Can I rewrite the 4-D covariant derivative using the induced 3-D covariant ?

[adsense yanyue]

The relationship is 

I tried the following method, but it didn't work. 

( CD  is the 4-D covariant derivative, n is the timelike vector used to induce the 3-D metric, gamma is the 3-D metric. V is a 4-D vector)

I am a beginer, and I am sorry to ask this "may stupied" question.

Thanks !

[Alex Vañó Viñuales]

Have you tried ToInducedDerivative ? 

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[adsense yanyue]

Hi Alex,

I tried ToInducedDerivative as : (psi is a scalar function and vv is a vector.)

However, as you can see, it just works for scalar.

It do nothing to a vector.

Thankyou very much.

​

[adsense yanyue]

Hi Alex, 

It's my mistake, it works for a pure spatial vector.

Thanyou again.

[Obinna Umeh]

Hi Adsense,

This may be coming late. Just in case you haven't resolved it yet.

You should define your vector vv^a such that it is orthogonal to n^a and also tell xTensor to project with metric3.

For example:

DefTensor[vv[-a]], {M}, OrthogonalTo -> {n[a]},  ProjectedWith -> {metric3[a, -b]}, PrintAs -> "v"] 

and then do

ToInducedDerivative[CD[-a][vv[-b]], CD, cd3]

It should work. Remember to automatic the normalization of your 4-vector.

Obinna