Warped Products and Conformal Rescalings
In case there's any confusion, the following computations belong to a setting where
and of course
Suppose you have a space with coordinates of the form
so that schematically
This is a so-called warped product. A common question is "What is the form of the Ricci tensor?"
Let the dimension of the x subspace - that is to say, how many values i can range over - be denoted by D. Then one obtains the result
where is the Ricci tensor of the metric and is the Ricci tensor of the metric . Note that is the covariant derivative with respect to the metric.
Observe that the form of the modification to is a multiple of the metric tensor for the x subspace.
Another very common computation involves the rescaling
A direct computation in local inertial coordinates for the new metric gives the result
where all covariant derivatives are measured in the unaltered metric , and all indices are raised and lowered using this metric too.