Showing papers on "Riemann curvature tensor published in 2017"

Aspects of general higher-order gravities

Abstract: We study several aspects of higher-order gravities constructed from general contractions of the Riemann tensor and the metric in arbitrary dimensions First, we use the fast-linearization procedure presented in [P Bueno and P A Cano, arXiv:160706463] to obtain the equations satisfied by the metric perturbation modes on a maximally symmetric background in the presence of matter and to classify L(Riemann) theories according to their spectrum Then, we linearize all theories up to quartic order in curvature and use this result to construct quartic versions of Einsteinian cubic gravity In addition, we show that the most general cubic gravity constructed in a dimension-independent way and which does not propagate the ghostlike spin-2 mode (but can propagate the scalar) is a linear combination of f(Lovelock) invariants, plus the Einsteinian cubic gravity term, plus a new ghost-free gravity term Next, we construct the generalized Newton potential and the post-Newtonian parameter γ for general L(Riemann) gravities in arbitrary dimensions, unveiling some interesting differences with respect to the four-dimensional case We also study the emission and propagation of gravitational radiation from sources for these theories in four dimensions, providing a generalized formula for the power emitted Finally, we review Wald’s formalism for general L(Riemann) theories and construct new explicit expressions for the relevant quantities involved Many examples illustrate our calculations

Soft gravitons and the memory effect for plane gravitational waves

Abstract: The “gravitational memory effect” due to an exact plane wave provides us with an elementary description of the diffeomorphisms associated with the analogue of “soft gravitons for this nonasymptotically flat system. We explain how the presence of the latter may be detected by observing the motion of freely falling particles or other forms of gravitational wave detection. Numerical calculations confirm the relevance of the first, second and third time integrals of the Riemann tensor pointed out earlier. Solutions for various profiles are constructed. It is also shown how to extend our treatment to Einstein-Maxwell plane waves and a midisuperspace quantization is given.

An effective formalism for testing extensions to General Relativity with gravitational waves

Abstract: The recent direct observation of gravitational waves (GW) from merging black holes opens up the possibility of exploring the theory of gravity in the strong regime at an unprecedented level. It is therefore interesting to explore which extensions to General Relativity (GR) could be detected. We construct an Effective Field Theory (EFT) satisfying the following requirements. It is testable with GW observations; it is consistent with other experiments, including short distance tests of GR; it agrees with widely accepted principles of physics, such as locality, causality and unitarity; and it does not involve new light degrees of freedom. The most general theory satisfying these requirements corresponds to adding to the GR Lagrangian operators constructed out of powers of the Riemann tensor, suppressed by a scale comparable to the curvature of the observed merging binaries. The presence of these operators modifies the gravitational potential between the compact objects, as well as their effective mass and current quadrupoles, ultimately correcting the waveform of the emitted GW.

Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness

Abstract: We develop foundational theory for the Laplacian flow for closed G2 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) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

Black holes in vector-tensor theories

Abstract: We study static and spherically symmetric black hole (BH) solutions in second-order generalized Proca theories with nonminimal vector field derivative couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann tensor. We find concrete Lagrangians which give rise to exact BH solutions by imposing two conditions of the two identical metric components and the constant norm of the vector field. These exact solutions are described by either Reissner-Nordstrom (RN), stealth Schwarzschild, or extremal RN solutions with a non-trivial longitudinal mode of the vector field. We then numerically construct BH solutions without imposing these conditions. For cubic and quartic Lagrangians with power-law couplings which encompass vector Galileons as the specific cases, we show the existence of BH solutions with the difference between two non-trivial metric components. The quintic-order power-law couplings do not give rise to non-trivial BH solutions regular throughout the horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead to BH solutions with a secondary hair. For all the solutions, the vector field is regular at least at the future or past horizon. The deviation from General Relativity induced by the Proca hair can be potentially tested by future measurements of gravitational waves in the nonlinear regime of gravity.

