Is the Cross Product of two vectors in General Relativity on a 3-Space the same as in “Non-Relativistic” Physics?

Considering the covariant tensor $$C = e_{ijk}A^j B^k = A \times B$$ in a 3 spherical-space diagonal metric $$ds^2 = g_{i,i}dx^{i}\cdot dx^i$$

Isn't $$C$$ the same as "Classical/Newtonian" physics? (Meaning $$C_r = A_{\theta} \cdot B_{\phi} - B_{\theta} \cdot A_{\phi} , C_\theta = ....$$ )

• I'm unclear what you are asking. In GR spacetime is 4D and the Levi-Civita symbol has correspondingly 4 indices. – jacob1729 Mar 26 at 17:12
• @jacob1729 First of all, Sorry for delayed answer. I didn't get notification (don't know why). I know its 4D but I'm referring to 3+1 General Relativity where you get to have a sub-manifold which refers the the surfaces of $t=C_0$ – billy Mar 27 at 11:42

If $$\pi_{ijk}$$ is the ordinary Levi-Civita symbol, then the Levi-Civita tensor has components $$\epsilon_{ijk}=\sqrt{g}\pi_{ijk}$$ (I am assuming the metric is positive definite, but if not then replace $$g$$ with $$|g|$$), where $$g$$ is the determinant of the metric tensor.
Therefore, we have for $$C_{i}=\epsilon_{ijk}A^j B^k$$ $$C_1=\sqrt g(A^2B^3-A^3B^2) \\ C_2=\sqrt g(A^3B^1-A^1B^3) \\ C_3=\sqrt g(A^1B^2-A^2B^3).$$
These are the covariant components however. If one wants the contravariant components, then one must raise the indices. If the metric is non-diagonal, this will cause a mixing of the components, but if the metric is diagonal, then we will simply have $$C^i=g^{ii}C_i$$ (no summation).
Also note that in vector calculus, the components of vectors in say spherical coordinates are usually taken for the orthonormal frame/dreibein $$e_r,e_\vartheta,e_\varphi$$ rather than the holonomic frame $$\partial_r,\partial_\vartheta,\partial_\varphi$$. Since the orthonormal basis vector fields are, well, orthonormal, they satisfy $$e_r\times e_\vartheta=e_\varphi \\ e_\vartheta\times e_\varphi=e_r \\ e_\varphi\times e_r=e_\vartheta,$$ essentially the same relation as the cartesian basis vectors, thus in the orthonormal frame, cross products are calculated the same.