Spacelike Hypersurfaces Research Articles

Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories

Modified gravity theories can accommodate exact solutions, for which the metric has the same form as the one in general relativity, i.e., stealth solutions. One problem with these stealth solutions is that perturbations around them exhibit strong coupling when the solutions are realized in degenerate higher-order scalar-tensor theories. We show that the strong coupling problem can be circumvented in the framework of the so-called U-DHOST theories, in which the degeneracy is partially broken in such a way that higher-derivative terms are degenerate only in the unitary gauge. In this sense, the scordatura effect is built-in in U-DHOST theories in general. There is an apparent Ostrogradsky mode in U-DHOST theories, but it does not propagate as it satisfies a three-dimensional elliptic differential equation on a spacelike hypersurface. We also clarify how this nonpropagating mode, i.e., the "shadowy" mode shows up at the nonlinear level.

Uniqueness and nullity of complete spacelike hypersurfaces immersed in the anti-de Sitter space

Uniqueness and nullity of complete spacelike hypersurfaces immersed in the anti-de Sitter space

Single-particle spectra in relativistic heavy-ion collisions within the thermal quantum field theory

Abstract A quantum generalization of the Cooper–Frye recipe is proposed. The calculation of the single-particle spectrum arising in relativistic collisions of particles and nuclei is carried out within the framework of thermal quantum field theory. The starting point of consideration is the solution of the initial-value problem of the radiation of particles from a space-like hypersurface. In the next steps, we obtain a single-particle spectrum using the ‘lesser’ Green’s function associated with the fireball environment. Based on this result, several specific conditions for particle emission are considered.

Cyclic spacetimes through singularity scattering maps. The laws of quiescent bounces

For spacetimes containing quiescent singularity hypersurfaces we propose a general notion of junction conditions based on a prescribed singularity scattering map, as we call it, and we introduce the notion of a cyclic spacetime (also called a multiverse) consisting of spacetime domains bounded by spacelike or timelike singularity hypersurfaces, across which our scattering map is applied. A local existence theory is established here while, in a companion paper, we construct plane-symmetric cyclic spacetimes. We study the singularity data space consisting of the suitably rescaled metric, extrinsic curvature, and matter fields which can be prescribed on each side of the singularity, and for the class of so-called quiescent singularities we establish restrictions that a singularity scattering map must satisfy. We obtain a full characterization of all scattering maps that are covariant and ultralocal, in a sense we define and, in particular, we distinguish between, on the one hand, three laws of bouncing cosmology of universal nature and, on the other hand, model-dependent junction conditions. The theory proposed in this paper applies to spacelike and timelike hypersurfaces and without symmetry restriction. It encompasses bouncing-cosmology scenarios, both in string theory and in loop quantum cosmology, and puts strong restrictions on their possible explicit realizations.

A family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry with Cauchy temporal functions

We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry that feature time coordinates, which are specific Cauchy temporal functions, i.e., the level sets of these time coordinates are smooth, asymptotically flat, spacelike Cauchy hypersurfaces. Coordinate systems of this kind are well suited for the study of the temporal evolution of matter and radiation fields in the joined exterior and interior regions of the Schwarzschild black hole geometry, whereas the associated foliations can be employed as initial data sets for the globally hyperbolic development under the Einstein flow. For their construction, we formulate an explicit method that utilizes the geometry of—and structures inherent in—the Penrose diagram of the Schwarzschild black hole geometry, thus relying on the corresponding metrical product structure. As an example, we consider an integrated algebraic sigmoid function as the basis for the determination of such a coordinate system. Finally, we generalize our results to the Reissner–Nordström black hole geometry up to the Cauchy horizon. The geometric construction procedure presented here can be adapted to yield similar coordinate systems for various other spacetimes with the same metrical product structure.

Entropy variation of rotating BTZ black hole under Hawking radiation

An axially symmetric rotating Banados-Teitelboim-Zanelli (BTZ) black hole is considered for comprehending its interior information. The largest space-like hyper-surface is chosen to estimate its maximal interior volume as a time-dependent quantity. Similarly, the quantum mode entropy of the scalar field associated with this volume is found to increase with Eddington time. An evolution relation between the variation of quantum mode entropy and Bekenstein-Hawking entropy is obtained for an infinitesimal time interval. On comparing to higher-dimensional black holes, the characteristic feature of this evolution relation is its divergent character.

