Finding znew for Covariant Conservation

[gentleboy]

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

 

[bcrowell]

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.

 

[gentleboy]

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.