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Is the covariant derivative of a vanishing tensor component necessarily zero?

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closed as unclear what you're asking by Brandon Enright, Jim, DavePhD, Qmechanic Apr 11 at 10:30

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Yes or no questions don't work well here. Do you have more to ask? –  Brandon Enright Apr 9 at 20:58
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It is not clear what you mean. A vanishing-tensor component or a vanishing component of a tensor? In the first case yes it is zero. In the second case the covariant derivative is not of a component. What you probably mean is if the component of the covariant derivative of the tensor. For example if the 11 component of rank 2 tensor is zero then is it true that the 11 component of the covariant derivative will also be zero. The answer is no, not always. –  MBN Apr 9 at 22:18
    
also, by vanishing, do you mean identically zero or do you mean zero at some spacetime coordinate? If the former, then it is what MBN said. If the latter then it need not be zero ever –  Jim Apr 10 at 13:57

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The question is somewhat ill-defined. This answer is based on what I gather the OP is asking.

No. Generically, the covariant derivative of a tensor is zero only if the entire tensor is zero. Note how the covariant derivative is defined (taking example of a 1-form) $$ (\nabla_\mu A)_\nu = \partial_\mu A_\nu - \Gamma^\lambda_{\mu\nu} A_\lambda $$ Now, suppose a paerticular component of the 1-form identically vanished, say $A_0$. We then find $$ (\nabla_\mu A)_0 = - \Gamma^\lambda_{\mu 0} A_\lambda \neq 0 $$

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