Tidal effect in ADM formulation under the foliations of spacetime

In this paper the tidal effect is explored by designing a mathematical model in the ADM formalism (Arnowitt et al., 1959) of foliation of spacetime into three dimensional space and time. We show critically that (i) when the time function coincides with the normal vector, i.e., the geodesics become hypersurface orthogonal then the shift vector vanishes; (ii) the tidal effect vanishes when the geodesics of a system of particles under the influence of a gravitational force become hypersurface orthogonal; (iii) the tidal effect arises only when the timelike flow is deviated from the normal direction to the spacelike hypersurface and (iv) when geodesics become non-orthogonal to the spacelike hypersurface at some time, the non-zero rotations (vorticity) come into play.

Chen inequalities on spacelike hypersurfaces of a GRW spacetime

In this paper, we establish Chen-Ricci inequality and Chen inequalities for spacelike hypersurfaces in GRW spacetimes. Considering these inequalities, we give some characterizations. Equality cases are also considered. We show that M is a totally geodesic or a totally umbilical in case of equalities. Moreover, we find some basic inequalities for shape operator of such hypersurfaces and obtain some inequalities for spacelike slices in GRW spacetimes.

Conformal geometry on a class of embedded hypersurfaces in spacetimes

In this work, we study various geometric properties of embedded spacelike hypersurfaces in 1 + 1 + 2 decomposed spacetimes with a preferred spatial direction, denoted e μ , which are orthogonal to the fluid flow velocity of the spacetime and admit a proper conformal transformation. To ensure non-vanishing and positivity of the scalar curvature of the induced metric on the hypersurface, we impose that the scalar curvature of the conformal metric is non-negative and that the associated conformal factor φ satisfies , where denotes derivative along the preferred spatial direction. Firstly, it is demonstrated that such hypersurface is either of Einstein type or the spatial twist vanishes on them, and that the scalar curvature of the induced metric is constant. It is then proved that if the hypersurface is compact and of Einstein type and admits a proper conformal transformation, then these hypersurfaces must be isomorphic to the three-sphere, where we make use of some well known results on Riemannian manifolds admitting conformal transformations. If the hypersurface is not of Einstein type and have nowhere vanishing sheet expansion, we show that this conclusion fails. However, with the additional conditions that the scalar curvatures of the induced metric and the conformal metric coincide, the associated conformal factor is strictly negative and the third and higher order derivatives of the conformal factor vanish, the conclusion that the hypersurface is isomorphic to the three-sphere follows. Furthermore, additional results are obtained under the conditions that the scalar curvature of a metric conformal to the induced metric is also constant. Finally, we consider some of our results in context of locally rotationally symmetric spacetimes and show that, if the hypersurfaces are compact and not of Einstein type, then under specified conditions the hypersurface is isomorphic to the three-sphere, where we constructed explicit examples of proper conformal Killing vector fields along e μ .

Relativistic extended uncertainty principle from spacetime curvature

The investigations presented in this study are directed at relativistic modifications of the uncertainty relation derived from the curvature of the background spacetime. These findings generalize previous work that is recovered in the nonrelativistic limit. Applying the $3+1$ splitting in accordance with the ADM formalism, we find the relativistic physical momentum operator and compute its standard deviation for wave functions confined to a geodesic ball on a spacelike hypersurface. Its radius can then be understood as a measure of position uncertainty. Under the assumption of small position uncertainties in comparison to background curvature length scales, we obtain the corresponding corrections to the uncertainty relation in flat space. Those depend on the Ricci scalar of the effective spatial metric, the particle is moving on, and, if there are nonvanishing time-space components of the spacetime metric, there are gradients of the shift vector and the lapse function. Interestingly, this result is applicable not only to massive but also to massless particles. Over all, this is not a covariant, yet a consistently general relativistic approach. We further speculate on a possible covariant extension.

