GTR FAQ

Covariant differentiation

Is the differential of a covariant scalar, dϕ, a covariant scalar?

 
Yes, it is.

Is the differential of a covariant vector, dA^a, a covariant vector?

 
Only in Minkowski space, where dg_ab=0. Generally dA_a is not a covariant vector, since in curvilinear coordinates dA_a=d(g_abA^b) ≠ g_abdA^b.

Is the covariant differential of a covariant vector, DA^a, a covariant vector?

 
Yes, it is, since DA_a=g_abDA^b=D(g_abA^b).

What are the definitions of ∂_a, D_a, _,a, and _;a?

 
∂_af ≡ f_,a ≡ [(df)/(dx^a)], D_af ≡ f_;a ≡ [(Df)/(dx^a)]

Is D_a a covariant thing?

 
Yes. In particular D_aϕ is a covariant vector, D_aA_b is covariant tensor and so forth.

Is ∂_a a covariant thing?

 
Only in Minkowski space. Generally, although ∂_aϕ is indeed a covariant vector, ∂_aA_b is NOT a covariant tensor. However, the antisymmetric combination ∂_aA_b−∂_bA_a=D_aA_b−D_bA_a is a covariant tensor.

What is the definition of D^a?

 
Since D_a is a covariant operator, the index in raised in the usual way, D^a=g^abD_b=D_bg^ba.

What is the definition of ∂^a?

 
Generally, since ∂_a is not a covariant operator, one cannot raise the index and therefore ∂^a is not defined. However, in Minkowski space, where the metric tensor is constant, g_ab=η_ab, this thing is defined as ∂^a=η^ab∂_b=∂_bη^ba.

What is the definition of the electromagentic tensor F_ab in curvilinear coordinates?

 
The electromagentic tensor F_ab is defined in a covariant way as F_ab=D_aA_b−D_bA_a ≡ A_b;a−A_a;b. However this particular combination can also be written with ordinary derivatives, D_aA_b−D_bA_a=∂_aA_b−∂_bA_a ≡ A_b,a−A_a,b.

And then what is F^a  _b and F^ab?

 
Since F_ab is a covariant tensor, the indexes are raised in the usual way, F^a  _b=g^acF_cb, F^ab=g^acF^a  _c.

But isn't F^ab=∂^aA^b−∂^bA^a?

 
Only in Minkowski space. Generally, since ∂^a is not defined, the tensor F^ab cannot be written this way in curvilinear coordinates.