Particle motion in circularly polarized vacuum pp waves

TL;DR

The paper addresses the problem of particle motion in exact circularly polarized gravitational waves and the trapping behavior proposed for such waves. By mapping Lukash spacetimes to a perturbed CPP problem through a time redefinition and transforming the transverse Sturm–Liouville system to constant coefficients via a log-time substitution and a rotation, the authors obtain analytic, chiral-decomposed solutions. They show that bounded geodesics arise for Lukash waves of $...$type $VI$, uncover a $7$-parameter conformal symmetry including a screw isometry that connects Lukash and CPP, and highlight that trapping can be coordinate-dependent. The work provides non-perturbative, exact insights into particle trapping by gravitational waves and clarifies the symmetry structure bridging CPP and Lukash spacetimes.

Abstract

Bialynicki-Birula and Charzynski argued that a gravitational wave emitted during the merger of a black hole binary may be approximated by a circularly polarized wave which may in turn trap particles [1]. In this paper we consider particle motion in a class of gravitational waves which includes, besides circularly polarized periodic waves (CPP) [2], also the one proposed by Lukash [3] to study anisotropic cosmological models. Both waves have a 7-parameter conformal symmetry which contains, in addition to the generic 5-parameter (broken) Carroll group, also a 6th isometry. The Lukash spacetime can be transformed by a conformal rescaling of time to a perturbed CPP problem. Bounded geodesics, found both analytically and numerically, arise when the Lukash wave is of Bianchi type VI. Their symmetries can also be derived from the Lukash-CPP relation. Particle trapping is discussed.

Figures (6)

• Figure 1: In terms of logarithmic position coordinates $\binom{\xi}{\eta}$\ref{['lnchange']} (or their rotated versions $\binom{\alpha}{\beta}$\ref{['Lukrotframe']}) one gets bounded Lukash trajectories in the parameter range \ref{['boundcond']}. The magenta curve near the origin is the contribution of the "CPP" term in \ref{['lnk']}, which is pushed outwards by the repulsive force. However the CPP term keeps the motion bounded.
• Figure 2: Trajectories shown in $\binom{\alpha}{\beta}$ coordinates unfolded in logarithmic time $T$ when close to the upper critical value $C_{crit}$. The motion of the guiding center${{\bf w}}_{+}$ is determined by $\lambda_+^2$ and the "epicycle" around it is described by ${\bf w}_{-}$, determined by $\textcolor{red}{\lambda_-^2}=\lambda_+^2+2\sqrt{C^{2}-\kappa^{2}}$. (a) when $\kappa<C<C_{crit}=\kappa^2+\hbox{$\frac{1}{4}$}$ (which has Bianchi type VI), the trajectory remains bounded ; (b) for $C=C_{crit}$ with $\kappa > \hbox{$\frac{1}{2}$}$ (which is Bianchi VI), the radius is constant in logarithmic time $T$ (grows linearly in $U$), (c) for $C>C_{crit}$ the trajectory escapes exponentially.
• Figure 3: Transverse trajectories in $(\xi,\eta)$ coordinates just below the upper critical value$C_{crit}$ for: (a) the CPP term alone, i.e., without the linear perturbation; (b) the full Lukash system. (a) and (b) show the components. Caveat: the CPP and Lukash scales in (a) and (b) repectively are different. (c) shows the transverse trajectories. The motion is bounded for Lukash but unbounded for CPP.
• Figure 4: The components of (a)CPP and (b)Lukash geodesics at the lower critical value$C = \kappa$ (which is Bianchi VII and Bianchi IV). The colors refer to the respective cases. Caveat: the CPP and Lukash scales in (a) and (b) are different. Fig.(c) shows both trajectories in the transverse plane.
• Figure 5: (a)CPP and (b)Lukash trajectories just above the lower critical value$C = \kappa$. The colors and labels refer to the respective cases. Caveat: the CPP and Lukash scales are different. (c) combines the two curves.