### 8.4 Third post-Newtonian equations of motion

By adding to Equation (171), we obtain our 3 PN equations of motion for two spherical compact stars whose representative points are defined by Equation (176),
in the harmonic gauge.

Now we list some features of our 3 PN equations of motion. In the test-particle limit, our 3 PN equations of motion coincide with a geodesic equation for a test-particle in the Schwarzschild metric in harmonic coordinates (up to 3 PN order). Suppose that star is a test particle, star is represented by the Schwarzschild metric, and . Then the geodesic equation for star becomes

in the harmonic gauge. Thus, in the test particle limit Equation (181) coincides with the geodesic equation for a test particle in the Schwarzschild metric up to 3 PN order.

With the help of the formulas developed in [28], we have checked the Lorentz invariance of Equation (181) (in the post-Newtonian perturbative sense). Also, we have checked that our 3 PN acceleration admits a conserved energy of the binary orbital motion (modulo the 2.5 PN radiation reaction effect). In fact, the energy of the binary associated with Equation (181) is

This orbital energy of the binary is computed based on that one found in Blanchet and Faye [27], the relation between their 3 PN equations of motion and our result described in Section 8.5 below, and Equation (167). (After constructing given as in Equation (183), we have checked that our 3 PN equations of motion make to be conserved.)

We note that Equation (171) as well gives a correct geodesic equation in the test-particle limit, is Lorentz invariant, and admits the conserved energy. These facts can be seen by the form of , Equation (175); it is zero when , is Lorentz invariant up to 3 PN order, and is the effect of the mere redefinition of the dipole moments which does not break energy conservation.

Finally, we here mention one computational detail. We have retained during our calculation -dependent terms with positive powers of or logarithms of . As stated below Equation (76), it is a good computational check to show that our equations of motion do not depend on physically. In fact, we found that the -dependent terms cancel each other out in the final result. There is no need to employ a gauge transformation to remove such an dependence. As for terms with negative powers of , we simply assume that those terms cancel out the dependent terms from the far zone contribution. Indeed, Pati and Will [129130], whose method we have adopted to compute the far zone contribution, have proved that all the -dependent terms cancel out between the far zone and the near zone contributions through all post-Newtonian orders.