Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case
Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case
复制标题
使用度量共形爱因斯坦场方程构建反德西特时空：真空情况
DOI：
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发表时间：
2018
影响因子：
3.5
通讯作者：
J. A. Valiente Kroon
中科院分区：
文献类型：
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作者：
D. A. Carranza;J. A. Valiente Kroon
我们利用共形爱因斯坦场方程的度量版本，通过适当提出的初始边值问题来构造反德西特时空。与该初始边值问题相关的演化系统由一组用于多个共形场的共形波动方程和共形度量组成。该公式利用广义波坐标，并允许通过共形规范源函数自由指定共形度量的 Ricci 标量。我们考虑了共形边界处演化方程的狄利克雷边界条件，并表明这些边界条件反过来可以根据共形边界的 3D 洛伦兹度量和由 Weyl 张量的某些分量测量的传入和传出辐射的线性组合来构造。为了表明共形演化方程的解意味着爱因斯坦场方程的解，我们还讨论了该初始边值问题的约束传播。初始超曲面和共形边界相交的角点邻域内初始边值问题的局部解的存在性受到初始数据和边界数据之间的兼容性条件的影响。所描述的结构适合于数值实施，并且应该允许对边界条件进行系统探索。
We make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. We consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3D Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions.