Electromagnetic Self-force Research Articles

A Classical Complete Action for a System of Small-Point Massive Charged Particles in General Relativity

A classical action which describes the motion of a system of small-point massive charged particles including the existence of the electromagnetic and gravitational self-forces, Maxwell equations and Einstein field equations is presented. The action possesses the particularity of being a functional of the variables zi (τi), the trajectory of the i-particle, Aα (x), the electromagnetic 4-potential, and gαβ(x), the metric tensor. It is also considered that the metric tensor gαβ (x) and the potential Aα (x) are not functions of the trajectory of each particle when the variations with respect to the trajectories of the particles are done. That is, the action is complete. The electromagnetic and the gravitational self-forces are analyzed.

Combined gravitational and electromagnetic self-force on charged particles in electrovac spacetimes

We consider the self-force on a charged particle moving in a curved spacetime with a background electromagnetic field, extending previous studies to situations in which gravitational and electromagnetic perturbations are comparable. The formal expression $f^{ret}_\alpha$ for the self-force on a particle, written in terms of the retarded perturbed fields, is divergent, and a renormalization is needed to find the particle's acceleration at linear order in its mass $m$ and charge $e$. We assume that, as in previous work in a Lorenz gauge, the renormalization for accelerated motion comprises an angle average and mass renormalization. Using the short distance expansion of the perturbed electromagnetic and gravitational fields, we show that the renormalization is equivalent to that obtained from a mode sum regularization in which one subtracts from the expression for the self-force in terms of the retarded fields a singular part field comprising only the leading and subleading terms in the mode sum. The most striking part of our result, arising from a remarkable cancellation, is that the renormalization involves no mixing of electromagnetic and gravitational fields. In particular, the renormalized mass is obtained by subtracting (1) the purely electromagnetic contribution from a point charge moving along an accelerated trajectory and (2) the purely gravitational contribution from a point mass moving along the same trajectory. In a mode-sum regularization, the same cancellation implies that the required regularization parameters are sums of their purely electromagnetic and gravitational values.

Classical formulations of the electromagnetic self-force of extended charged bodies

Several formulations of the classical electrodynamics of charged particles, as have been developed in the course of the twentieth century, are compared. The mathematical equivalence of the various dissimilar expressions for the electromagnetic self-force is demonstrated explicitly by deriving these expressions directly from one another. The new connections that are established present the previously published results on a common basis, thereby contributing to a coherent historical picture of the development of charged particle models.

Nonexistence of the self-accelerating dipole and related questions

We calculate the self-force of a constantly accelerating electric dipole, showing, in particular, that classical electromagnetism does not predict that an electric dipole could self-accelerate, nor could it levitate in a gravitational field. We also resolve a paradox concerning the inertial mass of a longitudinally accelerating dipole, showing that the combined system of dipole plus field can be assigned a well-defined energy-momentum four-vector, so that the Principle of Relativity is satisfied. We then present some general features of electromagnetic phenomena in a gravitational field described by the Rindler metric, showing in particular that an observer fixed in a gravitational field described by the Rindler metric will find any charged object supported in the gravitational field to possess an electromagnetic self-force equal to that observed by an inertial observer relative to which the body undergoes rigid hyperbolic motion. It follows that the Principle of Equivalence is satisfied by these systems.

The fields and self-force of a constantly accelerating spherical shell

We present a partial differential equation describing the electromagnetic potentials around a charge distribution undergoing rigid motion at constant proper acceleration, and obtain a set of solutions to this equation. These solutions are used to find the self-force exactly in a chosen case. The electromagnetic self-force for a spherical shell of charge of proper radius R undergoing rigid motion at constant proper acceleration a 0 is, to high-order approximation, , and this is conjectured to be exact.

Self-force as a cosmic censor

We examine Hubeny's scenario according to which a near-extremal Reissner-Nordstr\"om black hole can absorb a particle and be driven toward an over-extremal state in which the charge exceeds the mass, signaling the destruction of the black hole. Our analysis incorporates the particle's electromagnetic self-force and the energy radiated to infinity in the form of electromagnetic waves. With these essential ingredients, our sampling of the parameter space reveals no instances of an overcharged final state, and we conjecture that the self-force acts as a cosmic censor, preventing the destruction of a near-extremal black hole by the absorption of a charged particle. We argue, on the basis of the third law of black-hole mechanics, that this conclusion is robust and should apply to attempts to overspin a Kerr black hole.

