Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities

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In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild met- rics.

• We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively cor- respond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting.
• Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master quantities satisfying Regge- Wheeler type wave equations in each of the three portions.
• The scalar and co-vector quantities respectively generalize the Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four spacetime dimen- sions; beyond these quantities, we further discover a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector por- tion.