# Apparent Horizon and Gravitational Thermodynamics of Universe in the Eddington-Born-Infeld Theory

Authors: Jia-Cheng Ding; Qi-Qi Fan; Cong Li; Ping Li; Jian-Bo Deng
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

The thermodynamics of Universe in the Eddington-Born-Infeld (EBI) theory was restudied by utilizing the holographic-style gravitational equations that dominate the dynamics of the cosmical apparent horizonΥA and the evolution of Universe. We started in rewriting the EBI action of the Palatini approach into the Bigravity-type action with an extra metric qμν. With the help of the holographic-style dynamical equations, we discussed the property of the cosmical apparent horizon ΥA including timelike, spacelike, and null characters, which depends on the value of the parameter of state wm in EBI Universe. The unified first law for the gravitational thermodynamics and the total energy differential for the open system enveloped by ΥA in EBI Universe were obtained. Finally, applying the positive-heat-out sign convention, we derived the generalized second law of gravitational thermodynamics in EBI Universe.


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## Body

## 1. Introduction

Gravitational thermodynamics is quite an interesting question, which has attracted much attention. Recently, many studies have covered both the first and second laws of gravitational thermodynamics for the Friedmann-Robertson-Walker (FRW) Universe with a generic spatial curvature. The inspired work is the first law of thermodynamics for Universe by Cai and Kim [1], which is a part of the effort to seek the connections between thermodynamics and gravity [2] after discovering the black-hole thermodynamics [3, 4]. In [1, 5], Akbar and Cai reversed the formulation by rewriting the Friedmann equations into the heat balance equation and the unified first law of thermodynamics at the cosmical apparent horizon, for General Relativity (GR), Gauss-Bonnet, and Lovelock gravity. The results in [5] were soon generalized to other theories of gravity, such as the scalar-tensor gravity [6], f(R) gravity [7], braneworld scenarios [8–10], generic f(R,ϕ,∇μϕ∇μϕ) gravity [11], and Horava-Lifshitz gravity [12, 13], to construct the effective total energy differentials by the corresponding modified Friedmann equations.Inspired by the gravitational thermodynamics in these gravitational theories [5–14] and characteristics of the EBI action, we focused on generalizing the results to the EBI gravity. The Eddington-Born-Infeld action (EBI) was proposed in [15], which could mimic the presence of dark energy and dark matter in the expansion of Universe [15–17] so that EBI gravity can be regarded as a candidate for nonparticulate dark matter and dark energy, and it could also modify the Newton-Poisson equation that leads to flat rotation curves for galaxies. Generally, the EBI action is a Palatini-type action where the metric gμν is not associated with the connection Cμνλ. However, by defining an extra metric qμν to satisfy the condition in [15, 18], the EBI action can be rewritten as the Bigravity-type action.In this paper, we derived the holographic-style dynamical equations and discussed the properties of the cosmical apparent horizonΥA in EBI Universe, which rely on the contents inside the cosmical apparent horizon including the matter and the dark energy provided by the cosmological constant Λ and the spacetime self-coupling. Furthermore, we applied the Misner-Sharp energy, the Cai-Kim temperature T^A, and the Hawking-Bekenstein entropy SA to obtain the unified first law for the gravitational thermodynamics and the total energy differential for the open system enveloped by ΥA in EBI Universe. Finally, we derived the generalized second law of the nondecreasing entropy Seff(A) enclosed by ΥA in EBI Universe.This paper is organized as follows. In Section2, we reviewed the cosmical apparent horizon and derived the holographic-style dynamical equations in the EBI theory. Then we discussed the properties of the cosmical apparent horizon. In Section 3, the unified first law of gravitational thermodynamics and the Clausius equation on ΥA for an isochoric process in EBI Universe were discussed. And we derived the total energy differential enclosed by ΥA in EBI Universe. In Section 4, the generalized second law of gravitational thermodynamics in the EBI Universe was derived. Conclusions and discussion are given in Section 5.


## 2. Dynamics of the Cosmical Apparent Horizon in Eddington-Born-Infeld Gravity

### 2.1. Apparent Horizon

Physically, apparent horizons constitute the observable boundary which is the largest boundary of Universe in an instant. Mathematically, apparent horizons are many hypersurfaces where the outward expansion rateθ(l) or the inward expansion rate θ(n) is equal to zero. In general, the first kind of apparent horizons where θ(l)=0 and θ(n)≠0 usually locates near the black holes, and the another kind of apparent horizon where θ(n)=0 and θ(l)≠0 appears in the vicinity of the expanding boundary of Universe, called the cosmical apparent horizons. In this paper, we only discussed the cosmical apparent horizon via dynamic equations of Universe and thermodynamic methods.In order to calculate the apparent horizon of the cosmology, we use the FRW metric to describe the spatially homogeneous and isotropic Universe [1, 19](1)ds2=-dt2+at21-kr2dr2+at2r2dθ2+sin2⁡θdφ2where a(t) is the scale factor of the evolution of Universe and the index k denotes the normalized spatial curvature, with k = {+1,0,-1} corresponding to closed, flat, and open Universes, respectively. Using the spherical symmetry, the metric can be rewritten as(2)ds2=hαβdxαdxβ+Υ2dθ2+sin2⁡θdφ2where hαβ≔diag-1,a(t)2/1-kr2 represents the transverse 2-metric spanned by (x0=t,x1=r) and Υ≔a(t)r stands for the astronomical circumference/areal radius. Based on the FRW metric, one can structure the following null tetrad adapted to the spherical symmetry and the null radial flow:(3)lμ=121,1-kr2a,0,0nμ=12-1,1-kr2a,0,0mμ=12Υ0,0,1,isin⁡θm¯μ=12Υ0,0,1,-isin⁡θ,corresponding to the metric signature (-,+,+,+). By calculating the Newman-Penrose spin coefficients ρNP≔-mμm¯ν∇νlμ and μNP≔m¯μmν∇νnμ, the outward expansion rate θ(l)=-(ρNP+ρ¯NP) and the inward expansion rate θ(n)=-(μNP+μ¯NP) are, respectively, given by(4)θl=2H+Υ-11-kΥ2a2θn=2-H+Υ-11-kΥ2a2,where H≔a˙/a is the Hubble parameter of cosmic spatial expansion. The overdot denotes the derivative with respect to the comoving time t.For the expanding Universe(H>0), the cosmical apparent horizon is given by(5)ΥA=1H2+k/a2,derived from θ(n)=0 and θ(l)>0 corresponding to the unique marginally inner trapped horizon where ∂μΥ becomes a null vector with gμν∂μΥ∂νΥ=0 [20]. Then one derives the temporal derivative of (5)(6)Υ˙A=-HΥA3H˙-ka2that is a kinematic equation of the cosmical apparent horizon.


