Gravitational self-force by mode sum regularization

We propose a practical scheme for calculating the local gravitational self-force experienced by a test mass particle moving in a black hole spacetime. The method—equally effective for either weak or strong field orbits—employs the mode-sum regularization scheme previously developed for a scalar toy model. The starting point for the calculation, in this approach, is the formal expression for the regularized self-force derived by Mino et al. @Phys. Rev. D 55, 3457 ~1997!# „and, independently, by Quinn and Wald @Phys. Rev. D 56, 3381 ~1997!#..., which involves a worldline integral over the tail part of the retarded Green’s function. This force is decomposed into multipole ~tensor harmonic! modes, whose sum is subjected to a carefully designed regularization procedure. This procedure involves an analytical derivation of certain ‘‘regularization parameters’’ by means of a local analysis of the Green’s function. This paper contains the following main parts: ~1! The introduction of the mode sum scheme as applied to the gravitational case. ~2! Two simple cases studied: the test case of a static particle in flat spacetime, and the case of a particle at a turning point of a radial geodesic in Schwarzschild spacetime. In both cases we derive all necessary regularization parameters. ~3! An analytical foundation is set for applying the scheme in more general cases. ~In this paper, the mode sum scheme is formulated within the harmonic gauge. The implementation of the scheme in other gauges shall be discussed in a separate, forthcoming paper.!