We will work in the harmonic gauge,

where the semicolon means that the covariant derivative with respect to the background metric and the trace-reversed variable is defined as usual, Then the linearized Einstein equations take the following form: This can be solved formally as follows, where we have used the retarded tensor Green’s function defined by For the general tensor Green’s function and the definition of , , , , and , please refer to Mino et al. [121] and DeWitt and Brehme [68].Now we take the point particle limit. In this limit the above metric perturbation contains terms of different dependence which makes the calculation of the equations of motion simple. For example, terms with negative powers of appear from the Dirac delta distribution part of the Green’s function:

As explained below we only need the field on the boundary of the body zone which is the far zone of the body itself, and thus we may make use of the multipole expansion for the field. For this purpose we choose our coordinate system as follows,

where is the world line of the center of the object. The center is assumed to be always inside the body in the point particle limit, and thus there is no ambiguity for the choice of the center. In this coordinate system the volume element satisfies . Thus we have where is the retarded time of each point . Then the multipole expansion is obtained by expanding the above expression at the retarded time of the center of the object defined by . This can be easily done by noticing the condition . Then the difference between and is given by Using this we can expand around . In principle we can calculate arbitrarily high multipole moments in this way. Here we only calculate the leading term. Then we only need , and . By defining the mass as follows, we finally obtain the following expression for :Now we derive the equations of motion using this expression. First we define the dependent four-momentum of the object as the volume integral of the effective stress-energy tensor over the body zone ,

Since the effective stress-energy tensor satisfies the conservation law , the change of the four-momentum defined above may be expressed as the surface integral over the boundary of the body zone , where is the unit normal to the surface and is the four-acceleration. The equations of motion are obtained by taking the point particle limit of . Thus we need to calculate on the body zone boundary (field points) on which the multipole expansion of the self-field and the Taylor expansion of the nonsingular part of the field are available, and to pick up only terms of order in the expression. All the other terms but one term, which is proportional to and can be renormalized to the mass of the small object, vanish in the limit or the angular integration. Remembering that the LL tensor is bilinear in the Christoffel symbols and the Christoffel symbols are the derivatives of the metric tensor, one may realize that the only remaining terms come from the combination of the 0th order of the smooth part of the metric and part of the self-field which is given by Equation (252). The remaining part of the self-field is the so-called tail part that is regular at the object which is the only relevant point in our calculation. The field point is now on the body zone boundary, which is defined by andUsing this expression we have the following result in the point particle limit,

where the Christoffel symbols here are calculated in terms of the smooth part of the metric . Then the ADM mass is related to the four-momentum as follows, which is supported by the higher order post-Newtonian approximation [91, 95, 91]: Finally we have which is the geodesic equation on the geometry determined by the smooth part of the metric around the compact object.In fact, the spin effect on the equations of motion can be derived in a similar way and the standard result [152] can be obtained.

We have proved that a small compact object moves on the geodesic determined by only the smooth part of the geometry around the object. Thus the equations of motion are automatically obtained by determining the geometry around the object which is of course an implicit functional of the world line of the object. The smooth part contains the gravitational waves emitted by the orbital motion so that this equation includes the damping force due to radiation reaction. Our method avoids using a singular source in the first place by making use of the strong field point particle limit. All the quantities should be evaluated at the surface of the body zone boundary and thus we only need the dependence of the distance from the center of the object, namely the dependence of the field. In this way we are able to avoid using any divergent quantities in any part of our calculation. This strongly suggests that our method may be used to get unique equations of fast motion with radiation reaction. This will be investigated in future publications.

In this section, we have assumed spherical symmetry of the compact object except for the tidal effect. It is straightforward to generalize the case to multipole moments in our formalism. This will also be studied in future publications.

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