Simulations of gravitational collapse in null coordinates IV: evolving through the event horizon, with an application to the spherical charged scalar field

TL;DR

The paper develops and tests three horizon-penetrating formulations for gravitational collapse on null coordinates in spherical symmetry, decoupling formulation from radial gauge via a shift $B$. By evolving either the area radius $R$, the metric coefficient $G$, or both, and using the Raychaudhuri equation in flexible ways, the authors demonstrate stable evolution through event horizons and singularities in a charged scalar field context, with singularity excision and robust convergence across two domain types (regular centre and null rectangle). They report the first detailed, high-precision plots of the universal fine structure of mass $M$ and charge $Q$ scaling near type-II critical collapse for a spherical charged scalar field, finding a curvature exponent $\gamma\approx0.374$ and a charge exponent $\mu\approx0.883$, while observing potential non-universality in charge scaling. The results validate horizon-penetrating null-coordinates as a versatile tool for studying critical phenomena and black-hole interiors, and pave the way for axisymmetric extensions and investigations of extremal collapse.

Abstract

We consider line elements of the form $-2G\,du\,(dx+B\,du) + R^2(...)$, where $(...)$ does not contain $dx$. Surfaces of constant $u$ are then null surfaces, and their affinely parameterised generators have tangent vector $G^{-1}\partial_x$. Considering $u$ as the time coordinate, we can evolve either $R$ or $G$, with the other one found by solving the Raychaudhuri equation along the null generators, or we can evolve both. This choice of formulation is independent from the remaining gauge choice $x\to x'(u,x,...)$ in the line element above, which is fixed incrementally by the choice of $B$. For example, we can evolve $G$, in order to be able to evolve through an event horizon, and use $B$ in to adapt the coordinates to type-II critical collapse. As a demonstration of these ideas, we consider a charged scalar field in spherical symmetry. We consider two settings: a domain where the outgoing null cones of constant $u$ emanate from a regular centre $R=0$, and a domain where the outgoing null cones emanating from an ingoing-null boundary. We demonstrate convergence with resolution, within each formulation and between the three formulations in both settings. As testbeds, we compute the critical exponents and periodic fine-structures of the black hole charge and mass scaling laws in a one-parameter family of charged regular initial data, and examples of perturbed extremal Reissner-Nordström solutions.

Figures (9)

• Figure 1: Maximum curvature fine-structure for the $G=1$ family of initial data, in the eG formulation: we plot $-1/2\ln|\max T|-\gamma\ln(1-p/p_*)+A$ against $\ln(1-p/p_*)$. We use the approximate theoretical value $\gamma=0.374$, and the family-dependent offset $A=-0.23$ has been fitted by eye so that the average of the fine-structure is approximately zero at high fine-tuning. The range of $\ln(1-p/p_*)$ on the horizontal axes has been chosen as in Fig. 1 of Paper II for comparison. (As in Paper II, we have suppressed axis labels to make figures larger.) The thin dotted line has been fitted by eye to the first two peaks from the right and has slope $\delta\gamma=0.012$, meaning that the curvature critical exponent is slightly larger at low fine-tuning.
• Figure 2: Black hole mass fine-structure for the $G=1$ family of initial data, in the eG formulation, with $C_\text{thr}=1$: we plot $\ln M-\gamma\ln(p/p_*-1)+A$ against $\ln(p/p_*-1)$. $\gamma$ and $A$ and the horizontal range are as for the curvature scaling plot in Fig. \ref{['charged_iG_eG_finestructure_T']}. Compare also Fig. 2 of Paper II. The thin dotted line has been fitted by eye to the first two peaks from the right. It has slope $\delta\gamma=0.010$, meaning that the mass critical exponent is slightly larger at low fine-tuning, in line with that for that of the curvature to within a plausible fitting error.
• Figure 3: Black hole charge fine-structure for the $G=1$ family of initial data, in the eG formulation, with $C_\text{thr}=1$: we plot $\ln Q-\mu\ln(p/p_*-1)+A$ against $\ln(p/p_*-1)$. We have used the approximate expected critical exponent $\mu=0.883$, and $A$ as in the curvature scaling plot in Fig. \ref{['charged_iG_eG_finestructure_T']}. The thin dotted line has been fitted by eye to the first two peaks from right. It has slope $\delta\mu=-0.059$, meaning that the charge critical exponent is smaller at low fine-tuning. It is plausible from this figure that at high fine-tuning the expected value of $\mu$ is achieved. The fine-structure has approximately the same period as for the curvature and mass, as one expects.
• Figure 4: Maximum curvature fine-structure for the $G=1$ family of real initial data, in the eG formulation. We set $A=-0.58$, otherwise as in Fig. \ref{['charged_iG_eG_finestructure_T']}.
• Figure 5: Black hole mass fine-structure for the $G=1$ family of real initial data, in the eG formulation, with $C_\text{thr}=1$. We set $A=-0.58$, otherwise as in Fig. \ref{['charged_iG_eG_finestructure_M']}.