### 2.2. The Holographic-Style Dynamical Equations in Eddington-Born-Infeld Universe

The action of Eddington-Born-Infeld theory is given by [15–18](7)SEBIgμν,Cνρμ,Lm=116πG∫d4x-gR-2Λ+2αl2-det⁡gμν-l2KμνC+∫d4x-gLm,where R is the Ricci scalar for the metric gμν and g represents the determinant of gμν. Kμν is two-order Riemann curvature tensor dependent on the connection Cνρμ, provided by the Palatini approach. Λ is the cosmological constant and α is an arbitrary constant. G is the gravitational constant and Lm is the Lagrangian density of matter.Applying the Bigravity method [18] to replace the connection Cνρμ by the extra metric qμν in the EBI theory, the action (7) can be rewritten into the Bigravity-type action(8)SEBI=116πG∫d4x-gR-2Λ+-qK-2λ-1l2-qqαβgαβ+∫d4x-gLm,where(9)qμν=-1αgμν-l2Kμν.Kμν is the Ricci tensor for the extra metric qμν and K is the Ricci scalar for the extra metric qμν. λ is a constant (λ≡α/l2) corresponding to qμν and q is the determinant of qμν. Here, both gμν and qμν are innate metrics of spacetime and they are mutually independent. Hence, 1/l2-q(qαβgαβ) can be regarded as the term from self-coupling of spacetime.Varying the Bigravity-type action (8) with respect to the metric gμν, we get the field equations [15](10)Rμν-12Rgμν=8πGTμνm-Λgμν-1l2-q-ggμαqαβgβν,where -1/l2-q/-ggμαqαβgβν is the energy-momentum tensor of the spacetime self-coupling.The matter content of Universe is construed as the perfect fluid whose the energy-momentum tensor is(11)Tνμm=diag-ρm,pm,pm,pmwithpmρm≕wm,where wm refers to the Equation-of-State (EoS) parameter of the perfect fluid. In order to study the cosmological property of the EBI Universe, we made gμν be the FRW metric and assumed the extra metric qμν [15, 21] as(12)dsq2=-Udt2+at2V1-kr2dr2+at2Vr2dθ2+at2Vr2sin2⁡θdφ2,where U and V are two undetermined positive functions independent with t.Depending on the field equation and the two metrics, we get the first Friedmann equation(13)a˙2a2+ka2-Λ3=8πGρm3+13l2VUVand the second Friedmann equation(14)a˙2a2+ka2+2a¨a-Λ=-8πGpm+1l2VUU.Eq. (14) can be rewritten into(15)2H˙+3H2+ka2-Λ=-8πGpm+1l2VUU,which is equivalent to(16)ΥA-3Υ˙A-32HΥA=4πGpm-Λ2-12l2VUUH.With the help of (13) and (15), we obtain(17)H˙-ka2=-4πGρm+pm-12l2VUV-U.Based on (5) and (13), we get(18)ΥA-2=8πG3ρm+Λ3+13l2VUVand substituting (17) into (6), we get(19)Υ˙A=HΥA34πGρm+pm+12l2VUV-U.Eq. (16), (18), and (19) are the holographic-style dynamical equations of the cosmical apparent horizon [14], which means the evolution of Universe has a relation with the cosmical apparent horizon. If one takes U=V=1 and Λ=0, the holographic-style dynamical equations will return to the condition of the Einstein theory.Furthermore, from (13) and (14), we can obtain the acceleration equation of the EBI Universe(20)a¨a=-4πG3ρm+3pm+Λ3-12l2VU13V-U=-4πGρmwm+13-Λ12πGρm+18πGρml2VU13V-U.


