Schwarzschild black hole and generalized second law in phantom-dominated universe

doi:10.1016/j.physletb.2006.12.029

Physics Letters B

Volume 645, Issues 2–3, 8 February 2007, Pages 108-112

https://doi.org/10.1016/j.physletb.2006.12.029 Get rights and content

Abstract

In this Letter which is an extension of the work [G. Izquierdo, D. Pavón, Phys. Lett. B 639 (2006) 1], we study the conditions required for validity of the generalized second law in phantom-dominated universe in the presence of Schwarzschild black hole. Our study is independent of the origin of the phantom like behavior of the considered universe. We also discuss the generalized second law in the neighborhood of transition (from quintessence to phantom regime) time. We show that even for a constant equation of state parameter, the generalized second law may be satisfied provided that the temperature is not taken as de Sitter temperature. It is shown that in models with (only) a transition from quintessence to phantom regime the generalized second law does not hold in the transition epoch.

Introduction

Astrophysical data show that the universe is accelerating [2]. Based on some data, it is possible to consider an evolving equation of state parameter, ω, less than −1 at present time from $ω > - 1$ in the near past [3]. In this view we may assume that the universe is filled with a perfect fluid with a negative pressure and $ω < - 1$ , dubbed as phantom dark energy [4]. A candidate for phantom dark energy is a phantom scalar field with wrong sign for kinetic energy term [5]. Another method to study the present inflation is to use a running cosmological constant based on principles of quantum field theory (specially on the renormalization group) which can mimic the phantom like behavior of the universe [6].

This description of the universe may contain finite time future singularity accompanied with dark energy density singularity called big rip. The big rip may be avoided by the effect of gravitational backreactions which can end the phantom-dominated regime [7]. We can consider horizons for the accelerating universe and associate entropy (as a measure of our ignorance about what is going behind it) and temperature to them [8], [9], [10], [11], [12], [13], [14]. In this way one is able to study the thermodynamics of a system consisting of dark energy perfect fluid and the horizon.

In phantom-dominated universe black holes lose their masses by accreting phantom fluid [16]. Therefore their areas and consequently their entropies will decrease. So it may be of interest to know that if the generalized second law of thermodynamics (GSL) is satisfied in this situation. Indeed if the thermodynamics parameters assigned to the universe are the same as the ordinary thermodynamics parameters known in physical systems, then one expects that thermodynamics laws be also satisfied for the universe.

Thermodynamics of an accelerating universe has been studied in several papers [17]. In [8] and [9], the generalized second law for cosmological models that depart slightly from de Sitter space and also when the horizon shrinks, was studied respectively. The thermodynamics of super-accelerated universe in a de Sitter and quasi-de Sitter space–time was the subject of the paper [10].

In [11], it was shown that for a phantom-dominated universe with constant ω the total entropy is a constant and for time dependent ω, via two specific examples, the validity of GSL was verified. In [12] the conditions of validity of GSL in more general cases, including the transition epoch (from quintessence to phantom), and for temperatures proportional to de Sitter temperature were studied independently of the origin of dark energy.

In a recent paper the author of [1], using phantom scalar field model, showed that GSL is violated in the presence of a Schwarzschild black hole in the cases studied in [11] and in phantom-dominated era. In this Letter we try to study the same problem but by considering a temperature other than de Sitter temperature. Our study is independent of the origin of phantom like behavior of the universe. We also consider the possibility of transition from quintessence to phantom regime and discuss the validity of GSL in the neighborhood of transition time in the presence of the black hole.

We use the units $ℏ = c = G = k_{B} = 1$ .

Section snippets

GSL in the phantom-dominated FRW universe in the presence of a Schwarzschild black hole

We consider spatially flat Friedman–Robertson–Walker (FRW) metric with scale factor $a (t)$ : $d s^{2} = - d t^{2} + a^{2} (t) (d x^{2} + d y^{2} + d z^{2}) .$ The Hubble parameter is given by $H = \dot{a} / a$ . The overdot shows derivative with respect to the comoving time t. The equation of state of the universe which is assumed to behave as a perfect fluid at large scale is given by $p = ω ρ,$ where ω is the equation of state parameter. For an accelerating universe, i.e. $\ddot{a} > 0$ , we have $ω < - 1 / 3$ . The future event horizon, $R_{h}$ , is given by $R_{h} (t) = a (t) \int_{t}^{\infty} \frac{d t^{'}}{a}$

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Schwarzschild black hole and generalized second law in phantom-dominated universe

Abstract

Introduction

Section snippets

GSL in the phantom-dominated FRW universe in the presence of a Schwarzschild black hole

Phys. Lett. B

Class. Quantum Grav.

Phys. Rev. Lett.

J. Exp. Theor. Phys.

Nature (London)

Astron. J.

Astrophys. J.

Phys. Rev. D

Mon. Not. R. Astron. Soc.

JCAP

Phys. Lett. B

Phys. Rev. D

Phys. Rev. D

JCAP

Phys. Rev. D

Phys. Rev. D

Phys. Lett. B

Phys. Rev. D

Phys. Rev. D

Phys. Rev. Lett.

Class. Quantum Grav.

Phys. Lett. B

Phys. Rev. D

Phys. Rev. D

Int. J. Mod. Phys. D

Class. Quantum Grav.

Phys. Lett. B

Nucl. Phys. B

JCAP