Logarithmic Local Energy Decay for Scalar Waves on a General Class of Asymptotically Flat Spacetimes
Logarithmic Local Energy Decay for Scalar Waves on a General Class of Asymptotically Flat Spacetimes
复制标题
一般类渐近平坦时空上标量波的对数局域能量衰减
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Georgios Moschidis
中科院分区：
文献类型：
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作者：
Georgios Moschidis
证明了在一般维Lorentz流形的外传播域上，标量波动方程≥3的解的局部能量至少以逆对数速率衰减.这类洛伦兹流形包括(非极端的)黑洞时空，对囚禁集的性质没有限制。此外，这一类中的时空被允许有一个小的遍历区域，但必须满足能量有界性声明。在不作进一步假设的情况下，这种对数衰减率被证明是急剧的。我们的结果可以看作Burq关于紧致障碍物外平坦空间上波动方程的一个结果的推广，以及Rodnianski-Tao关于渐近锥积洛伦兹流形的结果的推广。这一证明将把罗德尼安斯基和陶渊明(见[58])的观点与达费尔莫斯和罗德尼安斯基(见[21，22])在黑洞环境中发展的技术联系起来。作为我们结果的一个软推论，我们将推导出在不存在遍历区的情况下所考虑的时空上的波动方程的渐近完备性声明。
This paper establishes that on the domain of outer communications of a general class of stationary and asymptotically flat Lorentzian manifolds of dimension $$d+1$$d+1, $$d\ge 3$$d≥3, the local energy of solutions to the scalar wave equation $$\square _{g}{\uppsi }=0$$□gψ=0 decays at least with an inverse logarithmic rate. This class of Lorentzian manifolds includes (non-extremal) black hole spacetimes with no restriction on the nature of the trapped set. Spacetimes in this class are moreover allowed to have a small ergoregion, but are required to satisfy an energy boundedness statement. Without making further assumptions, this logarithmic decay rate is shown to be sharp. Our results can be viewed as a generalisation of a result of Burq, dealing with the case of the wave equation on flat space outside compact obstacles, and results of Rodnianski–Tao for asymptotically conic product Lorentzian manifolds. The proof will bridge ideas from Rodnianski and Tao (see [58]) with techniques developed in the black hole setting by Dafermos and Rodnianski (see [21, 22]). As a soft corollary of our results, we will infer an asymptotic completeness statement for the wave equation on the spacetimes considered in the case where no ergoregion is present.
影响因子：
4.9
作者：
Andersson, Lars;Blue, Pieter
通讯作者：
Blue, Pieter