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Approximate Models for Gravitational Memory

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

Abstract

The large-distance development of a sandwich gravitational wave, consistent with Carroll symmetry, provides us with a surprisingly good analytic approximation of the motion of particles in a wave with Pöschl-Teller profile. The role of the 2nd solution of the Stern-Liouville equation is highlighted. Similar results hold for Gaussian profiles.

Approximate Models for Gravitational Memory

Abstract

The large-distance development of a sandwich gravitational wave, consistent with Carroll symmetry, provides us with a surprisingly good analytic approximation of the motion of particles in a wave with Pöschl-Teller profile. The role of the 2nd solution of the Stern-Liouville equation is highlighted. Similar results hold for Gaussian profiles.
Paper Structure (7 sections, 32 equations, 9 figures)

This paper contains 7 sections, 32 equations, 9 figures.

Table of Contents

  1. Introduction:
  2. Memory in plane gravitational wave
  3. An approximate toy model for the Pöschl - Teller profile
  4. Solutions of the Sturm-Liouville eqn.
  5. Carroll symmetry
  6. Gaussian
  7. Conclusion

Figures (9)

  • Figure 1: For large $|U|$, the approximate profile\ref{['extoy']} with amplitude $k\approx 2.4$ is close to that of Pöschl - Teller , ${\mathcal{A}}^{PT}$ in \ref{['PTellerPot']}. However ${\mathcal{A}}^{approx}$ in \ref{['extoy']} has a cusp at $U=0$ and the two profiles differ substantially.
  • Figure 3: DM toy trajectories (dashed) is obtained when the amplitude is a zero of the Bessel function $J_1$, eqn. \ref{['J1zero']}. The Wavezone contains an even number, $m=2\ell$ symmetrically positioned half-waves. These trajectories are symmetric w.r.t the origin (implying with no effective displacement) and approximate those for Pöschl - Teller .
  • Figure 4: When $k\neq k_{krit}$ then the positive and negative $U$-branches do not match smoothly : (a) either for antisymmetric fitting $X_{-}(0)\neq X_{+}(0)$ (b) for symmetric fitting $X_{-}'(0)\neq X_{+}'(0)$.
  • Figure 5: Increasing the amplitude $k$ shifts the trajectory along $U$ leftwards. DM is obtained when the far-most-left zero (in blue) or the far-most-left bottom (in purple) enters into the physical domain by reaching the $U=0$ axis.
  • Figure 6: The Sturm-Liouville solutions ${\bf P}$, the non-DM 2nd solution ${\bf Q=PS}$ and the Souriau matrix${\bf S}$ shown here for the toy model eqn \ref{['extoy']} with (a) DM amplitude $k=k_{crit}$ and (b) VM amplitude $k=4$.
  • ...and 4 more figures