### 2.3. The Characters of the Cosmical Apparent Horizon

In general, the cosmical apparent horizon is not null surface, which is different from the event and particle horizon. The equation of the cosmical apparent horizon in comoving coordinates is [22](21)Ft,r=atr-1H2+k/a2=0.Its normal has components(22)Nμ=∇μFAH=a˙r+HH˙-k/a2H2+k/a23/2δμ0+aδμ1AH=HΥA1+H˙-ka2ΥA2δμ0+aδμ1=HΥA3a¨aδμ0+aδμ1.The norm squared of the normal vector is(23)NaNa=1-krA2-H2ΥA6a¨a2=H2ΥA21-ΥA4a¨a2=H2ΥA6ΥA-2-a¨aΥA-2+a¨a,where rA=ΥA/a. Substituting (20) and (18), we get(24)NaNa=Hwm=-H2ΥA64πGρm2wm-13-Λ6πGρm-18πGρml2VU13V+Uwm+1+18πGρml2VUV-U,where we consider that NaNa is only the quadratic function H(wm) representing the inner product of the normal vector of the cosmical apparent horizon. The quadratic function H(wm) has two zero points, wm=1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U) and wm=-[1+1/8πGl2ρmV/U(V-U)].Considering the properties of the quadratic functionH(wm), we get three results as follows (considering the condition that Λ>-1/l2V/UV-8πGρm).(A) Whenwm=1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U) or wm=-[1+1/8πGl2ρmV/U(V-U)], NaNa=0 that shows the normal vector Na is a null vector and the apparent horizon ΥA is a null surface. It coincides with the cosmological event horizon ΥE=a∫t∞‍a-1dt^, which is a future-pointed null causal boundary [19, 22]. And it shares the signature of isolated black-hole horizons [23].(B) When-[1+1/8πGl2ρmV/U(V-U)]<wm<[1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U)], NaNa>0 that shows Na is a spacelike vector and ΥA is the timelike surface. ΥA has the signature (-,+,+) that shares the signature of a quasi-local timelike membrane in black-hole physics [20, 24].(C) When[1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U)]<wm or wm<-[1+1/8πGl2ρmV/U(V-U)], NaNa<0 that shows Na is a timelike vector and ΥA is the spacelike surface. Its signature is (+,+,+) that is same as the signature of the dynamical black-hole horizons [25].As we know, the present Universe is an accelerated expanding Universe that means the matter outside the cosmical apparent horizon may enter into the cosmical apparent horizon. Hence we considered that the timelike cosmical apparent horizon is reasonable and the range of the EoS parameter-[1+1/8πGl2ρmV/U(V-U)]<wm<[1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U)] is significative, which is similar to the range of the EoS parameter (-1<w<1/3) in Einstein Universe [14].


## 2.1. Apparent Horizon

Physically, apparent horizons constitute the observable boundary which is the largest boundary of Universe in an instant. Mathematically, apparent horizons are many hypersurfaces where the outward expansion rateθ(l) or the inward expansion rate θ(n) is equal to zero. In general, the first kind of apparent horizons where θ(l)=0 and θ(n)≠0 usually locates near the black holes, and the another kind of apparent horizon where θ(n)=0 and θ(l)≠0 appears in the vicinity of the expanding boundary of Universe, called the cosmical apparent horizons. In this paper, we only discussed the cosmical apparent horizon via dynamic equations of Universe and thermodynamic methods.In order to calculate the apparent horizon of the cosmology, we use the FRW metric to describe the spatially homogeneous and isotropic Universe [1, 19](1)ds2=-dt2+at21-kr2dr2+at2r2dθ2+sin2⁡θdφ2where a(t) is the scale factor of the evolution of Universe and the index k denotes the normalized spatial curvature, with k = {+1,0,-1} corresponding to closed, flat, and open Universes, respectively. Using the spherical symmetry, the metric can be rewritten as(2)ds2=hαβdxαdxβ+Υ2dθ2+sin2⁡θdφ2where hαβ≔diag-1,a(t)2/1-kr2 represents the transverse 2-metric spanned by (x0=t,x1=r) and Υ≔a(t)r stands for the astronomical circumference/areal radius. Based on the FRW metric, one can structure the following null tetrad adapted to the spherical symmetry and the null radial flow:(3)lμ=121,1-kr2a,0,0nμ=12-1,1-kr2a,0,0mμ=12Υ0,0,1,isin⁡θm¯μ=12Υ0,0,1,-isin⁡θ,corresponding to the metric signature (-,+,+,+). By calculating the Newman-Penrose spin coefficients ρNP≔-mμm¯ν∇νlμ and μNP≔m¯μmν∇νnμ, the outward expansion rate θ(l)=-(ρNP+ρ¯NP) and the inward expansion rate θ(n)=-(μNP+μ¯NP) are, respectively, given by(4)θl=2H+Υ-11-kΥ2a2θn=2-H+Υ-11-kΥ2a2,where H≔a˙/a is the Hubble parameter of cosmic spatial expansion. The overdot denotes the derivative with respect to the comoving time t.For the expanding Universe(H>0), the cosmical apparent horizon is given by(5)ΥA=1H2+k/a2,derived from θ(n)=0 and θ(l)>0 corresponding to the unique marginally inner trapped horizon where ∂μΥ becomes a null vector with gμν∂μΥ∂νΥ=0 [20]. Then one derives the temporal derivative of (5)(6)Υ˙A=-HΥA3H˙-ka2that is a kinematic equation of the cosmical apparent horizon.


