Frame dragging by rotating matter shells and gravitational-wave pulses beyond perturbation theory: the domain of validity of $\omega = 2L_z/R^3$

In general relativity, a slowly rotating thin spherical shell of angular momentum $L_z$ and radius $R$ drags the inertial frames in its (exactly flat) interior at the rigid rate $\omega = 2L_z/R^3$. Barker, Ledvinka, Lynden-Bell and Bičák (2017) showed that a time-symmetric, odd-parity (axial-sector) pulse of rotating gravitational waves — kept away from the origin by its own angular momentum — produces the same central dragging, $\omega \simeq 2L_z/R^3$ at the pulse radius, within second-order perturbation theory and for large spherical-harmonic index $l$; this is a quantitative realization of Mach's principle in which an 'immaterial' wave source acts on inertial frames like matter. Both derivations are perturbative: first order in the shell's rotation, second order in the wave amplitude, and restricted to special waveforms. The open problem is to establish frame dragging by the angular momentum of matter and of gravitational waves outside these perturbative regimes — finite shell rotation rates, nonlinear wave amplitudes with backreaction, and general waveform classes including $l=1$ — and thereby to decide in what sense the equality $\omega = 2L_z/R^3$ is exact, asymptotic, or corrected, including the physical resolution of the logarithmically divergent dragging integrals near the shell's unstable circular orbit.

Background

In general relativity, mass currents act on inertial frames the way electric currents act on magnets: a rotating body 'drags' the local inertial frames (the frames in which gyroscopes do not precess) around with it. This gravitomagnetic effect — the Lense–Thirring effect for weak, distant sources — is the relativistic descendant of Mach's idea that inertia arises from the distribution of mass-energy in the universe, not from absolute space. For a thin spherical shell of matter, the exterior and interior spacetimes are matched across the shell by the Israel junction conditions (the thin-shell matching conditions of general relativity); to linear order in the shell's angular velocity, with total angular momentum $L_z$ and radius $R$, the interior remains exactly flat but its inertial frames rotate rigidly, relative to observers at infinity, at the classic Thirring rate $\omega = 2L_z/R^3$ (first obtained by Hans Thirring in 1918 for slowly rotating shells).

Barker, Ledvinka, Lynden-Bell and Bičák (2017) studied a dynamical version of this system: a heavy spherical shell made of counter-rotating collisionless particles (zero net angular momentum, tangential pressure), which collapses from large radius and rebounds at a pericenter $R_P$ above its gravitational radius (the radius of order the Schwarzschild value $2M$, below which collapse to a black hole is unavoidable). They derived the shell's equation of motion from an energy conservation law, classified the trajectories (rebound from a pericenter, collapse to a black hole, bounded oscillation, and unstable circular orbits of the constituent particles), and then gave the shell a small net rotation $L_z$. To linear order in the rotation the interior stays flat and is rotationally dragged; the accumulated rotation of the interior frame during the whole collapse–rebound, $\Delta\varphi$, is given by an integral over the shell's radial motion. In the limiting case of asymptotic approach of the collapse to the unstable circular particle orbit, the radial velocity behaves as $\dot R \propto (R - R_P)$ instead of $\dot R \propto \sqrt{R - R_P}$, and both $\Delta\varphi$ and the elapsed time $\Delta t$ diverge logarithmically — a singular limit whose physical resolution (higher-order rotation effects, radiation reaction, or full numerical evolution) the paper does not provide.

The second ingredient is frame dragging by gravitational waves. Gravitational waves carry no local energy-density tensor in exact general relativity, but in perturbation theory about flat spacetime, $g = \eta + h^{(1)} + h^{(2)} + \dots$, the terms quadratic in the first-order perturbation $h^{(1)}$ act as an effective stress tensor that sources the second-order perturbation $h^{(2)}$. Bičák, Katz and Lynden-Bell (2008) treated rotating cylindrical wave pulses (a case with translational symmetry, not asymptotically flat), and Bičák, Katz, Ledvinka and Lynden-Bell (2012) treated time-symmetric (initial data invariant under $t \to -t$), odd-parity (axial-sector, the part of the spherical-harmonic decomposition of the perturbation that changes sign under spatial reflection) rotating wave pulses in an asymptotically flat vacuum spacetime: a wave shell falls in, bounces at radius $\sim A$ because of its angular momentum, and re-expands, leaving the central region nearly flat. The rotation of the central inertial frame, $\omega_0$, is determined by an elliptic constraint equation — it responds to the waves without time delay — and, after a gauge choice that fixes $\omega_0$ unambiguously with respect to spatial infinity, is given by an integral of the second-order Ricci component $R^{(2)}_{t\varphi}$ weighted by $1/r$.

