#### 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 horizon 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 . With the help of the holographic-style dynamical equations, we discussed the property of the cosmical apparent horizon including timelike, spacelike, and null characters, which depends on the value of the parameter of state in EBI Universe. The unified first law for the gravitational thermodynamics and the total energy differential for the open system enveloped by 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.

#### 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], gravity [7], braneworld scenarios [8–10], generic 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 is not associated with the connection . However, by defining an extra metric 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 horizon 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 and the spacetime self-coupling. Furthermore, we applied the Misner-Sharp energy, the Cai-Kim temperature , and the Hawking-Bekenstein entropy to obtain the unified first law for the gravitational thermodynamics and the total energy differential for the open system enveloped by in EBI Universe. Finally, we derived the generalized second law of the nondecreasing entropy enclosed by in EBI Universe.

This paper is organized as follows. In Section 2, 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 for an isochoric process in EBI Universe were discussed. And we derived the total energy differential enclosed by 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 rate or the inward expansion rate is equal to zero. In general, the first kind of apparent horizons where and usually locates near the black holes, and the another kind of apparent horizon where and 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]where is the scale factor of the evolution of Universe and the index denotes the normalized spatial curvature, with = corresponding to closed, flat, and open Universes, respectively. Using the spherical symmetry, the metric can be rewritten aswhere represents the transverse 2-metric spanned by () and 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:corresponding to the metric signature . By calculating the Newman-Penrose spin coefficients and , the outward expansion rate and the inward expansion rate are, respectively, given bywhere is the Hubble parameter of cosmic spatial expansion. The overdot denotes the derivative with respect to the comoving time .

For the expanding Universe , the cosmical apparent horizon is given byderived from and corresponding to the unique marginally inner trapped horizon where becomes a null vector with [20]. Then one derives the temporal derivative of (5)that 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]where is the Ricci scalar for the metric and represents the determinant of . is two-order Riemann curvature tensor dependent on the connection , provided by the Palatini approach. is the cosmological constant and is an arbitrary constant. is the gravitational constant and is the Lagrangian density of matter.

Applying the Bigravity method [18] to replace the connection by the extra metric in the EBI theory, the action (7) can be rewritten into the Bigravity-type actionwhere is the Ricci tensor for the extra metric and is the Ricci scalar for the extra metric . is a constant () corresponding to and is the determinant of . Here, both and are innate metrics of spacetime and they are mutually independent. Hence, can be regarded as the term from self-coupling of spacetime.

Varying the Bigravity-type action (8) with respect to the metric , we get the field equations [15]where 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 iswhere 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 be the FRW metric and assumed the extra metric [15, 21] aswhere and are two undetermined positive functions independent with .

Depending on the field equation and the two metrics, we get the first Friedmann equationand the second Friedmann equation

Eq. (14) can be rewritten intowhich is equivalent to

With the help of (13) and (15), we obtainBased on (5) and (13), we getand substituting (17) into (6), we get

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 and , 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

##### 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]Its normal has componentsThe norm squared of the normal vector iswhere . Substituting (20) and (18), we getwhere we consider that is only the quadratic function representing the inner product of the normal vector of the cosmical apparent horizon. The quadratic function has two zero points, and .

Considering the properties of the quadratic function , we get three results as follows (considering the condition that ).

(A) When or , that shows the normal vector is a null vector and the apparent horizon is a null surface. It coincides with the cosmological event horizon , which is a future-pointed null causal boundary [19, 22]. And it shares the signature of isolated black-hole horizons [23].

(B) When , that shows is a spacelike vector and is the timelike surface. has the signature that shares the signature of a quasi-local timelike membrane in black-hole physics [20, 24].

(C) When or , that shows is a timelike vector and 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 is significative, which is similar to the range of the EoS parameter () 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 (surface area and volume ): , which is the entire energy enveloped by the cosmical apparent horizon (take to represent the volume in order to distinguish the function ).

