Symmetric and Antisymmetric Tensor Contractions

The contraction of a symmetric tensor with an antisymmetric tensor is always zero. This is because the terms in the expansion of the product cancel each other out in pairs due to the definitions of symmetry and antisymmetry. A key application of this principle is seen when a tensor $T_{a b}$ is contracted with a vector-outer-product $v^a v^b$, which is inherently a symmetric tensor. Since any tensor can be uniquely broken down into its symmetric and antisymmetric components ( $T_{a b}=T_{\{a b\}}+T_{[a b]}$ ), the antisymmetric part ( $T_{[a b]}$ ) will vanish upon contraction with the symmetric $v^a v^b$. As a result, the expression $T_{a b} v^a v^b$ is solely dependent on the symmetric part of the tensor $T_{a b}$, with its antisymmetric component contributing nothing to the final value.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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The contraction of a symmetric tensor with an antisymmetric tensor is always zero. This fundamental property simplifies many tensor calculations.
Any tensor can be uniquely decomposed into a symmetric part and an antisymmetric part. This decomposition, $T_{a b}=T_{\{a b\}}+T_{[a b]}$, is a powerful tool for analyzing tensor properties.
When an arbitrary tensor $T_{a b}$ is contracted with a symmetric tensor formed by the outer product of a vector with itself, $v^a v^b$, the result depends solely on the symmetric part of $T_{a b}$. The antisymmetric part of the tensor does not contribute to the final value.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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