# Rigidity of asymptotically $AdS_2 \times S^2$ spacetimes

## TL;DR

This paper proves a rigidity theorem for asymptotically $AdS_2 imes S^2$ spacetimes satisfying energy conditions, showing they must be geometrically close or identical to the exact $AdS_2 imes S^2$ solution, supporting conjectures related to holography.

## Contribution

It establishes a rigidity result for asymptotically $AdS_2 imes S^2$ spacetimes under energy conditions, confirming a conjecture about their geometric uniqueness.

## Key findings

- Any such spacetime must resemble $AdS_2 imes S^2$ in structure.
- Under certain conditions, the spacetime is isometric to $AdS_2 imes S^2$.
- Supports the conjectural viewpoint of Maldacena regarding these geometries.

## Abstract

The spacetime $AdS_2 \times S^2$ 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 $AdS_2 \times S^2$ 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 $AdS_2 \times S^2$, and under certain circumstances must be isometric to $AdS_2 \times S^2$.