Extrinsic Curvature Tensor Research Articles

Asymptotic behavior of null geodesics near future null infinity. II. Curvatures, photon surface, and dynamically transversely trapping surface

Bearing in mind our previous study on asymptotic behavior of null geodesics near future null infinity, we analyze the behavior of geometrical quantities such as a certain extrinsic curvature and Riemann tensor in the Bondi coordinates. In the sense of asymptotics, the condition for an $r$-constant hypersurface to be a photon surface is shown to be controlled by a key quantity that determines the fate of photons initially emitted in angular directions. As a consequence, in four dimensions, such a non-expanding photon surface can be realized even near future null infinity in the presence of enormous energy flux for a short period of time. By contrast, in higher-dimensional cases, no such a photon surface can exist. This result also implies that the dynamically transversely trapping surface, which is proposed as an extension of a photon surface, can have an arbitrarily large radius in four dimensions.

Manifold curvature learning from hypersurface integral invariants

Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature tensor of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such invariants for a spherical volume component delimited by a hypersurface and for the hypersurface patch created by ball intersections, showing that the eigenvalues and eigenvectors can be used as multi-scale estimators of the principal curvatures and principal directions. This approach may be interpreted as performing statistical analysis on the underlying point-set of a submanifold in order to obtain geometric descriptors at scale with potential applications to Manifold Learning and Geometry Processing of point clouds.

Spherically symmetric double layers in Weyl+Einstein gravity

We study the matching conditions on singular hypersurfaces in Weyl[Formula: see text]Einstein gravity. Unlike General Relativity, the so-called quadratic gravity allows the existence of a double layer, i.e. the derivative of [Formula: see text]-function. This double layer is a purely geometrical phenomenon and it may be treated as the purely gravitational shock wave. The mathematical formalism was elaborated by Senovilla for generic quadratic gravity. We derived the matching conditions for the spherically symmetric singular hypersurface in the Weyl[Formula: see text]Einstein gravity. It was found that in the presence of the double layer, the matching conditions contain an arbitrary function. One of the consequences of such freedom is that a trace of the extrinsic curvature tensor of a singular hypersurface is necessarily equal to zero. We suggested that the [Formula: see text] and [Formula: see text] components of the surface matter energy–momentum tensor of the shell describe energy flow [Formula: see text] and momentum transfer [Formula: see text] of particles produced by the double layer itself. Moreover, the requirement of the zero trace of the extrinsic curvature tensor (mentioned above) implies that [Formula: see text], and this fact also supports our suggestion, because it means that for the observer sitting on the shell, particles will be seen created by pairs, and the sum of their momentum transfers must be zero. We found also that the spherically symmetric null double layer in the Weyl[Formula: see text]Einstein gravity does not exist at all.

Replacing dark energy by silent virialisation

Context.Standard cosmologicalN-body simulations have background scale factor evolution that is decoupled from non-linear structure formation. Prior to gravitational collapse, kinematical backreaction (QD) justifies this approach in a Newtonian context.Aims.However, the final stages of a gravitational collapse event are sudden; a globally imposed smooth expansion rate forces at least one expanding region to suddenly and instantaneously decelerate in compensation for the virialisation event. This is relativistically unrealistic. A more conservative hypothesis is to allow non-collapsed domains to continue their volume evolution according to theQDZel’dovich approximation (QZA). We aim to study the inferred average expansion under this “silent” virialisation hypothesis.Methods.We set standard (MPGRAFIC) EdS 3-torus (T3) cosmologicalN-body initial conditions. UsingRAMSES, we partitioned the volume into domains and called theDTFElibrary to estimate the per-domain initial values of the three invariants of the extrinsic curvature tensor that determine the QZA. We integrated the Raychaudhuri equation in each domain using theINHOMOGlibrary, and adopted the stable clustering hypothesis to represent virialisation (VQZA). We spatially averaged to obtain the effective global scale factor. We adopted an early-epoch–normalised EdS reference-model Hubble constantH1EDS= 37.7km s-1∕Mpc and an effective Hubble constantHeff,0= 67.7km s-1∕Mpc.Results.From 2000 simulations at resolution 2563, we find that reaching a unity effective scale factor at 13.8 Gyr (16% above EdS), occurs for an averaging scale ofL13.8= 2.5−0.4+0.1Mpc∕heff. Relativistically interpreted, this corresponds to strong average negative curvature evolution, with the mean (median) curvature functionalΩRDgrowing from zero to about 1.5–2 by the present. Over 100 realisations, the virialisation fraction and super-EdS expansion correlate strongly at fixed cosmological time.Conclusions.Thus, starting from EdS initial conditions and averaging on a typical non-linear structure formation scale, the VQZA dark-energy–free average expansion matchesΛCDM expansion to first order. The software packages used here are free-licensed.

