A simple metric question

Question

In their article Killing vector fields of standard static spacetimes, Dobarro and Unal derived the following simple identity.

Note that if $h:I→R$ is smooth and $Y,Z∈ {\frak{X}}\left(I\right)$, then $$L_{h∂_t}g_I(Y,Z) =Y(h)g_I(Z,∂_t) +Z(h)g_I(Y,∂_t)$$
where $I$ is an an open connected interval of the real line $\bf R$ furnished by the metric $g_I=-dt^2$ and $dt^2$ is the Euclidean metric.

My question is(it seems to be silly):
Is the following simplification of the identity right? $$\begin{align} L_{h∂_t}g_I(Y,Z) &=Y(h)g_I(Z,∂_t) +Z(h)g_I(Y,∂_t)\\ &=\dot{h}g_I(Z,Y) +\dot{h}g_I(Y,Z)\\ &=2\dot{h}g_I(Y,Z) \end{align}$$ If it is right, the authors didn't use it why? Thanks in advance.

Answer 1

Your simplification is correct. It shouldn't be too surprising to find such a simple expression - since $I$ is one-dimensional, every $(0,2)$-tensor on it is the product of $g_I$ with some smooth function.

I'm unsure why the authors didn't make this simplification - perhaps because the equation they give is slightly more general, while your simplification relies on the fact that $Y,Z$ must be multiples of $\partial_t$ in one dimension. Perhaps they just missed it.