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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?
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?