论文标题
Schwarzschild和Kerr指标的数学比较
A Mathematical Comparison of the Schwarzschild and Kerr Metrics
论文作者
论文摘要
最近，一些物理学家构建了Minkowski（M），Schwarzschild（S）和Kerr（K）指标的杀戮操作员的生成兼容条件（CC）。他们发现了二阶CC，CC，以M为众所周知，也是S和K的三阶CC。在最近的一篇论文中，我们研究了M和S的案例，没有使用Teukolski标量（例如Teukolski标量）或杀死YANO张量。但是，即使s（$ m $）和k（$ m，a $）也取决于持续的参数，以至于当$ m \ rightarrow 0 $ 0 $和k $ \ rightarrow $ \ rightArrow $ s时，当$ a \ rightarrow 0 $时，s cc of m \ rightarrow cc不提供$ m \ rightarrow y时， 0 $。在本文中，使用具有恒定或可变参数的操作员的棘手激励示例，我们解释了为什么CC取决于参数的选择。特别是，唯一可以定义的纯固有对象（即扩展模块）可能会发生巨大变化。由于代数支架与{\ it延长/投影}（pp）过程兼容，因此我们首次以固有方式提供了K的所有CC，这表明它们仅依赖于基本的杀戮代数，而Spencer操作员的作用至关重要。我们获得了k $ <$ s $ <$ m，杀死代数的$ 2 <4 <10 $，并解释为什么对M，S或K的CC正式搜索也有很大的不同，即使每个Spencer序列都与相应的lie algebra相应的poincaré序列的poincaré序列的张量相同。
A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M) , Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also third order CC for S and K. In a recent paper, we have studied the cases of M and S, without using specific technical tools such as Teukolski scalars or Killing-Yano tensors. However, even if S($m$) and K($m,a$) are depending on constant parameters in such a way that S $\rightarrow $ M when $m \rightarrow 0$ and K $\rightarrow$ S when $ a \rightarrow 0$, the CC of S do not provide the CC of M when $m \rightarrow 0$ while the CC of K do not provide the CC of S when $a\rightarrow 0$. In this paper, using tricky motivating examples of operators with constant or variable parameters, we explain why the CC are depending on the choice of the parameters. In particular, the only purely intrinsic objects that can be defined, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the {\it prolongation/projection} (PP) procedure, we provide for the first time all the CC for K in an intrinsic way, showing that they only depend on the underlying Killing algebras and that the role played by the Spencer operator is crucial. We get K$<$S$<$M with $2 < 4 < 10$ for the Killing algebras and explain why the formal search of the CC for M, S or K are strikingly different, even though each Spencer sequence is isomorphic to the tensor product of the Poincaré sequence for the exterior derivative by the corresponding Lie algebra.
