# Thermodynamics in Curved Space-Time and Its Application to Holography

^{*}

## Abstract

**:**

## 1. Introduction

_{1}, c

_{2}are numerical constants of order one which depend on the concrete properties of the particles. Such thermodynamic behaviors can easily be found by dimensional analysis. For the system to be gravitational stable, we must require E = c

_{1}zV T

^{4}≤ E

_{bh}~ L which means the energy of the system should be less than the energy of a black hole of the same size. This leads to the critical temperature T ≤ c

_{3}z

^{−}

^{1}

^{/}

^{4}L

^{−}

^{1}

^{/}

^{2}. Substituting it into the formula of entropy, one gets an entropy bound for conventional field theory

_{B}= 1 in order to simplify the expressions. If the number of particle species z is an unimportant constant, the maximum entropy will be ${\left(A/{l}_{p}^{2}\right)}^{3/4}$ by recovering units, where lp is the Planck length. Thus there is an entropy gap between the A

^{3/4}entropy bound for conventional filed theory and the holographic entropy [4–8]. The result is also verified by a direct examination of gravitational stable Hilbert space for bosonic and fermionic fields [6].

^{3/4}to A, one must resort to other mechanisms for extra degrees of freedom (DOFs). A fast observation is that, if one assumes a large number of particle species z, the entropy gap can be filled continuously. Especially, assuming $z~{L}^{2}/{l}_{p}^{2}$, the setting of E = E

_{bh}~ L leads to the temperature T = c

_{3}z

^{−}

^{1/4}L

^{−}

^{1/2}~ 1/L and the entropy

^{3}

^{/}

^{4}entropy bound may be only applicable to weak-gravitational systems, because the original derivation only takes E ≤ E

_{bh}as a global limitation to the system. When the system has such a large energy, the dominant self-gravitational interaction will lead to a strongly curved space-time metric. So a complete derivation should use the thermodynamics in a curved space-time rather than directly using the results in the Minkowski space-time.

## 2. Thermodynamics in Static Curved Space-Time

_{μ}k

^{μ}, where ${\xi}^{\mu}=(1,\overrightarrow{0})$ is the killing vector of the space-time and k

^{μ}is the four momentum of the particle. Using the fact that p

_{μ}p

^{μ}= m

^{2}, the energy for a single particle is calculated as $E=\sqrt{{g}_{00}}{({\gamma}^{\alpha \beta}{p}_{\alpha}{p}_{\beta}+{m}^{2})}^{\frac{1}{2}}$, where γ

_{αβ}is the spatial part of the metric. We also verify the energy formula from the lagrangian $L={g}_{\mu v}{\dot{x}}^{\mu}{\dot{x}}^{v}$ combined with a Legendre transformation. According to the uncertainty principle, the number of possible single-particle states with energy less than w is

^{α}dp

_{α}is the invariant phase space element and the Heaviside step function Θ(x) is used. Accordingly, the momentum integral requires to count all the momenta satisfying γ

^{αβ}p

_{α}p

_{β}≤ w

^{2}/g

_{00}−m

^{2}and gives a volume $\frac{4\pi}{3}\sqrt{\gamma}{({w}^{2}/{g}_{00}-{m}^{2})}^{3/2}$ in the momentum space [10]. It leads to

_{1}= π

^{2}/30. The temperature and energy of the system are measured by the Killing observer of the space-time, which can be seen from the definition of the single-particle energy. The entropy of the system is

_{2}= 2π

^{2}/45, and Ξ is the partition function of the thermodynamic system [12]. Obviously, the influence of the metric is reflected by the non-trivial factor $\sqrt{\gamma}/{\left(\sqrt{{g}_{00}}\right)}^{3}$.

_{1}VT

^{4}and S = c

_{2}VT

^{3}. When the system lives in a strong gravitational region close to the black hole, a large number z emerges and changes the thermodynamic behaviors of the system. It is extremely noticeable that z diverges when one edge of the system touches the event horizon, so one must choose some cutoff near the horizon.

^{3}and consequently the entropy is left unchanged. Actually according to Equation (10) there is always S ∼ U/T. The entropy is unchanged because of the simultaneous redshift of the energy and the temperature.

## 3. Thermodynamic Spheres and Holography

_{1}= ζ(3)/π ≃ 1.202/π. Notice the energy distribution $\frac{1}{{g}_{00}}\frac{{w}^{2}}{{e}^{\beta w}-1}dw$ was obtained in [14] by a standard quantization procedure for the photons on spheres, which verifies the present formulism we adopt. The corresponding entropy is

_{2}= 3ζ(3)/(2π) ≃ 1.803/π.

_{00}(R) is just a constant, so the formulae (13) and (14) can be directly calculated as

_{R}is the volume of the 2-dimensional system and z reflects the effect of the curved space-time metric. As in the 3-dimensional case, the emergence of z is a deduced result rather than an assumption. When the sphere is far from the black hole, z → 1, we recover the conventional thermodynamics U = c

_{1}A

_{R}T

^{3}, S = c

_{2}A

_{R}T

_{2}for a 2-dimensional bosonic gas in a flat space-time. In contrast, when the sphere is located in a strong gravitational region close to the horizon of the black hole, we get a very large z in Equation (15).

_{h}= 2M as the holographic screen for the black hole. However, on the event horizon g

_{00}= 0 leads to an infinite state density g(w)dw and further causes an embarrassing divergent thermodynamic behaviors according to Equations (13) and (14). So we should use the stretched horizon, which is located at a Planck proper distance away from the event horizon, as suggested in [11,13,15]. The concept of stretched horizon is a useful idea in the discussion of black hole thermodynamics [16]. Because of quantum uncertainty, one can not distinguish between a physical event on the stretched horizon and the one on the event horizon. The radius R of the stretched horizon can be calculated from the requirement

_{bh}∼ R, the thermodynamics (15) immediately leads to

## 4. Conclusions

## Acknowledgments

**PACS classifications:**04.70.Dy; 04.60.-m; 05.30.-d

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Xiao, Y.; Feng, L.-H.; Guan, L.
Thermodynamics in Curved Space-Time and Its Application to Holography. *Entropy* **2015**, *17*, 1549-1557.
https://doi.org/10.3390/e17041549

**AMA Style**

Xiao Y, Feng L-H, Guan L.
Thermodynamics in Curved Space-Time and Its Application to Holography. *Entropy*. 2015; 17(4):1549-1557.
https://doi.org/10.3390/e17041549

**Chicago/Turabian Style**

Xiao, Yong, Li-Hua Feng, and Li Guan.
2015. "Thermodynamics in Curved Space-Time and Its Application to Holography" *Entropy* 17, no. 4: 1549-1557.
https://doi.org/10.3390/e17041549