Dynamical variables and evolution of the universe

Abstract: One of the striking feature of inhomogeneous matter distribution under the effects of fourth-order gravity and electromagnetic field have been discussed in this manuscript. We have considered a compact spherical celestial star undergoing expansion due to the presence of higher curvature invariants of f(R) gravity and imperfect fluid. We have explored the dynamical equations and field equations in f(R) gravity. An explicit expression have been found for Weyl tensor and material variables under the dark dynamical effects. Using a viable f(R) model, some dynamical variables have been explored from splitting the Riemann curvature tensor. These dark dynamical variables are also studied for charged dust cloud with and without the constraint of constant Ricci scalar.

Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds

Abstract: We give a systematic derivation of the local expressions of the NS H-flux, geometric F- as well as non-geometric Q- and R-fluxes in terms of bivector β- and two-form B-potentials including vielbeins. They are obtained using a supergeometric method on QP-manifolds by twist of the standard Courant algebroid on the generalized tangent space without flux. Bianchi identities of the fluxes are easily deduced. We extend the discussion to the case of the double space and present a formulation of T-duality in terms of canonical transformations between graded symplectic manifolds. Thus, we find a unified description of geometric as well as non-geometric fluxes and T-duality transformations in double field theory. Finally, the construction is compared to the formerly introduced Poisson Courant algebroid, a Courant algebroid on a Poisson manifold, as a model for R-flux.

New torsion black hole solutions in Poincaré gauge theory

Abstract: We derive a new exact static and spherically symmetric vacuum solution in the framework of the Poincare gauge field theory with dynamical massless torsion. This theory is built in such a form that allows to recover General Relativity when the first Bianchi identity of the model is fulfilled by the total curvature. The solution shows a Reissner-Nordstrom type geometry with a Coulomb-like curvature provided by the torsion field. It is also shown the existence of a generalized Reissner-Nordstrom-de Sitter solution when additional electromagnetic fields and/or a cosmological constant are coupled to gravity.

Curvature aspects of graphs

Abstract: We prove the Lichnerowicz type lower bound estimates for finite connected graphs with positive Ricci curvature lower bound.

Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials

Abstract: In this paper, using consistent couple stress theory and Hamilton\'s principle, the free vibration analysis of Euler- Bernoulli nano-beams made of bi-directional functionally graded materials (BDFGMs) with small scale effects are investigated. To the best of the researchers\' knowledge, in the literature, there is no study carried out into consistent couple-stress theory for free vibration analysis of BDFGM nanostructures with arbitrary functions. In addition, in order to obtain small scale effects, the consistent couple-stress theory is also applied. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson\'s ratio are assumed to be graded in both axial and thickness directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of Hamilton principle. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of BDFG nano-beam. At the end, some numerical results are presented to study the effects of material length scale parameter, and inhomogeneity constant on natural frequency.

Beyond E 11

Abstract: We study the non-linear realisation of E 11 originally proposed by West with particular emphasis on the issue of linearised gauge invariance. Our analysis shows even at low levels that the conjectured equations can only be invariant under local gauge transformations if a certain section condition that has appeared in a different context in the E 11 literature is satisfied. This section condition also generalises the one known from exceptional field theory. Even with the section condition, the E 11 duality equation for gravity is known to miss the trace component of the spin connection. We propose an extended scheme based on an infinite-dimensional Lie superalgebra, called the tensor hierarchy algebra, that incorporates the section condition and resolves the above issue. The tensor hierarchy algebra defines a generalised differential complex, which provides a systematic description of gauge invariance and Bianchi identities. It furthermore provides an E 11 representation for the field strengths, for which we define a twisted first order self-duality equation underlying the dynamics.

Ricci flow with surgery on manifolds with positive isotropic curvature

Abstract: We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension $3$. In the second part, we develop a theory of ancient solutions which have bounded curvature; are $\kappa$-noncollapsed; are weakly PIC2; and are uniformly PIC. This is an extension of Perelman's work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly PIC1. In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension $n \geq 12$. This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman's surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension $n \geq 12$ which do not contain non-trivial incompressible $(n-1)$-dimensional space forms.

Heat Kernel Upper Bound on Riemannian Manifolds with Locally Uniform Ricci Curvature Integral Bounds

Abstract: This article shows that under locally uniformly integral bounds of the negative part of Ricci curvature, the heat kernel admits a Gaussian upper bound for small times. This provides general assumptions on the geometry of a manifold such that certain function spaces are in the Kato class. Additionally, the results imply bounds on the first Betti number.

