Showing papers on "Riemann curvature tensor published in 2015"

Generalized energy conditions in Extended Theories of Gravity

Abstract: In this work, we consider the further degrees of freedom related to curvature invariants and scalar fields in extended theories of gravity (ETG) These new degrees of freedom can be recast as ``effective fluids'' that differ in nature with respect to the standard matter fluids generally adopted as sources of the field equations It is, thus, somewhat misleading to apply the standard general relativistic energy conditions to this effective energy-momentum tensor, as the latter contains the matter content and a geometrical quantity, which arises from the specific ETG considered Here we explore this subtlety, extending our previous work, in particular, to cases with the contracted Bianchi identities with diffeomorphism invariance and to cases with generalized explicit curvature-matter couplings, which imply the nonconservation of the energy-momentum tensor Furthermore, we apply the analysis to specific ETGs, such as scalar-tensor gravity and $f(R)$ gravity Thus, in the context of ETGs, interesting results appear such as matter that may exhibit unusual thermodynamical features, for instance, gravity that retains its attractive character in the presence of large negative pressures; or alternatively, we verify that repulsive gravity may occur for standard matter

Generalized quadratic curvature, non-local infrared modifications of gravity and Newtonian potentials

Abstract: Metric theories of gravity are studied, beginning with a general action that is quadratic in curvature and allows infinite inverse powers of the d'Alembertian operator, resulting in infrared non-local extensions of general relativity. The field equations are derived in full generality and their consistency is checked by verifying the Bianchi identities. The weak-field limit is computed and a straightforward algorithm is presented to infer the post-Newtonian corrections directly from the action. We then apply this to various infrared gravity models including non-local $Rf(R/ \Box)$ cosmology and non-local dark energy and massive gravity models. Generically the Newtonian potentials are not identical and deviate from the $1/r$ behaviour at large distances. However, the former does not occur in a specific class of theories that does not introduce additional degrees of freedom in flat spacetime. A new nonlocal model within this class is proposed, defined by the exponential of the inverse d'Alembertian. This model exhibits novel features, such as weakening of the gravity in the infrared, suggesting de-gravitation of the cosmological constant.

Conformal supergravity in five dimensions: new approach and applications

Abstract: We develop a new off-shell formulation for five-dimensional (5D) conformal supergravity obtained by gauging the 5D superconformal algebra in superspace. An important property of the conformal superspace introduced is that it reduces to the super-conformal tensor calculus (formulated in the early 2000’s) upon gauging away a number of superfluous fields. On the other hand, a different gauge fixing reduces our formulation to the SU(2) superspace of arXiv:0802.3953, which is suitable to describe the most general off-shell supergravity-matter couplings. Using the conformal superspace approach, we show how to reproduce practically all off-shell constructions derived so far, including he supersymmetric extensions of R 2 terms, thus demonstrating the power of our formulation. Furthermore, we construct for the first time a supersymmetric completion of the Ricci tensor squared term using the standard Weyl multiplet coupled to an off-shell vector multiplet. In addition, we present several procedures to generate higher-order off-shell invariants in supergravity, including higher-derivative ones. The covariant projective multiplets proposed in arXiv:0802.3953 are lifted to conformal superspace, and a manifestly superconformal action principle is given. We also introduce unconstrained prepotentials for the vector multiplet, the $$ \mathcal{O}(2) $$ multiplet (i.e., the linear multiplet without central charge) and $$ \mathcal{O}\left(4+n\right) $$ multiplets, with n = 0, 1, . . . Superform formulations are given for the BF action and the non-abelian Chern-Simons action. Finally, we describe locally supersymmetric theories with gauged central charge in conformal superspace.

Exact solutions and spacetime singularities in nonlocal gravity

Abstract: We hereby study exact solutions in a wide range of local higher-derivative and weakly nonlocal gravitational theories. In particular, we give a list of exact classical solutions for two classes of gravitational theories both weakly nonlocal, unitary, and super-renormalizable (or finite) at quantum level. We prove that maximally symmetric spacetimes are exact solutions in both classes, while in dimension higher than four we can also have Anti-de Sitter solutions in the presence of positive cosmological constant. It is explicitly shown under which conditions flat and Ricci-flat spacetimes are exact solutions of the equation of motion (EOM) for the first class of theories not involving the Weyl tensor in the action. We find that the well-known physical spacetimes like Schwarzschild, Kerr, (Anti-) de Sitter serve as solutions for standard matter content, when the EOM does not contain the Riemann tensor alone (operators made out of only the Riemann tensor.) We pedagogically show how to obtain these exact solutions. Furthermore, for the second class of gravity theories, with terms in the Lagrangian written using Weyl tensors, the Friedmann-Robertson-Walker (FRW) spacetimes are also exact solutions (exactly in the same way like in Einstein theory), when the matter content is given by conformal matter (radiation). We also comment on rather inevitable presence and universality of singularities and possible resolution of them in finite and conformally invariant theories. “Delocalization” is proposed as a way to solve the black hole singularity problem in the first class. In order to solve the problem of cosmological singularities in the second class, it seems crucial to have a conformally invariant or asymptotically free quantum gravitational theory.

Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

Abstract: In the recent articles Paternain et al. (J. Differ Geom, 98:147–181, 2014, Invent Math 193:229–247, 2013), a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under the geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation.

Scattering amplitudes in super-renormalizable gravity

Abstract: We explicitly compute the tree-level on-shell four-graviton amplitudes in four, five and six dimensions for local and weakly nonlocal gravitational theories that are quadratic in both, the Ricci and scalar curvature with form factors of the d’Alembertian operator inserted between. More specifically we are interested in renormalizable, super-renormalizable or finite theories. The scattering amplitudes for these theories turn out to be the same as the ones of Einstein gravity regardless of the explicit form of the form factors. As a special case the four-graviton scattering amplitudes in Weyl conformal gravity are identically zero. Using a field redefinition, we prove that the outcome is correct for any number of external gravitons (on-shell n−point functions) and in any dimension for a large class of theories. However, when an operator quadratic in the Riemann tensor is added in any dimension (with the exception of the Gauss-Bonnet term in four dimensions) the result is completely altered, and the scattering amplitudes depend on all the form factors introduced in the action.

f-Minimal Surface and Manifold with Positive m-Bakry–Émery Ricci Curvature

Abstract: In this paper, we first prove a compactness theorem for the space of closed embedded f-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry–Emery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the f-Laplacian on a compact manifold with positive m-Bakry–Emery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the n-sphere, or the n-dimensional hemisphere. Finally, for a compact manifold with positive m-Bakry–Emery Ricci curvature and f-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if and only if the manifold is isometric to a Euclidean ball.

E 8 geometry

Abstract: We investigate exceptional generalised diffeomorphisms based on E (8(8)) in a geometric setting. The transformations include gauge transformations for the dual gravity field. The surprising key result, which allows for a development of a tensor formalism, is that it is possible to define field-dependent transformations containing connection, which are covariant. We solve for the spin connection and construct a curvature tensor. A geometry for the Ehlers symmetry SL(n + 1) is sketched. Some related issues are discussed.

Singular hermitian metrics on holomorphic vector bundles

Abstract: We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Paun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature Θh as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to Θh. Finally we define what it means for h to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.

Moduli Spaces of Riemannian Metrics

Abstract: Part I: Positive scalar curvature.- The (moduli) space of all Riemannian metrics.- Clifford algebras and spin.- Dirac operators and index theorems.- Early results on the space of positive scalar curvature metrics.- Kreck-Stolz invariants.- Applications of Kreck-Stolz invariants.- The eta invariant and applications.- The case of dimensions 2 and 3.- The observer moduli space and applications.- Other topological structures.- Negative scalar and Ricci curvature.- Part II: Sectional curvature.- Moduli spaces of compact manifolds with positive or non-negative sectional curvature.- Moduli spaces of compact manifolds with negative and non-positive sectional curvature.- Moduli spaces of non-compact manifolds with non-negative sectional curvature.- Positive pinching and the Klingenberg-Sakai conjecture.

Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature

Abstract: In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat equation. Based on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem, which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time. Using the backward pseudolocality theorem, we next establish a uniform $L^2$ curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity. We also obtain a stronger $\varepsilon$-regularity theorem for Ricci flows. This result is particularly useful in the study of Kahler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results.

Jacobi stability analysis of the Lorenz system

Abstract: We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi–Cartan–Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the turbulence, also representing a nontrivial testing object for studying nonlinear effects. The KCC theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach, we describe the evolution of the Lorenz system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a nonlinear connection and a Berwald type connection, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In order to apply the KCC theory, we reformulate the Lorenz system as a set of two second-order nonlinear differential equations. The geometric invariants associated to this system (nonlinear and Berwald connections), and the deviation curvature tensor, as well as its eigenvalues, are explicitly obtained. The Jacobi stability of the equilibrium points of the Lorenz system is studied, and the condition of the stability of the equilibrium points is obtained. Finally, we consider the time evolution of the components of the deviation vector near the equilibrium points.

