Evaluation of the self-force

1.10 Evaluation of the self-force

To concretely evaluate the self-force, whether it be for a scalar charge, an electric charge, or a point mass, is a difficult undertaking. The difficulty resides mostly with the computation of the retarded Green’s function for the spacetime under consideration. Because Green’s functions are known for a very limited number of spacetimes, the self-force has so far been evaluated in a rather limited number of situations.

The first evaluation of the electromagnetic self-force was carried out by DeWitt and DeWitt [41 ] for a charge moving freely in a weakly-curved spacetime characterized by a Newtonian potential $Φ ≪ 1$ . (This condition must be imposed globally, and requires the spacetime to contain a matter distribution.) In this context the right-hand side of Equation (33 ) reduces to the tail integral, since there is no external force acting on the charge. They found the spatial components of the self-force to be given by

where $M$ is the total mass contained in the spacetime, $r = |x|$ is the distance from the centre of mass, $ˆr = x∕r$ , and $g = − ∇ Φ$ is the Newtonian gravitational field. (In these expressions the bold-faced symbols represent vectors in three-dimensional flat space.) The first term on the right-hand side of Equation (50

) is a conservative correction to the Newtonian force $mg$ . The second term is the standard radiation-reaction force; although it comes from the tail integral, this is the same result that would be obtained in flat spacetime if an external force $mg$ were acting on the particle. This agreement is necessary, but remarkable!

A similar expression was obtained by Pfenning and Poisson [46 ] for the case of a scalar charge. Here

where $ξ$ is the coupling of the scalar field to the spacetime curvature; the conservative term disappears when the field is minimally coupled. Pfenning and Poisson also computed the gravitational self-force acting on a massive particle moving in a weakly curved spacetime. The expression they obtained is in complete agreement (within its domain of validity) with the standard post-Newtonian equations of motion.

The force required to hold an electric charge in place in a Schwarzschild spacetime was computed, without approximations, by Smith and Will [54 ]. As measured by a free-falling observer momentarily at rest at the position of the charge, the total force is

and it is directed in the radial direction. Here, $m$ is the mass of the charge, $M$ is the mass of the black hole, and $r$ is the charge’s radial coordinate (the expression is valid in Schwarzschild coordinates). The first term on the right-hand side of Equation (52

) is the force required to keep a neutral test particle stationary in a Schwarzschild spacetime; the second term is the negative of the electromagnetic self-force, and its expression agrees with the weak-field result of Equation (50

). Wiseman [62 ] performed a similar calculation for a scalar charge. He found that in this case the self-force vanishes. This result is not incompatible with Equation (51

), even for nonminimal coupling, because the computation of the weak-field self-force requires the presence of matter, while Wiseman’s scalar charge lives in a purely vacuum spacetime.

The intriguing phenomenon of mass loss by a scalar charge was studied by Burko, Harte, and Poisson [15 ] in the simple context of a particle at rest in an expanding universe. For the special cases of a de Sitter cosmology, or a spatially-flat matter-dominated universe, the retarded Green’s function could be computed, and the action of the scalar field on the particle determined, without approximations. In de Sitter spacetime the particle is found to radiate all of its rest mass into monopole scalar waves. In the matter-dominated cosmology this happens only if the charge of the particle is sufficiently large; for smaller charges the particle first loses a fraction of its mass, but then regains it eventually.

In recent years a large effort has been devoted to the elaboration of a practical method to compute the (scalar, electromagnetic, and gravitational) self-force in the Schwarzschild spacetime. This work originated with Barack and Ori [7 ] and was pursued by Barack [2 , 3 ] until it was put in its definitive form by Barack, Mino, Nakano, Ori, and Sasaki [6 , 9 , 11 , 38 ]. The idea is to take advantage of the spherical symmetry of the Schwarzschild solution by decomposing the retarded Green’s function $G (x,x ′) +$ into spherical-harmonic modes which can be computed individually. (To be concrete I refer here to the scalar case, but the method works just as well for the electromagnetic and gravitational cases.) From the mode-decomposition of the Green’s function one obtains a mode-decomposition of the field gradient $∇ αΦ$ , and from this subtracts a mode-decomposition of the singular field $∇α ΦS$ , for which a local expression is known. This results in the radiative field $∇ Φ α R$ decomposed into modes, and since this field is well behaved on the world line, it can be directly evaluated at the position of the particle by summing over all modes. (This sum converges because the radiative field is smooth; the mode sums for the retarded or singular fields, on the other hand, do not converge.) An extension of this method to the Kerr spacetime has recently been presented [44 , 34 , 10 ], and Mino [37 ] has devised a surprisingly simple prescription to calculate the time-averaged evolution of a generic orbit around a Kerr black hole.

The mode-sum method was applied to a number of different situations. Burko computed the self-force acting on an electric charge in circular motion in flat spacetime [12 ], as well as on a scalar and electric charge kept stationary in a Schwarzschild spacetime [14 ], in a spacetime that contains a spherical matter shell (Burko, Liu, and Soren [17 ]), and in a Kerr spacetime (Burko and Liu [16 ]). Burko also computed the scalar self-force acting on a particle in circular motion around a Schwarzschild black hole [13 ], a calculation that was recently revisited by Detweiler, Messaritaki, and Whiting [21 ]. Barack and Burko considered the case of a particle falling radially into a Schwarzschild black hole, and evaluated the scalar self-force acting on such a particle [4 ]; Lousto [33 ] and Barack and Lousto [5 ], on the other hand, calculated the gravitational self-force.