A Mathematical Comparison of the Schwarzschild and Kerr Metrics

Abstract

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 Poincar\'{e} sequence for the exterior derivative by the corresponding Lie algebra.