## 2.2. The Holographic-Style Dynamical Equations in Eddington-Born-Infeld Universe

The action of Eddington-Born-Infeld theory is given by [15–18](7)SEBIgμν,Cνρμ,Lm=116πG∫d4x-gR-2Λ+2αl2-det⁡gμν-l2KμνC+∫d4x-gLm,where R is the Ricci scalar for the metric gμν and g represents the determinant of gμν. Kμν is two-order Riemann curvature tensor dependent on the connection Cνρμ, provided by the Palatini approach. Λ is the cosmological constant and α is an arbitrary constant. G is the gravitational constant and Lm is the Lagrangian density of matter.Applying the Bigravity method [18] to replace the connection Cνρμ by the extra metric qμν in the EBI theory, the action (7) can be rewritten into the Bigravity-type action(8)SEBI=116πG∫d4x-gR-2Λ+-qK-2λ-1l2-qqαβgαβ+∫d4x-gLm,where(9)qμν=-1αgμν-l2Kμν.Kμν is the Ricci tensor for the extra metric qμν and K is the Ricci scalar for the extra metric qμν. λ is a constant (λ≡α/l2) corresponding to qμν and q is the determinant of qμν. Here, both gμν and qμν are innate metrics of spacetime and they are mutually independent. Hence, 1/l2-q(qαβgαβ) can be regarded as the term from self-coupling of spacetime.Varying the Bigravity-type action (8) with respect to the metric gμν, we get the field equations [15](10)Rμν-12Rgμν=8πGTμνm-Λgμν-1l2-q-ggμαqαβgβν,where -1/l2-q/-ggμαqαβgβν is the energy-momentum tensor of the spacetime self-coupling.The matter content of Universe is construed as the perfect fluid whose the energy-momentum tensor is(11)Tνμm=diag-ρm,pm,pm,pmwithpmρm≕wm,where wm refers to the Equation-of-State (EoS) parameter of the perfect fluid. In order to study the cosmological property of the EBI Universe, we made gμν be the FRW metric and assumed the extra metric qμν [15, 21] as(12)dsq2=-Udt2+at2V1-kr2dr2+at2Vr2dθ2+at2Vr2sin2⁡θdφ2,where U and V are two undetermined positive functions independent with t.Depending on the field equation and the two metrics, we get the first Friedmann equation(13)a˙2a2+ka2-Λ3=8πGρm3+13l2VUVand the second Friedmann equation(14)a˙2a2+ka2+2a¨a-Λ=-8πGpm+1l2VUU.Eq. (14) can be rewritten into(15)2H˙+3H2+ka2-Λ=-8πGpm+1l2VUU,which is equivalent to(16)ΥA-3Υ˙A-32HΥA=4πGpm-Λ2-12l2VUUH.With the help of (13) and (15), we obtain(17)H˙-ka2=-4πGρm+pm-12l2VUV-U.Based on (5) and (13), we get(18)ΥA-2=8πG3ρm+Λ3+13l2VUVand substituting (17) into (6), we get(19)Υ˙A=HΥA34πGρm+pm+12l2VUV-U.Eq. (16), (18), and (19) are the holographic-style dynamical equations of the cosmical apparent horizon [14], which means the evolution of Universe has a relation with the cosmical apparent horizon. If one takes U=V=1 and Λ=0, the holographic-style dynamical equations will return to the condition of the Einstein theory.Furthermore, from (13) and (14), we can obtain the acceleration equation of the EBI Universe(20)a¨a=-4πG3ρm+3pm+Λ3-12l2VU13V-U=-4πGρmwm+13-Λ12πGρm+18πGρml2VU13V-U.


## 2.3. The Characters of the Cosmical Apparent Horizon

In general, the cosmical apparent horizon is not null surface, which is different from the event and particle horizon. The equation of the cosmical apparent horizon in comoving coordinates is [22](21)Ft,r=atr-1H2+k/a2=0.Its normal has components(22)Nμ=∇μFAH=a˙r+HH˙-k/a2H2+k/a23/2δμ0+aδμ1AH=HΥA1+H˙-ka2ΥA2δμ0+aδμ1=HΥA3a¨aδμ0+aδμ1.The norm squared of the normal vector is(23)NaNa=1-krA2-H2ΥA6a¨a2=H2ΥA21-ΥA4a¨a2=H2ΥA6ΥA-2-a¨aΥA-2+a¨a,where rA=ΥA/a. Substituting (20) and (18), we get(24)NaNa=Hwm=-H2ΥA64πGρm2wm-13-Λ6πGρm-18πGρml2VU13V+Uwm+1+18πGρml2VUV-U,where we consider that NaNa is only the quadratic function H(wm) representing the inner product of the normal vector of the cosmical apparent horizon. The quadratic function H(wm) has two zero points, wm=1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U) and wm=-[1+1/8πGl2ρmV/U(V-U)].Considering the properties of the quadratic functionH(wm), we get three results as follows (considering the condition that Λ>-1/l2V/UV-8πGρm).(A) Whenwm=1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U) or wm=-[1+1/8πGl2ρmV/U(V-U)], NaNa=0 that shows the normal vector Na is a null vector and the apparent horizon ΥA is a null surface. It coincides with the cosmological event horizon ΥE=a∫t∞‍a-1dt^, which is a future-pointed null causal boundary [19, 22]. And it shares the signature of isolated black-hole horizons [23].(B) When-[1+1/8πGl2ρmV/U(V-U)]<wm<[1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U)], NaNa>0 that shows Na is a spacelike vector and ΥA is the timelike surface. ΥA has the signature (-,+,+) that shares the signature of a quasi-local timelike membrane in black-hole physics [20, 24].(C) When[1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U)]<wm or wm<-[1+1/8πGl2ρmV/U(V-U)], NaNa<0 that shows Na is a timelike vector and ΥA is the spacelike surface. Its signature is (+,+,+) that is same as the signature of the dynamical black-hole horizons [25].As we know, the present Universe is an accelerated expanding Universe that means the matter outside the cosmical apparent horizon may enter into the cosmical apparent horizon. Hence we considered that the timelike cosmical apparent horizon is reasonable and the range of the EoS parameter-[1+1/8πGl2ρmV/U(V-U)]<wm<[1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U)] is significative, which is similar to the range of the EoS parameter (-1<w<1/3) in Einstein Universe [14].


