Attractive Gravity Probe Surface (AGPS)

• Attractive Gravity Probe Surface (AGPS) is a closed, smooth two-dimensional surface on a spacelike hypersurface, defined by a lower bound on the normalized mean curvature derivative modulated by a parameter α.
• This concept provides a unified framework for quasi-local gravitational diagnostics, enabling derivation of sharp area–mass inequalities that generalize classical results like the Penrose inequality.
• AGPS leverages geometric flow techniques such as inverse mean curvature flow and conformal flow to rigorously connect surface properties with local and quasi-local mass measures in varied spacetime settings.

An attractive gravity probe surface (AGPS) is a closed, smooth, compact two-dimensional surface in a spacelike hypersurface that is designed to capture, in a quasilocal manner, the strength of the gravitational field—even in regions outside event horizons and deep trapped regions. The AGPS generalizes the concepts of minimal surfaces and loosely trapped surfaces by imposing refined geometric conditions on the mean curvature and its outward derivative, controlled by a tunable parameter. This concept provides a unified framework for quasi-local characterizations of both strong and weak gravitational fields, underpinning sharp area–mass inequalities and supporting the analysis of geometric inequalities beyond black hole horizons and in diverse spacetime settings.

1. Fundamental Definition and Key Properties

Let $S_\alpha$ be a compact, orientable two-surface embedded in a three-dimensional spacelike hypersurface $\Sigma$, with mean curvature $k$ (with respect to the outward normal $r^a$) strictly positive. The AGPS is defined by the pair of conditions

$$k > 0, \qquad \frac{r^a D_a k}{k^2} \ge \alpha, \qquad \textrm{with} \;\; \alpha > -\frac{1}{2},$$

where $D_a$ is the covariant derivative on $\Sigma$ and the parameter $\alpha$ encodes the "gravity probing intensity."

This requirement enforces a lower bound on the (normalized) rate of change of the mean curvature in the normal direction, which interpolates between the trapped (minimal surface) case ($k = 0$, $\alpha \to +\infty$), the loosely trapped surface ($\alpha = 0$), and weak-field regions ($\alpha \to -1/2$) (Izumi et al., 2021, Lee et al., 2021, Lee et al., 2022). In physically relevant spacetimes, such as Schwarzschild, the normalized rate $\frac{r^a D_a k}{k^2}$ monotonically decreases from $+\infty$ near the horizon through $0$ at the photon sphere, to $-1/2$ at spatial infinity.

By construction, the AGPS serves as a tuneable probe: as $\alpha$ is varied, these surfaces cover the transition from strong to weak gravitational fields. This foundational property enables quasi-local diagnostics of gravitational strength, and underpins the derivation of sharp area bounds.

2. Core Inequalities and Quasilocal Mass Relations

A central result for AGPS is the existence of an area bound, generalizing the Riemannian Penrose inequality. For asymptotically flat spacetimes with Arnowitt-Deser-Misner (ADM) mass $m$, the area $A_\alpha$ of any AGPS obeys: $$A_\alpha \leq 4\pi \left(\frac{3+4\alpha}{1+2\alpha}\right)^2 (Gm)^2.$$ Equality is achieved if the spacetime is isometric to a constant-$t$ Schwarzschild slice and $S_\alpha$ is the $r = \mathrm{const}$ surface with $\frac{r^a D_a k}{k^2} = \alpha$ (Izumi et al., 2021).

This bound interpolates between the minimal surface Penrose inequality ($\alpha \rightarrow \infty$, $A \leq 16\pi (Gm)^2$) and the loosely trapped surface area bound ($\alpha = 0$, $A \leq 36\pi (Gm)^2$). For $\alpha$ values nearer $-1/2$, the surfaces characterize weaker gravity regions.

AGPS-based inequalities can be further refined to account for angular momentum, gravitational wave content, and matter distributions: $$M_{ADM} - \left(m_{ext} + \frac{3+4\alpha}{1+2\alpha} m_{int} \right) \geq \frac{1}{1+2\alpha}\left(\frac{3+4\alpha}{R_{A0}} + \frac{2J^2_{min}}{3+4\alpha}R_{A0}^{-3}\right)A_0,$$ where $m_{int}, m_{ext}$ refer to quasi-local mass inside/outside the surface, $J_{min}$ is a quasilocal angular momentum, and $R_{A0}^2 = A_0/4\pi$ (Lee et al., 2021). In (axisymmetric) vacuum scenarios, the Komar angular momentum is involved via a sharper angular momentum–area bound.

