Near horizon linearized gravity and soft theorem

In this paper, we study the linearized gravity theory in the near horizon region of the Schwarzschild black hole in four dimensional spacetime. Under the Newman-Unti gauge, we derive the most general near horizon symmetry and solution space without any near horizon fall-off condition. There are four towers of surface charges that are generic functions on the horizon associated to the near horizon symmetry. With suitable near horizon fall-off conditions, we reveal a soft graviton theorem from the Ward identity of the near horizon supertranslation in both coordinates space and momentum space.