On a post-Galilean transformation appropriate to the post-Newtonian theory of Einstein, Infeld and Hoffmann

Abstract

It is shown that the line element, which forms the basis of the post-Newtonian theory of Einstein, Infeld and Hoffmann for the motion of mass points under their mutual gravitational attractions, is invariant in form to a certain post-Galilean transformation. It is necessary that the transformation, expressed as an expansion in inverse powers of c² (c is the velocity of light), include terms of O(c^-2) in the transformation for the spatial coordinate and terms of O(c^-4) in the transformation for the time coordinate. Comparison with the Lorentz transformation (between two frames in uniform relative motion with a velocity V), expanded similarly in inverse powers of c², shows (1) that the spatial part of the transformation includes the Lorentzian terms (up to O(c^-2)) and allows, in addition, an arbitrary rotation, a uniform motion, and a shift of the origin (all of O(c^-2)) and (2) that the temporal part of the transformation includes the Lorentzian terms (up to O(c^-4)) and an additional term of purely gravitational origin. It is the presence of this last gravitational term that gives to the transformation its non-Lorentzian character. For a suitable choice of the constants in the post-Galilean transformation, the parameter V can be interpreted as a velocity of relative motion between the two frames, even as in the Lorentz transformation. The invariance of the form of the line element to the transformation ensures that the equations of motion which follow are similarly invariant to the transformation. This fact is further verified by showing that the Lagrangians in the two frames differ by the total derivative of a function. The relations between the ten constants of the motion in the two frames are found. And the special case when the transformation can be regarded as one appropriate to a 'centre of mass system' is briefly considered.