The 2017 paper completed the comparison by computing the angular momentum of the spherical wave pulse, $L_z = (1/8\pi)\int R^{(2)}_{t\varphi}\,\mathrm{d}^3x$ (a conserved quantity of the second-order perturbation scheme, derived assuming $l > 1$ to simplify combinatorial factors, where $l$ and $m$ are the spherical-harmonic indices of the wave pattern). The exact second-order statements are the differential relation $\mathrm{d}\omega = 2\,\mathrm{d}L_z/r^3$ between the dragging contribution of each shell element and its angular momentum, and closed-form formulae for $L_z$ and $\omega_0(t)$. Because for $l \gg 1$ the pulse is localized on a thin shell of radius $R(t) \simeq \sqrt{A^2 + t^2}$, the factor $r^3$ can be pulled out of the integral and one obtains $\omega = 2L_z/R^3$ — the same law as for the slowly rotating material shell; at the moment of closest approach the exact result carries corrections, $\omega_0(t=0) = (L_z/A^3)(2 + 9/(2l) + 9/(16l^2) + \dots)$. The analogous cylindrical formulae agree with the material-cylinder dragging at the moment of time symmetry and at late times. The Machian conclusion is that 'immaterial' sources — gravitational waves via their effective stress tensor — drag inertial frames in exactly the same way as matter, so any formulation of Mach's principle must include the influence of waves and not only of material energy-momentum.

Both sides of this equivalence are, however, perturbative. The shell results hold to first order in the slow-rotation parameter $\mathcal{O}(R\omega)$; at finite rotation rates the interior is expected to cease being exactly flat (the classical stationary-rotating-shell literature, from Einstein 1912 and Thirring 1918 onward, shows curvature appearing inside rotating shells at second order in the angular velocity; see the historical reviews of Pfister). The wave results hold to second order in the wave amplitude, for time-symmetric, odd-parity, $l > 1$ data, and the headline equality $\omega = 2L_z/R^3$ is itself a large-$l$ localization approximation. Whether the quantitative matter–wave equivalence, and the simple dragging law itself, survive finite rotation, nonlinear amplitudes, and general waveforms is the open problem.

Problem Statement

Determine the rotation of inertial frames induced by the angular momentum of matter and of gravitational waves outside the perturbative regimes in which it has been established, with one overarching objective: to decide the generality of the quantitative Machian equivalence between matter-induced and wave-induced dragging, and in particular the domain of validity of the law $\omega = 2L_z/R^3$. The problem has four linked components.

(1) Finite rotation of the material shell. Extend the analysis of the collapsing–rebounding spherical shell (counter-rotating collisionless particles with tangential pressure) beyond first order in the slow-rotation parameter: determine the interior geometry and the frame dragging at $\mathcal{O}((R\omega)^2)$ and beyond — where the interior is no longer exactly flat and the rigid-dragging interpretation must be reformulated — together with the shell's own dynamics and the accumulated interior rotation $\Delta\varphi$ at finite rotation rates.

(2) Nonlinear rotating gravitational waves. Determine the central frame dragging and the angular momentum of rotating gravitational-wave pulses beyond second order in the wave amplitude, including the backreaction of the waves on their own propagation and on the conserved total angular momentum, and beyond the restrictions of the existing derivations (time-symmetric data, odd parity, $l > 1$): treat the low multipoles (in particular $l = 1$), general non-time-symmetric waveforms, and characterize the sense in which the large-$l$ expansions approximate the exact behaviour.

(3) Status of the dragging law. Decide for which classes of sources and waveforms the relation $\omega = 2L_z/R^3$ — or its differential form $\mathrm{d}\omega = 2\,\mathrm{d}L_z/r^3$ — is exact, asymptotic, or corrected: quantify the corrections already present at finite $l$ (e.g. the $9/(2l)$ term at $t = 0$), determine the law's behaviour for non-time-symmetric pulses and for interiors that are not nearly flat, and establish whether the matter–wave equality survives finite rotation and nonlinear amplitudes.

(4) Singular limit. Resolve the behaviour of $\Delta\varphi$ and $\Delta t$ in the limit in which the shell's collapse approaches the unstable circular particle orbit asymptotically, where the first-order integrals diverge logarithmically: determine whether and how higher-order rotation effects, radiation reaction, or the full shell dynamics regularize the divergence, and what the dragging does in that limit.

All four components are jointly required for a complete answer; a result that resolves only a proper subset of them settles the components it addresses but leaves the rest open. Component (3) is the general classification that the determinations of components (1) and (2) are together required to deliver, not a further independent computation beyond them.

Answers may be analytic (higher-order perturbation theory, matched junction expansions, or exact solutions), numerical (evolution of the shell or wave initial data), or a combination; an answer must treat the dragging of the interior or central inertial frames and the angular momentum of the source within the same calculation, so that the matter–wave equivalence and the dragging law can be judged.

References

1. W. Barker, T. Ledvinka, D. Lynden-Bell, J. Bičák, 'Rotation of inertial frames by angular momentum of matter and waves', Classical and Quantum Gravity 34 (2017) 205006. DOI: 10.1088/1361-6382/aa8a34, https://doi.org/10.1088/1361-6382/aa8a34; arXiv:1710.10360, https://arxiv.org/abs/1710.10360
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