Generalized Apparent Horizon

• Generalized apparent horizon is a quasi-local boundary defined by the vanishing expansion of outgoing null geodesics and negative ingoing expansion, serving as a key concept in gravitational collapse and cosmology.
• It plays a pivotal role in connecting geometric conditions, thermodynamics, and modified gravity theories by enabling the analysis of horizon entropy and cosmic evolution.
• Advanced numerical methods and non-linear elliptic equations are used to locate these horizons, providing insights into black hole dynamics, energy conditions, and the validity of the generalized second law.

A generalized apparent horizon refers to the marginally trapped, quasi-local boundary in spacetime—most commonly in cosmology or gravitational collapse—where geometric, causal, and thermodynamical notions unify. Distinguished from event horizons, generalized apparent horizons are defined by local or quasi-local geometric and physical conditions and are observable in finite spacetime regions. In current research, such horizons are central to discussions of cosmic evolution, gravitational thermodynamics, black hole formation, horizon entropy in modified gravity, and the viability of alternative entropy-area relations. They play a pivotal role in connecting geometric constraints, energy conditions, dynamical evolution equations, and the second law of thermodynamics in a range of classical and quantum-corrected gravitational theories.

1. Foundational Definition and Mathematical Characterization

In spherically symmetric or FLRW spacetimes, the generalized apparent horizon is the boundary where the expansion of outgoing null geodesics vanishes ($\theta_{(\ell)} = 0$) while the expansion of ingoing null geodesics is negative ($\theta_{(n)} < 0$). For an FRW universe with scale factor $a(t)$ and Hubble parameter $H$, the apparent horizon radius is given by

$r_A = \frac{1}{\sqrt{H^2 + k/a^2}}$

where $k$ is the spatial curvature parameter (Sheykhi, 2010). In spacetimes with less symmetry (e.g., Szekeres models), the AH may not be a sphere and requires a definition via the vanishing expansion of null congruences or a related condition for non-geodesic, maximally outward null paths (termed the absolute apparent horizon) (Krasiński et al., 2012).

Generalized apparent horizons also play a key role in time-dependent or nonstationary setups. In numerical relativity, locating the AH involves solving a nonlinear elliptic equation for the marginally trapped surface; efficient algorithms (e.g., multigrid methods) have been developed for this purpose, enabling searches in generic 3D slices without symmetry assumptions (Hui et al., 25 Apr 2024).

2. Thermodynamics at the Generalized Apparent Horizon

Thermodynamical laws are formulated at the apparent horizon by associating it with a temperature—often via the Hayward-Kodama surface gravity or the Hawking temperature $T = |\kappa|/(2\pi)$—and an entropy functional $S_h$ that is typically a function of the horizon area $A = 4\pi r_A^2$. The first law in this context takes the form

$dE = T_h dS_h + W dV$

where $E$ is the Misner–Sharp energy inside the horizon, $W$ is the work density, and $V$ is the horizon-bounded volume (Sheykhi, 2010, Li et al., 2013, Ding et al., 2018).

In GR, $S_h$ follows the area law, but quantum corrections, higher-curvature terms, or proposals from nonadditive statistical mechanics (e.g., Rényi, Tsallis, Sharma–Mittal, Kaniadakis, or Barrow entropies) lead to generalized forms: $$S_h = \frac{A}{4G} - \alpha \ln \frac{A}{4G} + \beta \frac{4G}{A}$$ for quantum corrections (Sheykhi, 2010), or

$S_K = \frac{S_{BH}}{1+\gamma S_{BH}}$

as a nonadditive generalization (Kruglov, 12 Feb 2025).

These generalized functional forms, derived from modified gravity or information-theoretic considerations, have direct implications for the Friedmann equations, horizon dynamics, and the universe's acceleration and inflationary phases.

3. Modified Gravity and Extended Theories

Generalized apparent horizons remain central in modified gravity:

• $f(R)$ gravity: The entropy is $S_{A} = (A f_R)/(4G)$, reflecting modifications to Einstein's theory. The horizon temperature is also generalized, and the GSL is respected only in certain parameter regimes, especially during early universe inflation (Karami et al., 2012).
• Einstein–Gauss–Bonnet gravity: The entropy functional is

$$S = \frac{A}{4}\left[1 + \frac{2\alpha(n-1)}{(n-3) R_h^2}\right]$$

leading to an apparent horizon radius $R_h$ defined via

$$\frac{1}{R_h^2} + \frac{\alpha}{R_h^4} = \beta \rho,$$

with thermodynamic response functions that introduce explicit dependence on the GB coupling and spacetime dimension (Sánchez et al., 14 Jul 2025).