Quasi-Local Mass Integrals and the Total Mass

Abstract: On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown–York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.

Gluing Eguchi-Hanson Metrics and a Question of Page

Abstract: In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kahler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with 16 orbifold points and resolves the orbifold singularities by gluing in 16 small Eguchi-Hanson manifolds that all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi-Hanson manifolds. This is a subtle question: if successful, this construction would produce Einstein metrics that are neither Kahler nor self-dual. In this paper, we focus on a configuration of maximal symmetry involving eight small Eguchi-Hanson manifolds of each orientation that are arranged according to a chessboard pattern. By analyzing the interactions between Eguchi-Hanson manifolds with opposite orientation, we identify a nonvanishing obstruction to the gluing problem, thereby destroying any hope of producing a metric of zero Ricci curvature in this way. Using this obstruction, we are able to understand the dynamics of such metrics under Ricci flow as long as the Eguchi-Hanson manifolds remain small. In particular, for the configuration described above, we obtain an ancient solution to the Ricci flow with the property that the maximum of the Riemann curvature tensor blows up at a rate of (−t)1/2, while the maximum of the Ricci curvature converges to 0. © 2016 Wiley Periodicals, Inc.

Tensor Completion with Side Information: A Riemannian Manifold Approach

Abstract: By restricting the iterate on a nonlinear manifold, the recently proposed Riemannian optimization methods prove to be both efficient and effective in low rank tensor completion problems. However, existing methods fail to exploit the easily accessible side information, due to their format mismatch. Consequently, there is still room for improvement. To fill the gap, in this paper, a novel Riemannian model is proposed to tightly integrate the original model and the side information by overcoming their inconsistency. For this model, an efficient Riemannian conjugate gradient descent solver is devised based on a new metric that captures the curvature of the objective. Numerical experiments suggest that our method is more accurate than the stateof-the-art without compromising the efficiency.

Black holes in vector-tensor theories

Abstract: We study static and spherically symmetric black hole (BH) solutions in second-order generalized Proca theories with nonminimal vector field derivative couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann tensor. We find concrete Lagrangians which give rise to exact BH solutions by imposing two conditions of the two identical metric components and the constant norm of the vector field. These exact solutions are described by either Reissner-Nordstrom (RN), stealth Schwarzschild, or extremal RN solutions with a non-trivial longitudinal mode of the vector field. We then numerically construct BH solutions without imposing these conditions. For cubic and quartic Lagrangians with power-law couplings which encompass vector Galileons as the specific cases, we show the existence of BH solutions with the difference between two non-trivial metric components. The quintic-order power-law couplings do not give rise to non-trivial BH solutions regular throughout the horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead to BH solutions with a secondary hair. For all the solutions, the vector field is regular at least at the future or past horizon. The deviation from General Relativity induced by the Proca hair can be potentially tested by future measurements of gravitational waves in the nonlinear regime of gravity.

How to obtain a cosmological constant from small exotic R^4

Abstract: In this paper we determine the cosmological constant as a topological invariant by applying certain techniques from low dimensional differential topology. We work with a small exotic $R^{4}$ which is embedded into the standard $\mathbb{R}^{4}$. Any exotic $R^4$ is a Riemannian smooth manifold with necessary non-vanishing curvature tensor. To determine the invariant part of such curvature we deal with a canonical construction of $R^4$ where it appears as a part of the complex surface $K3\#\overline{CP(2)}$. Such $R^{4}$'s admit hyperbolic geometry. This fact simplifies significantly the calculations and enforces the rigidity of the expressions. In particular, we explain the smallness of the cosmological constant with a value consisting of a combination of (natural) topological invariant. Finally, the cosmological constant appears to be a topologically supported quantity.

Sachs' free data in real connection variables

Abstract: We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs’ constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the first-order formulation is that Sachs’ propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null infinity in terms of a shear of a (non-geodesic) asymptotic null vector field in the physical spacetime. Finally, we compute the modification to the spin coefficients and the null congruence in the presence of torsion.

