Riemann Manifold: Choosing a Basis & Lie Algebra

[befj0001]

On the spacetime manifold in general relativity, one chooses a basis at a point and express it by the partial derivatives with respect to the four coordinates in the coordinate system. And then the basis vectors in the dual space will be the differentials of the coordinates. Why do one do that? I understand that by doing so it allows one to identify the vectorfields on the manifold by a Lie algebra.

But why do one choose to do so? And why is it so important for allowing a Lie algebra of the vector fields to be defined?

Could someone give a more intuitive explanation for this?

 

[WannabeNewton]

The choice of a coordinate basis is not a necessary one. Non-coordinate bases are used all the time in GR as they make direct contact with measurements made by observers or fields of observers. Coordinate bases are simply easier to handle computationally.

 

[Matterwave]

The nice thing about coordinate bases is that they don't have lie brackets with respect to each other:
$$\left[\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right]=0$$

This makes many computations (e.g. for the curvature tensors) simpler. It is not a requirement that one uses these bases though. One can use any linearly independent basis one wants, the other popular choice being a set of orthonormal bases (sometimes called a tetrad, frame field, or vierbien).