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Hi When contracting $T^{\mu \nu}$ with $ g_{\mu \nu}$ does one get $T^{\mu \nu}_{\mu \nu} = T$? is the metric tensor already a sum over its component, so it is effectively a trace of a matrix with its components. e.g $$g^{\mu\nu}=Tr A $$ if A is a matrix with the same components as $g^{\mu\nu}$. |
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Contraction implies a sum over indices, i.e. $T^{\mu\nu}g_{\mu\nu}=\sum_{\mu=0}^3\sum_{\nu=0}^3T^{\mu\nu}g_{\mu\nu}=T.$ An expression like $T^{\mu\nu}_{\mu\nu}$ makes no sense, since the amount of indices of $T^{\mu\nu}$ does not change. Furthermore, it does not make a statement about the trace of an object before it is summed. |
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