Title: Dynamics of Geometric Invariants in the Asymptotically Hyperboloidal Setting: Energy and Linear Momentum Show Chinese title

Title: 渐近双曲背景下几何不变量的动力学：能量与线性动量

Abstract: We investigate the evolution of geometric invariants, as defined by Michel \cite{Michel}, in the context of asymptotically hyperboloidal initial data sets. Our focus lies on the charges of energy and linear momentum, and we study their behavior under the Einstein evolution equations. We construct foliations describing the evolution of asymptotically hyperboloidal initial data sets using hyperboloidal time function. We define E-P chargeability as a property of the initial data set, and we show that it is preserved under the evolution for our choice of time function. This ensures that the charges are well-defined along the evolution, which is crucial for our approach. Along such foliations, we recover the same energy-loss and linear momentum-loss formulae as those derived by Bondi, Sachs, and Metzner \cite{Bondi-vanderBurg-Metzner} while operating under weaker asymptotic assumptions. Our approach is distinct from previous work as we do not utilize conformal compactifications and work directly at the level of the initial data set. Show Chinese abstract

Abstract: 我们研究了几何不变量的演化，这些不变量由米歇尔定义的\cite{Michel}在渐近双曲初始数据集的背景下。我们的重点是能量和线性动量的荷载，并研究它们在爱因斯坦演化方程下的行为。我们使用双曲时间函数构建描述渐近双曲初始数据集演化的叶层。我们将 E-P 充电性定义为初始数据集的一个性质，并证明在我们所选的时间函数下它是守恒的。这确保了沿演化过程这些荷载是良定义的，这是我们的方法的关键所在。沿着这些叶层，我们在较弱的渐近假设下，得到了与邦迪、萨克斯和梅茨纳 \cite{Bondi-vanderBurg-Metzner}所推导出的能量损失和线性动量损失公式相同的结果。我们的方法与之前的工作不同之处在于，我们没有使用共形紧化，在初始数据集的层面上直接工作。