The positivity of the Geroch and Hawking quasi-local masses on AGPSs is ensured in any spacetime dimension with appropriate energy conditions, without recourse to the Gauss–Bonnet theorem (Shiromizu et al., 2023).

3. Variants and Generalizations

Multiple generalizations of the AGPS have been introduced to probe a broader range of spacetime geometries and gravitational behaviors (Lee et al., 2022):

• Longitudinal AGPSs (LAGPS-$k$, LAGPS-$r$): Defined via constraints on the outward normal derivative of mean curvature, or, refined, involving the surface Ricci scalar.
• Transverse AGPSs (TAGPS-$k$, TAGPS-$r$): Defined using conditions on transverse null expansions and their derivatives, incorporating "intensity" parameters ($\beta$, $\gamma_T$).

All four types admit refined Penrose-inequality-type bounds, modified to incorporate matter, angular momentum, and gravitational radiation contributions.

The AGPS framework holds in higher dimensions $n\geq 3$, with the generalized area bound (Izumi et al., 2022): $$A \leq \omega_{n-1} \left[\frac{n+2(n-1)\alpha}{1+(n-1)\alpha} G m\right]^{\frac{n-1}{n-2}},$$ where $\omega_{n-1}$ is the area of the unit $(n-1)$-sphere and $\alpha > -1/(n-1)$. In four dimensions, this reduces to the previously stated area bound.

For spacetimes with a positive cosmological constant $\Lambda$, the AGPS definition is adapted to include the cosmological background curvature, replacing the defining inequality with $r^a D_a k \geq \alpha (k^2 + 4\Lambda/3)$. The associated areal bound becomes $A_\alpha \leq [12\pi/(3+4\alpha)\Lambda]$ (Shiromizu et al., 2023).

In the Einstein-Maxwell system, AGPS area inequalities incorporate electric/magnetic charge with extremality conditions: $$m_{ADM} - m_{ext} \geq \frac{1+2a}{3+4a} R_A + \frac{2(1+a)}{3+4a} \frac{q^2}{R_A} + ...,$$ reproducing the Reissner-Nordström black hole extremality bound when specialized to minimal surfaces (Lee et al., 30 Jul 2024).

4. Construction and Analytical Tools: IMCF and Conformal Flow

The mathematical core of AGPS-based inequalities is the use of geometric flow techniques:

• Inverse Mean Curvature Flow (IMCF): The flow equation

$(\partial x/\partial r)^a = r^a / k$

generates a foliation from a given AGPS, along which the Geroch energy

$$E(y) = (A^{1/2})/(64\pi^{3/2}G) \int_{S_y} [2R^{(2)} - k^2 - (4\Lambda/3)] dA$$

is non-decreasing if the three-manifold scalar curvature (and, for $\Lambda \neq 0$, its correction) satisfy suitable bounds (Izumi et al., 2021).

• Bray's Conformal Flow: For surfaces with multiple components or in gluing constructions, the conformal flow establishes the Penrose inequality by extending the manifold, matching parameters (notably $\alpha$), and performing smoothing. This method generalizes to AGPSs and permits the derivation of sharp area inequalities even for non-connected surfaces.

These techniques ensure that all intermediate steps in the derivation are controlled locally or quasilocally, making the resultant inequalities robust and broadly applicable.

5. Quasilocal and Local Mass Measures

The AGPS framework leverages both local and quasilocal mass measures:

• Geroch Quasilocal Mass:

$$m_G(S) = (1/16\pi G)\left(\frac{A}{4\pi}\right)^{1/2} \int_S [{^{(2)}R} - \frac{1}{2}k^2] dA.$$

• Hawking Quasilocal Mass:

$$m_H(S) = (1/16\pi G)\left(\frac{A}{4\pi}\right)^{1/2} \int_S \left[{^{(2)}R} + \frac{1}{2}\theta_+\theta_-\right] dA,$$

where $\theta_\pm$ are the null expansions (used in AGPS$^+$ variants that employ null geodesic congruences) (Shiromizu et al., 21 Oct 2025).

