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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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up vote 1 down vote accepted

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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sorry just realised i miss posted and had a mistake, making it unclear what i mean –  user21119 Feb 25 '13 at 11:56
    
thx sorted it out –  user21119 Feb 25 '13 at 12:51
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