Mirror Symmetry and Conformal Flatness in General Relativity

Using symmetry arguments only, we show that every spacetime with mirror-symmetric spatial sections is necessarily conformally flat. The general form of the Ricci tensor of such spacetimes is also determined. 1. Introduction. It is well known that the curvature tensor of any four-dimensional differentiable manifold has only 20 algebraically independent components. Ten out of these 20 components can be associated with its Weyl tensor, the remaining ten making up its Ricci tensor. When the four-dimensional manifold corresponds to an empty spacetime, its Ricci tensor becomes identically zero. The Weyl tensor can thus be seen as describing that part of the curvature of the spacetime which is not due to the presence of matter. The spacetime is said to be conformally flat when its Weyl tensor is identically zero (see, e.g., [4, Chapter 8]). In this note, we are interested in the conditions on the curvature tensor R of a space-time ᏹ 4 which follow from the assumption that ᏹ 4 has mirror-symmetric spatial sections. We will show that any such ᏹ 4 is conformally flat. We will also obtain the general form of the corresponding Ricci tensor.


Introduction.
It is well known that the curvature tensor of any four-dimensional differentiable manifold has only 20 algebraically independent components.Ten out of these 20 components can be associated with its Weyl tensor, the remaining ten making up its Ricci tensor.When the four-dimensional manifold corresponds to an empty spacetime, its Ricci tensor becomes identically zero.The Weyl tensor can thus be seen as describing that part of the curvature of the spacetime which is not due to the presence of matter.The spacetime is said to be conformally flat when its Weyl tensor is identically zero (see, e.g., [4,Chapter 8]).
In this note, we are interested in the conditions on the curvature tensor R of a spacetime ᏹ 4 which follow from the assumption that ᏹ 4 has mirror-symmetric spatial sections.We will show that any such ᏹ 4 is conformally flat.We will also obtain the general form of the corresponding Ricci tensor.

Mirror symmetry.
To mathematically translate the assumption concerning the existence of a mirror symmetry for the spatial sections of ᏹ 4 , we now introduce a system of coordinates on ᏹ 4 .Let x i , i = 0, 1, 2, 3, be a coordinate system such that the spatial sections of ᏹ 4 are described by x 0 = constant.We also consider a change of coordinate system for ᏹ 4 and designate by x i , i = 0, 1, 2, 3, the new coordinate system.Since we are only interested in the application of a mirror symmetry to the spatial sections of ᏹ 4 , we can assume that this change of coordinate system leaves invariant the time coordinate.Without loss of generality, we can also assume that the spatial mirror symmetry is defined with respect to the symmetry hyperplane x 1 = 0.This implies that the space coordinates transform according to the matrix The coordinates of ᏹ 4 then transform as where For the spatial sections of ᏹ 4 to be invariant under the transformation A, the curvature tensor R of the whole spacetime ᏹ 4 must be invariant under the change of coordinate (2.2).It follows that the algebraically independent components of R, at any given point of ᏹ 4 , will also be invariant under the same transformation.This property will then hold for the ten independent components of the Weyl tensor and the ten independent components of the Ricci tensor, which form the 20 independent components of the most general form of R.

Petrov index :
A, B = The matrix of the independent components of C can be simplified yet further if, instead of the fully covariant components C ijkl , one considers the mixed components C ij kl ↔ C A B .Here, we have C A B = G AC C CB , where the matrix (G AC ) = diag(I 3×3 , −I 3×3 ), and I 3×3 is the 3 × 3 identity matrix.The ten independent components of C are then given by where M = (m ij ) and N = (n ij ) are symmetric traceless 3 × 3 matrices.
To the coordinate transformation (2.2) corresponds a similarity transformation of the matrix Ꮿ = (C A B ). Denoting with an overbar the components of the Weyl tensor in the barred coordinate system x i , i = 0, 1, 2, 3, one indeed obtains (see [1, page 178]) where the sum is taken only over the pairs mn and pq corresponding to Petrov indices.If the Petrov indices A, B, C, D correspond, respectively, to the pairs of tensor indices ij, kl, mn, pq, then (3.3) is equivalent to where and the expressions S D B are the components of S, the inverse of the matrix S = (S A B ), when this inverse exists.The tensor C will be invariant under the transformation (2.2) if C A B = C A B for A, B = 1, 2,...,6.Equation (3.4) thus becomes which amounts to ᏯS = SᏯ. (3.7) We will now apply (3.7) to the case of a mirror symmetry with respect to the hyperplane x 1 = 0, that is, when the coordinate transformation is given by (2.3).The expression of the corresponding matrix S is It is then straightforward to show that all components of the matrices M and N in (3.2) vanish identically.This implies that C also vanishes identically, that is, that ᏹ 4 is four-dimensional conformally flat.

Ricci tensor.
To obtain a condition that the Ricci tensor (R ij ) of ᏹ 4 must satisfy in order for ᏹ 4 to be invariant under the transformation (2.2), we first observe that (R ij ) can be considered as the matrix realization of a bilinear form on ᏹ 4 .It follows that the change of coordinate (2.2) transforms (R ij ) according to Since the invariance of (R ij ) under the transformation (2.2) implies that (R ij ) = (R ij ), we obtain Substituting (2.3) into (4.2) directly leads to

Conclusion.
We have shown that every spacetime having mirror-symmetric spatial sections is conformally flat.This result applies in particular to spherically and cylindrically symmetric spacetimes, in rotation or not.It also applies to many of the simply or multiconnected spacetimes considered in cosmic crystallography [2,3].

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Weyl tensor C must satisfy for the corresponding spacetime to be invariant under the transformation (2.2) results from the Petrov matrix expression of the independent components of C. To obtain this condition, we use the following correspondence between pairs of tensor indices of C and single Petrov indices: Tensor indices : ijkl = 23, 31, 12, 10, 20, 30; C ijkl

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation