Echoes of the Maldacena-Milekhin-Popov traversable wormhole

Abstract

We study linear perturbations of massless scalar and vector gauge fields, treated as test fields, on the four-dimensional traversable wormhole of Maldacena, Milekhin, and Popov~\cite{Maldacena:2018gjk}. Both effective potentials vanish throughout the throat and rise in each mouth to a single barrier peaked at the extremal photon-sphere radius $r=2r_e$ for every multipole. In the tortoise coordinate these two barriers are only $\mathcal{O}(10^2)$ wide but sit a distance $2D\simeq \pi L_{\mathrm{wh}}\sim 10^{35}$ apart, which puts direct time-domain evolution out of reach. We instead compute the single-barrier scattering amplitudes by Numerov integration, validated against an exactly solvable barrier and against an independent time-domain evolution, and then sum the multiple reflections in closed form. The parabolic WKB formula always gives a transmission probability of $1/2$ at the barrier top, whereas the true value is $0.62$. It also fails below the top, where the transmission falls as $|\kappa {|}^2\propto {\omega }^{6.4}$ for $l=1$. Echoes emerge at $T_m=(2m+1)D$ as narrow-band wave packets near the light-ring frequency. Their frequency decreases with each reflection and the energy decays algebraically, $E_m\propto m^{-1}$. The quasinormal spectrum splits into two families. Modes trapped between the barriers are extraordinarily long-lived, with quality factors up to $10^{41}$, and carry the same astronomical time-scale as the echoes. The photon-sphere modes of a single mouth are damped $10^{36}$ times faster than cavity modes, with quality factors of order unity at low multipoles. Only this second family of quasinormal modes is observable, characterising the ringdown of a single mouth. The echoes and the trapped cavity modes instead characterise the late-time dynamics of the throat, not a detectable signal.

52 pages, 30 figures, v2 is substantially revised and extended. Corrects a numerical shortcut used in v1, adds electromagnetic perturbations as test field, and computes both the cavity and photon-sphere quasinormal mode spectra. Title changed from v1