Mars–Simon tensor and its smooth weighted extension

Use the preceding AIK-convention tensor and potential data. On $U^{\mathrm{\prime }}=\{\sigma \mathrm{\ne }1\}$ define ${\mathcal{Q}}_{abcd}=(1-\sigma {)}^{-1}({\mathcal{F}}_{ab}{\mathcal{F}}_{cd}-\frac{1}{3}{\mathcal{F}}^2{\mathcal{I}}_{abcd}),\mathcal{S}=\mathcal{R}+6\mathcal{Q}\mathrm{.}$ This $\mathcal{S}$ is the source's Mars–Simon tensor. Its weighted version is the globally smooth covariant four-tensor $\mathcal{W}=(1-\sigma )\mathcal{R}+6(\mathcal{F}\otimes \mathcal{F}-\frac{1}{3}{\mathcal{F}}^2\mathcal{I})\mathrm{.}$ On $U^{\mathrm{\prime }}$ it equals $(1-\sigma )\mathcal{S}$. At points where $\sigma =1$, $\mathcal{W}$ is defined by this polynomial formula, not by dividing by zero. No vanishing, Kerr-characterization, Weyl-symmetry or propagation theorem is inferred from these formulas alone.

Given: The smooth potential and tensor fields of [F1], the complex convention of [F2], and the real tensor operations of [F3].

1.1 For smooth real tensor fields $A,B,C,D$, define the complex tensor product by $(A+iB)\otimes (C+iD)=(A\otimes C-B\otimes D)+i(A\otimes D+B\otimes C)$. Bilinearity of the real tensor product makes this intrinsic and complex bilinear; [F3] makes each real and imaginary component smooth. Contraction extends by the same real and imaginary decomposition. Smooth scalar multiplication and sums are smooth componentwise, so these operations apply to all products and contractions in the Definition. [F1, F2, F3, algebra]

2.1 The set $U^{\mathrm{\prime }}$ is open because $\sigma$ is continuous. There $(1-\sigma {)}^{-1}$ is smooth: for $1-\sigma =x+iy$, its real and imaginary components are $x\mathrm{/}(x^2+y^2)$ and $-y\mathrm{/}(x^2+y^2)$, with nonzero denominator. Thus the displayed $\mathcal{Q}$ and $\mathcal{S}$ are smooth on $U^{\mathrm{\prime }}$. Multiplication by $1-\sigma$ gives exactly $\mathcal{W}$. Its polynomial formula uses only smooth products, contractions and sums, so it defines a smooth covariant four-tensor throughout the potential domain, including $\sigma =1$. This proves the weighted identity and well-definedness without assuming any rigidity conclusion. [F1, F2, F3, step 1.1, algebra] ∎