The set of local gauge invariant quantities for linearized gravity on the Kerr spacetime presented by two of the authors (Aksteiner and Bäckdahl in Phys Rev Lett 121:051104, 2018) is shown to be complete. In particular, any gauge invariant quantity for linearized gravity on Kerr that is local and of finite order in derivatives can be expressed in terms of these gauge invariants and derivatives thereof. The proof is carried out by constructing a complete compatibility complex for the Killing operator, and demonstrating the equivalence of the gauge invariants from Aksteiner and Bäckdahl (Phys Rev Lett 121:051104, 2018) with the first compatibility operator from that complex.

两位作者（Aksteiner和Bäckdahl在Phys Rev Lett 121：051104，2018）中提出的Kerr时空上的线性重力的局部规范不变量的集合已证明是完整的。特别地，可以根据这些量规不变量及其导数来表达在导数上局部且有限阶的克尔上的线性重力的量规不变量。证明是通过为Killing运算符构造一个完整的兼容性复合体并通过与该复合体的第一个兼容性运算符证明Aksteiner和Bäckdahl的规范不变式的等价性（Phys Rev Lett 121：051104，2018）来进行的。