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help1please
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Homework Statement
Homework Equations
The Attempt at a Solution
Does the Christoffel symbol [tex]\Gamma[/tex] have a dimension in physics? And if it does, what is its dimension?
Thank you!
physicus said:The Christoffel symbols are arrays of real numbers. They are dimensionless.
TSny said:To OP: Can you think of any equation which contains the Christoffel symbols? That should help you answer the question.
help1please said:Yes, I can think of this equation:
[tex]\frac{\partial e_i}{\partial U^j} = \Gamma_{ij}^k e_k[/tex]
But not really sure is that makes it easier for me. I can certainly think of the derivatives case for the christoffel symbols. I had decided before anyone posted here, that the symbol would have 1/length because of the metric derivatives. But the responses here seem to be... a bit mixed.
help1please said:The curvature tensor is made up completely of christoffel symbols, you'd expect that to have a dimension of length.
TSny said:I'm not familiar with the Uj notation, but your reasoning based on the derivatives of the metric is correct. The Christoffel symbols have dimension of 1/length.
help1please said:But,you seem to be telling me that the dimensions of the Christoffel symbols are indeed 1/length?
phyzguy said:Also, since the Christoffel symbols have dimension 1/length, the curvature tensor has dimensions 1/(length^2). This is exactly what you'd expect. Consider the Gaussian curvature of a 2D surface - it is the product of the two principle curvatures. Since each principle curvature is equal to 1/(radius of curvature), the Gaussian curvature has dimensions 1/(length^2).
TSny said:They have dimension of 1/length if all the coordinates xj have the dimension of length. Without thinking, that's what I was assuming .
However, generally the coordinates do not need to have dimension of length (for example, in spherical coordinates where some of the coordinates are angles.) For a given coordinate system it is possible for different Christoffel symbols to have different dimesions.
The Christoffel Symbol dimension is a mathematical concept in differential geometry that describes the number of independent components of a tensor. It is used to calculate the connection between points on a curved surface and is essential in understanding the curvature of a manifold.
The Christoffel Symbol dimension is calculated using the formula n(n+1)(n+2)/6, where n is the dimension of the surface or manifold. For example, a two-dimensional surface would have a Christoffel Symbol dimension of 3, while a three-dimensional manifold would have a dimension of 10.
The Christoffel Symbol dimension is significant because it helps us understand the geometry of curved surfaces and manifolds. It is a fundamental concept in the study of differential geometry and is used in various fields such as physics, engineering, and computer science.
The Christoffel Symbol dimension is related to the Riemann curvature tensor through the Ricci identity. The Riemann curvature tensor is a measure of the curvature of a manifold, and the Christoffel Symbol dimension is used to calculate its components.
Yes, the Christoffel Symbol dimension can be higher than the dimension of the manifold. This can occur when the manifold is not smooth or has singularities. In these cases, the Christoffel Symbol dimension can be higher than the actual dimension of the manifold, but it will still accurately describe the curvature of the surface or manifold.