## Appendix: Expansion of the 1PN vector potential under the standard PN gauge and rigidity approximationIn this appendix, we shall briefly derive Eq. (27) in Sect. 5. The standard PN gauge means Eq. (12) in Sect. 2. The rigidity approximation is just the following expression Under the gauge (12), the vector potential can be divided into three parts here comes from Eq. (6), and are gauge terms. We first calculate . According to the formula + , where , and applying (32) on and , we have Inserting (38) and (39) into (36), reads Then, we calculate . Using Eq. (32) and the formula = , we immediately have At last, we calculate . For convenience, a caret is attached to a STF tensor, e.g. , and so on. Now and only represent symmetric tensors. From (35) it is easy to obtain Making use of Doing with Then employing the following formulae After some calculation, we discover Inserting (47) into (42), and now letting and in place of and to represent the STF multipole moments, then reads as follows Finally, inserting (40), (41) and (48) into (33), we obtain the expression of the 1PN vector potential, which is expanded in terms of Cartesian multipole moments and , under the standard PN gauge and the assumption (32), which is exactly Eq. (27) in Sect. 5. © European Southern Observatory (ESO) 1998 Online publication: April 28, 1998 |