Local effective masses, defined as integrals of energy density, gravitational wave energy, and rotational pressures over the spatial region bounded by AGPSs, provide complementary constraints. For example,

$2G \Delta M_{eff} \leq \Delta L$

links the effective local mass in a region to the physical separation $\Delta L$ between enclosing AGPSs (Shiromizu et al., 21 Oct 2025).

6. Extensions and Applications

The AGPS and its variants (including AGPS$^+$, defined via outgoing and ingoing null expansions), serve not only for mathematical characterizations of geometric inequalities but also for physical diagnostics in a range of astrophysical and theoretical contexts (Shiromizu et al., 21 Oct 2025):

• Strong/Weak Gravity probes: AGPSs in both strong-field and weak-field regimes provide unified bounds and indicators, applicable up to spatial infinity.
• Black hole formation and cosmic censorship: The Penrose-like inequalities derived via AGPSs constrain mass/area relations, supporting cosmic censorship considerations.
• Gravitational waves and angular momentum: The explicit inclusion of these effects in refined inequalities is crucial for studying dynamic or rotating spacetimes.
• Higher-dimensional and nontrivial topology spacetimes: The AGPS area inequalities remain valid in higher dimensions and in asymptotically locally AdS contexts (Izumi et al., 2022, Lee et al., 30 Jul 2024).
• Numerical relativity and quasi-local energy diagnostics: AGPSs give computationally accessible geometric criteria for identifying regions of strong gravity outside horizons.

7. Tables: AGPS Variants and Associated Inequalities

AGPS Type	Defining Inequality	Main Area Bound or Inequality
LAGPS-$k$	$r^a D_a k \geq \alpha k^2$	$A \leq 4\pi[(3+4\alpha)/(1+2\alpha)]^2 (Gm)^2$
LAGPS-$r$	$r^a D_a k \geq -(^{(2)}R)(1-\gamma_L)$	$$m_{ADM}-... \geq (\gamma_L/3)\mathcal{R}_{A0} + ...$$
TAGPS-$k$	$\max(\bar{K}_{ab}k^a k^b) \leq -\beta k$	$$m_{ADM}+... \geq \frac{1+2\beta}{3+4\beta}\mathcal{R}_{A0} + ...$$
TAGPS-$r$	$$^{(3)}\mathcal{L}_n\kappa \leq (^{(2)}R)(1-\gamma_T)$$	$$m_{ADM}+... \geq \frac{\gamma_T}{3}\mathcal{R}_{A0} + ...$$
AGPS$^+$	$r^a\nabla_a\theta_+ \geq -\alpha\theta_+\theta_-$	$A \leq 4\pi[(4\alpha+3)/(2\alpha+1) G m_H]^2$

Key: $\mathcal{R}_{A0}$, area radius; $\theta_\pm$, null expansions; $m_H$, Hawking mass; $^{(2)}R$, Ricci scalar on $S$; $\bar{K}_{ab}$, extrinsic curvature tangent parts. Dots ($...$) denote additional terms (e.g., matter, GW, angular momentum).

• Area bound for surfaces in generic gravitational field (Izumi et al., 2021)
• Refined inequalities for loosely trapped surface/attractive gravity probe surface (Lee et al., 2021)
• Four types of attractive gravity probe surfaces (Lee et al., 2022)
• Attractive gravity probe surfaces in higher dimensions (Izumi et al., 2022)
• Attractive gravity probe surface, positivity of quasi-local mass and ADM expression (Shiromizu et al., 2023)
• Attractive gravity probe surface with positive cosmological constant (Shiromizu et al., 2023)
• Attractive gravity probe surface in Einstein-Maxwell system (Lee et al., 30 Jul 2024)
• Quasilocal inequalities for attractive gravity probe surface (Shiromizu et al., 21 Oct 2025)

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The AGPS, with its mean curvature condition parameterized by $\alpha$, provides a comprehensive and robust geometric probe of gravitational strength. Its broad applicability, adaptability to include physical effects (EM fields, rotation, gravitational wave energy), and deep connections to quasi-local mass and Penrose-type inequalities, position it as a key tool in the analysis of both mathematical and physical properties of general relativistic and modified gravity spacetimes.