Completing the Daboul-Jensen theory of radiation reaction for a rotating charged sphere: relativistic consistency, pole structure, nonsmooth forcing, and admissible initial data

In 1973 Daboul and Jensen derived an exact, causal self-torque, linear in the angular velocity, for a spherical shell with rigidly attached uniform surface charge rotating about a fixed axis, and analyzed the resulting convolution equation of motion, proving the absence of runaway solutions for non-negative mechanical moment of inertia. The issues that Daboul and Jensen (1973) and the follow-up literature leave open are: a fully relativistic treatment of the rotating extended charge; the numerical pole and residue structure of the response Green's function in the general case of nonzero mechanical moment of inertia; the self-consistent dynamics under discontinuous or sharply varying external torques, where the derivative expansion fails; the specification of physically admissible finite-energy initial fields and the role of the decaying torqueless mode; and the runaway regime of negative mechanical moment of inertia. One listed issue, the negative-moment-of-inertia runaway transition and its physical interpretation, was resolved by Daboul (1975) together with the pole-runaway correspondence of the companion scattering paper. The remaining four issues are unresolved in the literature to date: the rotational self-torque dynamics at relativistic rates, the general pole/residue map, the exact kernel under nonsmooth forcing, and the rotational initial-data admissibility question (only the translational analog is treated). They form one coherent objective: completing the Daboul-Jensen model into a relativistically consistent, well-posed theory of rotational radiation reaction.

Background

Radiation reaction is the back-action that the electromagnetic radiation emitted by an accelerated charge exerts on the charge's own motion. For an ideal point charge, the standard equation of motion including this effect, the Lorentz-Dirac equation, is third order in time and exhibits two classic pathologies: runaway solutions, in which the charge self-accelerates with no applied force, and preacceleration, in which motion begins before the force is applied. A long-standing alternative is to model the particle as an extended charge distribution: when the charge is spread over a sphere of radius $R$, the self-interaction at a given instant depends on the body's earlier configuration (retardation over the light-crossing time), and the equation of motion becomes a causal delay/integral equation rather than a local third-order equation. The translational version of this extended-charge program was developed by Petzold (1980), who founded the treatment of the radiating electron on an integral equation, by Grandy and Aghazadeh (1982), who computed radiative corrections for extended charges, and by Blanco (1987), who analyzed causality and the dynamics under linear forces for the translational problem.

Daboul and Jensen (1973) treated the rotational analog. A spherical shell carrying a rigidly attached uniform surface charge rotates about a fixed diameter; its inertia has a mechanical part $I_{\mathrm{mech}}$ (the non-electromagnetic moment of inertia of the shell) and an electromagnetic part $I_{\mathrm{el}}$ contributed by the electrostatic self-energy, which acts like an additional moment of inertia. The electromagnetic self-torque, the torque that the shell's own fields exert on it, is obtained exactly as an integral expression that is causal and linear in the angular velocity $\Omega(t)$: the torque at time $t$ depends on the values of $\Omega$ at earlier times through a response kernel whose temporal support is the interval $[0,2\tau]$, where $\tau=R/c$ is the light-travel time associated with the sphere of radius $R$. In the notation of Vlasov's open rederivation (1998), which Olbert and Belcher (2010) show to be equivalent to the source's own solution for this model ('a different but equivalent form'; the converted results 'demonstrate perfect agreement'), the exact self-torque in Gaussian units is $M_{\mathrm{self}}(t) = (Q^{2}c/6R^{2})\int_{0}^{2R/c}\mathrm{d}x\,(\mathrm{d}\Omega(t-x)/\mathrm{d}t)\,(x^{2}-2R^{2}/c^{2})$, and the linearized equation of motion is $\mathrm{d}[I_{\mathrm{mech}}\,\Omega(t)]/\mathrm{d}t = M_{\mathrm{self}}(t) + M_{\mathrm{ext}}(t)$, a linear Volterra-type convolution equation; the model is completely specified by the radius $R$, the total charge $Q$, and the mechanical moment of inertia $I_{\mathrm{mech}}$ (for a thin shell of mass $M$, $I_{\mathrm{mech}}=2MR^{2}/3$), and no further free parameter appears in any accessible formulation of it. Among the results: as long as no negative mechanical moment of inertia is added, no runaway solutions occur; the derivative expansion of the convolution (expanding in powers of $\tau$ times time derivatives of $\Omega$) reproduces an Abraham-Lorentz-type series for the self-torque, the rotational analog of the standard point-particle radiation-reaction force; and the solution is represented by contour integration in the complex frequency plane (the source's method), whose Green's function (resolvent) has poles at the complex eigenfrequencies of the homogeneous equation, that is, damped collective modes, each entering the mode expansion with its residue. The companion paper (Daboul and Jensen 1973b) treated electromagnetic scattering as the same radiation-reaction problem, fixed the $I_{\mathrm{mech}}=0$ normalization, where the scattering amplitude reduces to the standard P-wave amplitude of a perfectly conducting sphere, studied the poles of the scattering amplitude as functions of $I_{\mathrm{mech}}$, and established that a pole on the positive imaginary frequency axis corresponds to a runaway solution.