Equivalent and inequivalent canonical structures of higher order theories of gravity

Canonical formulation of higher order theory of gravity can only be accomplished associating additional degrees of freedom, which are extrinsic curvature tensor. Consequently, to match Cauchy data with the boundary data, terms in addition to the three-space metric, must also be fixed at the boundary. While, in all the three, viz. Ostrogradski's, Dirac's and Horowitz' formalisms, extrinsic curvature tensor is kept fixed at the boundary, a modified Horowitz' formalism fixes Ricci scalar, instead. It has been taken as granted that the Hamiltonian structure corresponding to all the formalisms with different end-point data are either the same or are canonically equivalent. In the present study, we show that indeed it is true, but only for a class of higher order theory. However, for more general higher order theories, e.g. dilatonic coupled Gauss-Bonnet gravity in the presence of curvature squared term, the Hamiltonian obtained following modified Horowitz' formalism is found to be different from the others, and is not related under canonical transformation. Further, it has also been demonstrated that although all the formalisms produce viable quantum description, the dynamics is different and not canonically related to modified Horowitz' formalism. Therefore it is not possible to choose the correct formalism which leads to degeneracy in Hamiltonian.

Hamiltonian analysis of curvature-squared gravity with or without conformal invariance

We analyze gravitational theories with quadratic curvature terms, including the case of conformally invariant Weyl gravity, motivated by the intention to find a renormalizable theory of gravity in the ultraviolet region, yet yielding general relativity at long distances. In the Hamiltonian formulation of Weyl gravity, the number of local constraints is equal to the number of unstable directions in phase space, which in principle could be sufficient for eliminating the unstable degrees of freedom in the full nonlinear theory. All the other theories of quadratic type are unstable -- a problem appearing as ghost modes in the linearized theory. We find that the full projection of the Weyl tensor onto a three-dimensional hypersurface contains an additional fully traceless component, given by a quadratic extrinsic curvature tensor. A certain inconsistency in the literature is found and resolved: when the conformal invariance of Weyl gravity is broken by a cosmological constant term, the theory becomes pathological, since a constraint required by the Hamiltonian analysis imposes the determinant of the metric of spacetime to be zero. In order to resolve this problem by restoring the conformal invariance, we introduce a new scalar field that couples to the curvature of spacetime, reminiscent of the introduction of vector fields for ensuring the gauge invariance.

Essential building blocks of dark energy

We propose a minimal description of single field dark energy/modified gravity within the effective field theory formalism for cosmological perturbations, which encompasses most existing models. We start from a generic Lagrangian given as an arbitrary function of the lapse and of the extrinsic and intrinsic curvature tensors of the time hypersurfaces in unitary gauge, i.e. choosing as time slicing the uniform scalar field hypersurfaces. Focusing on linear perturbations, we identify seven Lagrangian operators that lead to equations of motion containing at most two (space or time) derivatives, the background evolution being determined by the time-dependent coefficients of only three of these operators. We then establish a dictionary that translates any existing or future model whose Lagrangian can be written in the above form into our parametrized framework. As an illustration, we study Horndeski's — or generalized Galileon — theories and show that they can be described, up to linear order, by only six of the seven operators mentioned above. This implies, remarkably, that the dynamics of linear perturbations can be more general than that of Horndeski while remaining second order. Finally, in order to make the link with observations, we provide the entire set of linear perturbation equations in Newtonian gauge, the effective Newton constant in the quasi-static approximation and the ratio of the two gravitational potentials, in terms of the time-dependent coefficients of our Lagrangian.

