Memory effects in Kundt wave spacetimes

TL;DR

The paper studies memory effects for geodesics in exact Kundt wave spacetimes with non-flat constant- and variable-curvature transverse surfaces. It finds displacement memory for negative curvature, displacement plus velocity memory along \(x\) and a new frequency memory along \(y\) for positive curvature, with qualitatively similar behavior under variable curvature.

Key findings

• In \(H^2 \times M(1,1)\) (constant negative curvature), there is displacement memory along both \(x\) and \(y\): separation increases along \(x\) and decreases along \(y\), while no velocity memory is found because the velocity vanishes after the pulse departs.
• In \(S^2 \times M(1,1)\) (Plebański-Hacyan, constant positive curvature), there is displacement and velocity memory along \(x\); along \(y\), the separation shows periodic oscillations after the pulse, which the authors call a new “frequency memory effect.”
• Frequency memory is quantified in Table 1 for three initial configurations: \((x,y)=(2.0,3.0)\) gives troughs at \(5.98\) and \(7.13\), frequency \(0.869565\,u^{-1}\), wavelength \(1.15\,u\); \((0.1,0.1)\) gives \(28.82\), \(65.67\), frequency \(0.027137\,u^{-1}\), wavelength \(36.85\,u\); \((0.01,0.01)\) gives \(278.5\), \(647.9\), frequency \(0.002707\,u^{-1}\), wavelength \(369.4\,u\). Higher initial \(y\)-coordinate yields higher frequency and lower wavelength.
• For variable curvature, negative curvature (\(P=\operatorname{sech} y\), \(R=-2\operatorname{sech}^4 y\)) produces only displacement memory, while positive curvature (\(P=\sec y\), \(R=2\sec^4 y>0\)) produces monotonic displacement along \(x\) and oscillatory/frequency memory along \(y\), qualitatively agreeing with the constant-curvature cases.

Methods

• The authors use exact Kundt wave metrics of the form \(ds^2 = 2du\,dv + H\,du^2 + \frac{1}{P^2}(dx^2+dy^2)\), with \(H=\operatorname{sech}^2(u)(x^2-y^2)\), \(W_i=0\), and \(H\) independent of \(v\); they choose \(P\) to realize \(H^2 \times M(1,1)\), \(S^2 \times M(1,1)\), and variable-curvature backgrounds.
• They solve the geodesic equations for the transverse coordinates \(x(u), y(u)\) numerically using Mathematica 10, with \(u\) acting as an affine parameter, because the equations cannot be solved analytically.
• Initial transverse geodesic velocities are set to zero before the pulse; the authors compare geodesic separation and velocity before and after the pulse to identify memory effects, using a \(\operatorname{sech}^2\) pulse profile and checking a Gaussian profile as well.

Limitations

• Stated in the paper: the geodesic equations cannot be solved analytically and are solved numerically with Mathematica 10; for the \(S^2 \times M(1,1)\) case the authors explicitly state they were unable to find analytical solutions. The Gaussian pulse was checked but not shown.
• Evident from the text: the conclusions rely on numerical plots and selected initial conditions, and the frequency memory is quantified only for three initial configurations in Table 1; no analytic proof of the curvature-memory link or observational detection is provided.
• Stated model assumptions: gyratonic matter is excluded (\(W_i=0\)), \(H\) is independent of \(v\), and only the outside region containing gravitational radiation is considered.