Divergence of mixed II-order tensors

[enzomarino]

Dear friends,
How is the divergence in curvilinear coordinates of a second order mixed tensor defined? I mean, shall I contract the covariant or the contravariant index?? And for both cases which is the physical meaning?

$$\nabla_i N^i_j$$ or $$\nabla_j N^i_j$$?

Thanks a lot,
Enzo

 

[haushofer]

Dear friends,
How is the divergence in curvilinear coordinates of a second order mixed tensor defined? I mean, shall I contract the covariant or the contravariant index?? And for both cases which is the physical meaning?

$$\nabla_i N^i_j$$ or $$\nabla_j N^i_j$$?

Thanks a lot,
Enzo

You should always contract lower indices with upper indices in the first place. I think you're confused because the energy momentum tensor T is symmetric, so it doesn't mind with which index you contract.

The physical meaning of the divergence depends on what the tensor T represents; it depends on what the $$T^{\alpha\beta}$$ means for fixed $$\alpha$$ or fixed $$\beta$$. You could put other conserved quantities in some tensor T; the divergence of one of the indices means then covariant conservation of that quantity.