高维 Einstein-Klein-Gordon 系统中经典解的全局存在性与完备性
Global Existence and Completeness of Classical Solutions in Higher Dimensional Einstein-Klein-Gordon System
AI总结：
本文通过新 ansatz 将高维 Einstein-Klein-Gordon 系统约化为一阶积分-微分方程，并利用 Banach 压缩映射证明局部与全局经典解的存在性及相关时空完备性。
摘要：
本文研究高维中引力与标量场耦合系统（称为 Einstein-Klein-Gordon 系统）经典解的全局存在性与完备性。我们引入一个新的 ansatz 函数，将该问题约化为单个一阶积分-微分方程。随后，在适当的 Banach 空间中运用压缩映射。利用 Banach 不动点定理，我们证明存在唯一的不动点，即该理论的解。对于给定初始数据，我们证明了局部与全局经典解的存在性。我们还研究了时空的完备性性质。为此，我们在 Bondi 坐标中针对 $D\geq 4$ 引入一个类质量函数。时空沿未来指向的类时线向外延伸至一个类似黑洞事件视界区域的完备性得到了研究。
英文摘要：
In this paper we study the global existence and completeness of classical solutions of gravity coupled a scalar field system called Einstein-Klein-Gordon system in higher dimensions. We introduce a new ansatz function to reduce the problem into a single first-order integro-differential equation. Then, we employ the contraction mapping in the appropriate Banach space. Using Banach fixed theorem, we show that there exists a unique fixed point, which is the solution of the theory. For a given initial data, we prove the existence of both local and global classical solutions. We also study the completeness properties of the spacetime. Here, we introduce a mass-like function for $D\geq 4$ in Bondi coordinates. The completeness of spacetime along the future directed timelike lines outward to a region which resembles the event horizon of the black hole.
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