The continuous determination of spacetime geometry by the Riemann curvature tensor

1988 ◽  
Vol 5 (5) ◽  
pp. 695-705 ◽  
Author(s):  
A D Rendall
Keyword(s):  
Curvature Tensor ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Spacetime Geometry ◽  
Continuous Determination

scholarly journals Riemann curvature of a boosted spacetime geometry

2016 ◽  
Vol 13 (01) ◽  
pp. 1650002
Author(s):  
Emmanuele Battista ◽  
Giampiero Esposito ◽  
Paolo Scudellaro ◽  
Francesco Tramontano
Keyword(s):  
Curvature Tensor ◽  
Initial Conditions ◽  
Geodesic Equation ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Spacetime Geometry ◽  
Sitter Spacetime ◽  
First Time

The ultrarelativistic boosting procedure had been applied in the literature to map the metric of Schwarzschild–de Sitter spacetime into a metric describing de Sitter spacetime plus a shock-wave singularity located on a null hypersurface. This paper evaluates the Riemann curvature tensor of the boosted Schwarzschild–de Sitter metric by means of numerical calculations, which make it possible to reach the ultrarelativistic regime gradually by letting the boost velocity approach the speed of light. Thus, for the first time in the literature, the singular limit of curvature, through Dirac’s [Formula: see text] distribution and its derivatives, is numerically evaluated for this class of spacetimes. Moreover, the analysis of the Kretschmann invariant and the geodesic equation shows that the spacetime possesses a “scalar curvature singularity” within a 3-sphere and it is possible to define what we here call “boosted horizon”, a sort of elastic wall where all particles are surprisingly pushed away, as numerical analysis demonstrates. This seems to suggest that such “boosted geometries” are ruled by a sort of “antigravity effect” since all geodesics seem to refuse to enter the “boosted horizon” and are “reflected” by it, even though their initial conditions are aimed at driving the particles toward the “boosted horizon” itself. Eventually, the equivalence with the coordinate shift method is invoked in order to demonstrate that all [Formula: see text] terms appearing in the Riemann curvature tensor give vanishing contribution in distributional sense.

scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

scholarly journals Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space

2020 ◽  
Vol 72 (3) ◽  
pp. 427-432
Author(s):  
A. Sarkar ◽  
A. Sil ◽  
A. K. Paul
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Riemann Curvature ◽  
Sasakian Manifolds ◽  
Riemann Curvature Tensor ◽  
Order Parallel

UDC 514.7 The object of the present paper is to study three-dimensional trans-Sasakian manifolds admitting η -Ricci soliton. Actually, we study such manifolds whose Ricci tensor satisfy some special conditions like cyclic parallelity, Ricci semisymmetry, ϕ -Ricci semisymmetry, after reviewing the properties of second order parallel tensors on such manifolds. We determine the form of Riemann curvature tensor of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective spaces. We also give some classification results of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective spaces.

scholarly journals On The Formulation of Bianchi Identity from Action Principle

2021 ◽  
Author(s):  
Shiladittya Debnath
Keyword(s):  
Curvature Tensor ◽  
Basic Property ◽  
Covariant Derivative ◽  
Bianchi Identity ◽  
Action Principle ◽  
Lie Derivative ◽  
General Covariance ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor

Abstract In this letter, we investigate the basic property of the Hilbert-Einstein action principle and its infinitesimal variation under suitable transformation of the metric tensor. We find that for the variation in action to be invariant, it must be a scalar so as to obey the principle of general covariance. From this invariant action principle, we eventually derive the Bianchi identity (where, both the 1st and 2nd forms are been dissolved) by using the Lie derivative and Palatini identity. Finally, from our derived Bianchi identity, splitting it into its components and performing cyclic summation over all the indices, we eventually can derive the covariant derivative of the Riemann curvature tensor. This very formulation was first introduced by S Weinberg in case of a collision less plasma and gravitating system. We derive the Bianchi identity from the action principle via this approach; and hence the name ‘Weinberg formulation of Bianchi identity’.

scholarly journals Dislocation Density Tensor of Thin Elastic Shells at Finite Deformation

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Dislocation Density ◽  
Curvature Tensor ◽  
Finite Deformation ◽  
Middle Surface ◽  
Gauss Curvature ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor ◽  
Elastic Shells ◽  
Dislocation Density Tensor

The dislocation density tensors of thin elastic shells have been formulated explicitly in terms of the Riemann curvature tensor. The formulation reveals that the dislocation density of the shells is  proportional to KA3=2, where K is the Gauss curvature and A is the determinant of metric tensor ofthe middle surface.

scholarly journals Curvature Spinors in Locally Inertial Frame and the Relations with Sedenion

Universe ◽  
2020 ◽  
Vol 6 (3) ◽  
pp. 40
Author(s):  
In Ki Hong ◽  
Choong Sun Kim ◽  
Gyung Hyun Min
Keyword(s):  
Curvature Tensor ◽  
Inertial Frame ◽  
Riemann Curvature ◽  
Tensor Representation ◽  
Riemann Curvature Tensor ◽  
Chiral Structure ◽  
Multiplication Rule ◽  
Rank 2 ◽  
Spinor Formalism

In the 2-spinor formalism, the gravity can be dealt with curvature spinors with four spinor indices. Here we show a new effective method to express the components of curvature spinors in the rank-2 4 × 4 tensor representation for the gravity in a locally inertial frame. In the process we have developed a few manipulating techniques, through which the roles of each component of Riemann curvature tensor are revealed. We define a new algebra ‘sedon’, the structure of which is almost the same as sedenion except for the basis multiplication rule. Finally we also show that curvature spinors can be represented in the sedon form and observe the chiral structure in curvature spinors. A few applications of the sedon representation, which includes the quaternion form of differential Binanchi identity and hand-in-hand couplings of curvature spinors, are also presented.

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