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# Sturm–Liouville and Carroll: at the heart of the memory effect

Abstract : For a plane gravitational wave whose profile is given, in Brinkmann coordinates, by a $2\times 2$ symmetric traceless matrix K(U), the matrix Sturm–Liouville equation $\ddot{P}=KP$ plays a multiple and central rôle: (i) it determines the isometries, (ii) it appears as the key tool for switching from Brinkmann to BJR coordinates and vice versa, (iii) it determines the trajectories of particles initially at rest. All trajectories can be obtained from trivial “Carrollian” ones by a suitable action of the (broken) Carrollian isometry group.
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https://hal.archives-ouvertes.fr/hal-01768110
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Submitted on : Monday, April 16, 2018 - 10:46:42 PM
Last modification on : Tuesday, January 11, 2022 - 5:56:35 PM

### Citation

P.M. Zhang, M. Elbistan, G.W. Gibbons, P.A. Horvathy. Sturm–Liouville and Carroll: at the heart of the memory effect. Gen.Rel.Grav., 2018, 50 (9), pp.107. ⟨10.1007/s10714-018-2430-0⟩. ⟨hal-01768110⟩

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