Choptuik’s scaling in higher dimensions

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 $4\le D\le 11$ 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 $11\le D\le 13$. These results and their relation to the black hole–black string system are discussed.

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 three-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.

Figure 6

$D=4$: The maximal curvature on the axis as a function of $\log (p_{*}-p)$. The slope of the linear fit yields $\gamma =0.372$ which agrees well with the value cited in the literature ($\simeq 0.374$).

Figure 7

The logarithm of the maximal curvature on the axis as a function of $\log (p_{*}-p)$ for different $p$’s in various dimensions. The slope of the linear fit yields $-2\gamma$. Some wiggles, characteristic of the DSS solutions, occur about the linear fit.

Figure 8

The logarithm of the maximal curvature on the axis as a function of $\log (p_{*}-p)$ in 11D. Because of small number of wiggles and their large amplitude, the linear fitting approach fails to accurately estimate the slope. In this case we employ the fact that the period of wiggles ($\simeq 2.85$) is equal to $\Delta /(2\gamma )$ and use this to find $\gamma \simeq 0.44$ from the more accurate $\Delta$. This approach is seen in lower dimensions to be consistent with the regular way to obtain $\gamma$.

Figure 9

$D=8$: Normalized plot of $\log (R_{\max })+2\gamma \log (p_{*}-p)$ (stars) and a sine wave (solid line). The period of wiggles is about 6.18 and it agrees well with theoretical prediction $\Delta /\gamma \simeq 6.22$.

Figure 10

$\Delta$, including the errorbars, vs the spacetime dimension $D$. The solid line designates a second degree polynomial fitting, the dotted lines show the 95% prediction bounds. The speculative extrapolation to higher dimensions suggests that $\Delta$ reaches a minimum at $D\simeq 12$.

Figure 11

$\gamma$ and ${\gamma }_{\mathrm{mass}}$, with the errorbars, as a function of the spacetime dimension $D$. The dominant dimensional dependence of ${\gamma }_{\mathrm{mass}}=(D-3)\gamma$ is essentially linear increase due to the factor $D-3$, which masks the rather subtle variation of $\gamma$. The solid line designates a fourth order polynomial fit based on the data points in the range $4\le D\le 10$ (the $\gamma$ value for $D=11$ is not used). The dotted lines show the $95\%$ prediction bounds. The extrapolation for larger $D$’s indicates the existence of a maximum somewhere in between $10\le D\le 12$.