We study inverse problems for the Einstein equations with source fields in a general form. Under a microlocal linearization stability condition, we show that by generating small gravitational perturbations and measuring the responses near a freely falling observer, one can uniquely determine the background Lorentzian metric up to isometries in a region where the gravitational perturbations can travel to and return. We apply the result to two concrete examples when the source fields are scalar fields (i.e. Einstein-scalar field equations) and electromagnetic fields (i.e. Einstein-Maxwell equations).

中文翻译：

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从引力扰动确定时空结构

我们以一般形式研究具有源场的爱因斯坦方程的反问题。在微局部线性化稳定性条件下，我们表明，通过产生小的引力扰动并测量自由落体观察者附近的响应，人们可以唯一地确定背景洛伦兹度量，直到引力扰动可以传播到和返回的区域中的等距。当源场是标量场（即爱因斯坦-标量场方程）和电磁场（即爱因斯坦-麦克斯韦方程）时，我们将结果应用于两个具体的例子。