Choptuik’s critical phenomenon in Einstein-Gauss-Bonnet gravity

We investigate the effects of higher-order curvature corrections to Einstein’s gravity on the critical phenomenon near the black hole threshold, namely, the Choptuik phenomenon. We simulate numerically a five-dimensional spherically symmetric gravitational collapse of massless scalar field in Einstein-Gauss-Bonnet gravity towards a black hole formation threshold. When the curvature is sufficiently large, the additional higher-order terms affect the evolution of the whole system. Since high curvature characterizes the region when the critical behavior takes place, this critical behavior is destroyed. Both the self-similarity and the mass scaling relation disappear. Instead we find a different behavior near the black hole threshold, which depends on the coupling constant of the higher-order terms. The new features include a change of the sign of the Ricci scalar on the origin which indicates changes in the local geometry of spacetime, and never occurs in classical general relativity collapse, and oscillations with a constant rather than with a diminishing length scale.

Figure 1

The Penrose diagram in GR of the space time that is expected to form in a gravitational collapse of a shell of in-falling scalar field to a black hole. When the shell is far away from the origin, the self-gravitational effects are small. When it comes closer to the origin, the gravitational field becomes stronger. If the field does not collapses to a black hole, the diagram remains Minkowski-flat, and the event horizon or the singularity do not exist, of course. Light blue lines indicate the numerical domain of integration used in the current work. The initial hypersurface is a null ray. Since the field is massless, it behaves lightlike and it moves along null rays.

Figure 2

The field function $s$ in classical GR in five dimensions, for a slightly subcritical collapse. The field oscillates with a DSS pattern. The field pulsations decrease in temporal and spatial scales.

Figure 3

The domain of integration. The calculation employs the two previous lines, $L_1$ and $L_2$, in addition to the line, $L_0$, that is currently being solved. The boundary conditions on $r=0$ involving ${\partial }_r$ are implemented using the three-point derivatives along the diagonal line. Smoothing of some functions near $r=0$ (at a point marked by cross) is done using past light cone points (marked by circles). Figure taken from 3.

Figure 4

Convergence for a slightly subcritical run with $\alpha ={10}^{-4}$. The convergence rate—$n$—in panel (a) is derived from the field function $s(u,v)$, with $2^{16}$, $2^{17}$ and $2^{18}$ grid points in $u$ and $v$ directions [blue, red and green lines, respectively, in $s$ plot on panel (b)]. In panel (b), the divergence of $n$ around $v\sim 0.46$ and $v\sim 0.63$ is caused by $s$ lines crossing, as shown in the zoom window.

Figure 5

Outgoing null rays: $v$ vs $r$. Both plots have the same parameters except for the initial field amplitude: $\alpha ={10}^{-5}$, $v_{\max }=0.768$, the grid density is $2^{15}$ points in each direction.

Figure 6

The black hole’s Schwarzschild radius-$r_s$ vs the difference between the amplitude and the critical one-$|p-p_{*}|$, for supercritical cases with $\alpha ={10}^{-4}$. In the classical GR solution this relation is a power law. Here, in EGB gravity, a power law is not observed.

Figure 7

Contours of the scalar field function $s$ in EGB gravity with a coupling constant $\alpha ={10}^{-5}$, grid density $2^{15}$, $v_{\max }=0.768$, $v_c=0.22$. The initial field amplitude increases or decreases towards the critical amplitude: $p\to p_{*}$. Plots (a–d)—subcritical collapse: $p<p_{*}$. Plots (e–h)—supercritical collapse $p>p_{*}$. The inserts demonstrate the formation of black holes as seen in the $v-r$ diagram. The initial values of the field amplitude are: $p=0.1654(\mathrm{a})$, $0.16557(\mathrm{b})$, $0.1657(\mathrm{c})$, $0.165723(\mathrm{d})$, $0.1676(\mathrm{e})$, $0.166(\mathrm{f})$, $0.1658(\mathrm{g})$, $0.16573(\mathrm{h})$.

Figure 8

A slightly subcritical collapse with different values of the coupling constant $\alpha$. Panels (a1–3) correspond to a classical GR solution, $\alpha =0$. A self-similar behavior can be observed. Panels (b1–4) correspond to EGB gravity with $\alpha ={10}^{-6}$. The first row, panels (a1) and (b1), presents contour plots of the field function $s$. The second row, panels (a2) and (b2), shows the field function $s$ at the origin ($r=0$) vs $u$. The third row, panels (a3) and (b3), shows the Ricci scalar on the origin vs $u$ in a logarithmic scale. A red color indicates positive values and a purple color negative values of $R$. Panel (b4), displays contour plot of $|\alpha R|$, showing the regions where the higher order terms are significant, i.e., $|\alpha R|>1$.

Figure 9

A continuation of Fig. 8 with larger values of $\alpha$ in a slightly subcritical collapse. Panels (c1–4) correspond to EGB gravity solution with $\alpha ={10}^{-5}$. Panels (d1–4) correspond to EGB gravity solution with $\alpha ={10}^{-4}$.

Figure 10

The typical wavelength of scalar field oscillations in EGB gravity near the black hole threshold vs the coupling constant $\alpha$. Red line indicates a linear fit with a slope: $m=0.54\pm 0.05$. Namely, the wavelength is proportional to $\sqrt{\alpha }$ as expected from the dimensional analysis.