Self-force as probe of internal structure

The self-force acting on a (scalar or electric) charge held in place outside a massive body contains information about the body’s composition, and can therefore be used as a probe of internal structure. We explore this theme by computing the (scalar or electromagnetic) self-force when the body is a spherical ball of perfect fluid in hydrostatic equilibrium, under the assumption that its rest-mass density and pressure are related by a polytropic equation of state. The body is strongly self-gravitating, and all computations are performed in exact general relativity. The dependence on internal structure is best revealed by expanding the self-force in powers of r−10, with r0 denoting the radial position of the charge outside the body. To the leading order, the self-force scales as r−30 and depends only on the square of the charge and the body’s mass; the leading self-force is universal. The dependence on internal structure is seen at the next order, r−50, through a structure factor that depends on the equation of state. We compute this structure factor for relativistic polytropes, and show that for a fixed mass, it increases linearly with the body’s radius in the case of the scalar self-force, and quadratically with the body’s radius in the case of the electromagnetic self-force. In both cases we find that for a fixed mass and radius, the self-force is smaller if the body is more centrally dense, and larger if the mass density is more uniformly distributed.

Radiation Effects on Geodesics in de Sitter Space: A Classical Approach

In this work we first obtain a trajectory of a freely falling charged particle in de Sitter space and then in the classical approach, the effect of electromagnetic self-force on particle’s trajectory has been considered. Finally, some limits for the problem have been presented.

Rigorous derivation of electromagnetic self-force

During the past century, there has been considerable discussion and analysis of the motion of a point charge in an external electromagnetic field in special relativity, taking into account ``self-force'' effects due to the particle's own electromagnetic field. We analyze the issue of ``particle motion'' in classical electromagnetism in a rigorous and systematic way by considering a one-parameter family of solutions to the coupled Maxwell and matter equations corresponding to having a body whose charge-current density ${J}^{a}(\ensuremath{\lambda})$ and stress-energy tensor ${T}_{ab}(\ensuremath{\lambda})$ scale to zero size in an asymptotically self-similar manner about a worldline $\ensuremath{\gamma}$ as $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. In this limit, the charge, $q$, and total mass, $m$, of the body go to zero, and $q/m$ goes to a well-defined limit. The Maxwell field ${F}_{ab}(\ensuremath{\lambda})$ is assumed to be the retarded solution associated with ${J}^{a}(\ensuremath{\lambda})$ plus a homogeneous solution (the ``external field'') that varies smoothly with $\ensuremath{\lambda}$. We prove that the worldline $\ensuremath{\gamma}$ must be a solution to the Lorentz force equations of motion in the external field ${F}_{ab}(\ensuremath{\lambda}=0)$. We then obtain self-force, dipole forces, and spin force as first-order perturbative corrections to the center-of-mass motion of the body. We believe that this is the first rigorous derivation of the complete first-order correction to Lorentz force motion. We also address the issue of obtaining a self-consistent perturbative equation of motion associated with our perturbative result, and argue that the self-force equations of motion that have previously been written down in conjunction with the ``reduction of order'' procedure should provide accurate equations of motion for a sufficiently small charged body with negligible dipole moments and spin. (There is no corresponding justification for the non-reduced-order equations.) We restrict consideration in this paper to classical electrodynamics in flat spacetime, but there should be no difficulty in extending our results to the motion of a charged body in an arbitrary globally hyperbolic curved spacetime.

Electromagnetic self-forces and generalized Killing fields

Building upon previous results in scalar field theory, a formalism is developed that uses generalized Killing fields to understand the behavior of extended charges interacting with their own electromagnetic fields. New notions of effective linear and angular momenta are identified, and their evolution equations are derived exactly in arbitrary (but fixed) curved spacetimes. A slightly modified form of the Detweiler–Whiting axiom that a charge's motion should only be influenced by the so-called regular component of its self-field is shown to follow very easily. It is exact in some interesting cases and approximate in most others. Explicit equations describing the center-of-mass motion, spin angular momentum and changes in mass of a small charge are also derived in a particular limit. The chosen approximations—although standard—incorporate dipole and spin forces that do not appear in the traditional Abraham–Lorentz–Dirac or Dewitt–Brehme equations. They have, however, been previously identified in the test body limit.

