arXiv 2103.113282021-03-21hep-thgr-qc

广义相对论中的绝热解与边界对称性

Adiabatic Solutions in General Relativity and Boundary Symmetries

Emine Şeyma Kutluk

备注：: 100 pages, Thesis based on arXiv:1904.12869

摘要：

利用Manton近似研究了具有边界的空间切片时空的广义相对论绝热解。这种近似告诉我们，对于一个自然形式的拉格朗日理论，一个在真空静态解空间上被描述为慢运动的运动，使能量最小化，是一个很好的近似解。为了将这一点应用于广义相对论，我们首先通过分裂空间和时间并选择高斯法向坐标，将其转化为自然形式，其中时空由其空间切片上的度规描述。然后在Manton之后，我们提出了慢解，使得每个切片是参考切片的缓慢变化的仿射，因此每个解都由空间切片上的向量场描述。这些解具有这样的性质，即作用将成为空间切片边界上向量场的泛函。此外，使用Hodge-Morrey-Friedrichs分解，我们将表明，广义相对论的约束将确定一个唯一的解决方案，为一个给定的边界值。然后，我们评论的真空空间的结构，我们表明是一个（伪）-黎曼齐性空间。我们说明我们的程序为一个特定的参考切片，我们选择：三维欧几里德圆球。

英文摘要：

We investigate adiabatic solutions to general relativity for a spacetime with spatial slices with boundary, by Manton approximation. This approximation tells us for a theory with a Lagrangian in the natural form, a motion that is described as a slow motion on the space of vacua-static solutions that minimize the energy -- is a good approximate solution. To apply this to the case of general relativity we first bring it to the natural form by splitting space and time and choosing Gaussian normal coordinates, where a spacetime is described by the metric on its spatial slices. Then following Manton we propose slow solutions such that each slice is a slowly changing diffeomorphism of a reference slice, and thus each solution is described by a vector field on the spatial slice. These solutions will have the property that the action will become a functional of the vector fields on the boundaries of the spatial slices. Moreover using the Hodge-Morrey-Friedrichs decomposition we will show that the constraints of general relativity will identify a unique solution for a given boundary value. Then we comment on the structure of the space of vacua which we show to be a (pseudo)-Riemannian homogeneous space. We illustrate our procedure for a specific reference slice we choose: the 3d Euclidean round ball.

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