CURVATURES - RICCI TENSOR - BIANCHI IDENTITY - EINSTEIN EQUATIONS

2000 ◽  
pp. 323-330
Keyword(s):  
Ricci Tensor ◽  
Bianchi Identity ◽  
Einstein Equations

scholarly journals (Pseudo) generalized Kaluza–Klein G-spaces and Einstein equations

2014 ◽  
Vol 25 (12) ◽  
pp. 1450116 ◽  
Author(s):  
Constantin M. Arcuş ◽  
Esmaeil Peyghan
Keyword(s):  
Vector Bundle ◽  
Tangent Bundle ◽  
Ricci Tensor ◽  
Tensor Field ◽  
General Framework ◽  
Bianchi Type ◽  
Einstein Equations ◽  
Lie Algebroid ◽  
Linear Connections ◽  
Kaluza Klein

Introducing the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza–Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza–Klein G-spaces and we develop the Einstein equations in this general framework.

POINCARÉ ZERO-MASS REPRESENTATIONS

1994 ◽  
Vol 09 (01) ◽  
pp. 127-156 ◽  
Author(s):  
R. MIRMAN
Keyword(s):  
Gauge Invariance ◽  
Physical Meaning ◽  
Bianchi Identity ◽  
Einstein Equations ◽  
Matrix Elements ◽  
Poincare Group ◽  
Zero Mass ◽  
Finite Dimensional ◽  
Nonlinear Condition ◽  
Nonlinear Representation

The zero-mass, discrete-spin, finite-dimensional representations of the proper Poincaré group are discussed, using the nondecomposable — and so nonunitary — representations of the little group, SE(2); matrix elements are thus arbitrary functions. The physical meaning and significance of the results are emphasized. Matrices are not decomposable, so the bases are connections, not tensors, giving gauge invariance — a partial statement of Poincaré invariance for zero mass only. Gravitation, a zero-mass spin-2 field, obeys a nonlinear condition (unlike the zero-mass spin-1 electromagnetic A), the Bianchi identity, which follows from the nature of Γ, and its integrated form, the Einstein equations, resulting in a curvature of space. Gravitation must be and is nonlinear, and electromagnetism linear, because of restrictions on which massless objects can interact with massive ones, these resulting from the differences in their little groups. The nonlinear representation is equivalent to a curvature of space — which thus can be considered a consequence of nonlinearity.

scholarly journals GRAVITATION, ELECTROMAGNETISM AND THE COSMOLOGICAL CONSTANT IN PURELY AFFINE GRAVITY

2009 ◽  
Vol 18 (05) ◽  
pp. 809-829 ◽  
Author(s):  
NIKODEM J. POPŁAWSKI
Keyword(s):  
General Relativity ◽  
Electromagnetic Field ◽  
Cosmological Constant ◽  
Electromagnetic Fields ◽  
Ricci Tensor ◽  
Einstein Equations ◽  
Metric Tensor ◽  
Affine Metric ◽  
Affine Gravity

The Eddington Lagrangian in the purely affine formulation of general relativity generates the Einstein equations with the cosmological constant. The Ferraris–Kijowski purely affine Lagrangian for the electromagnetic field, which has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the Einstein–Maxwell Lagrangian in the metric formulation. We show that the sum of the two affine Lagrangians is dynamically inequivalent to the sum of the analogous Lagrangians in the metric–affine/metric formulation. We also show that such a construction is valid only for weak electromagnetic fields. Therefore the purely affine formulation that combines gravitation, electromagnetism and the cosmological constant cannot be a simple sum of terms corresponding to separate fields. Consequently, this formulation of electromagnetism seems to be unphysical, unlike the purely metric and metric–affine pictures, unless the electromagnetic field couples to the cosmological constant.

scholarly journals KALUZA–KLEIN THEORY WITH TORSION CONFINED TO THE EXTRA DIMENSION

2010 ◽  
Vol 25 (25) ◽  
pp. 2121-2130 ◽  
Author(s):  
KARTHIK H. SHANKAR ◽  
KAMESHWAR C. WALI
Keyword(s):  
Ricci Tensor ◽  
Vector Fields ◽  
Dynamical Variable ◽  
Einstein Equations ◽  
Cosmic Acceleration ◽  
Scalar And Vector Fields ◽  
Klein Theory ◽  
Cartan Theory ◽  
Torsion Free ◽  
Kaluza Klein

