ClausiusToPernull · section of the QIQT-H book

QIQTH.ClausiusToPernull

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ClausiusToPernull · entries 12–12 of 1000

Lemma 12 (BL_smul_sub).  source ↗

BL is linear in its tensor argument: BL(a·T − R) = a·BL T − BL R. The bilinear form ∑_{ij} C_{ij} v^i v^j distributes over the heat tensor a·T − Ric.

(λij↦a⋅T i j−R i j)(v,v)=a⋅(T)(v,v)−(R)(v,v)\href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({\lambda i j \mapsto a \cdot T\,i\,j - R\,i\,j})({v},{v})} = a \cdot \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({T})({v},{v})} - \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-bl}{({R})({v},{v})}

Proof. Immediate from the definitions. □\square

Used by bl_pernull_of_modular.

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