https://arxiv.org/pdf/astro-ph/0502031.pdf

Energy conditions and Segre classification of phantom fields

The algebraic classification of second-order symmetric tensors based on Segre type is used to give a systematic description of energy-momentum tensors in General Relativity. The uniqueness of the physical interpretation of a given energy-momentum tensor is discussed algebraically and a brief description of their “inheritance of symmetry” properties is also given.

Physical structure of the energy-momentum tensor in general relativity

Methods of classification of the Ricci tensor (and hence the energy-momentum) in general relativity are reviewed, and their geometric interpretation considered. It is shown that the Plebanski spinor represents the map of self-dual bivectors naturally induced from the map of vectors defined by the Ricci tensor. It is noted that in some cases the multiplicities for the Petrov classification of the Plebanski spinor and the eigenvalue equation of the Ricci spinor differ; as direct computation of roots is more difficult than finding their multiplicities, an algorithm is devised which exploits this difference. Details of the algorithm, including methods for distinguishing the sub-cases not separated by the multiplicities mentioned, are given, and the resulting computer program described. Test results are reported.

Computer-aided classification of the Ricci tensor in general relativity

This paper essentially deals with the classification of a symmetric tensor on a four‐dimensional Lorentzian space. A method is given to find the algebraic type of such a tensor. A system of concomitants of the tensor is constructed, which allows one to know the causal character of the eigenspace corresponding to a given eigenvalue, and to obtain covariantly their eigenvectors. Some algebraic as well as differential applications are considered.

/pdf/intrinsic-characterization-of-space-time-symmetric-tensors-1smlycxdvn.pdf

Intrinsic characterization of space-time symmetric tensors

A list is given of a canonical set of the Newman–Penrose quantities ΦAB, the tetrad components of the trace‐free Ricci tensor, for each Plebanski class according to Plebanski’s classification of this tensor. This comparative list can easily be extended to cover the classification in tetrad language of any second‐order, trace‐free, symmetric tensor in a space‐time. A fourth‐order tensor which is the product of two such tensors was defined by Plebanski and used in his classification. This has the same symmetries as the Weyl tensor. The Petrov classification of this tensor, here called the Plebanski tensor, is discussed along with the classification of the Ricci tensor. The use of the Plebanski tensor in a couple of areas of general relativity is also briefly discussed.

/pdf/the-classification-of-the-ricci-and-plebanski-tensors-in-30lzopf5gm.pdf

The classification of the Ricci and Plebański tensors in general relativity using Newman–Penrose formalism

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.

http://iopscience.iop.org/article/10.1088/0305-4470/12/1/016/pdf

W J Cormack

Papers

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