We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge--Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of nonlinear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.

中文翻译：

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Schwarzschild 解对引力扰动的线性稳定性

我们在本文中证明了广义相对论中著名的 Schwarzschild 黑洞家族的线性稳定性：由常规初始数据产生的围绕 Schwarzschild 度量的爱因斯坦真空方程线性化的解仍然全局有界于黑洞外部，实际上衰减到线性化克尔度量。我们用合适的双零点规范来表达方程。为了获得衰减，实际上必须添加一个残留的纯规范解，我们证明它本身可以从初始数据进行定量控制。我们的结果更重要的是包括 Teukolsky 方程的一般解的衰减陈述（满足规范不变零分解曲率分量）。后面的这些陈述实际上是在证明过程中通过利用显示的满足 Regge-Wheeler 方程的相关量推导出来的，通过调整先前关于线性标量波动方程的工作，可以很容易地获得适当的衰减。线性化克尔衰减率的界限是多项式的，这表明色散足以在未来非线性稳定性的潜在证明中控制爱因斯坦方程的非线性。这篇论文是独立的，包括从用双零位规表示的完整非线性爱因斯坦真空方程对 Schwarzschild 周围线性化重力方程的物理空间推导。线性化克尔衰减率的界限是多项式的，这表明色散足以在未来非线性稳定性的潜在证明中控制爱因斯坦方程的非线性。这篇论文是独立的，包括从用双零位规表示的完整非线性爱因斯坦真空方程对 Schwarzschild 周围线性化重力方程的物理空间推导。线性化克尔衰减率的界限是多项式的，这表明色散足以在未来非线性稳定性的潜在证明中控制爱因斯坦方程的非线性。这篇论文是独立的，包括从用双零位规表示的完整非线性爱因斯坦真空方程对 Schwarzschild 周围线性化重力方程的物理空间推导。