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We construct the marginal loosely trapped surface (marginal LTS) for the Kerr spacetime with a small Kerr parameter perturbatively, where the LTS condition is saturated. An LTS is a surface that specifies the strong gravity region, which is a generalization of the photon sphere in the Schwarzschild spacetime. It turns out that there are an infinite number of marginal LTSs. At the leading order of the small Kerr parameter, all of the marginal LTSs have the same area. However, one can see that the maximal marginal LTS among them is uniquely determined at the higher order. [ABSTRACT FROM AUTHOR]

Under certain conditions, it is shown that the positivity of the Geroch/Hawking quasi-local mass holds for attractive gravity probe surfaces in any higher dimensions than three. We also comment on the Arnowitt–Deser–Misner mass. [ABSTRACT FROM AUTHOR]

We show the well-posed variational principle in constraint systems. In a naive procedure of the variational principle with constraints, the proper number of boundary conditions does not match that of physical degrees of freedom , which implies that, even in theories with up to first-order derivatives, the minimal (or extremal) value of the action with the boundary terms is not a solution of the equation of motion in the Dirac procedure of constrained systems. We propose specific and concrete steps to solve this problem. These steps utilize the Hamilton formalism, which allows us to separate the physical degrees of freedom from the constraints. This reveals the physical degrees of freedom that are necessary to be fixed on boundaries, and also enables us to specify the variables to be fixed and the surface terms. [ABSTRACT FROM AUTHOR]

In 4D spacetimes with a positive cosmological constant, we introduce a new geometrical object associated with the cosmological horizon and then show the areal inequality. We also examine the attractive gravity probe surfaces as an indicator for the existence of attractive gravity. [ABSTRACT FROM AUTHOR]

A generalization of the Riemannian Penrose inequality in n -dimensional space (3 ≤ n < 8) is done. We introduce a parameter α (⁠|$-\frac{1}{n-1}\lt \alpha \lt \infty$|⁠) indicating the strength of the gravitational field, and define a refined attractive gravity probe surface (refined AGPS) with α. Then, we show the area inequality for a refined AGPS, |$A \le \omega _{n-1} \left[ (n+2(n-1)\alpha)Gm /(1+(n-1)\alpha) \right]^{\frac{n-1}{n-2}}$|⁠ , where A is the area of the refined AGPS, ω n − 1 is the area of the standard unit (n − 1)-sphere, G is Newton's gravitational constant, and m is the Arnowitt–Deser–Misner mass. The obtained inequality is applicable not only to surfaces in strong gravity regions such as a minimal surface (corresponding to the limit α → ∞), but also to those in weak gravity existing near infinity (corresponding to the limit |$\alpha \rightarrow -\frac{1}{n-1}$|⁠). [ABSTRACT FROM AUTHOR]

We discuss a field transformation from fields ψ a to other fields ϕ i that involves derivatives, |$\phi _i = \bar{\phi }_i(\psi _a, \partial _\alpha \psi _a, \ldots ;x^\mu)$|⁠ , and derive conditions for this transformation to be invertible, primarily focusing on the simplest case that the transformation maps between a pair of fields and involves up to their first derivatives. General field transformation of this type changes the number of degrees of freedom; hence, for the transformation to be invertible, it must satisfy certain degeneracy conditions so that additional degrees of freedom do not appear. Our derivation of necessary and sufficient conditions for invertible transformation is based on the method of characteristics, which is used to count the number of independent solutions of a given differential equation. As applications of the invertibility conditions, we show some non-trivial examples of the invertible field transformations with derivatives, and also give a rigorous proof that a simple extension of the disformal transformation involving a second derivative of the scalar field is not invertible. [ABSTRACT FROM AUTHOR]

We reexamine the Israel-type proof of the uniqueness theorem of the static spacetime outside the photon surface in the Einstein-conformal scalar system. We derive in a systematic fashion a new divergence identity which plays a key role in the proof. Our divergence identity includes three parameters, allowing us to give a new proof of the uniqueness. [ABSTRACT FROM AUTHOR]

