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suppose we have unit vector z=(0,0,0,1), we can use it to form a tensor [itex]z^\mu z^\nu [/itex],

it is easy to check that [itex]∂_\μ( z^\mu z^\nu[/itex]=0 in Minkowski spacetime,

now I want to generalize this equation to general curved spacetime, so that

∇μ ([itex] znew^\mu znew^\nu[/itex])=0.

But I am not sure how to find znew, which should be related to the original z.

Any one can help me? Thanks

it is easy to check that [itex]∂_\μ( z^\mu z^\nu[/itex]=0 in Minkowski spacetime,

now I want to generalize this equation to general curved spacetime, so that

∇μ ([itex] znew^\mu znew^\nu[/itex])=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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