But, we can always write and draw a geometrical picture of an incremental displacement (position) vector at any point on the sphere with the angular unit vectors. However, we cannot write a curved path position vector between two points on the surface of a sphere, as the directions of angular unit vectors change along the path. It can be easier to study a spherical body in spherical coordinates system. A change in coordinate system is only a change in the mathematical approach and does not change the physical nature either of the spherical body or the space. In the classical analysis, we can write a position vector in Cartesian coordinates or using the radial coordinate in a spherical coordinate system. For explaining the force at a distance, the space is assumed to be filled with a conceptual material called the field. Space in Classical Physics: In a classical picture, space is supposed to be empty and therefore has no characteristics assigned to it. Therefore, some curvature tensor components are on the suggested material and text books, I feel that there is a fundamental difference in the concept of space in the classical physics and general relativity:Ī. This symmetry leads to the symmetry of Christoffel symbols.īut assumption of this symmetry of position vector cannot guarantee symmetry of double derivatives of a general vector. This symmetry arises because we can take partial derivative of the incremental distance (position) vector. Note, writing a position vector expression in curved space is not necessary for suggesting that the double derivatives of the position vector are symmetric. Hence, the symmetry conditions for their corresponding vectors, general vector A and position vector s need not be same. (5)) and any other general incremental vector (eq. In a curved space, the expressions for the incremental distance vector (eq. The incremental distance vector has to be defined based on the Schwarzschild metric (eq. (4) in image) can be obtained by taking a covariant derivative of the vector.īut, there is no expression of position vector which can be differentiated to write the desired incremental distance vector. (b) In a curved space, an incremental vector of a general vector (eq. Therefore, the expressions of both the general incremental vector and the incremental distance vector satisfy same symmetry conditions in a flat space. (3)) can also be written by taking covariant derivative of the position vector. Similarly, an incremental distance (position) vector (eq. (1) in image) can be obtained by taking a covariant derivative of the vector. (a) In a flat space, an incremental vector of a general vector (eq. We compare incremental vectors in a flat space and curved space: The symmetry conditions seem to depend upon, how we define the vectors and their incremental vectors. I have tried to mathematically analyse the problem. Instead, vectors are elements of the tangent space and the basis vectors are the partial derivatives along the coordinate lines. Points can only be labelled by their coordinates and are not vectors. Kindly refer to the earlier answer: There is no position vector on a manifold.
Hope this clears up some of your confusion. This symmetry of position vector leads to symmetry of basis vectors leading to the Christoffel symbol symmetry with respect to the lower two indices You do not need to define any connection or covariant derivative to show this. In summary, the basis vectors commute because partial derivatives commute not because of the covariant derivative. $$\mathbf]$$īecause the extra terms cancel due to their symmetry. Instead, vectors are elements of the tangent space and the basis vectors are the partial derivatives along the coordinate lines: Points can only be labeled by their coordinates and are not vectors. There is no position vector on a manifold.