## 3. Thermodynamics of the Holographic-Style Dynamical Equations in the Eddington-Born-Infeld Universe

Based on the holographic-style dynamical equations (18), (19), and (16) in Section 2, we continue to investigate the thermodynamics about the cosmical apparent horizon. Firstly, we define the effective energy within a sphere of radius Υ (surface area A=4πΥ2 and volume V^=4/3πΥ3): Eeff=ρeffV^, which is the entire energy enveloped by the cosmical apparent horizon ΥA (take V^ to represent the volume in order to distinguish the function V).


### 3.1. Unified First Law of Thermodynamics

Applying the Misner-Sharp mass/energyEMS≔Υ/2G(1-hαβ∂αΥ∂βΥ) [26, 27] to be the effective energy Eeff and substituting hαβ=diag[-1,a2/1-kr2], one obtains(25)dE=-Υ˙AGΥ3ΥA3dt+32GΥ2ΥA2dΥand(26)dE=-1GΥ3ΥA3Υ˙A-32HΥAdt+32GΥ2ΥA2adr.For the EBI Universe, substituting (16) and (18) into (25), the total energy differential in the (t,Υ) coordinates is obtained(27)dE=Aρm+18πGl2VUV+Λ8πGdΥ-Aρm+pm+18πGl2VUV-U·HΥdt,where A=4πΥ2. Similarly, we can obtain(28)dE=Aρm+18πGl2VUV+Λ8πGadr-Apm-18πGl2VUU-Λ8πG·HΥdt,which is the total energy differential in the (t,r) coordinates. From the above two equations, we can know that the EoS parameter from spacetime self-coupling wac is negative, because of the energy density term “1/8πGl2V/UV” and the intensity of pressure “-1/8πGl2V/UU”.The unified first law of (equilibrium) thermodynamics is given by(29)dE=AΨ+WdV^,proposed by Hayward [28]. W is the work density, given by(30)W≔-12Tmαβhαβ,where hαβ=diag[-1,a(t)2/1-kr2]. Ψ is the energy supply covector, Ψ=Ψαdxα, where(31)Ψα≔Tαmβ∂βΥ+W∂αΥ.Here, W and Ψα are invariant. Moreover, the definitions of W and Ψα are valid for all spherically symmetric spacetimes and FRW spacetime.In the EBI theory, the field equations can be rewritten into(32)Rμν-12Rgμν=8πGTμνeff,where we define an effective energy-momentum tensor(33)Tμνeff=Tμνm-Λ8πGgμν-18πGl2VUUV·gμαqαβgβν.From the above equation, we can consider that the effective energy-momentum tensor includes the part of the dark energy corresponding to the terms with Λ and (U,V). Then we can generalize Hayward’s unified first law of (equilibrium) thermodynamics into the EBI Universe by taking Tμν(eff) to replace Tμν(m). Imitating the definitions of W and Ψα [28], we define(34)dE=AΨ~+W~dV^and(35)Ψ~=Ψ~αdxα,where(36)W~≔-12Teffαβhαβand(37)Ψ~α≔Tαeffβ∂βΥ+W~∂αΥ.We consider that the FRW metricgμν is physically subsistent, which is used to raise or descend the index here, and another metric qμν is an extra metric provided by the primordial mechanism of Universe. Based on this, we get(38)Tαeffβ=gμβTμαeff=gμβTμαm-Λ8πGδαβ-18πGl2VUUVqβμgμαand(39)Teffαβ=gμαgνβTμνeff=gμαgνβTμνm-Λ8πGgαβ-18πGl2VUUVqαβ.Substituting hαβ=diag[-1,a(t)2/1-kr2], we obtain(40)W~=-12Teff00h00+Teff11h11=12ρm-pm+Λ8πG+18πGl2VUU+V2;(41)Ψ~t=-12ρm+pm+18πGl2VUV-UHΥ;(42)Ψ~r=12ρm+pm+18πGl2VUV-Ua.Substituting Ψ~t, Ψ~r, and W~ into (34), we get(43)dE=AΨ~+W~dV^=AΨ~tdt+Ψ~rdr+W~dΥ=A-pm+Λ8πG+18πGl2VUUHΥdt+Aρm+Λ8πG+18πGl2VUVadr,which is the expression of dE in (t,r) coordinates.Naturally, because of the invariance ofW~ and Ψ~, we can rewrite these in the (t,Υ) coordinates, given by(44)dE=AΨ~+W~dV^=AΨ~t′dt+Ψ~Υ′dΥ+W~dΥ=-Aρm+pm+18πGl2VUV-UHΥdt+Aρm+Λ8πG+18πGl2VUVdΥ,where(45)Ψ~t′=-ρm+pm+18πGl2VUV-UHΥand(46)Ψ~Υ′=12ρm+pm+18πGl2VUV-U.It illustrates our hypothesis of replacingTμν(m) by Tμν(eff) is reasonable that (43) and (44) are, respectively, identical to (28) and (27). The unified first law for gravitational thermodynamics of Universe is totally different from the first law in the black-hole thermodynamics [14]. Eq. (43) and (44) are both called “the unified first law” for the EBI Universe’s gravitational thermodynamics.


