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In flat space, the classical vectors such as a position vector are bilocal (“point for head and point for tail”). The Schwarzschild metric suggests a four-dimensional curved space. This metric is mathematically similar to the metric of the surface of a sphere in a three-dimensional flat space, where we can write an incremental displacement vector at a point on sphere surface but cannot write a position vector. Similarly, in the curved space, we can write an incremental displacement vector based on curved space metric, even if writing a position vector is difficult. The mathematical analysis based on this classical incremental vector gives the desired results. We examine if this mathematical similarity between a curved space and sphere surface in a flat space can lead to the geometrical similarity. But, it becomes difficult to write unit vectors and draw a geometrical picture. Also, the incremental displacement vector and the incremental vector of any other vector show different symmetry properties. Though, mathematically the curved space metric and a sphere surface metric in a flat space are quite similar, their geometrical pictures differ. The curved space-time requires a different version of a vector and the difficulties in a classical version become basis for assigning peculiar characteristics to it. The geometrical picture could be redefined because the fourth dimension represents time and is distinct from the spatial dimensions. Changing the definition of vectors can overcome these difficulties but raises some new concerns. This article briefly mentions concerns with these local vectors in general relativity.
The Article Scheme:
(a) Initially, it is shown that the classical vector analysis gives all the desired results same as the tensor analysis based on scalar components of the tensor. (b) List all the geometrical difficulties associated with the classical picture. (c) Then examine, how the general relativity can overcome these difficulties by defining a new version of vectors (See Gravitation by C. W Misner, K. S. Thorne, J. A. Wheeler) (d) Concerns, if any, about this new definition based on our analysis.