Equivalent and inequivalent canonical structures of higher order theories of gravity

Abstract: The canonical formulation of higher-order theories of gravity can only be accomplished by introducing additional degrees of freedom, namely, the extrinsic curvature tensor ${K}_{ij}$. Consequently, to match Cauchy data with the boundary data, terms in addition to the three-space metric ${h}_{ij}$ must also be fixed at the boundary. While in the Ostrogradsky, Dirac, and Horowitz formalisms the extrinsic curvature tensor is kept fixed at the boundary, a modified Horowitz formalism fixes the Ricci scalar $R$ instead. It has been taken for granted that the Hamiltonian structures corresponding to all of the formalisms with different end-point data are either the same or are canonically equivalent. In the present study, we show that this indeed is true, but only for a class of higher-order theories. However, for more general higher-order theories---e.g., dilatonic coupled Gauss-Bonnet gravity in the presence of a curvature-squared term---the Hamiltonian obtained following the 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 only the modified Horowitz' formalism can produce a viable quantum description of the theory, since it only admits a classical analogue under an appropriate semiclassical approximation. Thus, fixing the Ricci scalar $R$ at the boundary appears to be a fundamental issue for a canonical formulation of higher-order theories of gravity.

Logamediate Inflation in f(T) Teleparallel Gravity

Abstract: We study logamediate inflation in the context of $f(T)$ teleparallel gravity. $f(T)$-gravity is a generalization of the teleparallel gravity which is formulated on the Weitzenbock spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor. We consider an $f(T)$-gravity model which is sourced by a canonical scalar field. Assuming a power-law $f(T)$ function in the action, we investigate an inflationary universe with a logamediate scale factor. Our results show that, although logamediate inflation is completely ruled out by observational data in the standard inflationary scenario based on Einstein gravity, it can be compatible with the 68\% confidence limit joint region of Planck 2015 TT,TE,EE+lowP data in the framework of $f(T)$-gravity.

Geometric aspects of extensions of hom-Lie superalgebras

Abstract: In this paper, we deal with (non-abelian) extensions of a given hom-Lie superalgebra and find a cohomological obstacle to the existence of extensions of hom-Lie superalgebras. Moreover, the setting of covariant exterior derivatives, super connection, curvature and the Bianchi identity in differential geometry has been studied.

Einstein manifolds with finite Lp-norm of the Weyl curvature☆

Abstract: Let ( M n , g ) ( n ≥ 4 ) be an n-dimensional complete Einstein manifold. Denote by W the Weyl curvature tensor of M. We prove that ( M n , g ) is isometric to a spherical space form if ( M n , g ) has positive scalar curvature and unit volume, and the L p ( p ≥ n 2 ) -norm of W is pinched in [ 0 , C ) , where C is an explicit positive constant depending only on n, p and S, which improves the isolation theorems given by [24] , [14] , [17] . This paper also states that W goes to zero uniformly at infinity if for p ≥ n 2 , the L p -norm of W of M with non-positive scalar curvature and positive Yamabe constant is finite. Assume that M has negative scalar curvature and the L α -norm of W is finite. As application, we prove that M is a hyperbolic space form if the L p -norm of W is sufficiently small, which generalizes an L n 2 -norm of W pinching theorem in [19] .

Curvature Tensor for the Spacetime of General Relativity

Abstract: In the differential geometry of certain F-structures, the role of W-curvature tensor is very well known. A detailed study of this tensor has been made on the spacetime of general relativity. The spacetimes satisfying Einstein field equations with vanishing W-tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing W-tensor have also been considered. The divergence of W-tensor is studied in detail and it is seen, among other results, that a perfect fluid spacetime with conserved W-tensor represents either an Einstein space or a Friedmann-Robertson-Walker cosmological model.

Complete Conformal Metrics of Negative Ricci Curvature on Euclidean Spaces

Abstract: We show the existence and nonexistence results of complete conformal metrics with prescribed symmetric functions of the eigenvalues of the Ricci tensor defined on negative cones on Euclidean spaces. Our results are sharp under certain decay conditions of the prescribed curvature functions.