The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part

In this paper, we study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d’Alembertian equation. In particular, we prove the solution decays at rate ${\tau }^{-1+\delta }$ to a linearlized Kerr solution.