Classical effective field theory and caged black holes

Matched asymptotic expansion is a useful technique in general relativity and other fields whenever interaction takes place between physics at two different length scales. Here matched asymptotic expansion is argued to be equivalent quite generally to classical effective field theory (CLEFT) where one (or more) of the zones is replaced by an effective theory whose terms are organized in order of increasing irrelevancy, as demonstrated by Goldberger and Rothstein in a certain gravitational context. The CLEFT perspective has advantages as the procedure is clearer, it allows a representation via Feynman diagrams, and divergences can be regularized and renormalized in standard field theoretic methods. As a side product we obtain a wide class of classical examples of regularization and renormalization, concepts which are usually associated with quantum field theories. We demonstrate these ideas through the thermodynamics of caged black holes, both simplifying the nonrotating case, and computing the rotating case. In particular we are able to replace the computation of six two-loop diagrams by a single factorizable two-loop diagram, as well as compute certain new three-loop diagrams. The results generalize to arbitrary compactification manifolds. For caged rotating black holes we obtain the leading correction for all thermodynamic quantities. The angular momentum is found to nonrenormalize at leading order.