We derive an expression for the second-order gravitational self-force that acts on a self-gravitating compact-object moving in a curved background spacetime. First we develop a new method of derivation and apply it to the derivation of the first-order gravitational self-force. Here we find that our result conforms with the previously derived expression. Next we generalize our method and derive a new expression for the second-order gravitational self-force. This study also has a practical motivation: The data analysis for the planned gravitational wave detector LISA requires construction of waveforms templates for the expected gravitational waves. Calculation of the two leading orders of the gravitational self-force will enable one to construct highly accurate waveform templates, which are needed for the data analysis of gravitational-waves that are emitted from extreme mass-ratio binaries.Comment: 35 page

Accurate calculation of the gradual inspiral motion in an extreme mass-ratio binary system, in which a compact object inspiral towards a supermassive black hole requires calculation of the interaction between the compact object and the gravitational perturbations that it induces. These metric perturbations satisfy linear partial differential equations on a curved background space-time induced by the supermassive black hole. At the point-particle limit the second-order perturbations equations have source terms that diverge as r ÿ4 , where r is the distance from the particle. This singular behavior renders the standard retarded solutions of these equations ill defined. Here we resolve this problem and construct well-defined and physically meaningful solutions to these equations. We recently presented an outline of this resolution [E. Rosenthal, Phys. Rev. D 72, 121503 (2005).]. Here we provide the full details of this analysis. These second-order solutions are important for practical calculations: the planned gravitational-wave detector LISA requires preparation of waveform templates for the potential gravitational waves. Construction of templates with desired accuracy for extreme mass-ratio binaries requires accurate calculation of the inspiral motion including the interaction with the second-order gravitational perturbations.

The equations for the second-order gravitational perturbations produced by a compact-object have highly singular source terms at the point particle limit. At this limit the standard retarded solutions to these equations are ill-defined. Here we construct well-defined and physically meaningful solutions to these equations. These solutions are important for practical calculations: the planned gravitational-wave detector LISA requires preparation of waveform templates for the expected gravitational-waves. Construction of templates with desired accuracy for extreme mass ratio binaries, in which a compact-object inspirals towards a supermassive black-hole, requires calculation of the second-order gravitational perturbations produced by the compact-object.

Accurate calculation of the motion of a compact object in a background spacetime induced by a supermassive black hole is required for the future detection of such binary systems by the gravitational-wave detector LISA. Reaching the desired accuracy requires calculation of the secondorder gravitational perturbations produced by the compact object. At the point particle limit the second-order gravitational perturbation equations turn out to have highly singular source terms, for which the standard retarded solutions diverge. Here we study a simplified scalar toy-model in which a point particle induces a nonlinear scalar field in a given curved spacetime. The corresponding second-order scalar perturbation equation in this model is found to have a similar singular source term, and therefore its standard retarded solutions diverge. We develop a regularization method for constructing well-defined causal solutions for this equation. Notably these solutions differ from the standard retarded solutions, which are ill-defined in this case.

We present a consistent extended-object approach for determining the self force acting on an accelerating charged particle. In this approach one considers an extended charged object of finite size ǫ, and calculates the overall contribution of the mutual electromagnetic forces. Previous implementations of this approach yielded divergent terms ∝ 1/ǫ that could not be cured by mass-renormalization. Here we explain the origin of this problem and fix it. We obtain a consistent, universal, expression for the extended-object self force, which conforms with Dirac's well known formula.When a charged particle is accelerated in a nonuniform manner, it exerts a force on itself. This phenomenon of self force (often called "radiation-reaction force") is known for almost a century, since the pioneer works by Abraham [1,2] and Lorentz [3] on the structure of the electron. The non-relativistic form of this force was obtained by Abraham and Lorentz, who found it to be proportional to the time-derivative of the acceleration. Later Dirac [4] derived the covariant relativistic expression for the self force acting on a point-like particle [Eq.The self-force is a remarkable phenomenon, because essentially it means that a charged particle may "exert a force on itself". A natural approach for comprehending this phenomenon within the framework of classical electrodynamics is the extended-object approach. In this approach one considers an extended charged object of finite size ǫ, and sum all the mutual electromagnetic forces that its various charge elements exert on each other. Then one applies the limit ǫ → 0, to obtain the self force in the point-particle limit . Obviously, if the charged object is static, the mutual forces will always cancel each other. However, if the charged object accelerates (under the influence of some external force), one generically finds that the sum of all mutual forces does not vanish. One would naturally be tempted to identify this nonvanishing "total force" as the self force acting on the particle. There is a problem, though: The resultant expression obtained for the "total force" usually includes a term that diverges like 1/ǫ. This divergent term must somehow be eliminated in order to obtain a physically meaningful notion of self-force. One would be tempted to apply the mass-renormalization procedure for this goal. In this procedure one re-defines the particle's rest mass so as to include the electrostatic energy E es ∝ ǫ −1 . This effectively adds a term E es a µ to the total force, where a µ is the particle's four-acceleration (we use c = 1 and signature (− + ++) throughout). Unfortunately the O(1/ǫ) term obtained in previous analyses was found to depend on the object's shape [5], and generally it does not have the form −E es a µ that would allow its elimination by mass-renormalization. In the special case of a charged spherical shell, Lorentz obtained an overall mutual force that diverges like −(4/3)E es a µ , which is 4/3 times larger than what required. Several authors later confirmed the presence of ...