论文标题
杀死Kerr指标的操作员
Killing Operator for the Kerr Metric
论文作者
论文摘要
当$ {\ cal {d}}：e \ rightArrow f $是订单$ q $的线性差异操作员，在verctor捆绑套件的各个部分上，在dimension $ n $的歧管$ x $上，它是由bundle map $φ定义的：bundle $φ：j_q（e） ... $。一个“直接问题”是以操作员$ {\ cal {d}} _ 1：f_0 \ rightarrow f_1 $的形式找到生成兼容条件（CC）。当$ {\ cal {d}} $是参与的时，那是当相应的系统$ r_q = ker（φ）$参与其中时，此过程提供连续的一阶参与参与操作员$ {\ cal {d}} _ 1，...，...，{\ cal {d}} _ n n $。虽然$ {\ cal {d}} _ 1 \ circ {\ cal {d}} = 0 $ nimnmim in Ingrize $ ad（{{\ cal {d}}）\ circ ad（{\ cal {d}}} _1） $ ad（{\ cal {d}} _ 1）$并测量此类“差距”导致在差分同源代数中引入扩展模块。它们也可能取决于参数。当$ r_q $不涉及时，标准{\ it延长/投影}（pp）程序通常允许在该预测的$ q+r $下的图像$ r^{（s）_ {q+r} $的图像$ r^{（s s）_ {（s s）_ {（s）_ {q+r} $ Q+r $ qulongation $ q+r $的延长$ q+r $ \ cap j_ {q+r+s}（e）\ subset j_ {r+s}（j_q（e））$是涉及的，但可能高度取决于参数。但是，有时所得系统不再取决于参数，并且扩展模块不取决于参数，因为知道它们不取决于其定义的差分序列。本文的目的是研究Kerr $（m，a）$，schwarzschild $（m，0）$和minkowski $（0，0，0）$参数的上述问题，同时计算夹杂物$ r^{（3）_ 1 \ subset r^subset r^subset r^{（2） J_1（t（x））$用于相应的杀戮操作员。
When ${\cal{D}}: E \rightarrow F$ is a linear differential operator of order $q$ between the sections of vector bundles over a manifold $X$ of dimension $n$, it is defined by a bundle map $Φ: J_q(E) \rightarrow F=F_0$ that may depend, explicitly or implicitly, on constant parameters $a, b, c, ...$. A "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1: F_0 \rightarrow F_1$. When ${\cal{D}}$ is involutive, that is when the corresponding system $R_q=ker(Φ)$ is involutive, this procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ . Though ${\cal{D}}_1 \circ {\cal{D}}=0 $ implies $ad({\cal{D}}) \circ ad({\cal{D}}_1)=0$ by taking the respective adjoint operators, then $ad({\cal{D}})$ may not generate the CC of $ad({\cal{D}}_1)$ and measuring such "gaps" led to introduce extension modules in differential homological algebra. They may also depend on the parameters. When $R_q$ is not involutive, a standard {\it prolongation/projection} (PP) procedure allows in general to find integers $r,s$ such that the image $R^{(s)}_{q+r}$ of the projection at order $q+r$ of the prolongation $ρ_{r+s}(R_q) = J_{r+s}(R_q) \cap J_{q+r+s}(E)\subset J_{r+s}(J_q(E)) $ is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr $(m, a)$, Schwarzschild $(m, 0)$ and Minkowski $(0, 0)$ parameters while computing the dimensions of the inclusions $R^{(3)}_1\subset R^{(2)}_1 \subset R^{(1)}_1 =R_1 \subset J_1(T(X))$ for the respective Killing operators.
