Strange plus sign in the Lie derivative of a 1-form

Question

While the coordinate expression for the Lie derivative of a vector field $Y$ with respect to another vector field $X$ is given by

$$ L_X Y = \sum_j \left\{X^j\left( \frac{\partial Y^i}{\partial x^j}\right) - Y^j \left(\frac{\partial X^i}{\partial x^j}\right)\right\} $$

there is a plus sign in the coordinate expression for the Lie derivative of a 1-form $\alpha$ with respect to $X$

$$ L_X \alpha = \sum_j \left\{X^j\left( \frac{\partial a^i}{\partial x^j}\right) + a^j \left(\frac{\partial X^i}{\partial x^j}\right)\right\}dx_i $$

The $a^i$ and $a^j$ are the coordinates of the 1-form.

Where does this plus sign come from?

Answer 1

Let $(\alpha,Y) = \sum_j \alpha_j Y^j$ denote the contraction of a $1$-form $\alpha = \sum_j \alpha_j dx^j$ with a vector field $Y = \sum_j X^j \tfrac{\partial}{\partial x^j}$. The Lie derivative $L_X\alpha$ of a $1$-form $\alpha$ with respect to a vector field $X$ is defined precisely so that for any other vector field $Y$, we have the following Leibniz rule $$ X(\alpha,Y) = (L_X\alpha,Y) + (\alpha,L_XY). $$ But now, $$ X(\alpha,Y) = \left(\sum_i X^i \frac{\partial}{\partial x^i} \right)\left(\sum_j \alpha_j Y^j\right) = \sum_i X^i \sum_j\left(\frac{\partial \alpha_j}{\partial x^i} Y^j + \alpha_j \frac{\partial Y^j}{\partial x^i} \right)\\ = \sum_j \left(\sum_i X^i \frac{\partial \alpha_j}{\partial x^i}\right)Y^j + \sum_j \alpha_j \sum_i X^i \frac{\partial Y^j}{\partial x^i}, $$ whilst $$ (\alpha,L_X Y) = \sum_j \alpha_j \sum_i \left(X^i \frac{\partial Y^j}{\partial x^i}-Y^i\frac{\partial X^j}{\partial x^i}\right) = \sum_j \alpha_j \sum_i X^i \frac{\partial Y^j}{\partial x^i} - \sum_j \left(\sum_i \alpha_i \frac{\partial X^i}{\partial x^j} \right) Y^j, $$ so that $$ (L_X\alpha,Y) = X(\alpha,Y) - (\alpha,L_X Y)\\ = \sum_j \left(\sum_i X^i \frac{\partial \alpha_j}{\partial x^i}\right)Y^j + \sum_j \alpha_j \sum_i X^i \frac{\partial Y^j}{\partial x^i} - \sum_j \alpha_j \sum_i X^i \frac{\partial Y^j}{\partial x^i} + \sum_j \left(\sum_i \alpha_i \frac{\partial X^i}{\partial x^j} \right) Y^j\\ = \sum_j \left(\sum_i \left(X^i \frac{\partial \alpha_j}{\partial x^i} + \alpha_i \frac{\partial X^i}{\partial x^j}\right) \right) Y^j. $$ Since this is true for arbitrary vector fields $Y$, it therefore follows that $$ L_X \alpha = \sum_j \left(\sum_i \left(X^i \frac{\partial \alpha_j}{\partial x^i} + \alpha_i \frac{\partial X^i}{\partial x^j}\right) \right) dx^j $$ with the plus sign you were wondering about, which is forced upon you precisely by that Leibniz rule above.