Video 83 - Riemann Normal Coordinates

This paragraph introduces the concept of Riemann normal coordinates in the context of tensor calculus. It explains that these coordinates are defined by two key properties: the metric tensor's diagonal elements are ones when the indices are equal and zeros when they are unequal at the origin point, and the partial derivatives of the metric tensor with respect to the coordinates are zero at the origin. Riemann normal coordinates simplify the analysis of the Riemann tensor and are likened to Cartesian coordinates at the point of origin but are not the same throughout the manifold. The paragraph also discusses the immediate consequence of this definition on the Christoffel symbols, which are zero at the origin point, and includes an illustration using a spherical surface to demonstrate the concept.

The second paragraph delves into the process of calculating the metric tensor for a new coordinate system defined on a spherical surface. It describes the transformation from standard spherical coordinates (theta and phi) to a new set of coordinates (u and v). The metric tensor in the unprimed system is used, along with the Jacobian factors derived from the transformation, to find the metric tensor in the primed system. The process involves identifying non-zero Jacobian factors and applying them to the metric tensor components. The result is a metric tensor for the new coordinate system, which is then expressed in terms of the new coordinates u and v, using trigonometric identities.

This paragraph focuses on deriving the partial derivatives of the metric tensor with respect to the new coordinates and evaluating them at the origin of the coordinate system. It explains that most of the partial derivatives are zero due to the constants in the metric tensor, with the exception of one derivative that involves sine and cosine functions. The evaluation at the origin shows that the metric tensor becomes the identity matrix and all partial derivatives are zero, confirming that the defined coordinates qualify as Riemann normal coordinates.

The fourth paragraph provides a geometric interpretation of Riemann normal coordinates on a spherical surface. It describes the point of origin as the intersection of two great circles generated by planes passing through the center of the sphere. The special nature of this point is highlighted, as it is where the planes intersect at right angles, forming the basis for the new coordinate system. The paragraph also discusses the arbitrariness of the rotational orientation of these planes and the generalization of this process to any point on any surface, emphasizing the use of the normal in constructing Riemann normal coordinates.

This paragraph discusses the mathematical process of establishing Riemann normal coordinates in higher dimensions. It involves taking partial derivatives of the metric tensor and applying the product rule, leading to a set of equations that must be satisfied for the coordinates to be considered Riemann normal. The paragraph explains that there are more independent variables than conditions, indicating an underdetermined system with one degree of freedom, which allows for the rotation of axes. The final part of the paragraph provides formulas for the number of conditions and degrees of freedom in any number of dimensions, confirming the possibility of establishing Riemann normal coordinates regardless of the dimension.

The final paragraph connects the concept of Riemann normal coordinates to the Riemann tensor, highlighting the simplification that occurs when the tensor is expressed in these coordinates. It explains that in Riemann normal coordinates, all Christoffel symbols evaluate to zero, which greatly simplifies the expression of the Riemann tensor. The paragraph concludes with a simplified form of the Riemann tensor that will be useful for evaluating its symmetries and identities in future discussions.