Echoes of Traversable Wormhole

TL;DR

The paper investigates linear scalar perturbations of the four-dimensional Maldacena-Milekhin-Popov traversable wormhole, formed by a near-extremal Reissner–Nordström mouth and an AdS2 × S2 throat supported by charged massless fermions. By constructing a global Schrödinger-like potential in tortoise coordinates that features two sharp, widely separated barriers, the authors identify a resonant cavity that can sustain wave trapping. Time-domain simulations of Gaussian perturbations inside the throat reveal a clear train of echoes whose amplitudes grow with the multipole number $l$, highlighting stronger trapping for higher angular momentum. These results link observable echo phenomenology to the wormhole's near-horizon geometry and holographic interpretation, offering avenues for gravitational-wave templates and further study of tensor perturbations and rotational effects.

Abstract

We study linear scalar perturbations of the four-dimensional, traversable wormhole solution of Maldacena, Milekhin, and Popov(arXiv:1807.04726). The geometry is constructed by matching an asymptotically flat, near-extremal Reissner--Nordström region to a throat described by $AdS_2 \times S^2$, supported by charged massless fermions. We derive the effective scalar potential governing wave dynamics, which when viewed in the tortoise coordinate, exhibits two extremely sharp and widely separated barriers. These barriers form a resonant cavity and are a direct consequence of the near-horizon geometry of the wormhole mouths. Using time-domain integration, we analyze the wormhole's response to an initial scalar wave packet inside the throat. We find that the late-time signal contains a distinct train of echoes whose amplitude depends on the angular momentum number $l$. We show that higher $l$ modes produce significantly stronger echoes, as the corresponding potential barriers are taller and more reflective, which results in more efficient trapping of the wave within the wormhole throat.

Figures (5)

• Figure 1: Comparison of the metric functions $f_{\text{mouth}}(r)$ and $f_{\text{throat}}(r)$ in the vicinity of the matching point $r=2.85397$. The near-coincidence illustrates smooth matching of the geometry.
• Figure 2: The scalar potential $V(r)$ for $l=3$ near the matching point $r=2.85397$, constructed by combining $V_{\text{throat}}(r)$ and $V_{\text{mouth}}(r)$. The potential is nearly continuous across the junction, indicating that wave propagation is well defined globally.
• Figure 3: The effective potential for a scalar field with angular momentum $l=3$ plotted against the tortoise coordinate $r_*$. The top panel shows the global double-barrier structure, emphasizing the vast separation between the peaks. The bottom panels show magnified views of the (a) left and (b) right barriers, revealing their smooth, bell-like shape.
• Figure 4: The effective Scalar Potential for angular modes $l=1$ through $l=6$.
• Figure 5: Time-domain profiles of the scalar field response, $|\psi(t, r_*= -303)|$, for angular modes $l=1$ through $l=6$. The waveform is measured by an observer at the center of the wormhole throat. The initial pulse is followed by a clear train of echoes, whose amplitude grows significantly with increasing $l$ due to the strengthening of the potential barriers.