On some properties of W -curvature tensor (original) (raw)

Symmetries of Energy-Momentum Tensor: Some Basic Facts

Communications in Theoretical Physics, 2007

It has been pointed by Hall et al. [1] that matter collinations can be defined by using three different methods. But there arises the question of whether one studies matter collineations by using the L ξ T ab = 0, or L ξ T ab = 0 or L ξ T b a = 0. These alternative conditions are, of course, not generally equivalent. This problem has been explored by applying these three definitions to general static spherically symmetric spacetimes. We compare the results with each definition.

Lie Symmetries of the Energy–Momentum Tensor for Plane Symmetric Static Spacetimes

International Journal of Modern Physics D, 2005

Matter collineations (MCs) are the vector fields along which the energy–momentum tensor remains invariant under Lie transport. Invariance of the metric, the Ricci and the Riemann tensors have been studied extensively and the vectors along which these tensors remain invariant are called Killing vectors (KVs), Ricci collineations (RCs) and curvature collineations (CCs), respectively. In this paper, plane symmetric static spacetimes have been studied for their MCs. Explicit form of MCs together with the Lie algebra admitted by them has been presented. Examples of spacetimes have been constructed for which MCs have been compared with their RCs and KVs. The comparison shows that neither of the sets of RCs and MCs contains the other, in general.

Space tensors in general relativity II: Physical applications

General Relativity and Gravitation, 1974

The general theory of space tensors is applied to the study of a space-time manifold ~24 carrying a distinguished timelike congruence F. The problem is to determine a physically relevant spatial tensor analysis (~,gT) over (~4,F), in order to proceed to a correct formulation of Relative Kinematics and Dynamics. This is achieved by showing that each choice of (~,9 T) gives rise to a corresponding notion of 'frame of reference' associated with the congruence F. In particular, the frame of reference (F,~Te) determined by the standard spatial tensor analysis (9n,9~ T) is shown to provide the most natural generalization of the concept of frame of reference in Classical Physics. The previous arguments are finally applied to the study of geodesic motion in ~4" As a result, the general structure of the gravitational fields in the frame of reference

Translational spacetime symmetries in gravitational theories

Classical and Quantum Gravity, 2006

How to include spacetime translations in fibre bundle gauge theories has been a subject of controversy, because spacetime symmetries are not internal symmetries of the bundle structure group. The standard method for including affine symmetry in differential geometry is to define a Cartan connection on an affine bundle over spacetime. This is equivalent to (1) defining an affine connection on the affine bundle, (2) defining a zero section on the associated affine vector bundle, and (3) using the affine connection and the zero section to define an 'associated solder form,' whose lift to a tensorial form on the frame bundle becomes the solder form. The zero section reduces the affine bundle to a linear bundle and splits the affine connection into translational and homogeneous parts; however it violates translational equivariance / gauge symmetry. This is the natural geometric framework for Einstein-Cartan theory as an affine theory of gravitation. The last section discusses some alternative approaches that claim to preserve translational gauge symmetry.

Gauging generalised symmetries in linear gravity

The theory of a free spin-2 field on Minkowski spacetime has 1-form and (d → 3)-form symmetries associated with conserved currents formed by contractions of the linearised Riemann tensor with conformal Killing-Yano 2-forms. We show that a subset of these can be interpreted as Noether currents for specific shift symmetries of the graviton that involve a Killing vector and a closed 1-form parameter. We give a systematic method to gauge these 1-form symmetries by coupling the currents to background gauge fields and introducing a particular set of counter-terms involving the background fields. The simultaneous gauging of certain pairs of 1-form and (d → 3)-form symmetries is obstructed by the presence of mixed 't Hooft anomalies. The anomalous pairs of symmetries are those which are related by gravitational duality. The implications of these anomalies are discussed. C Mixed 't Hooft anomalies between electric and magnetic symmetries 37 C.1 Four dimensions 37 C.2 d > 4 dimensions 39

Space tensors in general relativity I: Spatial tensor algebra and analysis

General Relativity and Gravitation, 1974

A pair (M,F) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence r. The spatial tensor algebra ~ associated with the pair (M,F) is discussed. A general definition of the concept of spatial tensor analysis over (M,r) is then proposed. Basically, this includes a spatial covariant differentiation ~ and a time-derivative 9 T, both acting on ~ and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair (V,VT) are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M,r) is finally established, and the resulting mathematical structure is examined in detail.

Matter Symmetries of Non-Static Plane Symmetric Spacetimes

2020

The matter collineations of plane symmetric spacetimes are studied according to the degenerate energy-momentum tensor. We have found many cases where the energy-momentum tensor is degenerate but the group of matter collineations is finite. Further we obtain different constraint equations on the energy-momentum tensor. Solving these constraints may provide some new exact solutions of Einstein field equations.

Geometrization of Lie and Noether symmetries with applications in Cosmology

Journal of Physics: Conference Series, 2013

We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in a n−dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the special projective and the homothetic vectors of the space. Therefore the Lie and the Noether symmetries for these equations are geometric symmetries or, equivalently, the geometry of the space is modulating the motion of dynamical systems in that space. We give two theorems which contain all the necessary conditions which allow one to determine the Lie and the Noether symmetries of a specific dynamical system in a given Riemannian space. We apply the theorems to various interesting situations covering Newtonian 2d and 3d systems as well as dynamical systems in cosmology.

Generalised symmetries in linear gravity

Linearised gravity has a global symmetry under which the graviton is shifted by a symmetric tensor satisfying a certain flatness condition. There is also a dual symmetry that can be associated with a global shift symmetry of the dual graviton theory. The corresponding conserved charges are shown to satisfy a centrally-extended algebra. We discuss the gauging of these global symmetries, finding an obstruction to the simultaneous gauging of both symmetries which we interpret as a mixed 't Hooft anomaly for the ungauged theory. We discuss the implications of this, analogous to those resulting from a similar structure in Maxwell theory, and interpret the graviton and dual graviton as Nambu-Goldstone modes for these shift symmetries.