### 3.2. Clausius Equation on the Cosmical Apparent Horizon for an Isochoric Process

Having obtained the unified first lawdE=AΨ~+W~dV^ in EBI Universe, we are interested in the region enclosed by the cosmical apparent horizon ΥA.Eq. (19) leads to(47)Υ˙AGdt=AAρm+pm+18πGl2VUV-UHΥA·dt,where AA=4πΥA2. The left-hand side of (47) can be manipulated into [14](48)Υ˙AGdt=12πΥA·2πΥAΥ˙AG·dt=12πΥAdπΥA2G.One applies the geometrically defined Hawking-Bekenstein entropy [4, 29] (in the units ħ=c=k[Boltzmannconstant]=1)(49)SA=πΥA2G=AA4G,and the Cai-Kim temperature [1, 30](50)T^A≡12πΥA,at the cosmical apparent horizon to simplify (48), given by Υ˙A/Gdt=T^AdSA (take “T^” on behalf of temperature not only the Cai-Kim temperature). With the help of (45), we get -[(ρm+pm)+1/8πGl2V/U(V-U)]HΥA=Ψ~t′|(Υ=ΥA)≡Ψ~tA′. From (44) and (47), we obtain(51)δQA=T^AdSA=-AAΨ~tA′=-dEAdΥ=0,where dEA is the specific condition of (44) when Υ=ΥA. Eq. (51) is actually the Clausius equation for equilibrium and reversible thermodynamic processes as same as the situation in GR [14]. After considering the EBI theory, the Clausius equation is generalized to include dark energy from the cosmological constant Λ and the effect the self-coupling of spacetime (U,V), which may explain the problems about the cosmic expansion.Finally, for the open system enveloped byΥA, we substitute the unified first law (44) and the Clausius equation (51) into the total energy differential(52)dEA=-T^AdSA+ρm+Λ8πG+18πGl2VUVdV^A,where the “-” sign shows the positive-heat-out sign convention that means heat emitted by the open system takes positive values (δQA=δQA(out)>0 ) rather than the traditional positive-heat-in thermodynamic sign convention [14].Comparing (52) with the total energy differential of the Einstein gravity, there are two extra terms Λ/8πGdV^A and 1/8πGl2V/UVdV^A, which originate from dark energy of the cosmological constant and the spacetime self-coupling, respectively. It means that there are the matter’s transition and the dark energy’s fluxion on both sides of the cosmical apparent horizon during the expansion of Universe.


## 3.1. Unified First Law of Thermodynamics

Applying the Misner-Sharp mass/energyEMS≔Υ/2G(1-hαβ∂αΥ∂βΥ) [26, 27] to be the effective energy Eeff and substituting hαβ=diag[-1,a2/1-kr2], one obtains(25)dE=-Υ˙AGΥ3ΥA3dt+32GΥ2ΥA2dΥand(26)dE=-1GΥ3ΥA3Υ˙A-32HΥAdt+32GΥ2ΥA2adr.For the EBI Universe, substituting (16) and (18) into (25), the total energy differential in the (t,Υ) coordinates is obtained(27)dE=Aρm+18πGl2VUV+Λ8πGdΥ-Aρm+pm+18πGl2VUV-U·HΥdt,where A=4πΥ2. Similarly, we can obtain(28)dE=Aρm+18πGl2VUV+Λ8πGadr-Apm-18πGl2VUU-Λ8πG·HΥdt,which is the total energy differential in the (t,r) coordinates. From the above two equations, we can know that the EoS parameter from spacetime self-coupling wac is negative, because of the energy density term “1/8πGl2V/UV” and the intensity of pressure “-1/8πGl2V/UU”.The unified first law of (equilibrium) thermodynamics is given by(29)dE=AΨ+WdV^,proposed by Hayward [28]. W is the work density, given by(30)W≔-12Tmαβhαβ,where hαβ=diag[-1,a(t)2/1-kr2]. Ψ is the energy supply covector, Ψ=Ψαdxα, where(31)Ψα≔Tαmβ∂βΥ+W∂αΥ.Here, W and Ψα are invariant. Moreover, the definitions of W and Ψα are valid for all spherically symmetric spacetimes and FRW spacetime.In the EBI theory, the field equations can be rewritten into(32)Rμν-12Rgμν=8πGTμνeff,where we define an effective energy-momentum tensor(33)Tμνeff=Tμνm-Λ8πGgμν-18πGl2VUUV·gμαqαβgβν.From the above equation, we can consider that the effective energy-momentum tensor includes the part of the dark energy corresponding to the terms with Λ and (U,V). Then we can generalize Hayward’s unified first law of (equilibrium) thermodynamics into the EBI Universe by taking Tμν(eff) to replace Tμν(m). Imitating the definitions of W and Ψα [28], we define(34)dE=AΨ~+W~dV^and(35)Ψ~=Ψ~αdxα,where(36)W~≔-12Teffαβhαβand(37)Ψ~α≔Tαeffβ∂βΥ+W~∂αΥ.We consider that the FRW metricgμν is physically subsistent, which is used to raise or descend the index here, and another metric qμν is an extra metric provided by the primordial mechanism of Universe. Based on this, we get(38)Tαeffβ=gμβTμαeff=gμβTμαm-Λ8πGδαβ-18πGl2VUUVqβμgμαand(39)Teffαβ=gμαgνβTμνeff=gμαgνβTμνm-Λ8πGgαβ-18πGl2VUUVqαβ.Substituting hαβ=diag[-1,a(t)2/1-kr2], we obtain(40)W~=-12Teff00h00+Teff11h11=12ρm-pm+Λ8πG+18πGl2VUU+V2;(41)Ψ~t=-12ρm+pm+18πGl2VUV-UHΥ;(42)Ψ~r=12ρm+pm+18πGl2VUV-Ua.Substituting Ψ~t, Ψ~r, and W~ into (34), we get(43)dE=AΨ~+W~dV^=AΨ~tdt+Ψ~rdr+W~dΥ=A-pm+Λ8πG+18πGl2VUUHΥdt+Aρm+Λ8πG+18πGl2VUVadr,which is the expression of dE in (t,r) coordinates.Naturally, because of the invariance ofW~ and Ψ~, we can rewrite these in the (t,Υ) coordinates, given by(44)dE=AΨ~+W~dV^=AΨ~t′dt+Ψ~Υ′dΥ+W~dΥ=-Aρm+pm+18πGl2VUV-UHΥdt+Aρm+Λ8πG+18πGl2VUVdΥ,where(45)Ψ~t′=-ρm+pm+18πGl2VUV-UHΥand(46)Ψ~Υ′=12ρm+pm+18πGl2VUV-U.It illustrates our hypothesis of replacingTμν(m) by Tμν(eff) is reasonable that (43) and (44) are, respectively, identical to (28) and (27). The unified first law for gravitational thermodynamics of Universe is totally different from the first law in the black-hole thermodynamics [14]. Eq. (43) and (44) are both called “the unified first law” for the EBI Universe’s gravitational thermodynamics.