Here we consider a variant of the five-dimensional Kaluza–Klein (KK) theory within the framework of Einstein–Cartan formalism that includes torsion. By imposing a set of constraints on torsion and Ricci rotation coefficients, we show that the torsion components are completely expressed in terms of the metric. Moreover, the Ricci tensor in 5D corresponds exactly to what one would obtain from torsion-free general relativity on a 4D hypersurface. The contributions of the scalar and vector fields of the standard KK theory to the Ricci tensor and the affine connections are completely nullified by the contributions from torsion. As a consequence, geodesic motions do not distinguish the torsion free 4D spacetime from a hypersurface of 5D spacetime with torsion satisfying the constraints. Since torsion is not an independent dynamical variable in this formalism, the modified Einstein equations are different from those in the general Einstein–Cartan theory. This leads to important cosmological consequences such as the emergence of cosmic acceleration.

Curvature of Cr Manifolds

2013 ◽  
Vol 59 (1) ◽  
pp. 43-72
Author(s):  
Aurel Bejancu ◽  
Hani Reda Farran
Keyword(s):  
Existence And Uniqueness ◽  
Ricci Tensor ◽  
Tensor Field ◽  
Differential Operators ◽  
Linear Connection ◽  
Horizontal Distribution ◽  
Einstein Equations ◽  
Space Forms ◽  
Torsion Free

Abstract We prove the existence and uniqueness of a torsion-free and h-metric linear connection ▽(CR connection) on the horizontal distribution of a CR manifold M. Then we define the CR sectional curvature of M and obtain a characterization of the CR space forms. Also, by using the CR Ricci tensor and the CR scalar curvature we define the CR Einstein gravitational tensor field on M. Thus, we can write down Einstein equations on the horizontal distribution of the 5-dimensional CR manifold involved in the Penrose correspondence. Finally, some CR differential operators are defined on M and two examples are given to illustrate the theory developed in the paper. Most of the results are obtained for CR manifolds that do not satisfy the integrability conditions

scholarly journals GENERAL RELATIVITY AS AN ÆTHER THEORY

2012 ◽  
Vol 21 (02) ◽  
pp. 1250011 ◽  
Author(s):  
MAURICE J. DUPRÉ ◽  
FRANK J. TIPLER
Keyword(s):  
General Relativity ◽  
Ricci Tensor ◽  
Maxwell Equations ◽  
Einstein Equations ◽  
Newtonian Gravity ◽  
Energy Tensor ◽  
Stress Energy Tensor ◽  
Spatial Curvature ◽  
Stress Energy ◽  
Special Case

Most early twentieth century relativists — Lorentz, Einstein, Eddington, for examples — claimed that general relativity was merely a theory of the æther. We shall confirm this claim by deriving the Einstein equations using æther theory. We shall use a combination of Lorentz's and Kelvin's conception of the æther. Our derivation of the Einstein equations will not use the vanishing of the covariant divergence of the stress–energy tensor, but instead equate the Ricci tensor to the sum of the usual stress–energy tensor and a stress–energy tensor for the æther, a tensor based on Kelvin's æther theory. A crucial first step is generalizing the Cartan formalism of Newtonian gravity to allow spatial curvature, as conjectured by Gauss and Riemann. In essence, we shall show that the Einstein equations are a special case of Newtonian gravity coupled to a particular type of luminiferous æther. Our derivation of general relativity is simple, and it emphasizes how inevitable general relativity is, given the truth of Newtonian gravity and the Maxwell equations.

Einstein equations for a Finsler-Larange space with canonical N-linear connections

2020 ◽  
Vol 4 (1) ◽  
pp. 240-247
Author(s):  
Roopa M. K ◽  
◽  
Narasimhamurthy S. K ◽  
Keyword(s):  
Einstein Equations ◽  
Linear Connections

The Kerr solution

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Kerr Black Hole ◽  
Geodesic Equation ◽  
Einstein Equations ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Axially Symmetric ◽  
Observable Universe ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum ◽  
The Many

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

Sign in / Sign up

Export Citation Format

Share Document