News from critical collapse: Bondi mass, tails, and quasinormal modes

We discuss critical gravitational collapse on the threshold of apparent horizon formation as a model both for the discussion of global aspects of critical collapse and for numerical studies in a compactified context. For our matter model we choose a self-gravitating massless scalar field in spherical symmetry, which has been studied extensively in the critical collapse literature. Our evolution system is based on Bondi coordinates, the mass-function is used as an evolution variable to ensure regularity at null infinity. We compute radiation quantities like the Bondi mass and news-function and find that they reflect the discretely self-similar (DSS) behavior. Surprisingly, the period of radiation at null infinity is related to the formal result for the leading quasinormal mode of a black hole with rapidly decreasing mass. Furthermore, our investigations shed some light on global versus local issues in critical collapse, and the validity and usefulness of the concept of null infinity when predicting detector signals.

Figure 1

A Penrose diagram of a typical collapse spacetime. Shown is our numerical null grid which extends to future null infinity ${\mathcal{I}}^{+}$. The grid consists of the null slices $u=\mathrm{const}$ and ingoing radial null geodesics $v=\mathrm{const}$. Evolution slows down in the vicinity of the future event horizon ${\mathcal{H}}^{+}$. We also indicate lines (a) $r=\mathrm{const}<2M_f$, (b) $r=\mathrm{const}=2M_f$, and (c) $r=\mathrm{const}>2M_f$, where $M_f$ is the final black-hole mass.

Figure 2

A representative null-paralellogram in the numerical grid made up by two $u=\mathrm{const}$ surfaces and two ingoing null geodesics $v=\mathrm{const}$.

Figure 3

The convergence of the error diagnostic ${\mathbf{E}}_{\mathbf{u}\mathbf{u}\mathbf{r}}$ with increasing grid resolution for two near-critical evolutions. Evolution (1) uses 5000 gridpoints and $p=p^{*}[5000]+{10}^{-10}$ and evolution (2) uses 10 000 gridpoints and $p=p^{*}[10000]+{10}^{-10}$. The data displayed as dots and circles have been sampled.

Figure 4

This figure shows the focussing of ingoing null geodesics by gravity in the late stages of a slightly supercritical evolution. The discretely self-similar dynamics causes the density of the geodesics to increase in a periodic manner.

Figure 5

This figure shows a surface plot of $|\psi (\tau ,x)|$ for the same near-critical evolution as in Figs. 7, 11, and 12. When the initial Gaussian reaches the origin, it is “instantly” (in retarded time $u$) radiated to future null infinity ${\mathcal{I}}^{+}$ (located at $x=1$) by interfering nonlinearly with the field that has not yet reached the origin. Once the evolution has come close to the critical solution, the matter field $\psi (u,x)=\varphi r$ decays exponentially; further self-similar features are thus not visible in this plot.

Figure 6

We compare the decay of the Bondi mass in supercritical, near-critical, and subcritical evolutions. In the subcritical case, the Bondi mass is found to decay for late times with a power-law exponent of approximately $-5$.

Figure 7

This figure plots the Bondi mass $m_{\mathrm{B}}$ against both $u_{\mathrm{B}}$ and the adapted time ${\tau }_{\mathrm{B}}$ for a barely supercritical evolution with final black-hole mass $M_f\approx 5\times {10}^{-6}$. The Bondi mass $m_{\mathrm{B}}$ and the mass at the past SSH, $m_{\mathrm{SSH}}$, are found to decrease exponentially in ${\tau }_{\mathrm{B}}$ (with an overlayed ${\tau }_{\mathrm{B}}$-periodic oscillation with period $\Delta /2$), once the evolution has sufficiently approached the critical solution near the center of spherical symmetry. We also show $m_{\mathrm{EXT}}$, the energy present outside of the SSH.

Figure 8

We show the news-function $N(u)$, first as a function of the natural time coordinate $u_{\mathrm{B}}$ of an asymptotic observer and also as a function of a suitably adapted time ${\tau }_{\mathrm{B}}=-\ln \frac{u_{\mathrm{B}}^{*}-u_{\mathrm{B}}}{u_{\mathrm{B}}}$ where $N({\tau }_{\mathrm{B}})$ is periodic with period $\Delta \simeq 3.44$ after the spacetime has come close to the critical solution. Even if the constant $u_{\mathrm{B}}^{*}$ is not known, it can be determined by a fit to periodicity in ${\tau }_{\mathrm{B}}$. Thus, it is possible to observe DSS at ${\mathcal{I}}^{+}$ and to extract the critical exponent $\Delta$.

Figure 9

The fine structure in $m_{\mathrm{BH}}$ after subtracting a linear fit. The measured period 4.6 is close to the value predicted by perturbation theory $\frac{1}{2}\Delta /\gamma \approx 4.61$.

Figure 10

A conformal diagram of a critical collapse spacetime. In the backwards light cone of the accumulation point the dynamics are close to the DSS critical solution. The lightlike boundary is also called the past self-similarity horizon (SSH). Depending on whether the initial data are sub- or supercritical, the spacetime will for late times be Minkowski or Schwarzschild.

Figure 11

This figure shows the scalar field monopole moment $c(u_{\mathrm{B}})$ for a near-critical (but barely supercritical) evolution. The half-periods measured from one extremum to the next roughly agree with the prediction of perturbation theory shown in Fig. 12.

Figure 12

The exponential decay of the QNM half-periods predicted by perturbation theory is shown with annotated values at the midpoints between the points of inflection for the same near-critical evolution as in Fig. 11.

Figure 13

This figure shows power-law exponents for a subcritical evolution and illustrates the domains of validity of the predictions of perturbation theory for the two zones: $-2$ near ${\mathcal{I}}^{+}$ and $-3$ near $i^{+}$.