We define an attractive gravity probe surface (AGPS) as a compact 2-surface |$S_\alpha$| with positive mean curvature |$k$| satisfying |$r^a D_a k / k^2 \ge \alpha$| (for a constant |$\alpha>-1/2$|⁠) in the local inverse mean curvature flow, where |$r^a D_a k$| is the derivative of |$k$| in the outward unit normal direction. For asymptotically flat spaces, any AGPS is proved to satisfy the areal inequality |$A_\alpha \le 4\pi [ (3+4\alpha)/(1+2\alpha) ]^2(Gm)^2$|⁠ , where |$A_{\alpha}$| is the area of |$S_\alpha$| and |$m$| is the Arnowitt–Deser–Misner mass. Equality is realized when the space is isometric to the |$t=$| constant hypersurface of the Schwarzschild spacetime and |$S_\alpha$| is an |$r=\mathrm{constant}$| surface with |$r^a D_a k / k^2 = \alpha$|⁠. We adapt the two methods of the inverse mean curvature flow and the conformal flow. Therefore, our result is applicable to the case where |$S_\alpha$| has multiple components. For anti-de Sitter (AdS) spaces, a similar inequality is derived, but the proof is performed only by using the inverse mean curvature flow. We also discuss the cases with asymptotically locally AdS spaces. [ABSTRACT FROM AUTHOR]

We examine the de Sitter entropy in the braneworld model with the Gauss–Bonnet/Lovelock terms. We find that the de Sitter entropy computed through the Euclidean action exactly coincides with the holographic entanglement entropy. [ABSTRACT FROM AUTHOR]

A dynamically transversely trapping surface (DTTS) is a new concept for an extension of a photon sphere that appropriately represents a strong gravity region and has close analogy with a trapped surface. We study formation of a marginally DTTS in time-symmetric, conformally flat initial data with two black holes, with a spindle-shaped source, and with a ring-shaped source, and clarify that |$\mathcal{C}\lesssim 6\pi GM$| describes the condition for the DTTS formation well, where |$\mathcal{C}$| is the circumference and |$M$| is the mass of the system. This indicates that an understanding analogous to the hoop conjecture for the horizon formation is possible. Exploring the ring system further, we find configurations where a marginally DTTS with the torus topology forms inside a marginally DTTS with the spherical topology, without being hidden by an apparent horizon. There also exist configurations where a marginally trapped surface with the torus topology forms inside a marginally trapped surface with the spherical topology, showing a further similarity between DTTSs and trapped surfaces. [ABSTRACT FROM AUTHOR]

We propose new concepts, a dynamically transversely trapping surface (DTTS) and a marginally DTTS, as indicators for a strong gravity region. A DTTS is defined as a two-dimensional closed surface on a spacelike hypersurface such that photons emitted from arbitrary points on it in transverse directions are acceleratedly contracted in time, and a marginally DTTS is reduced to the photon sphere in spherically symmetric cases. (Marginally) DTTSs have a close analogy with (marginally) trapped surfaces in many aspects. After preparing the method of solving for a marginally DTTS in the time-symmetric initial data and the momentarily stationary axisymmetric initial data, some examples of marginally DTTSs are numerically constructed for systems of two black holes in the Brill–Lindquist initial data and in the Majumdar–Papapetrou spacetimes. Furthermore, the area of a DTTS is proved to satisfy the Penrose-like inequality |$A_0\le 4\pi (3GM)^2$|⁠ , under some assumptions. Differences and connections between a DTTS and the other two concepts proposed by us previously, a loosely trapped surface [Prog. Theor. Exp. Phys. 2017 , 033E01 (2017)] and a static/stationary transversely trapping surface [Prog. Theor. Exp. Phys. 2017 , 063E01 (2017)], are also discussed. A (marginally) DTTS provides us with a theoretical tool to significantly advance our understanding of strong gravity fields. Also, since DTTSs are located outside the event horizon, they could possibly be related with future observations of strong gravity regions in dynamical evolutions. [ABSTRACT FROM AUTHOR]