In this paper, thermodynamic effects in modified (2+1)-dimensional FRW cosmology is explored. The impacts of fluctuations in the Hubble parameter and its derivatives on the universe's evolution and corresponding thermodynamic behavior are examined. Using the modified FRW metric as a starting point, and the conditions ∇μr∇μ=0, we obtained the adjusted apparent horizon radius rA-2 = H2 + k/a2 and matching surface gravity. The energy and generalized entropy at the horizon are obtained using the Misner–Sharp formalism. This is accomplished by using our modified (2+1)-dimensional FRW cosmology in conjunction with the unified first law of thermodynamics. The evolution equations were formulated for Ḣ and H2 based on modified Friedmann and acceleration equations with dimensionless constants α1, α2, β1 and β2. The dynamics of (2+1)-dimensional cosmology is improved by these changes. In order to guarantee conformity with the generalized first law, we apply requirements to the modified gravity parameters by examining the consistency of the thermodynamics equation Tds = dE +Wdv. These restrictions result in invariant relations between the corrections coefficients, namely α2/α1 = 1-β2/1-β1 and β2/β1 = 1-α2/1-α1, which are symmetric under the interchange αi ↔βi. This formulation establishes a connection between thermodynamics and gravitational dynamics in (2+1)-dimensions, thereby facilitating the systematic examination of the influence of modified gravity on cosmic evolution.

Cosmology, new entropy and thermodynamics of apparent horizon

Schoen–Yau [Comm. Math. Phys. 79 (1981), no. 2, 231–260] showed that the graph of a blowup solution to Jang’s equation is asymptotic to cylinders over apparent horizons. In this paper, we will give a simple proof of the sharp upper and lower estimates for asymptotic rate. Furthermore, this proof can be applied to general blowup solutions to Jang’s equation that do not necessarily come from the Schoen–Yau regularization procedure.

Apparent horizons associated with dynamical black hole entropy

We investigate the interaction between a nonrotating black hole and incoming gravitational waves, that is, a distorted black hole, using the characteristic formulation of the Einstein field equations, framed as a Bondi problem. By adopting retarded time as the null coordinate and recognizing that the final state is invariably a black hole, we show that an apparent horizon forms once sufficient mass accretes onto the black hole. We derive the evolution of the Bondi mass and compute its final value, enabling us to quantify the fraction of the incident mass absorbed by the black hole. Additionally, we establish a nonextensive relation for the absorbed mass as a function of initial parameters, such as the amplitude of the gravitational wave data.

Modified gravity theories with a nonminimal coupling between curvature and matter offer a compelling alternative to dark energy and dark matter by introducing an explicit interaction between matter and curvature invariants. Two of the main consequences of such an interaction are the emergence of an additional force and the non-conservation of the energy–momentum tensor, which can be interpreted as an energy exchange between matter and geometry. By adopting this interpretation, one can then take advantage of many different approaches in order to investigate the phenomenon of gravitationally induced particle creation. One of these approaches relies on the so-called irreversible thermodynamics of open systems formalism. By considering the scalar–tensor formulation of one of these theories, we derive the corresponding particle creation rate, creation pressure, and entropy production, demonstrating that irreversible particle creation can drive a late-time de Sitter acceleration through a negative creation pressure, providing a natural alternative to the cosmological constant. Furthermore, we demonstrate that the generalized second law of thermodynamics holds: the total entropy, from both the apparent horizon and enclosed matter, increases monotonically and saturates in the de Sitter phase, imposing constraints on the allowed particle production dynamics. Furthermore, we present brief reviews of other theoretical descriptions of matter creation processes. Specifically, we consider approaches based on the Boltzmann equation and quantum-based aspects and discuss the generalization of the Klein–Gordon equation, as well as the problem of its quantization in time-varying gravitational fields. Hence, gravitational theories with nonminimal curvature–matter couplings present a unified and testable framework, connecting high-energy gravitational physics with cosmological evolution and, possibly, quantum gravity, while remaining consistent with local tests through suitable coupling functions and screening mechanisms.

