Canonical energy and Hertz potentials for perturbations of Schwarzschild spacetime
Canonical energy and Hertz potentials for perturbations of Schwarzschild spacetime
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DOI：
10.1088/1361-6382/aae9ae
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发表时间：
2018-12-06
影响因子：
3.5
通讯作者：
Wald, Robert M.
中科院分区：
文献类型：
--
作者：
Prabhu, Kartik;Wald, Robert M.
正则能是分析黑洞时空线性稳定性的一个有价值的工具;正则能对所有扰动为正意味着模式稳定性，而对任何扰动不为正意味着不稳定性。然而，即使在四维史瓦西时空的情况下，这是已知的稳定的，明确的正的正则能量是很难建立，由于存在的约束条件的初始数据以及规范依赖的正则能量被积函数。考虑由赫兹势产生的微扰似乎是改善这种情况的一种有希望的方法，因为当正则能量用赫兹势表示时，约束和规范依赖性被消除了。证明了由赫兹势产生的史瓦西度规微扰的正则能为正。我们涉及的能量量中产生的线性稳定性证明的Dafermos，Holzegel和Rodnianski（DHR）的正则能量的相关度量扰动产生的赫兹潜在的。我们还将DHR的Regge-Wheeler变量与相关扰动的普通Regge-Wheeler和扭曲势变量联系起来。由于Hertz势形式可以推广到Kerr黑洞，我们的结果可能有助于分析Kerr的线性稳定性。
Canonical energy is a valuable tool for analyzing the linear stability of black hole spacetimes; positivity of canonical energy for all perturbations implies mode stability, whereas the failure of positivity for any perturbation implies instability. Nevertheless, even in the case of 4D Schwarzschild spacetime-which is known to be stable-manifest positivity of the canonical energy is difficult to establish, due to the presence of constraints on the initial data as well as the gauge dependence of the canonical energy integrand. Consideration of perturbations generated by a Hertz potential would appear to be a promising way to improve this situation, since the constraints and gauge dependence are eliminated when the canonical energy is expressed in terms of the Hertz potential. We prove that the canonical energy of a metric perturbation of Schwarzschild that is generated by a Hertz potential is positive. We relate the energy quantity arising in the linear stability proof of Dafermos, Holzegel and Rodnianski (DHR) to the canonical energy of an associated metric perturbation generated by a Hertz potential. We also relate the Regge-Wheeler variable of DHR to the ordinary Regge-Wheeler and twist potential variables of the associated perturbation. Since the Hertz potential formalism can be generalized to a Kerr black hole, our results may be useful for the analysis of the linear stability of Kerr.