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Gravitational wave memory: further examples

P. -M. Zhang, Q. -L. Zhao, M. Elbistan, P. A. Horvathy

TL;DR

The paper investigates gravitational wave memory by analyzing geodesics in a four-dimensional Brinkmann metric with memory profile $\mathcal{A}(U)$, comparing velocity memory ($VM$) and displacement memory ($DM$) across several profiles. It demonstrates that Gaussian and Pöschl–Teller profiles yield $DM$ in the attractive sector but not in the repulsive sector, while the $|U|^{-4}$ profile produces $DM$ in the attractive transverse direction with no $DM$ in the repulsive direction unless the trajectory is trivially zero; the flyby derivatives can yield $DM$ or $DM-2$ depending on gluing, and a double-square potential enables parameter-tuned $DM$ (e.g., $h a = m\pi + \pi/4$). The results refine Zel'dovich–Polnarev’s claim by showing that pure displacement memory is achievable only for specific wave parameters, whereas broader profiles can yield mixed memory (including $DM-2$). Overall, the work connects early expectations with concrete analytic and approximate solutions, clarifying how memory signatures depend on profile shape, gluing choices, and derivative order of the profile $\mathcal{A}(U)$.$

Abstract

Ehlers and Kundt [1] argued in favor of the velocity effect: particles initally at rest hit by a burst of gravitational waves should fly apart with constant velocity after the wave has passed. Zel'dovich and Polnarev [2] suggested instead that waves generated by flyby would be merely displaced. Their prediction is confirmed provided the wave parameters take some particular values.

Gravitational wave memory: further examples

TL;DR

The paper investigates gravitational wave memory by analyzing geodesics in a four-dimensional Brinkmann metric with memory profile , comparing velocity memory () and displacement memory () across several profiles. It demonstrates that Gaussian and Pöschl–Teller profiles yield in the attractive sector but not in the repulsive sector, while the profile produces in the attractive transverse direction with no in the repulsive direction unless the trajectory is trivially zero; the flyby derivatives can yield or depending on gluing, and a double-square potential enables parameter-tuned (e.g., ). The results refine Zel'dovich–Polnarev’s claim by showing that pure displacement memory is achievable only for specific wave parameters, whereas broader profiles can yield mixed memory (including ). Overall, the work connects early expectations with concrete analytic and approximate solutions, clarifying how memory signatures depend on profile shape, gluing choices, and derivative order of the profile .$

Abstract

Ehlers and Kundt [1] argued in favor of the velocity effect: particles initally at rest hit by a burst of gravitational waves should fly apart with constant velocity after the wave has passed. Zel'dovich and Polnarev [2] suggested instead that waves generated by flyby would be merely displaced. Their prediction is confirmed provided the wave parameters take some particular values.

Paper Structure

This paper contains 6 sections, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Memory effect
  3. Gaussian or Pöschl-Teller profile
  4. $|U|^{-4}$ profile
  5. Flyby (and beyond)
  6. Conclusion

Figures (4)

  • Figure 1: \ref{['even-X-U4']}: For $c_2=1$ as only nonzero parameter, the evenly-glued $X^2$ component \ref{['evenglue']} is symmetric w.r.t. $U$-reversal \ref{['Ureversal']} and thus $X(-\infty) = 1 = X(+\infty)$. \ref{['even-V-U4']}: At $U=\pm\infty$ the velocity goes to zero as it should.
  • Figure 2: \ref{['rLPP-displace-X-2-1']}: For $c_2=1$ as only nonzero parameter, the oddly-glued component \ref{['oddglue']} is antisymmetric w.r.t. $U$-reversal \ref{['Ureversal']} and is thus displaced from $X^2(-\infty) = 1$ to $X^2(+\infty) =-1$. \ref{['rLPP-displace-VX-2-1']}: At $U=\pm\infty$ the velocity goes to zero, consistently with DM.
  • Figure 3: Both of the $c_2=1$ odd \ref{['rLPP-ana-sol-Y-1']} and the $c_1=1$ even solution \ref{['rLPP-ana-sol-Y-2']} diverge at $U=0$ requiring $X^1(U)\equiv 0$ for all $U$.
  • Figure 4: Double-square approximation of the flyby profile.