Showing papers on "Riemann curvature tensor published in 2012"

General second-order scalar-tensor theory and self-tuning.

Abstract: Starting from the most general scalar-tensor theory with second order field equations in four dimensions, we establish the unique action that will allow for the existence of a consistent self-tuning mechanism on FLRW backgrounds, and show how it can be understood as a combination of just four base Lagrangians with an intriguing geometric structure dependent on the Ricci scalar, the Einstein tensor, the double dual of the Riemann tensor and the Gauss-Bonnet combination. Spacetime curvature can be screened from the net cosmological constant at any given moment because we allow the scalar field to break Poincar\'e invariance on the self-tuning vacua, thereby evading the Weinberg no-go theorem. We show how the four arbitrary functions of the scalar field combine in an elegant way opening up the possibility of obtaining non-trivial cosmological solutions.

On the Riemann tensor in double field theory

Abstract: Double field theory provides T-duality covariant generalized tensors that are natural extensions of the scalar and Ricci curvatures of Riemannian geometry. We search for a similar extension of the Riemann curvature tensor by developing a geometry based on the generalized metric and the dilaton. We find a duality covariant Riemann tensor whose contractions give the Ricci and scalar curvatures, but that is not fully determined in terms of the physical fields. This suggests that α′ corrections to the effective action require α′ corrections to T-duality transformations and/or generalized diffeomorphisms. Further evidence to this effect is found by an additional computation that shows that there is no T-duality invariant four-derivative object built from the generalized metric and the dilaton that reduces to the square of the Riemann tensor.

Geometric Action for Nongeometric Fluxes

Abstract: We give a geometrical interpretation of the non-geometric Q and R fluxes. To this end we consider double field theory in a formulation that is related to the conventional one by a field redefinition taking the form of a T-duality inversion. The R flux is a tensor under diffeomorphisms and satisfies a non-trivial Bianchi identity. The Q flux can be viewed as part of a connection that covariantizes the winding derivatives with respect to diffeomorphisms. We give a higher-dimensional action with a kinetic term for the R flux and a 'dual' Einstein-Hilbert term containing the connection Q.

Post-Newtonian, quasicircular binary inspirals in quadratic modified gravity

Abstract: We consider a general class of quantum gravity-inspired, modified gravity theories, where the Einstein-Hilbert action is extended through the addition of all terms quadratic in the curvature tensor coupled to scalar fields with standard kinetic energy. This class of theories includes Einstein-Dilaton-Gauss-Bonnet and Chern-Simons modified gravity as special cases. We analytically derive and solve the coupled field equations in the post-Newtonian approximation, assuming a comparable-mass, spinning black hole binary source in a quasicircular, weak-field/slow-motion orbit. We find that a naive subtraction of divergent piece associated with the point-particle approximation is ill-suited to represent compact objects in these theories. Instead, we model them by appropriate effective sources built so that known strong-field solutions are reproduced in the far-field limit. In doing so, we prove that black holes in Einstein-Dilaton-Gauss-Bonnet and Chern-Simons theory can have hair, while neutron stars have no scalar monopole charge, in diametrical opposition to results in scalar-tensor theories. We then employ techniques similar to the direct integration of the relaxed Einstein equations to obtain analytic expressions for the scalar field, metric perturbation, and the associated gravitational wave luminosity measured at infinity. We find that scalar field emission mainly dominates the energy flux budget, sourcing electric-type (even-parity) dipole scalar radiation and magnetic-type (odd-parity) quadrupole scalar radiation, correcting the General Relativistic prediction at relative $\ensuremath{-}1\mathrm{PN}$ and 2PN orders. Such modifications lead to corrections in the emitted gravitational waves that can be mapped to the parameterized post-Einsteinian framework. Such modifications could be strongly constrained with gravitational wave observations.

A covariant approach to general field space metric in multi-field inflation

Abstract: We present a covariant formalism for general multi-field system which enables us to obtain higher order action of cosmological perturbations easily and systematically. The effects of the field space geometry, described by the Riemann curvature tensor of the field space, are naturally incorporated. We explicitly calculate up to the cubic order action which is necessary to estimate non-Gaussianity and present those geometric terms which have not yet been known before.

Generalized quasi-Einstein manifolds with harmonic Weyl tensor

Abstract: In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n − 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.

Cylindrically symmetric relativistic fluids: a study based on structure scalars

Abstract: Applying the 1 + 3 formalism we write down the full set of equations governing the structure and the evolution of self-gravitating cylindrically symmetric dissipative fluids with anisotropic stresses, in terms of scalar quantities obtained from the orthogonal splitting of the Riemann tensor (structure scalars), in the context of general relativity. These scalars which have been shown previously (in the spherically symmetric case) to be related to fundamental properties of the fluid distribution, such as: energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux, active gravitational mass etc, are shown here to play also a very important role in the dynamics of cylindrically symmetric fluids. It is also shown that in the static case, all possible solutions to Einstein equations may be expressed explicitly through three of these scalars.