Entire spacelike hypersurfaces with constant $$\sigma _k$$ curvature in Minkowski space

In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant \(\sigma _k\) curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth, entire, strictly convex, spacelike, constant \(\sigma _k\) curvature hypersurfaces with prescribed Gauss map image. We also show that there doesn’t exist any entire, convex, strictly spacelike, constant \(\sigma _k\) curvature hypersurfaces. Moreover, we generalize the result in Ren et al. (Entire spacelike hypersufaces with constant \(\sigma _{n-1}\) curvature in Minkowski space. Preprint, arXiv:2005.06109) and construct strictly convex, spacelike, constant \(\sigma _k\) curvature hypersurface with bounded principal curvature, whose image of the Gauss map is the unit ball.

A note on the nonexistence of spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form

We obtain nonexistence results concerning complete noncompact spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form, under the assumption that the support functions with respect to a fixed nonzero vector are linearly related. Our approach is based on a suitable maximum principle recently established by Alías, Caminha and do Nascimento [3].

Locally constrained inverse mean curvature flow in GRW spacetimes

<p style='text-indent:20px;'>We obtain the long-time existence and smooth convergence results for the locally constrained inverse mean curvature flow of compact spacelike hypersurface in certain GRW spacetimes. As an application of the convergence result, we establish a weighted Minkowski type inequality for closed spacelike mean-convex hypersurface in de Sitter space under mild rigidity assumption.</p>

Complete spacelike hypersurfaces in the anti-de Sitter space: rigidity, nonexistence and curvature estimates

Our purpose is to investigate the geometry of complete spacelike hypersurfaces immersed in the anti-de Sitter space $\mathbb H_1^{n+1}$. We start by proving rigidity results for such hypersurfaces under suitable constraints on their higher order mean curv

Dressed asymptotic states and QED infrared physics

The dressed state formalisms, which incorporate interactions of soft particles into an asymptotic state, are known as the prescriptions expected to solve the problem of infrared (IR) divergence in the quantum field theory (QFT). A particularly famous example is the dressed state formalism proposed by Kulish and Faddeev in quantum electrodynamics (QED). As pointed out by Hirai and Sugishita, however, this formalism has problems in gauge invariance and the IR divergence. These problems are mainly caused by the existence of ghosts or unphysical photon modes. Therefore, we start by studying the asymptotic states in the Coulomb gauge, which excludes ghosts and/or unphysical photon modes. In this paper, we propose a formalism to construct the asymptotic states directly from the interaction of the theory by setting a sufficiently large time scale $T$. In this dressed state formalism, we define the asymptotic interaction remaining at $|t|>T$ in terms of some fixed order of $1/T$, and we are performing all calculations according to that order. We study the asymptotic states in QED specifically, but we can formally apply the dressed state formalism proposed in this paper to any perturbative QFT. We show that, at least in QED, we can construct divergence-free and unitary $S$-matrix using dressed states proposed in this paper. Furthermore, we discuss the transition rate to show that we can predict experimental results. We also show that the $\mathrm{U}(1)$ gauge symmetry of $S$-matrix leads to the QED large gauge symmetry, and deviation of the expectation values of the vector potential between initial and final spacelike hypersurfaces emerges as a QED memory effect. The dressed state formalism in this paper may give a unified and new insight into IR physics, including asymptotic symmetries, memory effects, and unitarity of the state evolution.

Complete spacelike hypersurfaces with constant scalar curvature: Descriptions and gaps

We provide sharp lower and upper bounds for the supremum of the norm of the total umbilicity tensor of complete spacelike hypersurfaces with constant scalar curvature immersed in a Lorentzian space form and satisfying a suitable Okumura-type inequality, which corresponds to a weaker hypothesis when compared with the geometric condition of the hypersurface having two distinct principal curvatures. Furthermore, we give a complete description and the gaps of the spacelike hypersurfaces which realize our estimates, obtaining as a consequence new characterizations of totally umbilical spacelike hypersurfaces and hyperbolic cylinders of Lorentzian space forms. Our approach is based on a version of Omori–Yau’s maximum principle for trace-type differential operators defined on a complete Riemannian manifold.

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) 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.

A Note on the Sagnac Effect in General Relativity as a Finslerian Effect

The geometry of the Sagnac effect in a stationary region of a spacetime is reviewed with the aim of emphasizing the role of asymmetry of a Finsler metric defined on a spacelike hypersurface associated to a stationary splitting and related to future-pointing null geodesics of the spacetime. We show also that an analogous asymmetry comes into play in the Sagnac effect for timelike geodesics.

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.