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Initial data in general relativity described by expansion, conformal deformation and drift
Communications in Analysis and Geometry ( IF 0.7 ) Pub Date : 2021-01-01 , DOI: 10.4310/cag.2021.v29.n1.a7
David Maxwell 1
Affiliation  

The conformal method is a technique for finding Cauchy data in general relativity solving the Einstein constraint equations, and its parameters include a conformal class, a conformal momentum (as measured by a densitized lapse), and a mean curvature. Although the conformal method is successful in generating constant mean curvature (CMC) solutions of the constraint equations, it is unknown how well it applies in the non-CMC setting, and there have been indications that it encounters difficulties there. We are therefore motivated to investigate alternative generalizations of the CMC conformal method. Introducing a densitized lapse into the ADM Lagrangian, we find that solutions of the momentum constraint can be described in terms of three parameters. The first is conformal momentum as it appears in the standard conformal method. The second is volumetric momentum, which appears as an explicit parameter in the CMC conformal method, but not in the non-CMC formulation. We have called the third parameter drift momentum, and it is the conjugate momentum to infinitesimal motions in superspace that preserve conformal class and volume form up to independent diffeomorphisms. This decomposition of solutions of the momentum constraint leads to extensions of the CMC conformal method where conformal and volumetric momenta both appear as parameters. There is more than one way to treat drift momentum, in part because of an interesting duality that emerges, and we identify three candidates for incorporating drift into a variation of the conformal method.

中文翻译:

广义相对论的初始数据,由膨胀,共形变形和漂移描述

保形方法是在广义相对论中求解爱因斯坦约束方程式的柯西数据的一种技术,其参数包括保形类,保形动量(通过密集化的时差测量)和平均曲率。尽管共形方法成功地生成了约束方程的恒定平均曲率(CMC)解,但未知的是它在非CMC设置中的应用效果如何,并且有迹象表明它在那里遇到了困难。因此,我们有动机去研究CMC保形方法的替代性概括。在ADM Lagrangian中引入了一个密集的失误,我们发现动量约束的解可以用三个参数来描述。第一个是在标准保形方法中出现的保形动量。第二个是体积动量,它在CMC保形方法中显示为一个明确的参数,但在非CMC公式中则不是。我们称第三个参数为漂移动量,它是超空间中无穷小运动的共轭动量,它保持共形的类和体积形式直至独立的微分形。动量约束解的这种分解导致CMC保形方法的扩展,其中保形动量和体积动量都作为参数出现。处理漂移动量的方法不止一种,部分原因是出现了有趣的对偶性,我们确定了三种将漂移合并到共形方法变体中的候选方法。我们称第三个参数为漂移动量,它是超空间中无穷小运动的共轭动量,它保持共形的类和体积形式直至独立的微分形。动量约束解的这种分解导致CMC保形方法的扩展,其中保形动量和体积动量都作为参数出现。处理漂移动量的方法不止一种,部分原因是出现了有趣的对偶性,我们确定了三种将漂移合并到共形方法变体中的候选方法。我们称第三个参数为漂移动量,它是超空间中无穷小运动的共轭动量,它保持共形的类和体积形式直至独立的微分形。动量约束解的这种分解导致CMC保形方法的扩展,其中保形动量和体积动量都作为参数出现。处理漂移动量的方法不止一种,部分原因是出现了有趣的对偶性,我们确定了三种将漂移合并到共形方法变体中的候选方法。动量约束解的这种分解导致CMC保形方法的扩展,其中保形动量和体积动量都作为参数出现。处理漂移动量的方法不止一种,部分原因是出现了有趣的对偶性,我们确定了三种将漂移合并到共形方法变体中的候选方法。动量约束解的这种分解导致CMC保形方法的扩展,其中保形动量和体积动量都作为参数出现。处理漂移动量的方法不止一种,部分原因是出现了有趣的对偶性,我们确定了三种将漂移合并到共形方法变体中的候选方法。
更新日期:2021-03-11
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