Lie derivative of two differnt size related tensors

Question

Let $\bar{M}=I\times M$ be a pseudo-Riemannian manifold equipped with metric $\bar g=-dt^2\oplus f^2g$ where $(M,g)$ is a Riemannian manifold, $I$ is an open connected interval and $f$ is a positive function on $I$.
The curvature tensor $\bar R$ on $\bar M$ is given by
$$\bar R(X,\partial_t,\partial_t,Y)=\frac{-\ddot f}{f}\bar g(X,Y)$$ where $X,Y\in {\frak X }( M)$. Now, I want to get the Lie derivative of $\bar R$ in direction of $h\partial_t$ where $h$ is a smooth function on $I$. Can I use $$(\bar {\frak{L}}_{h\partial _t}\bar R)(X,\partial_t,\partial_t,Y)= (\bar {\frak{L}}_{h\partial _t}\bar g)(\frac{-\ddot f}{f}X,Y)$$ If not, why?
I computed both sides, they are not equal. My calculations may be incorrect.

Answer 1

Why do you think that this would be correct? You are differentiating completely different objects on the left and right hand side. $g$ is not the same as $R$ (I'm omitting the bars).

Check this site (scroll down to 'Lie derivative' of tensor fields) to check how to calculate $${\frak{L}}_V (R(X, Z, Z, Y)) $$

(which results in $$({\frak{L}}_V R)(X, Z, Z, Y) + R({\frak{L}}_V X, Z, Z, Y)+\dots+R( X, Z, Z,{\frak{L}}_V Y) $$ where the first term of the resulting expression is what you are looking at tht left hand side in your eqaution. So to make use of your identity you have to solve for $({\frak{L}}_V R)(X, Z, Z, Y) $ first.)