of a strongly interacting gravitational system like black hole in terms of a weakly interacting quantum mechanical dual system.

^{2}, With this value of gL, the leading term in the expression of energy E

_{d}of the dual system becomes,

^{2}= 2Eπ/f′(a) thereby allowing us to determine the energy associated with the horizon

^{2}, T = f′(a)/4π. We stress that we have not done anything drastic or unconventional except to adhere to the duality prescription.

^{2}/r

^{2}) and the horizon is at $a=M+\sqrt{{M}^{2}-{Q}^{2}}$. The temperature associated with this outer horizon is T = κ/2π where κ is the surface gravity of the outer horizon. In this case it is easy to see that af′(a) = 2M/a − 2Q

^{2}/a

^{2}(so that af′(a) ≠ 1 unless Q = 0). If we require this analysis to be applicable for charged black holes also, the energy E should be given by Eq. (8):

^{2}/a

^{2}) da = dM which does not match dE. Therefore the energy expression in Eq. (9) does not have a proper thermodynamic interpretation.

^{2}= −N

^{2}dt

^{2}+ γ

_{µν}dx

^{µ}dx

^{ν}, where N and γ

_{µν}are independent of time t (Greek indices cover 1,2,3 and Latin indices cover 0-3). The comoving observer at x

^{µ}=constant have the four velocity ${u}_{i}=-N{\delta}_{i}^{0}$ and the four acceleration a

^{i}= (0, ∂

^{µ}N/N). If N → 0 on a two-surface and Na = (γ

_{µν}∂

^{µ}N∂

^{ν}N)

^{1/2}is finite (say κ, the surface gravity), then the coordinate system has a horizon. Regularity in the Euclidean sector requires the periodicity in Euclidean time with the period |β| = 2π/κ, allowing us to define a temperature T = |β

^{−1}| in terms of the derivatives of N, whenever there is a horizon. The expression for entropy associated with this horizon is given as [12],

^{i}, but is generally covariant. (For justification behind this definition of gravitational entropy, see ref.[12]). Now, in any spacetime, there is a differential geometric identity [14],

_{ab}is the extrinsic curvature of spatial hypersurfaces and K is its trace. In static spacetime we have K

_{ab}= 0 and when combined with Einstein’s equation we can write:

_{ab}. We next note that the source for gravitational acceleration is the covariant combination $({T}_{ab}-\frac{1}{2}T{g}_{ab}){u}^{a}{u}^{b}$, and the corresponding energy E is given by Tolman-Komar integral [19],

_{D}

_{−2}is the area of a (D − 2) dimensional unit sphere. The temperature of the horizon is still T = f′(a)/4π and let the relevant energy is E

^{(D)}. Then the dual transformations are,

_{αβ}is the surface element in (D − 2) dimensional hypersurface and ${\xi}_{\left(t\right)}^{\beta}$ is the spacetime’s timelike killing vector. We also have Stokes theorem for antisymmetric tensor field B

^{αβ}given by [14]

^{αβ}= ∇

^{α}${\xi}_{\left(t\right)}^{\beta}$. Then using killing equation we can write,

^{2}, and performing the reverse transformation, the entropy of the horizon is obtained as,

## 2. Conclusion

## Acknowledgements

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Sarkar, S.; Padmanabhan, T.
Thermodynamics of Horizons from a Dual Quantum System. *Entropy* **2007**, *9*, 100-107.
https://doi.org/10.3390/e9030100

**AMA Style**

Sarkar S, Padmanabhan T.
Thermodynamics of Horizons from a Dual Quantum System. *Entropy*. 2007; 9(3):100-107.
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**Chicago/Turabian Style**

Sarkar, Sudipta, and T. Padmanabhan.
2007. "Thermodynamics of Horizons from a Dual Quantum System" *Entropy* 9, no. 3: 100-107.
https://doi.org/10.3390/e9030100