论文标题
一般相对论中的lie-rinehart代数
A Lie-Rinehart algebra in general relativity
论文作者
论文摘要
我们在爱因斯坦方程的初始值场的无限范围内构建了一个lie-rinehart代数。该代数中的支架关系正是初始值问题的约束。 Lie-rinehart代数来自对谎言代数的轻微概括，其中代数由捆捆部分而不是矢量束组成。 （实际上是由Blohmann，Fernandes和Weinstein构建的。然后，lie-rinehart代数由变量的变化来构建。 BV-BFV方法的后果之一是证明约束集的共截相性质遵循的是时空差异下的爱因斯坦方程的不变性。
We construct a Lie-Rinehart algebra over an infinitesimal extension of the space of initial value fields for Einstein's equations. The bracket relations in this algebra are precisely those of the constraints for the initial value problem. The Lie-Rinehart algebra comes from a slight generalization of a Lie algebroid in which the algebra consists of sections of a sheaf rather than a vector bundle. (An actual Lie algebroid had been previously constructed by Blohmann, Fernandes, and Weinstein over a much larger extension.) The construction uses the BV-BFV (Batalin-Fradkin-Vilkovisky) approach to boundary value problems, starting with the Einstein equations themselves, to construct an $L_\infty$-algebroid over a graded manifold which extends the initial data. The Lie-Rinehart algebra is then constructed by a change of variables. One of the consequences of the BV-BFV approach is a proof that the coisotropic property of the constraint set follows from the invariance of the Einstein equations under space-time diffeomorphisms.