Effective action approach to cosmological perturbations in dark energy and modified gravity

Abstract: In light of upcoming observations modelling perturbations in dark energy and modified gravity models has become an important topic of research. We develop an effective action to construct the components of the perturbed dark energy momentum tensor which appears in the perturbed generalized gravitational field equations, ?G?? = 8?G?T??+?U?? for linearized perturbations. Our method does not require knowledge of the Lagrangian density of the dark sector to be provided, only its field content. The method is based on the fact that it is only necessary to specify the perturbed Lagrangian to quadratic order and couples this with the assumption of global statistical isotropy of spatial sections to show that the model can be specified completely in terms of a finite number of background dependent functions. We present our formalism in a coordinate independent fashion and provide explicit formulae for the perturbed conservation equation and the components of ?U?? for two explicit generic examples: (i) the dark sector does not contain extra fields, = (g??) and (ii) the dark sector contains a scalar field and its first derivative = (g??,,??). We discuss how the formalism can be applied to modified gravity models containing derivatives of the metric, curvature tensors, higher derivatives of the scalar fields and vector fields.

Nearly Round Spheres Look Convex

Abstract: We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains so M appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.

Instantons and Killing spinors

Abstract: We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kahler 6-manifolds, nearly parallel G 2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifolds which solves the instanton equation, and also show that the instanton equation implies the Yang-Mills equation, despite the presence of torsion. We then construct instantons on the cones over these manifolds, and lift them to solutions of heterotic supergravity. Amongst our solutions are new instantons on even-dimensional Euclidean spaces, as well as the well-known BPST, quaternionic and octonionic instantons.

Cylindrically Symmetric Relativistic Fluids: A Study Based on Structure Scalars

Abstract: The full set of equations governing the structure and the evolution of self--gravitating cylindrically symmetric dissipative fluids with anisotropic stresses, is written down in terms of scalar quantities obtained from the orthogonal splitting of the Riemann tensor (structure scalars), in the context of general relativity. These scalars which have been shown previously (in the spherically symmetric case) to be related to fundamental properties of the fluid distribution, such as: energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux, active gravitational mass etc, are shown here to play also a very important role in the dynamics of cylindrically symmetric fluids. It is also shown that in the static case, all possible solutions to Einstein equations may be expressed explicitly through three of these scalars.

Positivity of relative canonical bundles and applications

Abstract: Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the unique Kahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic equation to show that this metric is strictly positive on $\mathcal X$, unless the family is infinitesimally trivial. For degenerating families we show that the curvature form on the total space can be extended as a (semi-)positive closed current. By fiber integration it follows that the generalized Weil-Petersson form on the base possesses an extension as a positive current. We prove an extension theorem for hermitian line bundles, whose curvature forms have this property. This theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space. As an application the quasi-projectivity of the moduli space $\mathcal M_{\text{can}}$ of canonically polarized varieties follows. The direct images $R^{n-p}f_*\Omega^p_{\mathcal X/S}(\mathcal K_{\mathcal X/S}^{\otimes m})$, $m > 0$, carry natural hermitian metrics. We prove an explicit formula for the curvature tensor of these direct images. We apply it to the morphisms $S^p \mathcal T_S \to R^pf_*\Lambda^p\mathcal T_{\mathcal X/S}$ that are induced by the Kodaira-Spencer map and obtain a differential geometric proof for hyperbolicity properties of $\mathcal M_{\text{can}}$.

Weakly z symmetric manifolds

Abstract: We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named weakly Z-symmetric and is denoted by (WZS) n .I f theZ tensor is singular we give condi- tions for the existence of a proper concircular vector. For non singular Z tensors, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally har- monic case, and the general form of the Ricci tensor. For conformally flat (WZS) n manifolds, we derive the local form of the metric tensor.

Structure scalars for charged cylindrically symmetric relativistic fluids

Abstract: We investigate some structure scalars developed through Riemann tensor for self-gravitating cylindrically symmetric charged dissipative anisotropic fluid. We show that these scalars are directly related to the fundamental properties of the fluid. We formulate dynamical-transport equation as well as the mass function by including charge which are then expressed in terms of structure scalars. The effects of electric charge are investigated in the structure and evolution of compact objects. Finally, we show that all possible solutions of the field equations can be written in terms of these scalars.

Nullity conditions in paracontact geometry

Abstract: The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $% \tilde\kappa$ and $\tilde\mu$). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in \cite{MOTE}. In this paper we show in fact that there is a kind of duality between those manifolds and contact metric $(\kappa,\mu)$-spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric $(\kappa,\mu)$-structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under $% \mathcal{D}$-homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.

Curvature perturbation in multi-field inflation with non-minimal coupling

Abstract: In this paper we discuss a multi-field model of inflation in which generally all fields are non-minimally coupled to the Ricci scalar and have non-canonical kinetic terms. The background evolution and first-order perturbations for the model are evaluated in both the Jordan and Einstein frames, and the respective curvature perturbations compared. We confirm that they are indeed not the same - unlike in the single-field case - and also that the difference is a direct consequence of the isocurvature perturbations inherent to multi-field models. This result leads us to conclude that the notion of adiabaticity is not invariant under conformal transformations. Using a two-field example we show that even if in one frame the evolution is adiabatic, meaning that the curvature perturbation is conserved on super-horizon scales, in general in the other frame isocurvature perturbations continue to source the curvature perturbation. We also find that it is possible to realise a particular model in which curvature perturbations in both frames are conserved but with each being of different magnitude. These examples highlight that the curvature perturbation itself, despite being gauge-invariant, does not correspond directly to an observable. The non-equivalence of the two curvature perturbations would also be important when considering the addition of Standard Model matter into the system.

Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor

Abstract: We introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grassmannians G2(ℂm+2) from the Gauss equation. We then derive a new formula for the Ricci tensor of M in G2(ℂm+2). Finally, we prove that there does not exist any Hopf real hypersurface in complex two-plane Grassmannians G2(ℂm+2) with parallel Ricci tensor.

Post-Newtonian, quasicircular binary inspirals in quadratic modified gravity

Abstract: We consider a general class of quantum gravity-inspired, modified gravity theories, where the Einstein-Hilbert action is extended through the addition of all terms quadratic in the curvature tensor coupled to scalar fields with standard kinetic energy. This class of theories includes Einstein-Dilaton-Gauss-Bonnet and Chern-Simons modified gravity as special cases. We analytically derive and solve the coupled field equations in the post-Newtonian approximation, assuming a comparable-mass, spinning black hole binary source in a quasicircular, weak-field/slow-motion orbit. We find that a naive subtraction of divergent piece associated with the point-particle approximation is ill-suited to represent compact objects in these theories. Instead, we model them by appropriate effective sources built so that known strong-field solutions are reproduced in the far-field limit. In doing so, we prove that black holes in Einstein-Dilaton-Gauss-Bonnet and Chern-Simons theory can have hair, while neutron stars have no scalar monopole charge, in diametrical opposition to results in scalar-tensor theories. We then employ techniques similar to the direct integration of the relaxed Einstein equations to obtain analytic expressions for the scalar field, metric perturbation, and the associated gravitational wave luminosity measured at infinity. We find that scalar field emission mainly dominates the energy flux budget, sourcing electric-type (even-parity) dipole scalar radiation and magnetic-type (odd-parity) quadrupole scalar radiation, correcting the General Relativistic prediction at relative −1PN and 2PN orders. Such modifications lead to corrections in the emitted gravitational waves that can be mapped to the parameterized post-Einsteinian framework. Such modifications could be strongly constrained with gravitational wave observations.

Six-dimensional supergravity and projective superfields

Abstract: We propose a superspace formulation of $ \mathcal{N} = \left( {1,0} \right) $ conformal supergravity in six dimensions. The corresponding superspace constraints are invariant under super-Weyl transformations generated by a real scalar parameter. The known variant Weyl super-multiplet is recovered by coupling the geometry to a super-3-form tensor multiplet. Isotwistor variables are introduced and used to define projective superfields. We formulate a locally supersymmetric and super-Weyl invariant action principle in projective superspace. Some families of dynamical supergravity-matter systems are presented.

Nullity conditions in paracontact geometry

Abstract: The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition (1.2), for some real numbers κ ˜ and μ ˜ ). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in Cappelletti Montano (2010) [13] . In this paper we show in fact that there is a kind of duality between those manifolds and contact metric ( κ , μ ) -spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric ( κ , μ ) -structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under D -homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.

A curved brunn-minkowski inequality on the discrete hypercube, or: what is the ricci curvature of the discrete hypercube? ∗

Abstract: We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube $\{0,1\}^N$. While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm and the second author could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, including a curved Brunn--Minkowski inequality on the discrete hypercube.

Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

Abstract: The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.

The inflationary bispectrum with curved field-space

Abstract: We compute the covariant three-point function near horizon-crossing for a system of slowly-rolling scalar fields during an inflationary epoch, allowing for an arbitrary field-space metric. We show explicitly how to compute its subsequent evolution using a covariantized version of the separate universe or "delta-N" expansion, which must be augmented by terms measuring curvature of the field-space manifold, and give the nonlinear gauge transformation to the comoving curvature perturbation. Nonlinearities induced by the field-space curvature terms are a new and potentially significant source of non-Gaussianity. We show how inflationary models with non-minimal coupling to the spacetime Ricci scalar can be accommodated within this framework. This yields a simple toolkit allowing the bispectrum to be computed in models with non-negligible field-space curvature.

