Local anisotropy of space in a frame of reference co-moving with the   Earth
  We consider, in the framework of General Relativity, the linear approximation of the gravitational field of the Earth taking into account its mass, its quadrupole moment, its shape and its diurnal rotation.   We conclude that in the frame of reference co-moving with the Earth the local anisotropy of the space is of the order of $10^{-12}-10^{-13}$ and could be observed. 
Ll. Bel and A. Molina