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Modified Gravity and Thermodynamics of Time-Dependent Wormholes At Journal of High Energy Physics, Gravitation and Cosmology, 2017, 3, 708-714 http://www.scirp.org/journal/jhepgc ISSN Online: 2380-4335 ISSN Print: 2380-4327 Modified fR( ) Gravity and Thermodynamics of Time-Dependent Wormholes at Event Horizon Hamidreza Saiedi Department of Physics, Florida Atlantic University, Boca Raton, FL, USA How to cite this paper: Saiedi, H. (2017) Abstract Modified f(R) Gravity and Thermodynam- ics of Time-Dependent Wormholes at In the context of modified fR( ) gravity theory, we study time-dependent Event Horizon. Journal of High Energy wormhole spacetimes in the radiation background. In this framework, we at- Physics, Gravitation and Cosmology, 3, 708-714. tempt to generalize the thermodynamic properties of time-dependent worm- https://doi.org/10.4236/jhepgc.2017.34053 holes in fR( ) gravity. Finally, at event horizon, the rate of change of total Received: August 1, 2017 entropy has been discussed. Accepted: October 20, 2017 Published: October 23, 2017 Keywords Copyright © 2017 by author and fR( ) Gravity, Time-Dependent Wormholes, Thermodynamics, Scientific Research Publishing Inc. Event Horizon This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ 1. Introduction Open Access Wormholes, short cuts between otherwise distant or unconnected regions of the universe, have become a popular research topic since the influential paper of Morris and Thorne [1]. Early work was reviewed in the book of Visser [2]. The Morris-Thorne (MT) study was restricted to static, spherically symmetric space-times, and initially there were various attempts to generalize the definition of wormhole by inserting time-dependent factors into the metric. According to the Einstein field equations, the MT wormhole needs an exotic matter (matter violating the null and weak energy conditions (NEC and WEC)) which holds up the wormhole structure and keeps the wormhole throat open. Sushkov [3] and Lobo [4] independently, have shown that phantom energy could well be the class of exotic matter which is required to support traversable wormholes. Recent astronomical data indicate that the universe could be dominated by a DOI: 10.4236/jhepgc.2017.34053 Oct. 23, 2017 708 Journal of High Energy Physics, Gravitation and Cosmology H. Saiedi fluid which violates the null energy condition [5] [6] [7]. Although, an exotic matter is responsible for the early-time inflation and late-time acceleration, the modified theory of gravity ( fR( ) gravity), which can explain the present acceleration without introducing an exotic matter, has received intense attention. The literature on fR( ) gravity is vast, and the earlier reviews on this theory are given in [8] [9] [10] [11] [12]. Since the discovery of black hole thermodynamics [13] [14] [15], physicists have been speculating that there should be some relationship between thermodynamics and Einstein equations. Jacobson is the first one who was able to derive Einstein equations from the proportionality of entropy to the horizon area of the black hole together with the first law of thermodynamics [16]. In the cosmological setup, Cai and his collaborators made the major development by showing that the Einstein field equations evaluated at the apparent horizon can also be expressed in the form of the first law of thermodynamics in various theories of gravity. This connection between gravity and thermodynamics has also been extended in the braneworld cosmology [17] [18]. All these indicate that the thermal interpretation of gravity is to be generic, so we investigate this relation for a more general spacetimes. The idea that wormholes may show some characteristics and properties which are parallel to those already found in black holes, seems to be quite natural, including in particular wormhole thermodynamics [19]. Also, the authors [20] have discussed wormhole thermodynamics at apparent horizons in Einstein gravity. 