The alternative to classical mass renormalization for tube-based self-force calculations

To date, classical mass renormalization has been invoked in all tube-based self-force calculations, thus following the method introduced in Dirac's 1938 calculation of the electromagnetic self-force for the classical radiating electron. In this paper, a new tube method is described that does not rely on a mass renormalization procedure. As a result, exact self-force calculations become possible for classical radiating systems of finite size. A new derivation of the Lorentz–Dirac equation is given and the relationship between the new tube method and the classical mass renormalization procedure is explained. It is expected that a similar tube method could be used to obtain rigorous results in the gravitational self-force problem.

Multiscale analysis of the electromagnetic self-force in a weak gravitational field

We examine the motion of a charged particle in a weak gravitational field. In addition to the Newtonian gravity exerted by a large central body, the particle is subjected to an electromagnetic self-force that contains both a conservative piece and a radiation-reaction piece. This toy problem shares many of the features of the strong-field gravitational self-force problem, and it is sufficiently simple that it can be solved exactly with numerical methods, and approximately with analytical methods. We submit the equations of motion to a multiscale analysis, and we examine the roles of the conservative and radiation-reaction pieces of the self-force. We show that the radiation-reaction force drives secular changes in the orbit's semilatus rectum and eccentricity, while the conservative force drives a secular regression of the periapsis and affects the orbital time function; neglect of the conservative term can hence give rise to an important phasing error. We next examine what might be required in the formulation of a reliable secular approximation for the orbital evolution; this would capture all secular changes in the orbit and discard all irrelevant oscillations. We conclude that such an approximation would be very difficult to formulate without prior knowledge of the exact solution.

M-mode regularization scheme for the self-force in Kerr spacetime

We present a new, simple method for calculating the scalar, electromagnetic, and gravitational self-forces acting on particles in orbit around a Kerr black hole. The standard ``mode-sum regularization'' approach for self-force calculations relies on a decomposition of the full (retarded) perturbation field into multipole modes, followed by the application of a certain mode-by-mode regularization procedure. In recent years several groups have developed numerical codes for calculating black hole perturbations directly in $2+1$ dimensions (i.e., decomposing the azimuthal dependence into $m$-modes, but refraining from a full multipole decomposition). Here we formulate a practical scheme for constructing the self-force directly from the $2+1$-dimensional $m$-modes. While the standard mode-sum method is serving well in calculations of the self-force in Schwarzschild geometry, the new scheme should allow a more efficient treatment of the Kerr problem.

Electromagnetic and gravitational self-force on a relativistic particle from quantum fields in curved space

We provide a quantum field theoretical derivation of the Abraham-Lorentz-Dirac (ALD) equation, describing the motion of an electric point charge sourcing an electromagnetic field, which back-reacts on the charge as a self-force, and the Mino-Sasaki-Tanaka-Quinn-Wald (MSTQW) equation describing the motion of a point mass with self-force interacting with the linearized metric perturbations caused by the mass off an otherwise vacuous curved background spacetime. We regularize the formally divergent self-force by smearing the direct part of the retarded Green's function and using a quasilocal expansion. We also derive the ALD-Langevin and the MSTQW-Langevin equations with a classical stochastic force accounting for the effect of the quantum fluctuations in the field, which causes small fluctuations on the particle trajectory. These equations will be useful for studying the stochastic motion of charges and small masses under the influence of both quantum and classical noise sources, derived either self-consistently or put in by hand phenomenologically. We also show that history-dependent noise-induced drift motions could arise from such stochastic sources on the trajectory that could be a hidden feature of gravitational wave forms hitherto unknown.

Optical geometry analysis of the electromagnetic self-force

We present an analysis of the behavior of the electromagnetic self-force for charged particles in a conformally static spacetime, interpreting the results with the help of optical geometry. Some conditions for the vanishing of the local terms in the self-force are derived and discussed.

