We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition B_L of order L=l yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions B_L with L<l, which include the widely used freezing-Psi_0 boundary condition that imposes the vanishing of the Newman-Penrose scalar Psi_0.
The Einstein equations for generalized theories of gravity and the thermodynamic relation $δQ = T δS$ are equivalent
证明了广义引力理论的运动方程与热力学关系式$δQ = T δS$是等价的。我们的证明依赖于扩展以前的参数，使用一个更一般的定义的Noether电荷熵。这样，我们就完成了雅各布森提出的将爱因斯坦方程表示为热力学状态方程的建议的实现。此外，我们还发现，如果物质能量动量张量满足零能条件，则Noether荷熵服从热力学第二定律。我们的结果支持这样一种观点，即宏观尺度上的引力是真空热力学的一种表现。