Black holes in the brane world: Some exact solutions

We briefly review methods for and results from constructing multidimensional solutions with an event horizon generated by matter localized on a brane. We investigate a particular exact solution of the black star type with the Schwarzschild-Tangherlini geometry in the bulk. We study the extrinsic curvature tensor, which predicts the matter distribution on the brane in accordance with the Israel conditions. We calculate all components of the four-dimensional Riemann tensor describing the intrinsic geometry on the brane surface and also investigate their asymptotic forms at a large distance from the center, limits on the horizon, and the asymptotic forms in the singularity neighborhood of a five-dimensional black hole in the case where the brane intersects it.

The supermultiplet of boundary conditions in supergravity

Boundary conditions in supergravity on a manifold with boundary relate the bulk gravitino to the boundary supercurrent, and the normal derivative of the bulk metric to the boundary energy-momentum tensor. In the 3D N=1 setting, we show that these boundary conditions can be stated in a manifestly supersymmetric form. We identify the Extrinsic Curvature Tensor Multiplet, and show that boundary conditions set it equal to (a conjugate of) the boundary supercurrent multiplet. Extension of our results to higher-dimensional models (including the Randall-Sundrum and Horava-Witten scenarios) is discussed.

Erratum: Quantum mechanics of a constrained electrically charged particle in the presence of electric currents [Phys. Rev. A80, 052109 (2009)

In our article, we used the following definition of the extrinsic curvature tensor Kij [1] (following the notation in our article): ∇||iNj = −Kij . However, in our article, after Eq. (17), the sign of the extrinsic curvature tensor followed the convention ∇||iNj = Kij but did not correct for the change in the sign of the extrinsic curvature. When we correct for this missed minus sign, the divergence of the electromagnetic vector potential A in Eqs. (18), (23), and (26) reads ∇ · Aψ = ( ∇|| · A|| + ∂3A − 2A3M)ψ, where ψ denotes the wave function and M is the mean curvature in the surface M ≡ 2 TrKij . If we choose not to fix the electromagnetic gauge, the last term in this expression apparently cancels a corresponding term that appears in the Aj∇jψ term in the Schrodinger equation. This cancellation apparently removes a possible unphysical contribution to the orbital magnetic moment of the particle. This leaves us with the same expression for the Schrodinger equation in an unfixed electromagnetic gauge as the one in Ref. [2], thus leaving the impression that there is no coupling between curvature and electromagnetic field. A nonessential typo also appeared in Eq. (39). This equation should read

Singularities in Horava–Lifshitz theory

Singularities in (3+1)-dimensional Horava–Lifshitz (HL) theory of gravity are studied. These singularities can be divided into scalar, non-scalar curvature, and coordinate singularities. Because of the foliation-preserving diffeomorphisms of the theory, the number of scalars that can be constructed from the extrinsic curvature tensor Kij, the 3-dimensional Riemann tensor and their derivatives is much large than that constructed from the 4-dimensional Riemann tensor and its derivatives in general relativity (GR). As a result, even for the same spacetime, it may be singular in the HL theory but not in GR. Two representative families of solutions with projectability condition are studied, one is the (anti-)de Sitter Schwarzschild solutions, and the other is the Lü–Mei–Pope (LMP) solutions written in a form satisfying the projectability condition – the generalized LMP solutions. The (anti-)de Sitter Schwarzschild solutions are vacuum solutions of both HL theory and GR, while the LMP solutions with projectability condition satisfy the HL equations coupled with an anisotropic fluid with heat flow. It is found that the scalars K and KijKij are singular only at the center for the de Sitter Schwarzschild solution, but singular at both the center and r=(3M/|Λ|)1/3 for the anti-de Sitter Schwarzschild solution. The singularity at r=(3M/|Λ|)1/3 is absent in GR. In addition, all the generalized LMP solutions have two scalar curvature singularities, located at either r=0 and r=rs>0, or r=r1 and r=r2 with r2>r1>0, or r=rs>0 and r=∞, depending on the choice of the free parameter λ.

