Scalar fields in black hole spacetimes

The time evolution of matter fields in black hole exterior spacetimes is a well-studied subject, spanning several decades of research. However, the behavior of fields in the black hole interior spacetime has only relatively recently begun receiving some attention from the research community. In this paper, we numerically study the late-time evolution of scalar fields in both Schwarzschild and Kerr spacetimes, including the black hole interior. We recover the expected late-time power-law “tails” on the exterior (null infinity, timelike infinity, and the horizon). In the interior region, we find an interesting oscillatory behavior that is characterized by the multipole index $\ell$ of the scalar field. In addition, we also study the extremal Kerr case and find strong indications of an instability developing at the horizon.

Figure 1

Constant-$\tilde{t}$ slices of Schwarzschild spacetime plotted on a background of Kruskal coordinates $u$ and $v$ as defined, in terms of Schwarzschild $(t,r)$ coordinates, in Eqs. (31.18) and as pictured in Fig. 31.4(b) of Ref. [16]. The diagonal line on the right is the event horizon at $r=2M$. The curves plotted show how the constant $\tilde{t}$ slices extend from the black hole interior to exterior regions.

Figure 2

Penrose diagram of the computational domain for the Kerr spacetime case, adapted from Ref. [9]. The constant $\rho$ (dotted lines) and constant $\tau$ (solid lines) slices are clearly marked. In addition, on the initial Cauchy slice $\mathrm{\Sigma }$ we also mark (thick black line) where the compactly supported initial data are nonzero.

Figure 3

Late-time scalar field tails for different multipoles $\ell$ at $\rho =1$ for the case of Schwarzschild black hole interior. The power laws agree with the expression ${\tau }^{-2\ell -3}$.

Figure 4

Late-time scalar field radial $\rho$ profile for multipole $\ell =2$ for a Schwarzschild black hole. Note that even at late stages, the field exhibits oscillatory behavior but only in the black hole interior region.

Figure 5

Late-time scalar field radial $\rho$ profile for multipole $\ell =4$ for a Schwarzschild black hole. Note that even at late stages the field exhibits oscillatory behavior but only in the black hole interior region. The number of cycles is proportional to $\ell$.

Figure 6

Late-time scalar field radial $\rho$ profile for multipole $\ell =4$ for a Schwarzschild black hole. The initial field has compact support. At late stages the field exhibits oscillatory behavior but only in the black hole interior region. The number of cycles is proportional to $\ell$.

Figure 7

Late-time scalar field tails for different multipoles ${\ell }^{\prime }$ at $\rho =1$ of a Kerr black hole interior. The power laws agree with Eq. (11). Data are presented from computations using two different grid sizes [$8000(\rho )\times 32(\theta )\times 1665972(\tilde{t})$ and $16000(\rho )\times 64(\theta )\times 3331944(\tilde{t})$], to demonstrate convergence of the numerical results.

Figure 8

The local power index (LPI), defined as $n=\tau \dot{\mathrm{\Psi }}/\mathrm{\Psi }$, for different multipoles ${\ell }^{\prime }$ at $\rho =1$ of a Kerr black hole interior. The late-time index values agree with Eq. (11).

Figure 9

Late-time scalar field radial $\rho$ profile for multipole ${\ell }^{\prime }=1$ for a Kerr black hole. Note that even at late stages the field exhibits oscillatory behavior but only in the black hole interior region.

Figure 10

Late-time scalar field radial $\rho$ profile for multipole ${\ell }^{\prime }=4$ for a Kerr black hole. There are no oscillations at late times because of the dominance of the $\ell =0$ multipole.

Figure 11

Late-time scalar field tails for ${\ell }^{\prime }=4$ at different locations (horizon, $\rho =3.0$, and null infinity) for an extremal Kerr black hole. The horizon decay rate is slower than the $\rho =3.0$ case and matches the rate at null infinity.

Figure 12

Late-time scalar field radial $\rho$ profile for multipole ${\ell }^{\prime }=3$ for an extremal Kerr black hole. Note that the field’s gradient exhibits unbounded growth at the horizon.