Scalar, electromagnetic, and gravitational self-forces in weakly curved spacetimes

Abstract

We calculate the self-force experienced by a point scalar charge q, a point electric charge e, and a point mass m moving in a weakly curved spacetime characterized by a time-independent Newtonian potential ⌽. We assume that the matter distribution responsible for this potential is bounded, so that ⌽ϳϪM /r at large distances r from the matter, whose total mass is M; otherwise, the Newtonian potential is left unspecified. ͑We use units in which Gϭcϭ1.͒ The self-forces are calculated by first computing the retarded Green's functions for scalar, electromagnetic, and ͑linearized͒ gravitational fields in the weakly curved spacetime, and then evaluating an integral over the particle's past world line. The self-force typically contains both a conservative and a nonconservative ͑radiation-reaction͒ part. For the scalar charge, the conservative part of the self-force is equal to 2q 2 M r/r 3 , where is a dimensionless constant measuring the coupling of the scalar field to the spacetime curvature, and r is a unit vector pointing in the radial direction. For the electric charge, the conservative part of the self-force is e 2 M r/r 3 . For the massive particle, the conservative force vanishes. For the scalar charge, the radiation-reaction force is 1 3 q 2 dg/dt, where gϭϪ"⌽ is the Newtonian gravitational field. For the electric charge, the radiation-reaction force is 2 3 e 2 dg/dt. For the massive particle, the radiation-reaction force is Ϫ 11 3 m 2 dg/dt. Our result for the gravitational self-force is disturbing: a radiation-reaction force should not appear in the equations of motion at this level of approximation, and it should certainly not give rise to radiation antidamping. In the last section of the paper we prove that while a massive particle in a vacuum spacetime is subjected only to its self-force, it is also subjected to a matter-mediated force when it moves in a spacetime that contains matter; this force originates from the changes in the matter distribution that are induced by the presence of the particle. We show that the matter-mediated force contains a radiation-damping term that precisely cancels out the antidamping contribution from the gravitational self-force. When both forces are combined, the equations of motion are conservative, and they agree with the appropriate limit of the standard post-Newtonian equations of motion.

Quantum weak energy inequalities for the Dirac field in flat spacetime

“…Comparison with the quantum inequality for the scalar and electromagnetic fields in Minkowski space derived in [8,11] shows that the massless Dirac bound is weaker than the corresponding scalar bound by a factor of 4 3 , and stronger than the electromagnetic bound by a factor of 2 3 . Since none of these bounds are optimal, it is difficult to draw definite conclusions from this beyond the observation that the bounds are of comparable magnitude.…”

“…Bounds of this type have been obtained by various means [4,5,6,7,8,9,10,11,12,13,14] for the scalar, electromagnetic and Proca fields, in both flat and curved spacetimes, leading to results of great generality. Taking four-dimensional Minkowski space as a specific example, and averaging the energy density along an inertial worldline, the QWEI satisfied by these three theories may be written in the form [8,9,11,12] dt : T 00 : ψ (t, x 0 )g(t) for any smooth, real-valued g vanishing outside a compact region, where m is the particle's mass, S denotes its helicity (S = 1 for scalars, 2 for photons and 3 for massive spin-1 particles), while…”

“…Taking four-dimensional Minkowski space as a specific example, and averaging the energy density along an inertial worldline, the QWEI satisfied by these three theories may be written in the form [8,9,11,12] dt : T 00 : ψ (t, x 0 )g(t) for any smooth, real-valued g vanishing outside a compact region, where m is the particle's mass, S denotes its helicity (S = 1 for scalars, 2 for photons and 3 for massive spin-1 particles), while…”

Scalar self-force on a static particle in Schwarzschild spacetime using the massive field approach

“…Here, one of the main difficulties is the non-local nature of the selfforce in curved spacetime (i.e., the self-force value at each point on the particle's trajectory generically depends on entire past history of the particle). For a weakly curved spacetime the above mentioned explicit self-forces expressions were derived by DeWitt and DeWitt [5], and by Pfenning an Poisson [6]. The self-force on a static particle was investigated analytically by several authors: Smith and Will have obtained a non-vanishing result for the electromagnetic self-force on a static particle in Schwarzschild [7].…”

Radiative falloff of a scalar field in a weakly curved spacetime without symmetries

“…Methods to compute Green's functions for these fields were introduced in Refs. [43,44] in the context of a weakly curved spacetime with vanishing vector potential A, and these methods could easily be generalized to the situation at hand. While we can be reasonably sure that these fields will give rise to a very similar formulation of the generalized falloff theorem, a complete justification of this statement must await future work.…”