Abstract
We prove that sufficiently regular solutions to the wave equation \({\square_g\phi=0}\) on the exterior of the Schwarzschild black hole obey the estimates \({|\phi|\leq C_\delta v_+^{-\frac{3}{2}+\delta}}\) and \({|\partial_t\phi|\leq C_{\delta} v_+^{-2+\delta}}\) on a compact region of r, including inside the black hole region. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite r and along the event horizon.
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Communicated by Piotr T. Chrusciel.
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Luk, J. Improved Decay for Solutions to the Linear Wave Equation on a Schwarzschild Black Hole. Ann. Henri Poincaré 11, 805–880 (2010). https://doi.org/10.1007/s00023-010-0043-6
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DOI: https://doi.org/10.1007/s00023-010-0043-6