Schwarzschild度规引力扰动的守恒律和通量界
Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric
摘要：
本文导出了双零规范下史瓦西时空中引力微扰系统的能量守恒定律。由此产生的身份只涉及一阶导数的度量扰动。利用出现的通量边界项的规范不变性，我们能够建立任何输出零超曲面上的通量到初始数据的未来的正性。这使我们能够根据初始数据来约束通过任何这样的超曲面的总能量通量，包括事件视界。同样，我们将辐射的总能量限制在零无穷大。我们的估计提供了一个直接的方法，以弱形式的稳定性，从而补充证明的完全线性稳定的Schwarzschild解决方案最近得到的[M。达费莫斯湾霍尔泽格尔和我。Rodnianski \emph{The linear stability of the Schwarzschild solution to gravitational perturbations}，arXiv：1601.06467].
英文摘要：
We derive an energy conservation law for the system of gravitational perturbations on the Schwarzschild spacetime expressed in a double null gauge. The resulting identity involves only first derivatives of the metric perturbation. Exploiting the gauge invariance up to boundary terms of the fluxes that appear, we are able to establish positivity of the flux on any outgoing null hypersurface to the future of the initial data. This allows us to bound the total energy flux through any such hypersurface, including the event horizon, in terms of initial data. We similarly bound the total energy radiated to null infinity. Our estimates provide a direct approach to a weak form of stability, thereby complementing the proof of the full linear stability of the Schwarzschild solution recently obtained in [M. Dafermos, G. Holzegel and I. Rodnianski \emph{The linear stability of the Schwarzschild solution to gravitational perturbations}, arXiv:1601.06467].
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