SPHERICALLY SYMMETRIC GRAVITATIONAL COLLAPSE

In this paper, we discuss gravitational collapse of spherically symmetric spacetimes. We derive a general formalism by taking two arbitrary spherically symmetric spacetimes with g00 = 1. Israel's junction conditions are used to develop this formalism. The formulas for extrinsic curvature tensor are obtained. The general form of the surface energy–momentum tensor depending on extrinsic curvature tensor components is derived. This leads us to the surface energy density and the tangential pressure. The formalism is applied to two known spherically symmetric spacetimes. The results obtained show the regions for the collapse and expansion of the shell.

Low-energy effective theory for a Randall-Sundrum scenario with a moving bulk brane

We derive the low-energy effective theory of gravity for a generalized Randall-Sundrum scenario, allowing for a third self-gravitating brane to live in the 5D bulk spacetime. At zero order the 5D spacetime is composed of two slices of anti-de Sitter spacetime, each with a different curvature scale, and the 5D Weyl tensor vanishes. Two boundary branes are at the fixed points of the orbifold whereas the third brane is free to move in the bulk. At first order, the third brane breaks the otherwise continuous evolution of the projection of the Weyl tensor normal to the branes. We derive a junction condition for the projected Weyl tensor across the bulk brane, and combining this constraint with the junction condition for the extrinsic curvature tensor, allows us to derive the first-order field equations on the middle brane. The effective theory is a generalized Brans-Dicke theory with two scalar fields. This is conformally equivalent to Einstein gravity and two scalar fields, minimally coupled to the geometry, but nonminimally coupled to matter on the three branes.

Extrinsically flat static spacetimes

We consider static spacetimes whose spatial part admits foliations with the extrinsic curvature tensor Kab = 0. There are two complementary cases when the gradient of the lapse function points (1) to the direction of foliation or (2) orthogonally to it. Case (1) gives a generalization of metrics like Bertotti–Robinson or Nariai. In case (2), the matter source violates the null energy condition at least on part of the manifold, having in this sense a phantom nature. We also demonstrate that for the manifolds under discussion the horizon can be naked in the sense that certain Weyl components diverge in the free-falling frame, although the Kretschmann scalar is finite. The Petrov type is D or O. Explicit solutions for (i) the linear anisotropic equation of state, (ii) Chaplygin gas and (iii) uniform energy density are found.

New minimal distortion shift gauge

Based on the recent understanding of the role of the densitized lapse function in Einstein's equations and of the proper way to pose the thin sandwich problem, a slight readjustment of the minimal distortion shift gauge in the 3+1 approach to the dynamics of general relativity allows this shift vector to serve as the vector potential for the longitudinal part of the extrinsic curvature tensor in the new approach to the initial value problem, thus extending the initial value decomposition of gravitational variables to play a role in the evolution as well. The new shift vector globally minimizes the changes in the conformal 3-metric with respect to the spacetime measure rather than the spatial measure on the time coordinate hypersurfaces, as the old shift vector did.

Second variation of the Helfrich–Canham Hamiltonian and reparametrization invariance

A covariant approach towards a theory of deformations is developed to examine both the first and second variation of the Helfrich-Canham Hamiltonian -- quadratic in the extrinsic curvature -- which describes fluid vesicles at mesoscopic scales. Deformations are decomposed into tangential and normal components; At first order, tangential deformations may always be identified with a reparametrization; at second order, they differ. The relationship between tangential deformations and reparametrizations, as well as the coupling between tangential and normal deformations, is examined at this order for both the metric and the extrinsic curvature tensors. Expressions for the expansion to second order in deformations of geometrical invariants constructed with these tensors are obtained; in particular, the expansion of the Hamiltonian to this order about an equilibrium is considered. Our approach applies as well to any geometrical model for membranes.

