Matched asymptotic expansion for caged black holes: Regularization of the post-Newtonian order

The 'dialogue of multipoles' matched asymptotic expansion for small black holes in the presence of compact dimensions is extended to the post-Newtonian order for arbitrary dimensions. Divergences are identified and are regularized through the matching constants, a method valid to all orders and known as Hadamard's partie finie. It is closely related to 'subtraction of self-interaction' and shows similarities with the regularization of quantum field theories. The black hole's mass and tension (and the 'black hole Archimedes effect') are obtained explicitly at this order, and a Newtonian derivation for the leading term in the tension is demonstrated. Implications for the phase diagram are analysed, finding agreement with numerical results and extrapolation shows hints for Sorkin's critical dimension - a dimension where the transition turns second order.