Finding znew for Covariant Conservation

gentleboy
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suppose we have unit vector z=(0,0,0,1), we can use it to form a tensor z^\mu z^\nu,
it is easy to check that ∂_\μ( z^\mu z^\nu=0 in Minkowski spacetime,
now I want to generalize this equation to general curved spacetime, so that
∇μ (znew^\mu znew^\nu)=0.
But I am not sure how to find znew, which should be related to the original z.
Any one can help me? Thanks
 
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Welcome to PF! Could you please mark up your equations using LaTeX so they're more readable? Here's an example: z^\mu. To see how I did this, click on the QUOTE button underneath mypost.
 
each time when I tried to submit the revision, the web kept saying it is too short, need to lengthen it to at least 4 characters, i do not know what does that mean.
 

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