Wada Boundaries in Generic Polynomial PP-Wave Spacetimes

TL;DR

This work addresses unpredictability in geodesic scattering within pp-wave spacetimes that feature polynomial harmonic profiles of degree $n$. By mapping the problem to a two-dimensional Hamiltonian with potential $V_n(x,y)=\frac{1}{2}\Re[(x+iy)^n]$, the authors compute escape basins for $3\le n\le 10$ and verify the Wada property using a grid-based test, while quantifying uncertainty with basin entropy $S_b$ and boundary entropy $S_{bb}$. They find that all basins are Wada for the studied range, with $S_b$ increasing with $n$ and $S_{bb}$ exceeding $\ln 2$ for $n>3$, indicating fractal, highly unpredictable boundaries; a monotonic decrease of the uncertainty exponent $\alpha$ with $n$ is also reported. The study broadens the connection between exact gravitational wave solutions and chaotic scattering, provides robust numerical diagnostics for Wada boundaries, and conjectures the persistence of Wada properties for all integer degrees $n$ after $n>10$.

Abstract

We study the dynamics of the geodesics of pp-wave spacetimes with polynomial profiles of degrees $3\leq n\leq10$, which are dynamically equivalent to the motion of a classical particle in a two-dimensional harmonic polynomial potential. By analysing the escape basins associated with different asymptotic outcomes, we show that all basins exhibit the Wada property for every polynomial degree considered. We further compute the basin entropy $S_{b}$, finding that it increases monotonically with the polynomial degree, indicating enhanced unpredictability of the final state of the system. The boundary basin entropy $S_{bb}$ is also evaluated, and the $\ln(2)$ criterion confirms that the basin boundaries are fractal for $n>3$. We conjecture that the Wada property persists for polynomial degrees $n>10$.

Figures (6)

• Figure 1: (a) Trajectory of a ring of particles equally spaced around the circle $x^2+y^2=1$, initially at rest. (b) Potential \ref{['potential']} for $n=5$.
• Figure 2: One-dimensional escape basin for $n=5$ and for particles distributed along a unit circle with zero initial velocity.
• Figure 3: Exit basins for the $p_x$-$p_y$ plane for the polynomial degree values of (a) $n=4$ and (b) $n=5$. The region outside of the disk leads to unphysical initial conditions. The color gradient indicates the escape channels, with dark blue being the first channel and yellow being the last channel.
• Figure 4: Three zoom levels of the exit basin for $n=5$. Each successive zoom level gives a magnification of $10$ times. The color gradient indicates the escape channels, with dark blue being the first channel and yellow being the fifth channel.
• Figure 5: Grid method applied to the $n=5$ basins. (a) For the angular position space, displayed in Figure \ref{['fig:basin_n=5_1D']}, and (b) for the momentum plane basin, displayed in Figure \ref{['fig:grid_wada']}(b). For both boundaries, the grid method concludes that the boundaries have the Wada property.