I know that the lie derivative of a metric $g_{ab}$ along a vector field $X$ can be written as $L_X g_{ab} = \nabla_a X_b + ∇_b X_a$. Now my question is, how would you use this to actually compute the lie derivative?
Say for example, if $X = (X^1, X^2) = (-y, x)$
Apologies for being unable to insert the correct notation.
Best Answer
Assume that $n=2$. First, you have to find the metric components $g_{ab}$ and the connection components $\Gamma^c_{ab}$. Then, you may use the following lowering index to find $X_a$ $$X_a=g_{ab}X^b$$For example $$X_1=g_{11}X^1+g_{12}X^2$$ Then, $$\nabla_aX_b=\frac{\partial X_b}{\partial x^a}-X_c\Gamma^c_{ab}$$ For example, $$\nabla_1X_1=\frac{\partial X_1}{\partial x^1}-X_1\Gamma^1_{11}-X_2\Gamma^2_{11}$$