SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

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

[EQUATION]

Under the gauge (12), the vector potential can be divided into three parts

[EQUATION]

here [FORMULA] comes from Eq. (6), [FORMULA] and [FORMULA] are gauge terms. We first calculate [FORMULA]. According to the formula [FORMULA] [FORMULA] + [FORMULA], where [FORMULA] [FORMULA], and applying (32) on [FORMULA] and [FORMULA], we have

[EQUATION]

Inserting (38) and (39) into (36), [FORMULA] reads

[EQUATION]

Then, we calculate [FORMULA]. Using Eq. (32) and the formula [FORMULA] = [FORMULA], we immediately have

[EQUATION]

At last, we calculate [FORMULA]. For convenience, a caret is attached to a STF tensor, e.g. [FORMULA], [FORMULA] and so on. Now [FORMULA] and [FORMULA] only represent symmetric tensors. From (35) it is easy to obtain

[EQUATION]

Making use of

[EQUATION]

we have

[EQUATION]

Doing with

[EQUATION]

we obtain

[EQUATION]

and

[EQUATION]

Then employing the following formulae

[EQUATION]

After some calculation, we discover

[EQUATION]

Inserting (47) into (42), and now letting [FORMULA] and [FORMULA] in place of [FORMULA] and [FORMULA] to represent the STF multipole moments, then [FORMULA] reads as follows

[EQUATION]

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 [FORMULA] and [FORMULA], under the standard PN gauge and the assumption (32), which is exactly Eq. (27) in Sect. 5.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

helpdesk.link@springer.de