Universality of massive scalar field late-time tails in black-hole spacetimes

TL;DR

The paper investigates the late-time tails of a massive scalar field in black-hole spacetimes, focusing on Schwarzschild and Kerr geometries. It employs numerical time-domain evolutions—1+1D double-null for Schwarzschild and 2+1D penetrating Teukolsky for Kerr—to test analytic predictions. The main finding is a universal, oscillatory tail decaying as $t^{-5/6}$ with a time-dependent frequency $\omega(t)$ that approaches the mass $\mu$, independent of the multipole index $\ell$ and spin parameter $a/M$, with a preceding flat-spacetime-like decay $t^{-3/2}$ at early times and a broken power-law crossover for intermediate masses. This universality, matching and extending prior analytic results, has implications for wave propagation and tail behavior in curved spacetimes, while highlighting open issues in nonaxisymmetric Kerr evolutions and potential late-time modulations.

Abstract

The late-time tails of a massive scalar field in the spacetime of black holes are studied numerically. Previous analytical results for a Schwarzschild black hole are confirmed: The late-time behavior of the field as recorded by a static observer is given by $ψ(t)\sim t^{-5/6}\sin [ω(t)\times t]$, where $ω(t)$ depends weakly on time. This result is carried over to the case of a Kerr black hole. In particular, it is found that the power-law index of -5/6 depends on neither the multipole mode $\ell$ nor on the spin rate of the black hole $a/M$. In all black hole spacetimes, massive scalar fields have the same late-time behavior irrespective of their initial data (i.e., angular distribution). Their late-time behavior is universal.

Figures (14)

• Figure 1: Convergence test for the 1+1D code. We find the times at which the field has zeros for three different resolutions, high $t({\rm h})$, medium $t({\rm m})$, and low $t({\rm l})$, and compute the ratio $[t({\rm h})-t({\rm m})]/[t({\rm m})-t({\rm l})]$ for each zero. We then plot this ratio as a function of time $t$. Here, we used $N=5,10$, and 20 grid points per $M$ for the three resolutions. Second order convergence corresponds to the ratio equaling 0.25.
• Figure 2: Field of a massive scalar field along $r_*=0$. The initial data are for $\ell=0$, $u_1=20M$ and $u_2=60M$, and $N=1$. The solid curve is the massive scalar field (with $\mu M=10^{-3}$), and the dashed line is proportional to $(t/M)^{-3/2}$.
• Figure 3: Field of a massive scalar field along $r_*=0$. The initial data are for $\ell=0$, $u_1=20M$ and $u_2=60M$, and $N=4$. The solid curve is the massive scalar field (with $\mu M=1$), and the dashed line is proportional to $(t/M)^{-5/6}$.
• Figure 4: Field of a massive scalar field along $r_*=0$. The initial data are for $\ell=0$, $u_1=20M$ and $u_2=60M$, and $N=4$. The solid curve is the massive scalar field (with $\mu M=2.5\times 10^{-2}$). The dashed line (a) is proportional to $(t/M)^{-3/2}$, and the dotted line (b) is proportional to $(t/M)^{-5/6}$.
• Figure 5: Same as Fig. \ref{['high_mu']}, but with $\ell=1$.