On Choptuik's scaling in higher dimensions

TL;DR

The paper investigates Choptuik-type scaling in higher-dimensional GR by numerically solving the Einstein-scalar system in $D$-dimensional spherical symmetry ($4 \le D \le 11$) using double-null coordinates. It demonstrates that near-threshold collapse remains discretely self-similar, extracting universal quantities $\\gamma$ and $\\Delta$ across dimensions and revealing a smooth, dimension-dependent trend: $\\Delta$ decreases with $D$ while $\\gamma$ rises (with hints of extrema around $11$–$13$). The study also expresses $\\gamma_{\\rm mass}=(D-3)\\gamma$, highlighting how black hole mass scales grow with dimension, and discusses connections to the black-string/black-hole phase transition, offering insights into the dimensional origin of critical phenomena in GR. These results motivate further high-D investigations to identify potential sharp dimensional thresholds and their physical implications for horizon topology changes in higher-dimensional spacetimes.

Abstract

We extend Choptuik's scaling phenomenon found in general relativistic critical gravitational collapse of a massless scalar field to higher dimensions. We find that in the range 4 <= D <= 11 the behavior is qualitatively similar to that discovered by Choptuik. In each dimension we obtain numerically the universal numbers associated with the critical collapse: the scaling exponent gamma and the echoing period Delta. The behavior of these numbers with increasing dimension seems to indicate that gamma reaches a maximum and Delta a minimum value around 11 <= D <= 13. These results and their relation to the black hole--black string system are discussed.

Figures (11)

• Figure 1: The domain of integration. At any moment, in addition to the current outgoing hypersurface, $L_0$ we keep two preceding levels $L_1$ and $L_2$. Boundary conditions involving ${\partial}/{\partial} r$ are implemented using 3-point derivatives along the shown diagonal line. Mesh refinement is illustrated in the topmost rows. The smoothing of $z$ (or $d$) near the axis at a point marked by a cross is done using the values of $z$ (or $d$) at points on the past light cone of that point marked with circles.
• Figure 2: $D=6$: Contours of the scalar field profile in slightly subcritical collapse. After a short transient, the field oscillates, approaches the accumulation point where the curvature is maximal and then disperses.
• Figure 3: $D=5$: The scalar field on the axis as a function of $\log(T_*-T)$. The period of osculations is $\Delta\simeq 3.19$. The actual data is designated by points. The distance between the points increases close to $T_*$ indicating the decrease of resolution.
• Figure 4: $D=5$: The metric function $\alpha$ on the axis decays fast as the accumulation point $T_*$ is approached. Its evolution is accompanied by oscillations whose period ($\simeq 1.6$) is half the period of the scalar field ($\simeq 3.19$).
• Figure 5: $D=7$: Behavior of the curvature as the accumulation point $T_*$ is approached. The evolution of the curvature, like other metric functions, is accompanied by oscillations. After each pulsation $\log(1-R)$ increases by $\Delta$. The period of the last six echoes is approximately constant and equal to $\Delta/2 \simeq 1.41$. Similar behavior (with a different echoing period) is observed for other $D$'s as well.