General effective theories of black holes in the large D limit

A bstract We derive the general form of the effective equations governing black hole dynamics in the limit of a large number of dimensions D . These split into a universal soap-bubble embedding condition for stationary configurations and a set of nonlinear dynamical evolution equations describing near-horizon fluctuations of O (1/ D ) amplitude over horizon scales of $$ O\left(1/\sqrt{D}\right) $$ O 1 / D . We obtain these equations in full generality, including arbitrary asymptotic sources in the near-horizon region, and we show that they form a parabolic system with a well-posed initial value problem. To connect the various approaches to large- D black hole dynamics, we also show that both the embedding and dynamical equations can be derived from the covariant membrane formalism. We clarify the intrinsic scope of the large- D approach, emphasizing that it yields a well-posed dynamical evolution only on horizon scales of $$ O\left(1/\sqrt{D}\right) $$ O 1 / D , which is the range where the most relevant horizon dynamics occur. Our results highlight the versatility of these effective theories for studying a wide class of black hole phenomena.