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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 u

^{i}= g

^{ij}u

_{j}

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?