Rigidity of asymptotically $Ad{S}_2\times S^2$ spacetimes

Abstract

The spacetime $Ad{S}_2\times S^2$ is well known to arise as the 'near horizon' geometry of the extremal Reissner-Nordstrom solution, and for that reason it has been studied in connection with the AdS/CFT correspondence. Here we consider asymptotically $Ad{S}_2\times S^2$ spacetimes that obey the null energy condition (or a certain averaged version thereof). Supporting a conjectural viewpoint of Juan Maldacena, we show that any such spacetime must have a special geometry similar in various respects to $Ad{S}_2\times S^2$, and under certain circumstances must be isometric to $Ad{S}_2\times S^2$.

26 pages, v2: minor changes to the introduction and abstract