Introduction

C.1 Introduction

The idea that an apple and the Moon fall with the same acceleration led Newton to the discovery of the law of gravity. This is not trivial because the Moon is self-gravitating and an apple is not. That any test particle moves on a geodesic of an external gravitational field is called the weak equivalence principle (WEP). Including self-gravitating objects in this statement is leads to the strong equivalence principle (SEP). The SEP is now experimentally verified with an accuracy of better than $1.5 × 10−13$ [4 ]. The verification of WEP and SEP is regarded as one of the most important experiments in physics because it plays a fundamental role in any theory of gravity. In fact, Einstein regarded the WEP as the starting point of his theory of gravity. However it is not obvious that a theory constructed that way respects the SEP, and there have been investigations to confirm this. It is then natural to ask how far one can generalize the principle within the framework of general relativity and other theories of gravity.

This is not only of academic interest but also has practical importance. There are already several detectors of gravitational waves around the world and we are expecting to directly detect gravitational waves from astronomical sources in the near future and hopefully to open a new window to the universe by gravitational waves. In order to use gravitational waves as a practical tool in astronomy, it is definitely necessary to have a good understanding of the equations of motion for systems with more general situations such as small compact objects like a neutron star/black hole moving at an arbitrary speed in an arbitrary external field. In such a situation the perturbation of the external field including gravitational waves generated by the orbital motion is not negligible. This is exactly the situation we have in mind here and for which we would like to generalize the equivalence principle. In this respect it should be mentioned that Mino et al. and others derived the equations of motion for a point particle with mass $m$ which is represented by a Dirac delta distribution source in an arbitrary background. The equation is interpreted as the geodesic equation on the geometry determined by the external field and the so-called tail part of the self-field of the particle in the first order in $m$ [67 , 121 , 133 ]. Furthermore, Mino et al. used another approach, the matched asymptotic expansion, to obtain the equations of motion without employing the concept of a point particle and thus avoiding divergences in their derivation.

We avoid using a singular source and make use of the point particle limit to derive directly the geodesic equations on the smooth part of the geometry around the object. Here the point particle limit is the strong field point particle limit [81 ]. The smooth part includes the gravitational waves emitted by the orbital motion of the object, and thus the equivalence principle is generalized to including the emission of gravitational waves. We believe that our approach simplifies the proof that the Mino–Sasaki–Tanaka equations of motion are applicable to a nonsingular source where Mino et al. used the matched asymptotic expansion for their proof.