On the hyperbolicity of Einstein's and other gauge field equations
On the hyperbolicity of Einstein's and other gauge field equations
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关于爱因斯坦和其他规范场方程的双曲性
DOI：
10.1007/bf01217728
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发表时间：
1985
影响因子：
2.4
通讯作者：
H. Friedrich
中科院分区：
文献类型：
--
作者：
H. Friedrich
证明了在坐标系形式下的爱因斯坦真空场方程（分别为共形真空场方程）对于任何坐标系和坐标系场（和共形因子）的选择都意味着一个“简化”传播方程的对称双曲系统。在简化方程中出现的某些可自由指定的“规范源”函数反映了规范的选择。与初始数据一起，它们确定了唯一的量规。它们的选择不影响初值问题解的等距类（共形类）。用同样的方法从其他规范场方程得到对称双曲传播方程，而不考虑规范。利用源函数的概念，人们发现爱因斯坦的场方程，作为度规系数的二阶方程，在任何坐标系中都是波动方程。
It is shown that Einstein's vacuum field equations (respectively the conformal vacuum field equations) in a frame formalism imply a symmetric hyperbolic system of “reduced” propagation equations for any choice of coordinate system and frame field (and conformal factor). Certain freely specifiable “gauge source” functions occurring in the reduced equations reflect the choice of gauge. Together with the initial data they determine the gauge uniquely. Their choice does not affect the isometry class (conformal class) of a solution of an initial value problem. By the same method symmetric hyperbolic propagation equations are obtained from other gauge field equations, irrespective of the gauge. Using the concept of source functions one finds that Einstein's field equation, considered as second order equations for the metric coefficients, are of wave equation type in any coordinate system.