Rotating stratified ellipsoids of revolution and their effects on the dragging of inertial frames

We determine here the off-diagonal metric components (the Lense-Thirring terms) resulting from the rotation of an oblate ellipsoid of revolution stratified into similar concentric elliptic layers of equal density. To find the unique function H involved in these off-diagonal components, we use a set of characteristic properties which may be compared with the well-known characteristic properties of the Newtonian potentials. We get H in terms of usual functions when the spinning ellipsoid is homogeneous. Applying this result to the Galaxy, we show that the dragging of inertial frames induced by the galactic rotation is too small to be presently detected. On the other hand, we perform the multipole expansion of H outside the stratified ellipsoid. Assuming then the terrestrial spheroid to be layered into similar ellipsoids following Roche's density law, we calculate the preponderant relativistic multipole term arising in the development of the function H of the earth. The value of this term agrees with what we have previously obtained from Bullen's model A