##### 3.1. Unified First Law of Thermodynamics

Applying the Misner-Sharp mass/energy [26, 27] to be the effective energy and substituting , one obtainsandFor the EBI Universe, substituting (16) and (18) into (25), the total energy differential in the coordinates is obtainedwhere . Similarly, we can obtainwhich is the total energy differential in the coordinates. From the above two equations, we can know that the EoS parameter from spacetime self-coupling is negative, because of the energy density term “” and the intensity of pressure “”.

The unified first law of (equilibrium) thermodynamics is given byproposed by Hayward [28]. is the work density, given bywhere . is the energy supply covector, , whereHere, and are invariant. Moreover, the definitions of and are valid for all spherically symmetric spacetimes and FRW spacetime.

In the EBI theory, the field equations can be rewritten intowhere we define an effective energy-momentum tensorFrom the above equation, we can consider that the effective energy-momentum tensor includes the part of the dark energy corresponding to the terms with and . Then we can generalize Hayward’s unified first law of (equilibrium) thermodynamics into the EBI Universe by taking to replace . Imitating the definitions of and [28], we defineandwhereand

We consider that the FRW metric is physically subsistent, which is used to raise or descend the index here, and another metric is an extra metric provided by the primordial mechanism of Universe. Based on this, we getandSubstituting , we obtainSubstituting , , and into (34), we getwhich is the expression of in coordinates.

Naturally, because of the invariance of and , we can rewrite these in the coordinates, given by

whereand

It illustrates our hypothesis of replacing by 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 law in EBI Universe, we are interested in the region enclosed by the cosmical apparent horizon .

Eq. (19) leads towhere . The left-hand side of (47) can be manipulated into [14]One applies the geometrically defined Hawking-Bekenstein entropy [4, 29] (in the units )and the Cai-Kim temperature [1, 30]at the cosmical apparent horizon to simplify (48), given by (take “” on behalf of temperature not only the Cai-Kim temperature). With the help of (45), we get . From (44) and (47), we obtainwhere is the specific condition of (44) when . 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 and the effect the self-coupling of spacetime , which may explain the problems about the cosmic expansion.

Finally, for the open system enveloped by , we substitute the unified first law (44) and the Clausius equation (51) into the total energy differentialwhere the “-” sign shows the positive-heat-out sign convention that means heat emitted by the open system takes positive values ( ) 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 and , 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 () can be rewritten intowhere . Then we can rewrite the Friedmann equations intoandwhere we define that . Based on two above equations, one can obtain the continuity equation in the EBI theory

In many papers [31–33], the entropy of the cosmic energy-matter content with temperature is always determined by the traditional Gibbs equation . 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 [14], given byFrom , we obtainBased on the continuity equation (56), we obtainWhen , one can getwhere . With the help of (19), (60) yieldsthat is the evolution of the effective inner entropy enclosed by .

In order to discuss more concretely, we assume that and , which are two auxiliary parameters. Following , we setandwhere and are the parameters of state like . And (61) can be simplified as

Based on the above assumptions, we can obtainandThen we obtainAs a result, can be rewritten intoPhysically, temperature is positive () and the present Universe is expanding ().

If satisfies the precondition () shown in Section 2, we can exactly obtain that . And (68) illustrates that(A)when , ;(B)when or , ;(C)when or , .

On the other hand, from the acceleration equation of the EBI Universe (20), we know that when , the present Universe is an accelerated expanding Universe. Hence, if , 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 entropy inside the cosmical apparent horizon satisfies for the stage of accelerated expansion () when . Noteworthily, 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 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 () coordinates and () 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 by redefining the effective energy-momentum tensor in Hayward’s approach [28].

Thirdly, we derived the total energy differential for the open system enveloped by , , where the two extra terms and 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 with the expansion of Universe.

Finally, we discussed the properties of the effective entropy enclosed by the cosmical apparent horizon in EBI Universe. The results show that when and , the generalized second law of the nondecreasing entropy is obtained. If satisfies the condition , 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.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

We would like to thank the National Natural Science Foundation of China (Grant No. 11571342) for supporting us on this work.