Flat tori in three-dimensional space and convex integration

Abstract: It is well-known that the curvature tensor is an isometric invariant of C2 Riemannian manifolds. This invariant is at the origin of the rigidity observed in Riemannian geometry. In the mid 1950s, Nash amazed the world mathematical community by showing that this rigidity breaks down in regularity C1. This unexpected flexibility has many paradoxical consequences, one of them is the existence of C1 isometric embeddings of flat tori into Euclidean three-dimensional space. In the 1970s and 1980s, M. Gromov, revisiting Nash’s results introduced convex integration theory offering a general framework to solve this type of geometric problems. In this research, we convert convex integration theory into an algorithm that produces isometric maps of flat tori. We provide an implementation of a convex integration process leading to images of an embedding of a flat torus. The resulting surface reveals a C1 fractal structure: Although the tangent plane is defined everywhere, the normal vector exhibits a fractal behavior. Isometric embeddings of flat tori may thus appear as a geometric occurrence of a structure that is simultaneously C1 and fractal. Beyond these results, our implementation demonstrates that convex integration, a theory still confined to specialists, can produce computationally tractable solutions of partial differential relations.

Riemann compatible tensors

Abstract: Derdzinski and Shen’s theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new “Codazzi deviation tensor”, with a geometric significance. The general properties are studied, with their implications on Pontryagin forms. Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mappings. Compatibility is extended to generalized curvature tensors, with an application to Weyl’s tensor and general relativity.

Bianchi identities for non‐geometric fluxes from quasi‐Poisson structures to Courant algebroids

Abstract: Starting from a (non-associative) quasi-Poisson structure, the derivation of a Roytenberg-type algebra is presented. From the Jacobi identities of the latter, the most general form of Bianchi identities for fluxes (H, f, Q, R) is then derived. It is also explained how this approach is related to the mathematical theory of quasi-Lie and Courant algebroids.

Conformal couplings of a scalar field to higher curvature terms

Abstract: We present a simple way of constructing conformal couplings of a scalar field to higher order Euler densities. This is done by constructing a four-rank tensor involving the curvature and derivatives of the field, which transforms covariantly under local Weyl rescalings. The equation of motion for the field and its energy–momentum tensor are shown to be of second order. The field equations for the spherically symmetric ansatz are integrated, and for generic non-homogeneous couplings, the solution is given in terms of a polynomial equation, in close analogy with Lovelock theories.

Heterotic domain wall solutions and SU(3) structure manifolds

Abstract: We examine compactifications of heterotic string theory on manifolds with SU(3) structure. In particular, we study $ \mathcal{N} = {{1} \left/ {2} \right.} $ domain wall solutions which correspond to the perturbative vacua of the 4D, $ \mathcal{N} = 1 $ supersymmetric theories associated to these compactifications. We extend work which has appeared previously in the literature in two important regards. Firstly, we include two additional fluxes which have been, heretofore, omitted in the general analysis of this situation. This allows for solutions with more general torsion classes than have previously been found. Secondly, we provide explicit solutions for the fluxes as a function of the torsion classes. These solutions are particularly useful in deciding whether equations such as the Bianchi identities can be solved, in addition to the Killing spinor equations themselves. Our work can be used to straightforwardly decide whether any given SU(3) structure on a six-dimensional manifold is associated with a solution to heterotic string theory. To illustrate how to use these results, we discuss a number of examples taken from the literature.

Gravitational radiation reaction and second-order perturbation theory

Abstract: A point particle of small mass m moves in free fall through a background vacuum spacetime metric g_ab and creates a first-order metric perturbation h^1ret_ab that diverges at the particle. Elementary expressions are known for the singular m/r part of h^1ret_ab and for its tidal distortion determined by the Riemann tensor in a neighborhood of m. Subtracting this singular part h^1S_ab from h^1ret_ab leaves a regular remainder h^1R_ab. The self-force on the particle from its own gravitational field adjusts the world line at O(m) to be a geodesic of g_ab+h^1R_ab. The generalization of this description to second-order perturbations is developed and results in a wave equation governing the second-order h^2ret_ab with a source that has an O(m^2) contribution from the stress-energy tensor of m added to a term quadratic in h^1ret_ab. Second-order self-force analysis is similar to that at first order: The second-order singular field h^2S_ab subtracted from h^2ret_ab yields the regular remainder h^2R_ab, and the second-order self-force is then revealed as geodesic motion of m in the metric g_ab+h^1R+h^2R.