By Larissa Borissova and Dmitri Rabounski,; edited by Chifu Ebenezer Ndikilar
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Additional info for Fields, Vacuum, and the Mirror Universe
44) This implies that the vector product of two vectors (i. e. an antisymmetric tensor of the 2nd rank) is a pad, oriented in the space according to the directions of its forming vectors. Contraction of an antisymmetric tensor Vαβ with any symmetric tensor Aαβ = Aα Aβ is zero, because Vαα = 0 and Vαβ = −Vβα so that we have Vαβ Aα Aβ = V00 A0 A0 + V0i A0 Ai + Vi0 Ai A0 + Vik Ai Ak = 0 . 32) is zero, because in any antisymmetric tensor all diagonal components are zeroes. Physical observable components V ik (the projections of V αβ on the observer’s spatial section) are analogous to a vector product in a threeV ·i dimensional space, while the quantity √g0·00 , which is the space-time V ik = (mixed) projection of the tensor V αβ , has no equivalent among components of a regular three-dimensional vector product.
Indices in a geometric object, marking its axial components, are found not in tensors only, but in other geometric objects as well. For this reason, if we come across a quantity in by-component notation, this is not necessarily a tensor quantity. In practice, to know whether a given object is a tensor or not, we need to know a formula for this object in a reference frame and to transform it to any other reference frame. For instance, we consider this classic question: are Christoffel’s symbols (i.
160) we obtain the formula g 00 = 1 1− w c2 2 1− 1 vi v i , c2 vi v i = hik v i v k = v 2 . 162) where, in contrast to the regular operators, ∗ is the d’Alembert chr. 163) ∗ 2 ∂ . 164) ∂xi ∂xk Now, we apply d’Alembert operator to an arbitrary four-dimensional vector field Aα Aα = g µν ∇µ ∇ν Aα . 166) B i = hiσ Aσ = hiσ g µν ∇µ ∇ν Aσ . -form for a vector field in a pseudo-Riemannian space is not a trivial task, because the Christoffel symbols are not zeroes, so formulae for projections of the second derivatives take dozens of pages∗.