A Ricci flow with a vector field

Question

Let $(M,g)$ be a riemannian manifold with a vector field $X$. Over the manifold ${\bf R}.M$, I consider the metric:

$$g_X=g+ X^* \otimes X^* + dt \otimes X^*+ X^* \otimes dt + dt \otimes dt$$

Then, the Ricci curvature decomposes following $M$:

$$Ric(g_X)=Ric_M (g_X)+ \tilde Ric (g_X)$$

I consider the Ricci flow:

$$\frac{\partial g}{\partial t}=-2Ric_M (g_X)$$

We can take:

$$\frac{\partial X}{\partial t}= dr^*$$

With $r$, the scalar curvature.

Have we solutions of the Ricci flow for short time?