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...