Christoffel Symbols

dynamic_master

I'm having trouble understanding what Christoffel symbols are. In simple language, what are they? What are they used for?

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WannabeNewton

Consider a smooth manifold $M$ with affine connection $\nabla$. Let $(U,\varphi)$ be a chart on $M$ and denote by $\{\partial_i\}$ the coordinate basis associated with this chart. Then the Christoffel symbols $\Gamma^{k}{}{}_{ij}$ associated with $\{\partial_i\}$ are simply given by $\nabla_{\partial_i}\partial_j = \Gamma^{k}{}{}_{ij}\partial_k$. In other words, they represent the coefficients of $\nabla$ in the chart $(U,\varphi)$ and allow one to do coordinate computations using $\nabla$ such as calculating the covariant derivative $\nabla_{l}V^{m} = \partial_l V^m + \Gamma^{m}{}{}_{lk}V^{k}$.

Chestermiller

Mentor
I'm having trouble understanding what Christoffel symbols are. In simple language, what are they? What are they used for?
Hi dynamic_master. Welcome to physics forums.

If you have a vector quantity (field) expressed in component form and you want to find out how the vector changes with spatial position, then, in cartesian coordinates, you just take the derivatives of the components. However, if you are using curvilinear coordinates, the coordinate basis vectors (or unit vectors) change with spatial position, and you need to take this into account. This is where the christoffel symbols come in. The partial derivatives of the coordinate basis vectors with respect to spatial position can be expressed as a linear summation of the coordinate basis vectors times the christoffel symbols.

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