Limitations of the adiabatic approximation to the gravitational self-force

A small body moving in the field of a much larger black hole and subjected to its own gravity moves on an accelerated world line in the background spacetime of the large black hole. The acceleration is produced by the body's gravitational self-force, which is constructed from the body's retarded gravitational field. The adiabatic approximation to the gravitational self-force is obtained instead from the half-retarded minus half-advanced field; it is known to produce the same dissipative effects as the true self-force. We argue that the adiabatic approximation is limited, because it discards important conservative terms which lead to the secular evolution of some orbital elements. We argue further that this secular evolution has measurable consequences; in particular, it affects the phasing of the orbit and the phasing of the associated gravitational wave. Our argument rests on a simple toy model involving a point electric charge moving slowly in the weak gravitational field of a central mass; the charge is also subjected to its electromagnetic self-force. In this simple context the true self-force is known explicitly and it can cleanly be separated into conservative and radiation-reaction pieces. We observe that the conservative part of the self-force produces a secular regression of the orbit's periapsis. We explain how the conclusions reached on the basis of the toy model can be extended to the gravitational self-force, and to fast motions and strong fields. While the limitations of the adiabatic approximation are quite severe in a post-Newtonian context in which the motion is slow and the gravitational field weak, they appear to be less so for rapid motions and strong fields.

Gravitational self-force on a particle orbiting a Kerr black hole.

We present a practical method for calculating the gravitational self-force, as well as the electromagnetic and scalar self-forces, for a particle in a generic orbit around a Kerr black hole. In particular, we provide the values of all the regularization parameters needed for implementing the (previously introduced) mode-sum regularization method. We also address the gauge-regularization problem, as well as a few other issues involved in the calculation of gravitational radiation reaction in Kerr spacetime.

Regularization parameters for the self-force in Schwarzschild spacetime. II. Gravitational and electromagnetic cases

We obtain all ``regularization parameters'' (RPs) needed for calculating the gravitational and electromagnetic self-forces for an arbitrary geodesic orbit around a Schwarzschild black hole. These RP values are required for implementing the previously introduced mode-sum method, which allows a practical calculation of the self-force by summing over contributions from individual multipole modes of the particle's field. In the gravitational case, we provide here full details of the analytic method and results briefly reported in a recent Letter [Phys. Rev. Lett. 88, 091101 (2002)]. In the electromagnetic case, the RPs are obtained here for the first time.

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

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 $\ensuremath{\Phi}.$ We assume that the matter distribution responsible for this potential is bounded, so that $\ensuremath{\Phi}\ensuremath{\sim}\ensuremath{-}M/r$ at large distances r from the matter, whose total mass is $M;$ otherwise, the Newtonian potential is left unspecified. (We use units in which $G=c=1.)$ The self-forces are calculated by first computing the retarded Green's functions for scalar, electromagnetic, and (linearized) 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-reaction) part. For the scalar charge, the conservative part of the self-force is equal to $2\ensuremath{\xi}{q}^{2}M\mathit{r}^{/r}^{3},$ where $\ensuremath{\xi}$ is a dimensionless constant measuring the coupling of the scalar field to the spacetime curvature, and $\mathit{r}^$ is a unit vector pointing in the radial direction. For the electric charge, the conservative part of the self-force is ${e}^{2}M\mathit{r}^{/r}^{3}.$ For the massive particle, the conservative force vanishes. For the scalar charge, the radiation-reaction force is $\frac{1}{3}{q}^{2}d\mathit{g}/dt,$ where $\mathit{g}=\ensuremath{-}\mathbf{\ensuremath{\nabla}}\ensuremath{\Phi}$ is the Newtonian gravitational field. For the electric charge, the radiation-reaction force is $\frac{2}{3}{e}^{2}d\mathit{g}/dt.$ For the massive particle, the radiation-reaction force is $\ensuremath{-}\frac{11}{3}{m}^{2}d\mathit{g}/dt.$ 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.

The electromagnetic self-force on a charged spherical body slowly undergoing a small, temporary displacement from a position of rest

The self-force of classical electrodynamics on a charged "rigid" body of radius R is evaluated analytically for the body undergoing a slow (i.e., with a speed v<<c), slight (i.e., small compared to R), and temporary displacement from an initial position of rest. The results are relevant to the Bohr-Rosenfeld analysis of the measurability of the electromagnetic field, which has been the subject of a recent controversy.