We investigate the dynamic shadows of a black hole with a self-interacting massive complex scalar hair. The complex scalar field evolves with time t, and its magnitude on the apparent horizon starts from zero, undergoes a sharp rise followed by rapid oscillations, and eventually converges to a constant value. The variation in the photon sphere radius is similar to that of the magnitude . Owing to the emergence of the complex scalar hair , the apparent horizon radius starts increasing sharply and then smoothly approaches a stable value eventually. The shadow radius of the black hole with an accretion disk increases with time at the observer's position. In the absence of an accretion disk, the shadow radius is larger and also increases as increases. Furthermore, we slice the dynamical spacetime into spacelike hypersurfaces for all time points . For the case with an accretion disk, the variation in is similar to that in the apparent horizon , because the inner edge of the accretion disk extends to the apparent horizon. In the absence of an accretion disk, the variation in is similar to that in the photon sphere radius , because the black hole shadow boundary is determined by the photon sphere. As the variation in is induced by , it can be stated that the variation in the size of the shadow is similarly caused by the change in . Regardless of the presence or absence of the accretion disk, the emergence of the complex scalar hair causes the radius of the shadow to start changing. Moreover, we investigate the time delay of light propagating from light sources to the observer. These findings not only enrich the theoretical models of dynamic black hole shadows but also provide a foundation for testing black hole spacetime dynamics.

We consider the strong field behavior of the Wang-Yau quasi-local energy. In particular, we examine the limit of the Wang-Yau quasi-local energy as the defining spacelike 2-sphere Σ approaches an apparent horizon from the exterior. We find that if the apparent horizon Σ cannot be isometrically embedded into R3, the Wang-Yau quasi-local energy diverges at the horizon. Further, in this situation, the optimal embedding equation does not admit any solution that is smooth up to the horizon.

Abstract This study proposes new entropy of the apparent horizon , where is the Bekenstein–Hawking entropy. As parameter one comes to the Bekenstein–Hawking entropy. This allows to consider the generalized Friedmann–Lemaître–Robertson–Walker (FLRW) equations for the barotropic matter fluid with for arbitrary equation of state parameter . The modified Friedmann's equations are found. The addition term in the second modified Friedmann's equation plays the role of a dynamical cosmological constant. The dark energy density, pressure, and the deceleration parameter are found. It is shown that at some parameters and , two phases can be obtained, acceleration and deceleration or the eternal inflation. The model under consideration by using the holographic principle can describe the universe inflation and late time of the universe acceleration. Thus, the holographic dark energy model with the generalized entropy of the apparent horizon is considered. It is shown that entropic cosmology with the entropy proposed is equivalent to cosmology based on the teleparallel gravity with the function . New cosmology based on the generalized entropy can be of interest for a description of inflation and late time of the universe evolution.

Abstract We compute the holographic entanglement entropy via the Ryu-Takayanagi prescription in the three-dimensional Friedmann-Lemaître-Robertson-Walker universe. We consider two types of holographic scenarios analogous to the static patch holography and half de Sitter holography, in which the holographic boundary is timelike and placed in the bulk. We find in general that strong subadditivity can be satisfied only in the former scenario, and, in addition, the holographic boundary must lie within the apparent horizon. Also, for the universe filled with an ideal fluid with constant equation of state w < –1, the condition is sharpened as that the holographic boundary must lie within the event horizon instead. These constraints provide necessary conditions for the dual quantum field theory to be standard and compatible with strong subadditivity.

In this study, we establish the corrected first law of thermodynamics for dynamical regular black holes on both the event and apparent horizons. We found that the temperature of dynamical regular black holes derived from the traditional first law differs from that obtained through other approaches. This indicates that, similar to static cases, the first law of thermodynamics requires correction. We derived the corrected first law of thermodynamics from the Einstein field equations. Our analysis reveals that the corrected factor originates from the fact that the component of the energy-momentum tensor depends on the black hole mass. This dependence implies that the mass of a regular black hole can no longer be directly identified as the internal energy, leading to corrections of the first law of thermodynamics.

In this study, we consider FRW universe filled with matter, non-minimally coupling (NMC) scalar field under V(ϕ)=V0ϕ2 potential and holographic vacuum energy. Dark energy is contributed from both holographic vacuum energy and the NMC scalar field. NMC effective gravitational constant Geff(ϕ), is naturally defined at the action level. Therefore, the gravitational constant in the holographic vacuum density is an effective one, i.e. ρΛ=3c2/8πGeffL2. Apparent horizon is chosen as IR holographic cutoff scale as it is a trapped null surface. There are nine fixed points in this dynamical system with four independent dimensionless parameters. We consider flat case and find that viable cosmological evolution follows the sequence: an initial stiff-fluid-dominated phase, transitioning through a nearly dust-dominated era, and eventually reaching a stable dark energy-dominating state. Stability analysis requires that ξ<0 and 0<c<1 for the theory to be physically valid. Since zero NMC coupling, ξ=0, is not allowed in the autonomous system, the model can not completely recover canonical scalar field case. That is to say, as ξ→0- and c→0+, the model can only approach the canonical scalar case but can not completely recover it. To approach dust or stiff fluid dominations, both magnitudes of the NMC coupling and the holographic parameter must be small. Numerical integration shows that for any allowed values of ξ and c, weff approaches -1 at late times. Increasing of c does not change shape of the weff, but larger c increases weff. As ξ becomes stronger, dust era gradually disappears. Good behaviors of the dynamics require -1≪ξ<0 and 0<c≪1.

We present a family of exact, singularity-free solutions describing the collapse of baryonic matter characterized by a barotropic equation of state whose coefficient varies in both radius and time. By matching these interior solutions to the Husain exterior metric, we obtain a self-consistent, dynamical spacetime representing a regular black hole. Although the pressure profile of our models grows with the radius and eventually violates the dominant energy condition beyond a critical surface, it necessitates an external junction for ensuring a globally well-defined spacetime, and the interior solution remains non-singular throughout the collapse. We further analyze the optical properties of these regular black holes and find that both the photon sphere radius and corresponding shadow radius increase monotonically as the local equation of state parameter α is raised. Moreover, the matching interface between the interior and exterior metrics naturally suggests a phase transition in the collapsing fluid, which can postpone the formation of an apparent horizon. Taken together, our results not only highlight novel physical features of horizon formation in regular collapse models but also identify characteristic shadow signatures that can be tested by future observations.

We discuss the notion of generating a cosmic inflation without any big bang singularity. It has recently been proved by Good and Linder (arXiv:2503.02380 [gr-qc]) that such an expansion of the universe can be driven by quantum fluctuations embedded in vacuum. The rate of expansion is guided by a cosmological sum rule defined through the Schwarzian derivative. We explore the thermodynamic roots of Schwarzian and connect it with the surface gravity associated with an apparent horizon. In General Relativity the cosmological sum rule can be enforced only if the early universe is a Milne vacuum. We show that this restriction can be removed by considering an entropic source term in the Einstein–Hilbert action.

In this paper, we consider the nonlinear radiation defined by power law and rational nonlinear electrodynamics (NLE) Lagrangian in homogeneous and isotropic background. These forms of NLE Lagrangian have been studied by using the numerical solution technique to explore the cosmic dynamics in models with matter and dark energy. The universe composed of decelerating and accelerating phases in these models has been examined to understand the evolution of total entropy enclosed by the apparent horizon. The validity of generalized second law of thermodynamics (GSLT) in different cosmic eras of NLE models with and without dark energy has been studied with their corresponding differences.

