Late-time tails in the Reissner-Nordström spacetime revisited

We propose that the late-time tail problem in the Reissner-Nordström (RN) spacetime is dual to a tail problem in the Schwarzschild spacetime with a different initial data set: at a fixed observation point the asymptotic decay rate of the fields are equal. This duality is used to find the decay rate for tails in RN. This decay rate is exactly as in Schwarzschild, including the case of the extremely-charged RN spacetime (ERN). The only case where any deviation from the Schwarzschild decay rate is found is the case of the tails along the event horizon of an ERN spacetime, where the decay rate is the same as at future null infinity. As observed at a fixed location, the decay rate in ERN is the same as in Schwarzschild. We verify these expectations with numerical simulations.

Figure 1

The effective potential for $\ell =1$ in ERN (solid curve) and Schwarzschild (dashed curve) as a function of $r_{*}$.

Figure 2

A schematic diagram of scattered waves. The initial pulse is initially outgoing (and is therefore shown on the incoming leg of the characteristic hypersurface), with the primary waves (p) arriving to ${\mathrm{scri}}_{+}$. The waves which cross the worldline of an observer at $r=\mathrm{const}$ at late times are scattered waves: the secondary waves (s) undergo one scattering event, and quaternary waves (q) undergo three scattering events. We only display the waves approaching the observer from the right, hence no tertiary waves are shown. The abbreviation “EH” stands for the event horizon.

Figure 3

The magnitude of the tail in Schwarzschild at a fixed value of time as a function of $A$ for different values of $r_0$. Same initial data and evaluation point were used in all cases. $*$: $r_0=100M$; □: $r_0=125M$; $\times$: $r_0=150M$; ⋄: $r_0=175M$; ▵: $r_0=200M$. The circles (○) are the Richardson extrapolations of the data at finite values of $r_0$ to $r_0\to \infty$, and the solid line is a best fit line of the extrapolated data (circles). The squared correlation coefficient for the solid line is $R^2=0.9995$. The dotted curves are $3\sigma$ confidence curves of the extrapolated data.

Figure 4

The spacetime diagram with three curves corresponding to three different values of $r_0$, dividing spacetime into three regions I, II, and III. See text for more information.

Figure 5

Same as Fig. 3 for the ERN spacetime. The squared correlation coefficient for the solid line is $R^2=0.9987$.

Figure 6

The magnitude of the tail in ERN at a fixed evaluation point as a function of $A$ for different values of $t$, for the same choice of $r_0$. Same initial data and evaluation point ($r_{*}=0$) were used in all cases. Here, $\ell =1$, and $r_0=100M$. The evaluation times are *: $t=300M$; $+$: $t=400M$; □: $t=500M$; ○: $t=600M$; $\times$: $t=800M$; and ▵: $t=1000M$.

Figure 7

The duality transformations of the tail problem for ERN with initial data of compact support. See text for details.

Figure 8

The duality transformations of the tail problem for ERN with initial data of an initial static moment. See text for details.

Figure 9

The local power index $n(t)$ as a function of $m/t$ in Schwarzschild, for initial data given by $\psi \sim r_{*}^{-\ell }$ for $r_{*}>0$ and $\psi =0$ for $r_{*}<0$, for a scalar field ($s=0$), for $\ell =1$ (upper panel), $\ell =2$ (middle panel), and $\ell =3$ (lower panel). See text for details.

Figure 10

The field along $r_{*}=0$ as a function of $t$, for initial data with an initially static moment, for a scalar field ($s=0$), for $\ell =1$ (upper panel), $\ell =3$ (middle panel), and $\ell =4$ (lower panel). See text for details.

Figure 11

The local power index $n$ along $r_{*}=0$ as a function of $M/t$, for initial data with an initially static moment, for a scalar field ($s=0$), for $\ell =1$ (left upper panel), $\ell =2$ (right upper panel), $\ell =3$ (left lower panel), and $\ell =4$ (right lower panel). See text for details.

Figure 12

The field along $r_{*}=0$ (solid curve), along the event horizon (dotted curve), and along future null infinity (dashed curve). Upper panel: initial data of a pure initially static moment. Lower panel: initial data of an initially incoming pulse of compact support between $-10M<r_{*}<10M$. For both cases $\ell =1$. The “bending” of the curves along the event horizon and along future null infinity for late advanced and retarded times, correspondingly, occurs because these two asymptotic regions of spacetimes are represented by large (but finite) retarded and advanced times, respectively.

Figure 13

Convergence test for a scalar field ($s=0$) and dipole mode ($\ell =1$), for Schwarzschild (solid) and for ERN (dashed) for initial data with an initial static moment. Shown are the convergence order $N$ as a function of time along $r_{*}=0$. Oscillations of the convergence order at the quasinormal mode epoch are typical of oscillatory data.