Displacement memory in regular black hole spacetimes

TL;DR

This paper investigates displacement memory in regular black hole spacetimes using a restricted Bondi–Sachs framework with a pulse $H(u)$. By analyzing timelike geodesics through geodesic separation and geodesic deviation, the authors show a persistent, pulse-induced memory that depends on the regularization parameter $g$ and the pulse amplitude, with distinct signatures across Bardeen, Hayward, and Kar–Kar regular BH models. The Schwarzschild limit ($g=0$) yields the largest memory, suggesting memory as a potential observational probe to distinguish regular from singular spacetimes and to differentiate among regular BH families. The work connects to asymptotic Bondi quantities by highlighting how $g$ would influence Bondi mass-related terms, and lays groundwork for extensions to nonlinear memory and realistic astrophysical sources in regular spacetimes.

Abstract

Displacement memory, induced by a wave pulse in a regular black hole spacetime, is studied using geodesic (timelike) separation and geodesic deviation. The presence of the wave pulse in such a black hole is modeled via a function $H(u)$ appearing in a restricted version of a generic Bondi-Sachs type line element. Choosing a sech-squared profile for $H(u)$, we first study (numerically) geodesic separation and geodesic deviation in a flat background. Thereafter, similar investigations are carried out in the presence of the black hole, but in regions far away from the vicinity of the horizon. Our results suggest the presence of a distinct displacement memory effect, which depends on the value of the regularisation parameter $g$ as well as the pulse height. Between different types of regular black holes, one notices parameter-dependent changes in the net displacement memory. Further, a clear difference in the magnitude of displacement memory (at large $u$) in regular and singular black holes is also visible in our numerical results.

Figures (21)

• Figure 1: Pulse profile ($H(u)$) with the parameters $A=1$, $u_0=0$, $w=1$ and its double derivative.
• Figure 2: Top panel: Geodesic components $r(u)$ and $\phi(u)$ with and without the GW pulse in flat spacetime. Bottom panel:$dr/du$ and $d\phi/du$ as functions of $u$, both in the presence and absence of GW pulse for flat metric. The values of the parameters are: $L_c=0.005$, and $A=3$.
• Figure 3: Variation of $\Delta r$ and $\Delta \phi$ with $u$ for the flat metric, shown both in the presence and absence of a GW pulse. The chosen parameters are $L_c = 0.005$ and $A = 3$.
• Figure 4: Left panel: Trajectories of two geodesics have been shown in the presence of the GW with $L_c= 0.005$ and $A=1.5$. The red dot represents $u=-100$ and the blue dot marks $u=+100$. Right panel: Trajectories of the same pair of test particles shown in the region close to the GW pulse: $u \in [-10, 10]$.
• Figure 5: Top panel:$r (u)$ and $\phi (u)$ both in the presence and absence of the pulse for Bardeen Profile. Bottom panel: Variation of $dr/du$ and $d\phi/ du$ with $u$ in the presence of pulse and without pulse for Bardeen Profile. The parameters are: $L_c= 0.1$, $g=0.1$ and $A=1$.