修正引力的f（R）理论的数学有效性
Mathematical Validity of the f(R) Theory of Modified Gravity
摘要：
我们建立了修正引力的f（R）理论的适定性理论，它是爱因斯坦引力理论的推广。时空的数量曲率R，它产生于爱因斯坦-希尔伯特泛函的被积函数，被函数f=f（R）所代替。场方程涉及到时空度规的四阶导数，挑战在于理解这些高阶项的结构。当初始数据被指定在超曲面上时，我们提出了修正重力学中的初值问题的一个公式。除了初始切片和初始物质含量的诱导度规和第二基本型外，初始数据集还必须提供时空数量曲率及其时间导数。我们引入一个增广的共形公式，我们称之为，其中时空曲率是一个自变量。在所谓的波规范中，修正引力场方程简化为具有散焦势的非线性波-Klein-Gordon方程耦合系统，其未知量为保角变换的度规、数量曲率和物质场。当物质用标量场表示时，我们能够建立最大全局双曲发展的存在性。我们分析了所谓的约旦耦合和工作与爱因斯坦度规，这是共形等效的物理度量-共形因子取决于数量曲率。我们的分析使我们严格验证了修正引力理论。我们得到了关于非线性函数f（R）的一致的定量估计，并证明了当修正引力作用泛函中的函数f（R）“接近”Einstein-Hilbert被积函数R时，修正引力的渐近平坦时空“接近”Einstein时空.
英文摘要：
We establish a well-posedness theory for the f(R) theory of modified gravity, which is a generalization of Einstein's theory of gravitation. The scalar curvature R of the spacetime, which arises in the integrand of the Einstein-Hilbert functional, is replaced by a function f=f(R). The field equations involve up to fourth-order derivatives of the spacetime metric, and the challenge is to understand the structure of these high-order terms. We propose a formulation of the initial value problem in modified gravity when the initial data are prescribed on a hypersurface. In addition to the induced metric and second fundamental form of the initial slice and the initial matter content, an initial data set must also provide the spacetime scalar curvature and its time-derivative. We introduce an augmented conformal formulation, as we call it, in which the spacetime curvature is an independent variable. In the so-called wave gauge, the field equations of modified gravity reduce a coupled system of nonlinear wave-Klein-Gordon equations with defocusing potential, whose unknowns are the conformally-transformed metric and the scalar curvature, as well as the matter fields. We are able to establish the existence of maximal globally hyperbolic developments when the matter is represented by a scalar field. We analyze the so-called Jordan coupling and work with the Einstein metric, which is conformally equivalent to the physical metric --the conformal factor depending upon the scalar curvature. Our analysis leads us to a rigorous validation of the theory of modified gravity. We derive quantitative estimates, which are uniform in terms of the nonlinearity f(R), and we prove that asymptotically flat spacetimes of modified gravity are `close' to Einstein spacetimes, when the function f(R) in the action functional of modified gravity is `close' to the Einstein-Hilbert integrand R.
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