Curvature collineations for type-N Robinson-Trautman space-times

The metric of type-N Robinson-Trautman space-times is generated by a real functionP satisfying certain field equations. Canonical forms forP are obtained under the assumption that at least one curvature collineation exists. In order to give an example of the improper subgroup structure of a group of curvature collineations all the curvature collineations are determined for the space-times corresponding to one of the two canonical forms.     

Curvature collineations in certain gravitational space-times

It has been shown that the space-times formed from the product of two surfaces and from a thick gravitational plane wave sandwiched between two flat spacetimes admit proper curvature collineation in general. The curvature collineation vectors have been determined explicitly. For the space-time formed from the product of two surfaces conditions are obtained for it to admit motion. It has also been pointed out that the spacetime formed from a thick plane gravitational wave belongs to the class (III_b) of pure gravitational radiation and admits five- and six-parameter groups of motion in the two possible cases. Conservation laws given by Sachs and Katzin-Levine-Davis in terms of curvature collineation vectors are satisfied identically in the case of the plane gravitational wave solution, and Sachs' conservation law can be deduced in this case as a consequence of the theorem given by Katzin and others.     

Curvature and conformal collineations in presence of matter

The necessary and sufficient conditions for the existence of curvature and conformai collineations, when they are not conformal motions, are applied in order to obtain some solutions of Einstein's equations in the presence of spherical symmetric distributions of matter.     

Curvature and Weyl collineations of Bianchi type V spacetimes

The objective of this paper is twofold: (a) First the curvature collineations of the Bianchi type V spacetimes are studied using rank argument of curvature matrix. It is found that the rank of the 6×6 curvature matrix is 3, 4, 5 or 6 for these spacetimes. In one of the rank 3 cases the Bianchi type V spacetime admits proper curvature collineations which form infinite dimensional Lie algebra. (b) Then the Weyl collineations of the Bianchi type V spacetimes are investigated using rank argument of the Weyl matrix. It is obtained that the rank of the 6×6 Weyl matrix for Bianchi type V spacetimes is 0, 4 or 6. It is further shown that these spacetimes do not admit proper Weyl collineations, except in the trivial rank 0 case, which obviously form infinite dimensional Lie algebra. In some special cases it is found that these spacetimes admit Weyl collineations in addition to the Killing vectors, which are in fact proper conformal Killing vectors. The obtained conformal Killing vectors form four-dimensional Lie algebra.     

Curvature collineations and the determination of the metric from the curvature in general relativity

It is shown that for a very general class of space-times, the componentsR _bcd ^a of the curvature tensor determine the metric components up to a constant conformal factor. This general class contains most of those cases which are usually considered to be interesting from the point of view of Einstein's general relativity theory. The connection between the above result and the existence of proper curvature collineations is given.     

Perfect fluid spacetimes admitting curvature collineations

Some basic concepts about curvature collineations are reviewed and the existing results on this topic are applied to the case of perfect fluids, giving a characterization of those amongst them which admit proper curvature collineations.     

Affine collineations in Einstein-Maxwell space-times

The existence of an affine vector field in an Einstein-Maxwell space-time is discussed. We first consider the non-null electromagnetic field case, and show that there are no solutions of the Einstein-Maxwell equations admitting a proper affine collineation. In the case of a null electromagnetic field case, we characterize all the possible solutions with such property.On leave from Universidad de Los Andes, Facultad de Ciencias, Departamento de Física, Mérida 5101, Venezuela     

Ricci tensor with six collineations

In classifying Ricci tensors in terms of their collineations, an interesting case possessing six collineations arises. These collineations are worked out and discussed.On leave from Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.     

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.     

Bianchi types of a three-parameter group of curvature collineations

The Bianchi types of the three-parameter group of curvature collineations admitted by a previously discussed family of typeN Robinson-Trautman empty space-times are obtained.     

Ricci collineations of the Bianchi type II,VIII, and IX space-times

Ricci and contracted Ricci collineations of the Bianchi type II, VIII, and IX space-times, associated with the vector fields of the form (i) one component of ^a (x ^b ) is different from zero and (ii) two components of ^a (x ^b ) are different from zero, fora, b=1, 2, 3, 4, are presented. In subcase (i.b), which is ^a = (0, ¹(x ^a ),0,0), some known solutions are found, and in subcase (i.d), which is ^a =(0, 0, 0, ⁴(x ^a )), choosingS(t) = const.×R(t), the Bianchi type II, VIII, and IX spacetime is reduced to the Robertson-Walker metric.     

Conservation laws in general relativity based upon the existence of preferred collineations

The conservation laws based upon the existence of curvature and Ricci collineations are investigated and the results given recently by Katzin, Levine and Davis are reinterpreted and generalized. The concept of a Maxwell collineation is introduced and corresponding conservation laws are found.     

Curvature from spin

We show that, considering the dislocation defect induced by torsion in spacetime, which behaves like a string with tension, we are lead also to defect angle and then to curvature of spacetime. The space with torsion and curvature is then equivalent to an elastic continuum which has undergone plastic deformations and, following Sakharov idea of the spacetime as a elastic continuum, we are lead to a gravitational constant, which occurs in the Einstein action, as the metrical elasticity of spacetime with the exact value without introducing any arbitrary cutoff, when also torsion is considered.     

The Robertson-Walker metric and the symmetries belong to the family of contracted Ricci collineations

It is shown that the general form of the Robertson-Walker cosmological metric admits symmetry properties that are members of the symmetry family of contracted Ricci collineations. A particular form for the conservation law generator given by _j [(–g)^1/2(T _i ^j –1/2 _i ^jT ) ⁱ ] = 0 following in consequence of these symmetries is obtained and interpreted.     

Extrinsic Curvature Embedding Diagrams

Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the extrinsic curvature (instead of the intrinsic curvature). Such an extrinsic curvature embedding diagram, when used together with the usual kind of intrinsic curvature embedding diagram, carries the information of how a surface is embedded in the higher dimensional curved space. Simple examples are given to illustrate the idea.     

Gravity as Spacetime Curvature

Einstein's general theory of relativity conceives the phenomena of gravity as manifestations of the curvature of the spacetime manifold in which physical events take place. I sketch the line of thought that led Einstein to this conception, and I briefly discuss proposals by Jeffreys and Feynman for retaining Einstein's gravitational field equations while discarding their purportedly geometrical meaning.     

曲率与运动界面发展

对界面传播速度依赖于曲率的界面发展问题进行研究,传播速度包括法向和切向,并且,在界面传播过程中全变差的变化仅依赖在曲率为零处的法向速度对曲率的导数,切向速度对全变差的变化没有影响.     

Curvature calculations with GEOCALC

A new method for calculating the curvature tensor has been recently proposed by D. Hestenes. This method is a particular application of geometric calculus, which has been implemented in an algebraic programming language on the form of a package called GEOCALC. We show how to apply this package to the Schwarzchild case and we discuss the different results.     

Quantum Sensing of Curvature

We address the problem of sensing the curvature of a manifold by performing measurements on a particle constrained to the manifold itself. In particular, we consider situations where the dynamics of the particle is quantum mechanical and the manifold is a surface embedded in the three-dimensional Euclidean space. We exploit ideas and tools from quantum estimation theory to quantify the amount of information encoded into a state of the particle, and to seek for optimal probing schemes, able to actually extract this information. Explicit results are found for a free probing particle and in the presence of a magnetic field. We also address precision achievable by position measurement, and show that it provides a nearly optimal detection scheme, at least to estimate the radius of a sphere or a cylinder.