Astron. Astrophys. 333, 1100-1106 (1998)

Appendix: Expansion of the 1PN vector potential under the standard PN gauge and rigidity approximation

In 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

we have

Doing with

we obtain

and

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

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