2. Thermodynamics of Time-Dependent Wormholes in fR( ) Gravity In this section, we generalize the thermodynamics of evolving wormholes within the modified fR( ) gravity. Hayward first introduced a formalism for defining thermal properties of black holes in terms of measurable quantities. This formalism also works for the dynamical black holes which consistently recover the results obtained by global considerations using the event horizon as in the static case. A fascinating and rather surprising feature emerges if one recognize that the static wormhole reveals thermodynamics properties analogous to the black holes if one considers the local quantities. It is important to note that the non-vanishing surface gravity at the wormhole throat characterized by a non-zero temperature for which one would expect that wormhole should emit some sort of thermal radiation. Let us start with the evolving wormhole metric [21] 2 2 2,Φ(tr) 2 2 dr 22 ds=− e d t + at( ) +Ω rd,2 (1) br( ) 1− r 2 2 22 where dΩ=2 dθ + sin θφ d is the line element of a two-dimensional unit sphere, at( ) is the scale factor of the universe, br( ) and Φ(tr, ) are the DOI: 10.4236/jhepgc.2017.34053 709 Journal of High Energy Physics, Gravitation and Cosmology H. Saiedi arbitrary shape and redshift functions of the evolving wormhole, respectively. We rewrite the spherically symmetric metric (1) in the following form (by considering Φ=(tr,0) ): 2 ab 22 ds= hab d x d x +Ω r d2 ,( ab , = 0,1) (2) where x01= tx, = r, and r = atr( ) represents the radius of the sphere while the two-dimensional metric hab is written as −1 2 br( ) hab =−− diag1, a( t) 1 . (3) r The surface gravity is defined as [22] [23] 1 ab κ = ∂−∂ab( hh r), (4) 2 −h where h is the determinant of metric hab (3). So, the surface gravity at the wormhole horizon rh can be written as r 1 κ =−++h 2 −′ (H2, H) 2 ( ab( r) rh b( r)) (5) 2 4rh where b′ =∂∂ br and the overdot denotes differentiation with respect to time. H= aa is the Hubble parameter. The horizon temperature is defined as Th = κ 2π . So r 1 =−++h 2 −′ Thh( H2. H) 2 ( ab( r) r b( r)) (6) 4π 8πrh The area of the wormhole horizon is defined as 2 Ar= 4πh (7) One can relate the entropy with the surface area of the horizon through A Sh = . In the fR( ) theory of gravity, the entropy has the following rela- 4G rh tion to the horizon area 2 AF πrFh 22 Shh= = = 8π rF (8π G=1,) (8) 4GGrh where F=dd fR ≠ 0. Therefore 22 2 d8Sh=π r hd F + 16π rF hhd. r (9) Using the above equation and the relation (6), we obtain 2 ab r− r b′ r rHh ( + 2 H) 22 2 ( ) h ( ) Thhd8 S=( π r hd F+− 16π rF hd. r h) (10) 8πr 2 4π h Now we consider the Gibbs equation [24] ThId S= d E I + pV d, (11) where SI is the entropy of the matter bounded by the horizon and EI is the DOI: 10.4236/jhepgc.2017.34053 710 Journal of High Energy Physics, Gravitation and Cosmology H. Saiedi 4 energy of the matter distribution. The volume V is defined as Vr= π3 , and 3 p denotes the average pressure inside the horizon which is pp=( rt + 23 p) . Where pr ( tr, ) and pt ( tr, ) are the radial pressure and tangential pressure, respectively. Here for thermodynamical equilibrium, the temperature of the matter inside the horizon is assumed to be the same as on the horizon i.e. Th . Now starting with 44 V= π r33, EV=ρρ = π r, (12) 33hI h and the continuity equation ρρ +H(3 ++ pprt 20) =, which can be achieved µ from the energy conservation equation Tνµ; , the Gibbs equation leads to 2 4πrrhhρ′ TShId= 3ρ ++pr 2 p t +( d rh − Hr h d, t) (13) 3 a where ρ (tr, ) is the energy density and ρρ′ =∂∂r . So, by combining Equations (10) and (13), one can easily reach the following equation for the variation of total entropy. 3 ρ′ 4πHrhhr TSh tot= T h( S I + S h ) =−32ρ ++pr p t + 3 a 2 ab( r) − r b′( r) rHh ( + 2 H) +−8π22rF h h 8πr 2 4π h (14) 2 ab( r) − r b′( r) rHh ( + 2 H) +−16π2rFh r hh8πr 2 4π h 2 ρ′ 4πrrhh +32ρ ++pprt + r h . 3 a 3. Thermodynamics at Event Horizon Now we shall find out the event horizon radius and then analyze the above equation for evolving wormholes. By considering br( ) = r0 , event horizon 22 radius rE can be found from the relation(i.e. ds = 0d = Ω2 ) ar0 rEE= rH −−1, (15) rE or r E ddrt∞ ∫∫a = . (16) 0 r t a 1− 0 r Therefore, the event horizon radius can be found from the above equation. In the fR( ) theory of gravity, the author and Nasr Esfahani [25] [26] have shown that the energy density (ρ (tr, )) , radial pressure ( pr ( tr, )) and tangential pressure ( pt ( tr, )) can be written as DOI: 10.4236/jhepgc.2017.34053 711 Journal of High Energy Physics, Gravitation and Cosmology H. Saiedi ρ =−+F 3, HF2 (17) rF p=−+−23 HF HF H2 F −0 , (18) r ar23 rF p=−+−23 HF HF H2 F +0 , (19) t 2ar23 Now by substituting relations (17), (18), (19) and (15) into the Equation (14), the variation of total entropy can be written as 2 ar 4πrE 0 TSh tot =−(3HF −− 6 HF 31 F ) − 3 rE 2 (H+ 2 Hr) E 2 ar0 +8π rEE(2 rHF +− rF E) (20) 8πr 2 4π E 2 ar ar (H+ 2 Hr) E −16π2rF 1. −−00 E r 8πr 2 4π E E Thus, the expression in Equation (20) for the rate of change of total entropy at event horizon shows that the validity of the generalized second law of thermodynamics depends on the right hand side (r.h.s.) of the equation.
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