Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n(n2 − 4)/3 for the first time. Subsequently, we show its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw and Templeton. For each class examples are given. Finally, we investigate the relation between the Cotton tensor and the energy–momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.

https://iopscience.iop.org/article/10.1088/0264-9381/21/4/024/pdf

The Cotton tensor in Riemannian spacetimes

A new approach to the solution of certain differential equations, the double complex function method, is developed, combining ordinary complex numbers and hyperbolic complex numbers. This method is applied to the theory of stationary axisymmetric Einstein equations in general relativity. A family of exact double solutions, double transformation groups, and n‐soliton double solutions are obtained.

Generation of new solutions of the stationary axisymmetric Einstein equations by a double complex function method

The hole argument was developed by Einstein in 1913 while he was searching for a relativistic theory of gravitation. Einstein used the language of coordinate systems and coordinate invariance, rather than the language of manifolds and diffeomorphism invariance. He formulated the hole argument against covariant field equations and later found a way to avoid it using coordinate language. In this paper we shall use the invariant language of categories, manifolds and natural objects to give a coordinate-free description of the hole argument and a way of avoiding it. Finally we shall point out some important implications of further extensions of the hole argument to sets and relations for the problem of quantum gravity.

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The hole argument for covariant theories

We apply the Newman–Janis algorithm and its Azreg–A¨inou formulation to obtain the rotating models from a seed static Einstein–Euler–Heisenberg electrically charged black hole solution. Both algorithms generate solutions whose geometric part do not correspond to the energy–momentum tensor of the Euler-Heisenberg electromagnetic field. Only for slow rotations one has a Lense– Thirring–like slow rotating approximate solution to the field equations. According to the Azreg– A¨inou method, the obtained rotating metric is a solution with a kind of ad hoc constructed electromagnetic imperfect fluid as a source. This explanation is artificial, since one is not working with the source at all, instead one is working only with the geometry via Einstein tensor. Both methods lead to the rotating G¨urses and G¨ursey metric, that we consider as a geometrical model. The corresponding energy conditions are analyzed, its event horizons, ergoregions and test particle circular orbits are studied, and its shadow is shown as well.

Rotating black holes in the Einstein–Euler–Heisenberg theory

In this paper we give a complete classification of spherically symmetric static space-times by their Noether symmetries. The determining equations for Noether symmetries are obtained by using the usual Lagrangian of a general spherically symmetric static spacetime which are integrated for each case. In particular we observe that spherically symmetric static spacetimes are categorized into six distinct classes corresponding to Noether algebra of dimensions 5, 6, 7, 9, 11 and 17. Using Noether`s theorem we also write down the first integrals for each class of such spacetimes corresponding to their Noether symmetries. Some new spherically symmetric static solutions have also been obtained.

A. Barnes

Papers

Exact solution of einstein's field equations