Smooth perturbations of Schwarzschild black holes whose initial data has compact support outside the horizon are shown to die in time along the trajectory of the asymptotically timelike Killing vector tα. A gravitational or other zero-rest-mass perturbation of a Schwarzschild black hole can be expressed in terms of radial derivatives of a scalar field Φ that satisfies a wave equation with positive potential. A theorem due to Wilcox is used to show that the pointwise limit as t→±∞ of Φ and all its derivatives vanishes. This result strengthens previous work that bounds the perturbation in time.

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