The model has well-known physical limits. The assumption of geometrical rigidity, the whole shell rotating with one instantaneous angular velocity, requires effectively instantaneous propagation of internal forces and so conflicts with special relativity; and the non-relativistic approximation permits formally superluminal rim speeds for sufficiently large applied torques. The relativistically consistent notion of a rigid body, Born rigidity (constant proper distances between neighboring material points), already forbids rotation beyond the light cylinder where the rim speed reaches the speed of light, so a consistent treatment must either accept that restriction or introduce relativistic elasticity for the internal forces. Daboul and Jensen (1973) identify five unresolved issues: (a) a fully relativistic treatment of the rotating extended charge; (b) the numerical pole locations and residues in the general case of nonzero mechanical moment of inertia ($I_{\mathrm{mech}}\neq0$); (c) the dynamics under discontinuous or very sharp external torques, where the derivative expansion is not uniformly valid and the exact convolution kernel must be used, together with the boundary between smooth and nonsmooth regimes, characterized by the analytic-continuation radius of the motion relative to the kernel's temporal support $2\tau$; (d) the physical interpretation and initial-condition content of the decaying torqueless radiating mode, the homogeneous solution of the exact equation of motion at zero external torque, under which the shell radiates while its rotation settles toward a constant rate, raising questions of realistic preparation of initial states and of the field energy carried by such modes; and (e) the runaway regime of negative $I_{\mathrm{mech}}$, deferred to the companion works (paper I's own reference list announces papers II and III as 'closely connected with the present article'). Issue (e) was later resolved by Daboul (1975); issues (a)-(d) structure the open problem stated below.

Problem Statement

Complete the Daboul-Jensen non-relativistic theory of radiation reaction for a rigidly rotating spherical shell with uniform surface charge into a consistent, well-posed physical theory, by resolving the four issues that remain open of the five identified by Daboul and Jensen (1973) (the fifth, the runaway regime of negative mechanical moment of inertia, was settled by Daboul 1975). The common objective uniting them is the exact self-torque kernel of the rotating shell: issue (a) replaces this kernel by a relativistic reformulation and analyzes the resulting equation of motion, while (b)-(d) concern the Daboul-Jensen non-relativistic kernel, determining its pole structure (b), its behavior under nonsmooth forcing (c), and its admissible initial data (d). The four issues are independent sub-problems. Concretely:

(a) Relativistic consistency. Construct a fully relativistic treatment of the rotational dynamics of an extended charged body: determine the self-torque including retardation consistently with relativistic elasticity or another explicitly specified, relativistically consistent model of the internal forces (or with Born-rigid kinematics within its domain of validity), for prescribed external torques and relativistic angular speeds, and analyze the well-posedness of the resulting equation of motion (existence, uniqueness, causality), including whether the absence of runaway solutions survives. The domain of validity of the internal-force model adopted is part of the problem: Born-rigid kinematics below the light cylinder, and relativistic elasticity or another explicitly specified, relativistically consistent model of the internal forces for relativistic rotation approaching the light cylinder, where rigid rotation ceases to be physically preparable; the runaway-survival question is posed on that same domain.

(b) Spectral structure of the kernel. For the general case of nonzero mechanical moment of inertia ($I_{\mathrm{mech}}\neq0$; the $I_{\mathrm{mech}}=0$ normalization is fixed by the companion scattering paper), compute and characterize numerically, as functions of $I_{\mathrm{mech}}$ (equivalently, of the dimensionless inertia ratio $I_{\mathrm{mech}}/I_{\mathrm{el}}$), the complete pole structure of the Green's function (resolvent) of the exact equation of motion: all complex poles, that is, the eigenfrequencies of the homogeneous problem, giving the damped-mode frequencies and decay widths, together with the residues (mode weights) with which they enter the Green's-function expansion, with controlled truncation and error bounds.

(c) Nonsmooth forcing. Using the exact convolution kernel rather than the truncated derivative expansion, determine the self-consistent dynamics under discontinuous or very sharp external torques: for which torque profiles solutions exist and are unique, and where the boundary between the smooth regime (in which the derivative expansion is valid, characterized by the motion's analytic-continuation radius exceeding the kernel's temporal support $2\tau$) and the nonsmooth regime lies.

(d) Admissible initial data. Specify the physically admissible initial-field configurations for the rotational problem, consistent with finite total field energy (with the admissibility criterion itself to be determined, subject to the requirements that it be explicit, exclude configurations whose field energy diverges in the infinite past, and be realizable by a stated preparation protocol), and determine the initial-condition content of the decaying torqueless radiating mode, the homogeneous solution of the exact equation of motion at zero external torque, under which the angular acceleration decays exponentially while the shell radiates and its rotation settles toward a constant rate (in Vlasov's equivalent rederivation, $\mathrm{d}\Omega/\mathrm{d}t\propto e^{bt}$ with $b<0$, where $\nu=bR/c<-1$ solves a transcendental characteristic equation and the arbitrary amplitude can be fixed only by an additional condition whose physical motivation is unclear), thereby resolving the questions of how such modes are prepared, whether they carry divergent field energy in the infinite past, and which initial data are physically admissible.

References

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