Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operatorvalued) stress-energy bi-tensor which describes the fluctuations of quantum matter fields in curved spacetimes. In the first part, we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity, showing the link from the mean value of the stress-energy tensor to their correlation functions. The functional approach uses the Feynman-Vernon influence functional and the Schwinger-Keldysh closed-time-path effective action methods which are convenient for computations. It also brings out the open systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise, and decoherence. We then focus on the properties of the stress-energy bi-tensor. We obtain a general expression for the noise kernel of a quantum field defined at two distinct points in an arbitrary curved spacetime as products of covariant derivatives of the quantum field's Green function. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime. We offer an analytical solution of the Einstein-Langevin equation and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. Second, we discuss structure formation from the stochastic gravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, we discuss the backreaction of Hawking radiation in the gravitational background of a quasi-static black hole (enclosed in a box). We derive a fluctuation-dissipation relation between the fluctuations in the radiation and the dissipative dynamics of metric fluctuations.

中文翻译：

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随机引力：理论与应用。

半经典引力基于半经典爱因斯坦方程，其源由量子场的应力-能量张量的期望值给出，而随机半经典引力基于爱因斯坦-朗之万方程，由于噪声核，该方程还具有额外的源。噪声核是（算子值）应力-能量双张量的真空期望值，它描述了弯曲时空中量子物质场的涨落。在第一部分中，我们通过两种方法描述了这一新理论的基础：公理化和泛函化。公理化方法有助于从半经典引力的框架中查看理论结构，显示应力-能量张量的平均值与其相关函数之间的联系。泛函法采用了便于计算的Feynman-Vernon影响泛函和Schwinger-Keldysh闭时路径有效作用方法。它还提出了开放系统概念和理论的统计和随机内容，例如耗散、波动、噪声和退相干。然后我们关注应力-能量双张量的性质。我们获得了在任意弯曲时空中的两个不同点处定义的量子场噪声核的一般表达式，作为量子场格林函数的协变导数的乘积。在第二部分中，我们描述了随机引力理论的三个应用。首先，我们考虑 Minkowski 时空中的度量扰动。我们提供了爱因斯坦-朗之万方程的解析解，并计算了线性化爱因斯坦张量和度量扰动的两点相关函数。其次，我们从随机引力的角度讨论结构形成，它可以通过结合暴胀子涨落的全量子效应来超越标准处理。第三，我们讨论了霍金辐射在准静态黑洞（封闭在盒子中）的引力背景中的反向反应。我们推导出辐射波动与公制波动的耗散动力学之间的波动-耗散关系。通过结合暴胀子波动的全部量子效应，它可以超越标准治疗。第三，我们讨论了霍金辐射在准静态黑洞（封闭在盒子中）的引力背景中的反向反应。我们推导出辐射波动与公制波动的耗散动力学之间的波动-耗散关系。通过结合暴胀子波动的全部量子效应，它可以超越标准治疗。第三，我们讨论了霍金辐射在准静态黑洞（封闭在盒子中）的引力背景中的反向反应。我们推导出辐射波动与公制波动的耗散动力学之间的波动-耗散关系。