A classical vector method for the curved space-time analysis

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Referee this paper: issn = 3041-8984 by Nitin Ramchandra Gadre

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Answer 1

The Philosophy, in short, is:

5.2    Significance of the vector method

Finally, we discuss the significance of the vector method used in Sec. 3.2, Sec. 3.3 and Sec. 3.4 to obtain the desired results. An element of an arc can be written in the form of a tensor: ds² = g_ij dxⁱ dx^j  .This tensor is called a metric tensor because all the metric properties of space should be completely determined by this tensor. Success of a curved space multidimensional analysis very much depends upon the intelligent choice of the metric tensor and this may not easily enlighten us about the geometrical aspects of the physical problem. However, an incremental displacement vector  ds = ε _i dxⁱ   corresponding to this multidimensional metric can be written in terms of basis vectors such that,  g_ij = ε_i . ε_j   . The basis vector symmetry condition is true for zero torsion condition. Suitable conditions will have to be written if we wish to study non-zero torsion situations. In Section 3, we have shown that the Riemann-Christoffel tensor is related to the second partial derivatives while the Bianchi identity is related to the third partial derivatives of these basis vectors. Describing the classical vectors encounters various geometrical difficulties (Sec. 2.2, Sec. 2.3, Sec. 4). The standard general relativity has tried to overcome these difficulties by (Sec. 5.1) redefining the vector as a local vector but this definition also raises some new concerns.  But, note that, the vector method, based on the classical incremental vectors gives all the desired mathematical results including various identities, similar to the conventional tensor analysis based on scalar components of tensors. A vector analysis is inherently superior to a scalar analysis as a vector has both magnitude and direction. This vector method, based on a classical geometrical picture, is an easier mathematical alternative to the tensor analysis in the curved multidimensional space and also throws light on the geometrical complications, if any, in describing such curved space. This method is generic in nature as it can be applied to any metric of any dimensions. We can also establish relationship between the three-dimensional Lagrangian method and the four-dimensional Geodesic analysis, both giving the same results.