## 3.2. Clausius Equation on the Cosmical Apparent Horizon for an Isochoric Process

Having obtained the unified first lawdE=AΨ~+W~dV^ in EBI Universe, we are interested in the region enclosed by the cosmical apparent horizon ΥA.Eq. (19) leads to(47)Υ˙AGdt=AAρm+pm+18πGl2VUV-UHΥA·dt,where AA=4πΥA2. The left-hand side of (47) can be manipulated into [14](48)Υ˙AGdt=12πΥA·2πΥAΥ˙AG·dt=12πΥAdπΥA2G.One applies the geometrically defined Hawking-Bekenstein entropy [4, 29] (in the units ħ=c=k[Boltzmannconstant]=1)(49)SA=πΥA2G=AA4G,and the Cai-Kim temperature [1, 30](50)T^A≡12πΥA,at the cosmical apparent horizon to simplify (48), given by Υ˙A/Gdt=T^AdSA (take “T^” on behalf of temperature not only the Cai-Kim temperature). With the help of (45), we get -[(ρm+pm)+1/8πGl2V/U(V-U)]HΥA=Ψ~t′|(Υ=ΥA)≡Ψ~tA′. From (44) and (47), we obtain(51)δQA=T^AdSA=-AAΨ~tA′=-dEAdΥ=0,where dEA is the specific condition of (44) when Υ=ΥA. Eq. (51) is actually the Clausius equation for equilibrium and reversible thermodynamic processes as same as the situation in GR [14]. After considering the EBI theory, the Clausius equation is generalized to include dark energy from the cosmological constant Λ and the effect the self-coupling of spacetime (U,V), which may explain the problems about the cosmic expansion.Finally, for the open system enveloped byΥA, we substitute the unified first law (44) and the Clausius equation (51) into the total energy differential(52)dEA=-T^AdSA+ρm+Λ8πG+18πGl2VUVdV^A,where the “-” sign shows the positive-heat-out sign convention that means heat emitted by the open system takes positive values (δQA=δQA(out)>0 ) rather than the traditional positive-heat-in thermodynamic sign convention [14].Comparing (52) with the total energy differential of the Einstein gravity, there are two extra terms Λ/8πGdV^A and 1/8πGl2V/UVdV^A, which originate from dark energy of the cosmological constant and the spacetime self-coupling, respectively. It means that there are the matter’s transition and the dark energy’s fluxion on both sides of the cosmical apparent horizon during the expansion of Universe.


