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  What can be the mathematical justification for double derivatives of the position vector and any other general vector to obey different rules?

+ 1 like - 0 dislike

Kindly refer to the earlier question: Why in the curved space-time, the double derivatives of the position vector are symmetric but any other vector are not symmetric?

This problem can be mathematically analysed as follows:

The symmetry conditions seem to depend upon, how we define the vectors and their incremental vectors. We compare incremental vectors in a flat space and curved space:

(a) In a flat space, an incremental vector of a general vector (eq. (1) in image) can be obtained by taking a covariant derivative of the vector. Similarly, an incremental distance (position) vector (eq. (3)) can also be written by taking covariant derivative of the position vector.

Therefore, the expressions of both the general incremental vector and the incremental distance vector satisfy same symmetry conditions in a flat space.

(b) In a curved space, an incremental vector of a general vector (eq. (4) in image) can be obtained by taking a covariant derivative of the vector.

But, there is no expression of position vector which can be differentiated to write the desired incremental distance vector. The incremental distance vector has to be defined based on the Schwarzschild metric (eq. (5) in image).

In a curved space, the expressions for the incremental distance vector (eq. (5)) and any other general incremental vector (eq. (4)) are not similar.

Hence, the symmetry conditions for their corresponding vectors, general vector A and position vector s need not be same. Note, writing a position vector expression in curved space is not necessary for suggesting that the double derivatives of the position vector are symmetric. This symmetry arises because we can take partial derivative of the incremental distance (position) vector. This symmetry leads to the symmetry of Christoffel symbols. 

But assumption of this symmetry of position vector cannot guarantee symmetry of double derivatives of a general vector. Therefore, some curvature tensor components are non-zero.

Reference Article:


asked Jul 14, 2021 in Theoretical Physics by NSRG [ revision history ]
edited Jul 14, 2021

2 Answers

+ 0 like - 0 dislike

The standard GR interpretation doesn't allow a position vector. Therefore we have to define the incremental distance vector from the Schwarzschild metric. But, this leads to the contradictory unit vector differentiation values. Therefore we cannot even write the unit vectors which can give these contradictory results after differentiation. Hence, it is not possible to suggest even a coordinate system for the curved space.  

Kindly refer to the question: 

Is it incorrect to assume the Christoffel symbol symmetry (with respect to the lower indices) for a curved space-time?

answered Jul 16, 2021 by NSRG [ no revision ]
+ 0 like - 0 dislike

I have uploaded a preprint titled "Vector analysis in the curved space-time: The problems, the remedies and some concerns" on Research Gate.

Preprint Vector analysis in the curved space-time: The problems, the ...

This article discusses the geometrical difficulties associated with the classical definition of vectors and how general relativity proposes to overcome them by giving a new definition of vectors in the curved space-time.

The article discusses both these definitions and their geometrical concerns.

I shall be happy to receive specific comments. I have been raising these issues in my questions but now I have consolidated them in an article form.

answered Jan 17, 2022 by NSRG (0 points) [ no revision ]

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