Recently Hollands proposed a new formula for the entropy of a dynamical black hole for an arbitrary theory of gravity obtained from a diffeomorphism covariant Lagrangian via the Noether charge method []. We present an alternative, pedagogical derivation of the dynamical black hole entropy for f(R) gravity as well as canonical scalar-tensor theory by means of conformal transformations. First, in general relativity we generalize Visser and Yan’s pedagogical proof of the nonstationary physical process first law to black holes that may not be in vacuum, and give a pedagogical derivation of the second-order behavior of the dynamical black hole entropy for vacuum perturbations by considering the second-order variation of the Raychaudhuri equation. Second, we apply the derivation for general relativity to theories in the Einstein frames, and then recast the conclusions in their original frames. We show that the dynamical black hole entropy formulas of these theories satisfy both the nonstationary physical process first law and the nonstationary comparison first law via the Einstein frame. We further study the second-order behavior of the dynamical black hole entropy for vacuum perturbations, and observe that the second law is obeyed at second order in those theories. Using the Einstein frame, we also determine the relationship between the dynamical black hole entropy and the Wald entropy of the generalized apparent horizon in the original frame. Published by the American Physical Society 2025

In our study, we delved into the profound ramifications of string tension on the collapse of a string fluid, culminating in the formation of a black hole within the realm of [Formula: see text] gravity. A string fluid is a perfect fluid model, each whose particle admits a radially stretched string. The outer region for the collapsing object is assumed to be the Schwarzschild spacetime, while the fluid collapses in the internal region are supposed to be Friedman–Robertson–Walker spacetime. The junction conditions are developed leading to gravitational mass and energy. The time frame for apparent horizon and singularity formation was estimated, showing a black hole as the outcome. String tension emerged as a pivotal factor in prolonging horizon creation. Our findings illuminate the relation among [Formula: see text] gravity, string tension, and collapse dynamics, enriching our understanding of cosmic phenomena.

New physics beyond General Relativity impacts black-hole spacetimes. The effects of new physics can be investigated in a largely theory-agnostic way by following the principled-parameterized approach. In this approach, a classical black-hole metric is upgraded by following a set of principles, such as regularity, i.e., the absence of curvature singularities. We expect these principles to hold in many theories beyond General Relativity. In the present paper, we implement this approach for time-dependent spacetimes describing gravitational collapse. We find that the Vaidya spacetime becomes regular through the same modification of the spacetime metric as stationary black-hole spacetimes [1–3]. We investigate null geodesics and find indications that the modification is even sufficient to render null geodesics future complete. Finally, we find that the modification of the spacetime structure results in violations of the null energy condition in a finite region inside the apparent horizon of the black hole that forms. Null geodesics are attracted to the boundary of this region, such that the new-physics effects are shielded from asymptotic observers. An exception occurs, if the classical spacetime has a naked singularity. Then, the upgraded spacetime is singularity-free and null geodesics from the regular core can escape towards asymptotic observers.

This study investigates back-reaction effects from matter accretion onto a cylindrically symmetric black hole using a perturbative scheme, focusing on cases where accretion reaches a quasi-steady state. We examine three distinct models by deriving corrections to the metric coefficients and obtaining expressions for the mass function. We analyze energy conditions and the self-consistency of the corrected solution and present formulas for the corrected apparent horizon and discussed thermodynamic properties. Our results align with the Vaidya form near the apparent horizon, regardless of the energy-momentum tensor form. Furthermore, we show that for a charged cylindrically symmetric black hole, the corrected mass term resembles that of a static case, indicating that charge does not alter the corrected metric form in this perturbative approach.

Abstract This paper deals with gravitational thermodynamics on the dynamical apparent horizon of an Friedmann–Lemaitre–Robertson–Walker (FLRW) Universe with dissipation. The dissipation is assumed to arise due to adiabatic gravitational particle creation. For the thermodynamic study, we consider the Bekenstein–Hawking formalism and also assume a nonzero curvature κ for a general study. In particular, we study the unified first law, the generalized second law, and thermodynamic stability in our model. The specific heat capacities are taken into account for the study of thermodynamic stability. Our study reveals a nice result! The ratio of the specific heat capacity at constant pressure and that at constant volume in a flat FLRW Universe with dissipation is nothing but the negative of the deceleration parameter. In classical thermodynamics, this ratio is known as the isentropic expansion factor or (for ideal gases) the adiabatic index. A more interesting fact that has come to light is that this relation is independent of the cosmological model used. So, this is actually a generic result in big bang cosmology. We discuss the implications of this result on the evolution of the Universe. Finally, we determine the constraints on the effective equation of state and the particle creation rate which guarantees thermodynamic stability in our model.