## 4. Generalized Second Laws of Thermodynamics in Eddington-Born-Infeld Universe

With the help of the first holographic-style dynamical equation (18), the effective energy (EMS=Υ3/2GΥA2) can be rewritten into(53)EMS=ρeff43πΥ3=ρeffV^,where ρeff=ρm+Λ/8πG+1/8πGl2V/UV. Then we can rewrite the Friedmann equations into(54)a˙a+ka=8πGρeff3and(55)a˙a+ka+2a¨a=-8πGpeff,where we define that peff=pm-Λ/8πG-1/8πGl2V/UU. Based on two above equations, one can obtain the continuity equation in the EBI theory(56)ρ˙eff+3a˙aρeff+peff=0.In many papers [31–33], the entropy Sm of the cosmic energy-matter content with temperature T^m is always determined by the traditional Gibbs equation dE=T^mdSm-pmdV^. In order to generalize to the EBI theory, we keep the same form and redefine it into the positive-heat-out sign convention for consistency with the horizon entropy SA [14], given by(57)dEeff=-T^effdSeff-peffdV^.From dEeff=ρeffdV^+V^dρeff, we obtain(58)T^effdSeff=-V^dρeff-ρeff+peffdV^.Based on the continuity equation (56), we obtain(59)dρeff=-3Hρeff+peffdt.When Υ=ΥA, one can get(60)T^effdSeffA=AAρeff+peff·HΥA-Υ˙Adt,where AA=3/2Gρeff. With the help of (19), (60) yields(61)S˙effA=-94G·HΥAT^eff·1ρeff2ρeff+peff13ρeff+peffthat is the evolution of the effective inner entropy enclosed by ΥA.In order to discussS˙eff(A) more concretely, we assume that ρaux≡ρm+1/8πGl2V/UV and paux≡pm-1/8πGl2V/UU, which are two auxiliary parameters. Following pm=wmρm, we set(62)peff=εeffρeffand(63)paux=σauxρaux,where εeff and σaux are the parameters of state like wm. And (61) can be simplified as(64)S˙effA=-94G·HΥAT^eff·εeff+1εeff+13.Based on the above assumptions, we can obtain(65)σaux=wm+V/UwmV+U8πGl2ρm+V/UVand(66)εeff=σaux-σaux+1Λ8πGρaux+Λ.Then we obtain(67)εeff=8πGl2ρm8πGl2ρm+V/UV+Λl2·wm-V/UU+Λl28πGl2ρm+V/UV+Λl2.As a result, S˙eff(A) can be rewritten into(68)S˙effA=-94G·HΥAT^eff·8πGl2ρm8πGl2ρm+V/UV+Λl22·wm+1+V/UV-U8πGl2ρm·wm+13+V/U1/3V-U8πGl2ρm-Λ12πGρm.Physically, temperature is positive (T^eff>0) and the present Universe is expanding (H>0).IfΛ satisfies the precondition (Λ>-1/l2V/UV-8πGρm) shown in Section 2, we can exactly obtain that [1+V/U(V-U)/8πGl2ρm]>[1/3+V/U1/3V-U/8πGl2ρm-Λ/12πGρm]. And (68) illustrates that(A)
when-[1+V/U(V-U)/8πGl2ρm]<wm<-[1/3+V/U(1/3V-U)/8πGl2ρm-Λ/12πGρm], S˙eff(A)>0;(B)
when-[1+V/U(V-U)/8πGl2ρm]>wm or -[1/3+V/U(1/3V-U)/8πGl2ρm-Λ/12πGρm]<wm, S˙eff(A)<0;(C)
whenwm=-[1+V/U(V-U)/8πGl2ρm] or -[1/3+V/U(1/3V-U)/8πGl2ρm-Λ/12πGρm], S˙eff(A)=0.On the other hand, from the acceleration equation of the EBI Universe (20), we know that when wm<-[1/3-Λ/12πGρm+V/U(1/3V-U)/8πGl2ρm], the present Universe is an accelerated expanding Universe. Hence, if V<3U-8πGl2ρmU/V, wm has the possibility to be a positive number to produce an accelerated expanding Universe, which is different from the result of Einstein Universe [14].In a word, the physical effective entropySeff(A) inside the cosmical apparent horizon satisfies S˙eff(A)>0 for the stage of accelerated expansion (a¨>0) when -[1+V/U(V-U)/8πGl2ρm]<wm<-[1/3+V/U(1/3V-U)/8πGl2ρm-Λ/12πGρm]. Noteworthily, Seff(A) is composed of the matter’s entropy and the dark energy’s entropy.


## 5. Conclusions and Discussion

In this paper, we obtained the gravitational dynamics in the EBI Universe. Firstly, we derived the holographic-style dynamical equations. Because of the present accelerated expanding Universe that means the outer matter can enter into the cosmical apparent horizon, we considered the timelike cosmical apparent horizon is reasonable and-[1+1/8πGl2ρmV/U(V-U)]<wm<[1/3+Λ/6πGρm+1/8πGl2ρmV/U(1/3V+U)] is a rational range of the matter’s EoS parameter.Secondly, based on the holographic-style dynamical equations, we obtained two forms of the total energy differential in the (t,Υ) coordinates and (t,r) coordinates. And we proved that these two forms of the total energy differential can be derived from the unified first laws of the gravitational dynamics dE=AΨ~+W~dV^ by redefining the effective energy-momentum tensor Tμν(eff) in Hayward’s approach [28].Thirdly, we derived the total energy differential for the open system enveloped byΥA, dEA=-T^AdSA+[ρm+Λ/8πG+1/8πGl2V/UV]dV^A, where the two extra terms Λ/8πGdV^A and 1/8πGl2V/UVdV^A are, respectively, corresponding to the dark energy and the spacetime self-coupling in EBI Universe. It illustrates that not only the matter’s transition but also the dark energy’s fluxion arises on the cosmical apparent horizon ΥA with the expansion of Universe.Finally, we discussed the properties of the effective entropySeff(A) enclosed by the cosmical apparent horizon ΥA in EBI Universe. The results show that when -[1+V/U(V-U)/8πGl2ρm]<wm<-[1/3+V/U(1/3V-U)/8πGl2ρm-Λ/12πGρm] and a¨>0, the generalized second law of the nondecreasing entropy Seff(A) is obtained. If V satisfies the condition V<3U-8πGl2ρmU/V, wm can be a positive number to generate an accelerated expanding Universe, which is different from the results in Einstein Universe.In addition to all the above we would like to point out that the method of the theories we mentioned in our paper satisfying the equilibrium thermodynamics does not mean this method fits all possible theories. Actually, nonequilibrium thermodynamics might be also a good way in finding new gravitational theories [34–36], which would be a further task for us to study.



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*Source: 7801854-2018-11-18.xml*