Effective theory for close limit of two branes
  We discuss the effective theory for the close limit of two branes in a covariant way. To do so we solve the five dimensional Einstein equation along the direction of the extra dimension. Using the Taylor expansion we solve the bulk spacetimes and derive the effective theory describing the close limit. We also discuss the radion dynamics and braneworld black holes for the close limit in our formulation. 
Tetsuya Shiromizu, Kazuya Koyama, Keitaro Takahashi