Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories

We study U-DHOST theories, i.e., higher-order scalar-tensor theories which are degenerate only in the unitary gauge and yield an apparently unstable extra mode in a generic coordinate system. We show that the extra mode satisfies a three-dimensional elliptic differential equation on a spacelike hypersurface, and hence it does not propagate. We clarify how to treat this “shadowy” mode at both the linear and the nonlinear levels.

Symmetries and conformal bridge in Schwarschild-(A)dS black hole mechanics

We show that the Schwarzschild-(A)dS black hole mechanics possesses a hidden symmetry under the three-dimensional Poincaré group. This symmetry shows up after having gauge-fixed the diffeomorphism invariance in the symmetry-reduced homogeneous Einstein-Λ model and stands as a physical symmetry of the system. It dictates the geometry both in the black hole interior and exterior regions, as well as beyond the cosmological horizon in the Schwarzschild-dS case. It follows that one can associate a set of non-trivial conserved charges to the Schwarzschild-(A)dS black hole which act in each causally disconnected regions. In T-region, they act on fields living on spacelike hypersurface of constant time, while in R-regions, they act on time-like hypersurface of constant radius. We find that while the expression of the charges depend explicitly on the location of the hypersurface, the charge algebra remains the same at any radius in R-regions (or time in T-regions). Finally, the analysis of the Casimirs of the charge algebra reveals a new solution-generating map. The $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) $$ sl 2 ℝ Casimir is shown to generate a one-parameter family of deformation of the black hole geometry labelled by the cosmological constant. This gives rise to a new conformal bridge allowing one to continuously deform the Schwarzschild-AdS geometry to the Schwarzschild and the Schwarzschild-dS solutions.

Complete CSC Hypersurfaces Satisfying an Okumura-Type Inequality in Ricci Symmetric Manifolds

We investigate the spacelike hypersurface with constant scalar curvature (SCS) immersed in a Ricci symmetric manifold obeying standard curvature constraints. By supposing these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Meléndez, which is a weaker hypothesis than to assume that they have two distinct principal curvatures, we obtain a series of umbilicity and pinching results. In particular, when the Ricci symmetric manifold is an Einstein manifold, then we further obtain some rigidity classifications of such hypersufaces.

Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1

In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

Main Theorem

This chapter presents the main theorem, its main conclusions, as well as a full strategy of its proof, divided in nine supporting intermediate results, Theorems M0–M8. The chapter specifies the closeness to Schwarzschild of the initial data in the context of the Characteristic Cauchy problem. The conclusions of the main theorem can be immediately extended to the case where the data are specified to be close to Schwarzschild on a spacelike hypersurface Σ‎. The chapter then outlines the main bootstrap assumptions. It also provides a short description of the results concerning the General Covariant Modulation procedure, which is at the heart of the proof.

On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance [Formula: see text] from a spacelike hypersurface [Formula: see text] is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure, compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.

Transversely trapping surfaces: Dynamical version

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.

Generalized Uncertainty Principle in Quantum Cosmology

The effects of gravity which manifest themselves when performing the simultaneous measurement of two non-commuting observables in the quantum theory are discussed. Matter and gravity are considered as quantum fields. The Schr¨odinger-type time equation is given for the case of a finite number of degrees of freedom: one for the matter field and one for geometry. For a spatially closed system filled with dust and radiation being in definite quantum states, the solutions to the quantum equations are found, and the existence of the minimum measurable length and the minimum momentum is shown. It appears that the simultaneous measurement of fluctuations of the intrinsic and extrinsic curvatures of the spacelike hypersurface in spacetime cannot be performed with an accuracy exceeding the Planck constant. Unruh’s and Bronstein’s uncertainty relations are discussed.

The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities

We consider the wave equation, [Formula: see text], in fixed flat Friedmann–Lemaître–Robertson–Walker and Kasner spacetimes with topology [Formula: see text]. We obtain generic blow up results for solutions to the wave equation toward the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to the solutions that blow up in an open set of the Big Bang hypersurface [Formula: see text]. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate [Formula: see text]-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the [Formula: see text] norms of the solutions blow up toward the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates, respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

Generalized Uncertainty Principle in Quantum Cosmology for the Maximally Symmetric Space

The new uncertainty relation is derived in the context of the canonical quantum theory with gravity in the case of the maximally symmetric space. This relation establishes a connection between fluctuations of the quantities, which determine the intrinsic and extrinsic curvatures of the spacelike hypersurface in spacetime and introduces the uncertainty principle for quantum gravitational systems. The generalized time-energy uncertainty relation taking gravity into account gravity is proposed. It is shown that known Unruh’s uncertainty relation follows, as a particular case, from the new uncertainty relation. As an example, the sizes of fluctuations of the scale factor and its conjugate momentum are calculated within an exactly solvable model. All known modifications of the uncertainty principle deduced previously from different approaches in the theory of gravity and the string theory are obtained as particular cases of the proposed general expression.