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

2. BD moments expansion of the 1PN Earth's gravitational potential

In this section, we first introduce some relevant results in Damour et al. (1991).

The Earth's local coordinates are taken as DSX coordinates , . The corresponding metric tensor is . The coordinates satisfy an algebraic coordinate condition: . Harmonic coordinates and standard PN coordinates are the special cases of DSX coordinates. Let denote the 1PN Earth's gravitational potential, the Earth's mass density and mass current density, here , , and is the stress-energy tensor of the gravitational source. satisfies the continuity equation: . According to DSX theory, satisfies the linearized gravitational field equations

where d'Alembertian . Eq. (1) is gauge invariant under a gauge transformation:

here is an arbitrary function. Eq. (2) is equivalent to a coordinate transformation

To further fix the time coordinate T, it is necessary to add a coordinate gauge condition to the field equations (1). Usually one would choose the harmonic gauge or the standard PN gauge . If we take

Eq. (3) will turn the harmonic coordinates into the standard PN coordinates. If we take the harmonic gauge, the field equations (1) becomes very simple

Its time-symmetric solution is

In Eqs. (4) and (6), the subscript E means the Earth. Solution (6) and the gauge term in Eq. (2) construct the general solution of the field equations (1). It can be expanded in terms of the 1PN mass multipole moments and the spin multipole moments as follows

where , and

where and are multipole-like moments, i.e. so-called "bad moments". They and have the following expressions

The 1PN mass multipole moments and the spin multipole moments defined by Eq. (8) were first introduced by Blanchet and Damour (1989), so they are also called BD moments.

The BD moments expansion of the 1PN gravitational potential, as shown by Eq. (7), depends on the choice of the gauge function . Here we only consider two specific gauges. They respectively belong to the harmonic gauge and the standard PN one.

(i) , this belongs to the harmonic gauge. Damour et al. (1991) called it the skeletonized-body harmonic gauge. Under the gauge, the expansion of the 1PN gravitational potential is fully expressed in terms of BD moments, no "bad moments" and in it. Besides, the expansion of the vector potential is the simplest one in form. Eq. (7) can be written as

and

Eq. (10) shows that appearing in the expression of W breaks the simplicity of the expansion of W in classical mechanics.

(ii) If we take

then it is easy to find that Eq. (12) belongs to the standard PN gauge. The primary advantage of this gauge lies in that the expansion of the 1PN scalar potential is formally identical with that of the Newtonian gravitational potential. This 1PN scalar potential can be viewed as a natural generalization of the Newtonian gravitational potential in the 1PN case. But the expansion of the vector potential is more complicated under this gauge than the one shown by Eq. (11). The reason for that is the gauge transformation will result in some additional terms. After some computation, we can write out the expansion of under the gauge (12)

and

From Eqs. (13) and (14), we can find that these expansions include BD moments and their time-derivatives, and are not involved with "bad moments" and .

The 1PN gravitational effects are gauge-independent, but the gravitational potential is gauge-dependent. The choice of gauge should be decided according to the specific problem under consideration. As for the expansion of the 1PN potential , one of the principles for choosing gauge is to avoid the "bad moments" and appearing in the expansion, which is in accordance with that and have been absorbed into the definition of . Gauge (i) and (ii) both follow this principle. Another is that the choice of gauge should make the expansion of the scalar potential as simple as possible for the reason that the vector potential only produces the 1PN gravitational effect and is negligible in many practical cases. In this paper, we will adopt the standard PN gauge shown by Eq. (12) (hereafter the standard PN gauge always means this gauge unless otherwise stated). This gauge complies with two principles mentioned above. Especially, under this gauge the expansion of the scalar potential is formally identical with that of the Newtonian gravitational potential. This means that the 1PN mass multipole moments generate the 1PN scalar potential in a similar way to that in classical mechanics.

© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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