Defining perturbations on submanifolds

We study the definition of perturbations in the presence of a submanifold, like e.g. a brane. In the standard theory of cosmological perturbations, one compares quantities at the same coordinate points in the non-perturbed and the perturbed manifolds, identified via a (non-unique) mapping between the two manifolds. In the presence of a physical submanifold one needs to modify this definition in order to evaluate perturbations of quantities at the submanifold location. As an application, we compute the perturbed metric and the extrinsic curvature tensors at the brane position in a general gauge.

Numerical tests of evolution systems, gauge conditions, and boundary conditions for 1D colliding gravitational plane waves

We investigate how the accuracy and stability of numerical relativity simulations of 1D colliding plane waves depends on choices of equation formulations, gauge conditions, boundary conditions, and numerical methods, all in the context of a first-order 3+1 approach to the Einstein equations, with basic variables some combination of first derivatives of the spatial metric and components of the extrinsic curvature tensor. Hyperbolic schemes, specifically variations on schemes proposed by Bona and Masso and Anderson and York, are compared with variations of the Arnowitt-Deser-Misner formulation. Modifications of the three basic schemes include raising one index in the metric derivative and extrinsic curvature variables and adding a multiple of the energy constraint to the extrinsic curvature evolution equations. Redundant variables in the Bona-Masso formulation may be reset frequently or allowed to evolve freely. Gauge conditions which simplify the dynamical structure of the system are imposed during each time step, but the lapse and shift are reset periodically to control the evolution of the spacetime slicing and the longitudinal part of the metric. We show that physically correct boundary conditions, satisfying the energy and momentum constraint equations, generically require the presence of some ingoing eigenmodes of the characteristic matrix. Numerical methods are developed for the hyperbolic systems based on decomposing flux differences into linear combinations of eigenvectors of the characteristic matrix. These methods are shown to be second-order accurate, and in practice second-order convergent, for smooth solutions, even when the eigenvectors and eigenvalues of the characteristic matrix are spatially varying.

Self-adjoint wave equations for dynamical perturbations of self-gravitating fields

It is shown that the dynamical evolution of linear perturbations on a static space-time is governed by a constrained wave equation for the extrinsic curvature tensor. The spatial part of the wave operator is manifestly elliptic and self-adjoint. In contrast to metric formulations, the curvature-based approach to gravitational perturbation theory generalizes in a natural way to self-gravitating matter fields. It is also demonstrated how to obtain symmetric pulsation equations for self-gravitating non-Abelian gauge fields, Higgs fields and perfect fluids. For vacuum fluctuations on a vacuum space-time, the Regge-Wheeler and Zerilli equations are rederived.

Constant crunch coordinates for black hole simulations

We reinvestigate the utility of time-independent constant mean curvature foliations for the numerical simulation of a single spherically symmetric black hole. Each spacelike hypersurface of such a foliation is endowed with the same constant value of the trace of the extrinsic curvature tensor K. Of the three families of K-constant surfaces possible (classified according to their asymptotic behaviors), we single out a subfamily of singularity-avoiding surfaces that may be particularly useful, and provide an analytic expression for the closest approach such surfaces make to the singularity. We then utilize a nonzero shift to yield families of K-constant surfaces which (1) avoid the black hole singularity, and thus the need to excise the singularity, (2) are asymptotically null, aiding in gravity wave extraction, (3) cover the physically relevant part of the spacetime, (4) are well behaved (regular) across the horizon, and (5) are static under evolution, and therefore have no ``grid stretching/ sucking'' pathologies. Preliminary numerical runs demonstrate that we can stably evolve a single spherically symmetric static black hole using this foliation. We wish to emphasize that this coordinatization produces K-constant surfaces for a single black hole spacetime that are regular, static, and stable throughout their evolution.