Tensor In General Relativity Research Articles

ORTOCARTAN — a new computer program for algebraic calculations

The paper is intended to announce the appearance of a new computer program for algebraic calculations named ORTOCARTAN. It is readily applicable for calculating the curvature tensors in general relativity, and, with more effort from the user, can be applied to many other kinds of calculation. The paper describes shortly the main abilities of the program, its availability, and its speed in comparison with other systems. An example of a simple calculation is given.

Spinors, algebraic geometry, and the classification of second-order symmetric tensors in general relativity

Two approaches to the problem of classifying second-order symmetric tensors in space-time given by Ludwig and Scanlon and by Penrose are discussed. Ludwig and Scanlon use both spinor and tensor algebra in their approach, whereas Penrose uses spinors and the properties of certain curves in complex projective 3-space. These approaches yield essentially identical classifications, and this paper points out the connections between them in detail and tabulates the results.

Energy-momentum tensors in locally isotropic space-times

This paper discusses the restrictions imposed on the energy-momentum tensor in general relativity when the space-time admits certain types of symmetries and considers, in particular, the case of a G 4 of motions acting on null or non-null hypersurfaces.

Invariant two-spaces and canonical forms for the Ricci tensor in general relativity

An algebraic classification of the Ricci tensor is given in terms of its invariant two-space structure. The method involves classifying a complex fourth-order tensor which is algebraically equivalent to the trace-free Ricci tensor and which has all the algebraic symmetries of the (complex) Riemann tensor.

Some remarks on the energy-momentum tensor in general relativity

In general relativity, the energy-momentum tensor of a classical tensor field can be constructed by varying the action of the field with respect to the background metric. This paper suggests an alternative interpretation of the construction which also makes sense for spinor fields, and which gives some insight into the locality of energy-momentum operators in generally covariant quantum field theory.

Space tensors in general relativity III: The structural equations

A self-consistent theory of spatial differential forms over a pair (M,Γ)is proposed. The operators d(spatial exterior differentiation), dT (temporal Lie derivative) andL (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis $$(\tilde \nabla ,\tilde \nabla T)$$ (M,Γ). The structural equations of $$(\tilde \nabla ,\tilde \nabla T)$$ and the corresponding spatial Bianchi identities are discussed. The special case $$(\tilde \nabla ,\tilde \nabla T) = (\tilde \nabla *,\tilde \nabla T*)$$ is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resultingstructure of Eintein's equations over a pair (ν4,Γ)is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.

Space tensors in general relativity II: Physical applications

The general theory of space tensors is applied to the study of a space-time manifoldsV 4 carrying a distinguished time-like congruence Γ. The problem is to determine a physically relevant spatial tensor analysis\(\left( {\tilde \nabla ,\tilde \nabla _T } \right)\) over (V 4, Γ), in order to proceed to a correct formulation of Relative Kinematics and Dynamics.

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

A pair (M, Γ) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence Γ. The spatial tensor algebraD associated with the pair (M, Γ) is discussed. A general definition of the concept of spatial tensor analysis over (M, Γ) is then proposed. Basically, this includes a spatial covariant differentiation\(\tilde \nabla \) and a time-derivative\(\tilde \nabla _T \), both acting onD and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair\(\left( {\tilde \nabla ,\tilde \nabla _T } \right)\) are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M, Γ) is finally established, and the resulting mathematical structure is examined in detail.

The Algebra of the Riemann Curvature Tensor in General Relativity: Preliminaries

In a four‐dimensional curved space‐time it is well‐known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley‐Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.