Vector analysis in the curved space-time: The problems, the remedies and some concerns

Question

Referee this paper: Research Gate by Nitin Ramchandra Gadre

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

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Preprint Address: 

https://www.researchgate.net/publication/357875376_Vector_analysis_in_the_curved_space-time_The_problems_the_remedies_and_some_concerns

Abstract: 

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.

Comments

The link is broken for me.

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The article has been slightly revised to further explain some issues. 

Answer 1

Philosophy behind writing the article:

The modern theories like quantum mechanics, general relativity give results which match the experimental observations. However, the problem with the modern theories seems to be that, there is no attempt to draw a classical picture of the physical reality and mathematics (with some postulates) itself is supposed to represent the reality.

Therefore, the author has attempted to draw a classical picture of the basic particle like electron in the earlier works published in AIP Advances in 2011 and Canadian Journal of Physics in 2017, based on the mathematics and postulates of these theories.

In this article, it is attempted to understand the classical geometrical picture of the curved space-time. The space-time is curved can be initially treated as a mathematical assumption. Once, you assume that this is also the actual physical picture, then you have to change the geometrical definition of vectors and the geometrical picture of space is modified.

It may be noted that the “bilocal vectors” based classical treatment can give all the desired results but this picture has some geometrical difficulties. It becomes impossible to write the conventional vectors including the unit vectors, making it difficult to even draw a coordinate system. Therefore, it is required to suggest a new definition of “local vectors” as discussed in Ref. 1 (Gravitation by Misner...) of the article. But, this definition raises some new concerns.

Therefore, drawing a classical geometrical picture of curved space-time still remains difficult.