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Sturm-Liouville and Carroll: at the heart of the Memory Effect

P. -M. Zhang, M. Elbistan, G. W. Gibbons, P. A. Horvathy

TL;DR

This work shows that the memory effect in exact plane gravitational waves is governed by the matrix Sturm-Liouville equation $\ddot{P}=K(U)P$ for the wave profile $K(U)$. This equation simultaneously controls the spacetime isometries, the Brinkmann$\Leftrightarrow$BJR coordinate switch, and the trajectories of particles; a broken Carroll symmetry underlies the full solution, with Carrollian motion serving as the seed from which all geodesics are generated via isometries. In the BJR frame, the evolution is encapsulated by $H(u)$ with $a(u)=P^T P$, and in Brinkmann coordinates by $Q=PH$, both obeying the same SL dynamics; the generic isometry group is the Carroll subgroup without rotations. A polarized sandwich-wave example demonstrates that even simple initial data yield nontrivial Brinkmann motion, while BJR coordinates reveal the memory through time-dependent translations, emphasizing that solving $\ddot{P}=K(U)P$ (numerically in general) is the key to the memory effect.

Abstract

For a plane gravitational wave whose profile is given, in Brinkmann coordinates, by a $2\times2$ 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.

Sturm-Liouville and Carroll: at the heart of the Memory Effect

TL;DR

This work shows that the memory effect in exact plane gravitational waves is governed by the matrix Sturm-Liouville equation for the wave profile . This equation simultaneously controls the spacetime isometries, the BrinkmannBJR coordinate switch, and the trajectories of particles; a broken Carroll symmetry underlies the full solution, with Carrollian motion serving as the seed from which all geodesics are generated via isometries. In the BJR frame, the evolution is encapsulated by with , and in Brinkmann coordinates by , both obeying the same SL dynamics; the generic isometry group is the Carroll subgroup without rotations. A polarized sandwich-wave example demonstrates that even simple initial data yield nontrivial Brinkmann motion, while BJR coordinates reveal the memory through time-dependent translations, emphasizing that solving (numerically in general) is the key to the memory effect.

Abstract

For a plane gravitational wave whose profile is given, in Brinkmann coordinates, by a symmetric traceless matrix , the matrix Sturm-Liouville equation 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.

Paper Structure

This paper contains 6 sections, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Killing vectors and isometries
  3. Brinkmann $\Leftrightarrow$ BJR
  4. Trajectories
  5. Illustration: polarized sandwich waves
  6. Conclusion

Figures (3)

  • Figure 1: Polarized sandwich wave with Gaussian envelope as given in (\ref{['polgaussprof']}). The colors refer to the $\textcolor{blue}{\mathcal{A}_{+}}$ and the ${\mathcal{A}_{\times}}$ polarisation components.
  • Figure 2: The images under the B $\Leftrightarrow$ BJR map (\ref{['Bsimplegeo']}) of the simple trajectories (\ref{['simpleBJRtraj']}) initially at rest for $u_0<0$ at ${\bf x}_0^1={\bf X}_0^{(1)}=(\textcolor{blue}{1},\textcolor{rgb(0,89,0)}{0})$ and at ${\bf x}_0^{(2)}={\bf X}_0^{(2)}=(\textcolor{red}{0},\textcolor{cyan}{1})$, respectively, are, B coordinates, the two columns of the $P$ matrix. The motion is complicated in the inside-zone but follows straight lines with constant velocity in the after-zone.
  • Figure 3: In BJR coordinates boosts act according to (\ref{['genCarr']}). The implementation differs substantially from the Galilean one (dashed). We took here ${\bf b}=(1,1)$.