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## Main Question or Discussion Point

This question is probably silly, but suppose I have a contraction of the form ##g_{\mu \nu} C^{\mu \nu} = 0## where ##C^{\mu \nu}## is a tensor* and ##g_{\mu \nu}## is the metric tensor. Can I say that it must vanish for any ##g_{\mu \nu}##, and since in the most general case all ##g_{\mu \nu}## are non zero, then necessarely ##C^{\mu \nu} = 0##?

*##C^{\mu \nu}## is a symmetric tensor.

*##C^{\mu \nu}## is a symmetric tensor.

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