On Lovelock analogues of the Riemann tensor

Abstract: It is possible to define an analogue of the Riemann tensor for $N$th order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative yields the corresponding analogue of the Einstein tensor. Interestingly there exist two parallel but distinct such analogues and the main purpose of this note is to reconcile both these formulations. In addition we will show that any pure Lovelock vacuum in odd $d = 2N + 1$ dimensions is Lovelock flat, i.e. any vacuum solution of the theory has vanishing Lovelock-Riemann tensor. Further, in presence of cosmological constant it is the Lovelock-Weyl tensor that vanishes.

A discerning gravitational property for gravitational equ ation in higher dimensions

Abstract: It is well-known that Einstein gravity is kinematic (no non-trivial vacuum solution;i.e. Riemann vanishes whenever Ricci does so) in 3 dimension because Riemann is entirely given in terms of Ricci. Could this property be universalized for all odd dimensions in a generalized theory? The answer is yes, and this property uniquely singles out pure Lovelock (it has only one Nth order term in action) gravity for which Nth order Lovelock Riemann tensor is indeed given in terms of corresponding Ricci for all odd d = 2N+ 1 dimensions. This feature of gravity is realized only in higher dimensions and it uniquely picks out pure Lovelock gravity from all other generalizations of Einstein gravity. It serves as a good discerning and guidi ng criterion for gravitational equation in higher dimensions.

Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness

Abstract: We develop foundational theory for the Laplacian flow for closed G_2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on $\Lambda(x,t)=\left(| abla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on all covariant derivatives of Rm and T. (2). We show that $\Lambda(x,t)$ will blow up at a finite-time singularity, so the flow will exist as long as $\Lambda(x,t)$ remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2). (5). Finally, we study compact soliton solutions of the Laplacian flow.

Massive graviton on arbitrary background: derivation, syzygies, applications

Abstract: We give the detailed derivation of the fully covariant form of the quadratic action and the derived linear equations of motion for a massive graviton in an arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]). Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free massive gravities and using a simple model of this family, we are able to express this action and these equations of motion in terms of a single metric in which the graviton propagates, hence removing in particular the need for a “reference metric' which is present in the non perturbative formulation. We show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. This last constraint involves powers and combinations of the curvature of the background metric. The 5 constraints are obtained for a background metric which is unconstrained, i.e. which does not have to obey the background field equations. We then apply these results to the case of Einstein space-times, where we show that the 5 constraints become trivial, and Friedmann-Lemaitre-Robertson-Walker space-times, for which we correct in particular some results that appeared elsewhere. To reachmore » our results, we derive several non trivial identities, syzygies, involving the graviton fields, its derivatives and the background metric curvature. These identities have their own interest. We also discover that there exist backgrounds for which the dRGT equations cannot be unambiguously linearized.« less

Instability of meridional axial system in f( R) gravity

Abstract: We analyze the dynamical instability of a non-static reflection axial stellar structure by taking into account the generalized Euler equation in metric f(R) gravity. Such an equation is obtained by contracting the Bianchi identities of the usual anisotropic and effective stress-energy tensors, which after using a radial perturbation technique gives a modified collapse equation. In the realm of the \(R+\epsilon R^n\) gravity model, we investigate instability constraints at Newtonian and post-Newtonian approximations. We find that the instability of a meridional axial self-gravitating system depends upon the static profile of the structure coefficients, while f(R) extra curvature terms induce the stability of the evolving celestial body.

Sectional curvature for Riemannian manifolds with density

Abstract: In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of Cartan–Hadamard, Synge, and Bonnet–Myers as well as a generalization of the (non-smooth) 1/4-pinched sphere theorem. The main idea is to modify the radial curvature equation and second variation formula and then apply the techniques of classical Riemannian geometry to these new equations.

How to lift positive Ricci curvature

Abstract: Remark Various definitions of lower Ricci curvature bounds on metric spaces are proposed in Kuwae and Shioya [23], Lott and Villani [25], Ohta [28], Sturm [40; 41] and Zhang and Zhu [50]. Our proof only requires that the quotient space of the principal orbits, M reg=G , has Ricci curvature greater than or equal to 1, and since M reg=G is a Riemannian manifold, it does not matter which definition we choose.

On projective K\"ahler manifolds of partially positive curvature and rational connectedness

Abstract: In a previous paper, we proved that a projective Kahler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollar-Miyaoka-Mori that a projective Kahler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of $k$-positive Ricci curvature and prove that for a projective $n$-dimensional Kahler manifold of $k$-positive Ricci curvature the MRC fibration has generic fibers of dimension at least $n-k+1$. We also establish an analogous result for projective Kahler manifolds of semi-positive holomorphic sectional curvature based on an invariant which records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. In particular, the latter result confirms a conjecture of S.-T. Yau in the projective case.

Proof of the averaged null energy condition in a classical curved spacetime using a null-projected quantum inequality

Abstract: Quantum inequalities are constraints on how negative the weighted average of the renormalized stress-energy tensor of a quantum field can be. A null-projected quantum inequality can be used to prove the averaged null energy condition (ANEC), which would then rule out exotic phenomena such as wormholes and time machines. In this work we derive such an inequality for a massless minimally coupled scalar field, working to first order of the Riemann tensor and its derivatives. We then use this inequality to prove ANEC on achronal geodesics in a curved background that obeys the null convergence condition.

Scattering amplitudes in super-renormalizable gravity

Abstract: We explicitly compute the tree-level on-shell four-graviton amplitudes in four, five and six dimensions for local and weakly nonlocal gravitational theories that are quadratic in both, the Ricci and scalar curvature with form factors of the d'Alembertian operator inserted between. More specifically we are interested in renormalizable, super-renormalizable or finite theories. The scattering amplitudes for these theories turn out to be the same as the ones of Einstein gravity regardless of the explicit form of the form factors. As a special case the four-graviton scattering amplitudes in Weyl conformal gravity are identically zero. Using a field redefinition, we prove that the outcome is correct for any number of external gravitons (on-shell $n-$point functions) and in any dimension for a large class of theories. However, when an operator quadratic in the Riemann tensor is added in any dimension (with the exception of the Gauss-Bonnet term in four dimensions) the result is completely altered, and the scattering amplitudes depend on all the form factors introduced in the action.

Stability of the effective potential of the gauge-less top-Higgs model in curved spacetime

Abstract: We investigate stability of the Higgs effective potential in curved spacetime. To this end, we consider the gauge-less top-Higgs sector with an additional scalar field. Explicit form of the terms proportional to the squares of the Ricci scalar, the Ricci tensor and the Riemann tensor that arise at the one-loop level in the effective action has been determined. We have investigated the influence of these terms on the stability of the scalar effective potential. The result depends on background geometry. In general, the potential becomes modified both in the region of the electroweak minimum and in the region of large field strength.

Multi-disformal invariance of nonlinear primordial perturbations

Abstract: We study disformal transformations of the metric in the cosmological context. We first consider the disformal transformation generated by a scalar field $\phi$ and show that the curvature and tensor perturbations on the uniform $\phi$ slicing, on which the scalar field is homogeneous, are non-linearly invariant under the disformal transformation. Then we discuss the transformation properties of the evolution equations for the curvature and tensor perturbations at full non-linear order in the context of spatial gradient expansion as well as at linear order. In particular, we show that the transformation can be described in two typically different ways: one that clearly shows the physical invariance and the other that shows an apparent change of the causal structure. Finally we consider a new type of disformal transformation in which a multi-component scalar field comes into play, which we call a "multi-disformal transformation". We show that the curvature and tensor perturbations are invariant at linear order, and also at non-linear order provided that the system has reached the adiabatic limit.

Avoidance of singularities in asymptotically safe Quantum Einstein Gravity

Abstract: New general spherically symmetric solutions have been derived with a cosmological “constant” Λ as a source. This Λ term is not constant but it satisfies the properties of the asymptotically safe gravity at the ultraviolet fixed point. The importance of these solutions comes from the fact that they may describe the near to the centre region of black hole spacetimes as this is modified by the Renormalization Group scaling behaviour of the fields. The consistent set of field equations which respect the Bianchi identities is derived and solved. One of the solutions (with conventional sign of temporal-radial metric components) is timelike geodesically complete, and although there is still a curvature divergent origin, this is never approachable by an infalling massive particle which is reflected at a finite distance due to the repulsive origin. Another family of solutions (of both signatures) range from a finite radius outwards, they cannot be extended to the centre of spherical symmetry, and the curvature invariants are finite at the minimum radius.

Some rigidity results on critical metrics for quadratic functionals

Abstract: In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some curvature conditions, a partial converse is true. In particular, for a class of quadratic curvature functionals, we prove that every critical metric with non-negative sectional curvature must be Einstein.

Stability of the effective potential of the gauge-less top-Higgs model in curved spacetime

Abstract: We investigate stability of the Higgs effective potential in curved spacetime. To this end, we consider the gauge-less top-Higgs sector with an additional scalar field. Explicit form of the terms proportional to the squares of the Ricci scalar, the Ricci tensor and the Riemann tensor that arise at the one-loop level in the effective action has been determined. We have investigated the influence of these terms on the stability of the scalar effective potential. The result depends on background geometry. In general, the potential becomes modified both in the region of the electroweak minimum and in the region of large field strength.