If we look at the rotational transformation (specifically in the x-y plane) we get x' = cos(wt)*x - sin(wt)*y y' = cos(wt)*y + sin(wt)*x z' = z t' = t and we can write dx' = cos(wt)*dx - sin(wt)*dy -w*(y*cos(wt)+x*sin(wt))*dt dy' = cos(wt)*dy + sin(wt)*dx + w*(x*cos(wt)-y*sin(wt))*dt dz' = dz dt' = dt We can intepret dx,dy,dz, and dt as one-forms, the basis of the cotangent space. When we substitute x=y=z=t=0, we find that dx' = dx dy' = dy and of course dz' = dz dt' = dt This means that rotation in x and y leaves the basis one-forms invariant at the origin (the origin of the rotation). Letting u_1 = dx, u_2=dy, u_3 = dz, and u_4 = dt, we can ask what the basis vectors of the tangent space are. These will be just ui = gijuj Because the basis one-forms are not changed at the origin by rotation, and the above equation converts the basis one-forms into basis vectors, the basis vectors are also not changed at the origin by rotation. The argument appears to me to be completely general - it says that tranforming a tensor quantity into a spatially rotating coordinate system does not have and can never have any effect on a tensor quantity at the origin of the rotation. Does anyone see any flaws to this argument?