Abstract
Plane gravitational waves can admit a sixth ‘screw’ isometry beyond the usual five. The same is true of plane electromagnetic waves. From the point of view of integrable systems, a sixth isometry would appear to over-constrain particle dynamics in such waves; we show here, though, that no effect of the sixth isometry is independent of those from the usual five. Many properties of particle dynamics in a screw-symmetric gravitational wave are also seen in a (non-plane-wave) electromagnetic vortex; we make this connection explicit, showing that the screw-symmetric gravitational wave is the classical double copy of the vortex.
Highlights
The plane wave approximation provides a simplified setting in which to investigate signatures of gravitational waves [1, 2], such as the velocity memory effect [3, 4, 5], in which particles initially at rest acquire a constant, nonzero velocity after the wave has passed over them
Using this we can see that the screw-symmetric gravitational plane wave in Brinkmann coordinates is the classical double copy [11] of the electromagnetic vortex, as follows
Classical particle dynamics in plane waves has long been known to be exactly solvable. This is a rare example of a superintegrable system
Summary
The plane wave approximation provides a simplified setting in which to investigate signatures of gravitational waves [1, 2], such as the velocity memory effect [3, 4, 5], in which particles initially at rest acquire a constant, nonzero velocity after the wave has passed over them. Each Killing vector implies the existence of a conserved quantity in the particle motion. To analyse this we use the Hamiltonian formalism, which requires gauging the reparameterisation invariance of the particle action as usual, and we take u as time [25]. A test particle in a gravitational plane wave is a superintegrable system; to show this, and noting that the Hamiltonian is time-dependent, we follow the standard method of converting to an autonomous system; we expand phase space to eight dimensions by promoting u to a coordinate with conjugate momentum pu, and use a new Hamiltonian K = H − pu, for a review see [32]. The solution of the equations of motion proceeds algebraically from here: the three momenta are conserved, Q4 and Q5 determine {x1, x2} as functions of time u, while Q7 determines v
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