• Teleparallel gravity: Modifications lead to further corrections in the entropy–area relation via torsional degrees of freedom and their impact on Friedmann-like evolution (Kruglov, 12 Feb 2025).
• Energy-momentum-squared gravity: Entropy and heat capacity are expressed in terms of horizon quantities, with positivity constraints on specific heat and temperature restricting viable model parameters (Rudra et al., 2020).

Generalized apparent horizons in these settings help discriminate between physically acceptable and unacceptable solutions—e.g., the GSL can be used to discard Brans–Dicke cosmological solutions where the horizon entropy decreases (Faraoni, 2011).

4. Generalized Second Law and Entropy Evolution

The generalized second law (GSL) in cosmology demands that the total entropy—the sum of the generalized horizon entropy and the entropy of matter (and other fields) inside the horizon—never decreases: $$\frac{d}{dt}(S_{\text{horizon}} + S_{\text{matter}}) \geq 0.$$ For standard Bekenstein–Hawking entropy, the GSL holds universally under the dominant energy condition (Sheykhi, 2010, Karami et al., 2011). However, for deformed or nonadditive entropies (MRE, SME, DKE, Barrow), the GSL can be conditionally violated, especially for large nonextensive parameters or at small cosmological redshifts in $\Lambda$CDM models (Abreu et al., 2021). Thus, the validity of the GSL provides phenomenological constraints on the size and form of nonstandard entropy parameters.

In Einstein–Gauss–Bonnet gravity, enforcing the first law at the apparent horizon implies a universal dark energy equation of state $p + \rho = 0$ for the horizon—corresponding to $w = -1$—which is independent of both the Gauss–Bonnet coupling and the spacetime dimension (Sánchez et al., 14 Jul 2025).

5. Observability and Physical Role in Astrophysics and Cosmology

Unlike event horizons, generalized apparent horizons are detectable via quasi-local measurements of curvature and expansion—in practice, by evaluating the local Misner–Sharp mass or geometric invariants. This makes them the proper tool for characterizing evolving or dynamical black holes and the causal structure of the observable universe (Visser, 2014, Melia, 2018). In scenarios such as black hole mergers, gravitational collapse, and dynamical spacetimes, the location and dynamics of the apparent horizon provide critical information about trapped regions, the causal structure, and entropy flow.

In cosmology, the apparent/gravitational horizon sets the causal boundary for cosmological observations: light emitted beyond the horizon never reaches the observer. The proper radius of the apparent horizon, evolving with $H(t)$, is essential for understanding the visible universe, the horizon problem, and the dynamics of cosmic expansion (Melia, 2018).

6. Topology, Dimension, and Nontrivial Generalizations

Generalized apparent horizons admit nontrivial topology in higher dimensions. Numerical relativity studies confirm the existence of MOTS (marginally outer trapped surfaces) with topologies such as $S^2 \times S^2$ or $S^1 \times S^3$ in six-dimensional spacetime—behavior not present in four dimensions—and provide support for the hyperhoop conjecture (Kurata et al., 2012). In non-spherical or inhomogeneous models (e.g., Szekeres), distinctions emerge between the "true" apparent horizon (trapped region boundary), the absolute apparent horizon (based on non-geodesic, optimally outgoing null curves), and the light collapse region boundary (Krasiński et al., 2012).

7. Numerical Methods and Algorithmic Advances

Efficient identification of generalized apparent horizons in numerical simulation has become critical in high-resolution studies of black hole mergers and dynamical spacetimes. The reformulation of the nonlinear horizon equation as a Poisson-type equation with a nonlinear source, coupled with a fourth-order compact finite difference method and an efficient multigrid V-cycle with line Gauss–Seidel relaxation, enables robust and scalable horizon finding on arbitrary slices. These techniques outperform Newton-based methods (e.g., AHFinderDirect), especially at high resolutions or with poor initial guesses, and are highly suitable for searching for complex and asymmetric horizons (Hui et al., 25 Apr 2024).

8. Outlook and Open Questions

Generalized apparent horizons have established themselves as foundational geometric objects linking gravitational dynamics, thermodynamics, information-theoretic concepts, and cosmic acceleration. Their role in selecting physically viable modified gravity models (via the GSL), the consequences of horizon entropy quantization (e.g., the discretization of coupling constants in Lovelock gravity), and the causal structure of observable universes remain active research areas. The interplay between local (apparent horizon) and global (event horizon) structures continues to inform debates on black hole information, quantum gravity, and the thermodynamic origin of gravitational field equations.