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##T_{\ \ \ \nu}^\mu=g_{\nu\xi}T^{\mu\xi}## and

##T^{\ \nu}_\mu=g_{\mu\xi}T^{\xi\nu}##

are different in general. Obviously when T is symmetric

##T^\mu_{\ \ \ \nu}=T^{\ \nu}_\mu## and they can be denoted as ##T^{\nu}_\mu##.

##T^{\ \nu}_\mu=g_{\mu\xi}T^{\xi\nu}##

are different in general. Obviously when T is symmetric

##T^\mu_{\ \ \ \nu}=T^{\ \nu}_\mu## and they can be denoted as ##T^{\nu}_\mu##.

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