Critical phenomena in a gravitational collapse with a competing scalar field and gravitational waves in $4+1$ dimensions

In the gravitational collapse of matter beyond spherical symmetry, gravitational waves are necessarily present. On the other hand, gravitational waves can collapse to a black hole even without matter. One might therefore wonder how the interaction and competition between the matter fields and gravitational waves affects critical phenomena at the threshold of black hole formation. As a toy model for this, we study the threshold of black-hole formation in $4+1$ dimensions, where we add a massless minimally coupled scalar matter field to the gravitational wave ansatz of Bizón, Chmaj, and Schmidt (in a nutshell, Bianchi IX on $S^3\times \text{radius}\times \mathrm{time}$). In order to find a stable discretization of the equation governing the gravitational waves in $4+1$ physical dimensions, which has the same principal part as the spherical wave equation in $9+1$ dimensions, we first revisit the problem of critical spherical scalar field collapse in $n+2$ dimensions with large $n$. Returning to the main problem, we find numerically that weak gravitational wave perturbations of the scalar field critical solution decay, while weak scalar perturbations of the gravitational wave critical solution also decay. A dynamical systems picture then suggests the existence of a codimension-2 attractor. We find numerical evidence for this attractor by evolving mixed initial data and fine-tuning both an overall amplitude and the relative strength of the two fields.

Figure 1

The scaled error $\mathrm{\Delta }{b}_k$ for $k=0\dots 4$, represented for $0\le x\le 0.6$ and at a particular time instance $u=0.444$ for centered Gaussian pure gravitational wave initial data with $b(0,x)=13.88\exp [-(x/0.25{)}^2]$. While the curves progressively coincide for $x\gtrsim 0.2$, they differ slightly at the first grid points, although some (slower) convergence is still noticeable.

Figure 2

Scaling laws for the radius $R_{\mathrm{ah}}$ of apparent horizon formation, and the global maximum of the Ricci scalar $R_{\mathrm{\Psi }}$ and of $R_B$ for the case $q=1-ϵ$ and Gaussian initial data. These two last quantities are rescaled by $-1/2$ in the log plot to account for their dimension ${\text{length}}^{-2}$. The black lines represent the linear fits to each curve. The slope of the lines fitted against $R_{\mathrm{ah}}$ and $R_B^{-1/2}$ are 0.1638, and 0.133 for $R_{\mathrm{\Psi }}^{-1/2}$.

Figure 3

Residuals of the linear fit to Fig. 2 for the radius $R_{\mathrm{ah}}$ of apparent horizon formation for $q=1-ϵ$. The scaling exponent is $\gamma =0.1638$, and the fitted period of the residuals is ${\mathrm{\Delta }}_{\mathrm{res}}=1.235$, which is related to the echoing period of the critical solution by ${\mathrm{\Delta }}_{\mathrm{res}}={\mathrm{\Delta }}_B/(\ln (10)\gamma )$, resulting in ${\mathrm{\Delta }}_B\simeq 0.47$, consistent with Fig. 8.

Figure 4

Residuals of the linear fit to Fig. 2 for $R_B$ for $q=1-ϵ$. The scaling exponent is $\gamma =0.1638$, and the fitted period of the residuals is ${\mathrm{\Delta }}_{\mathrm{res}}=1.24$, resulting in ${\mathrm{\Delta }}_B\simeq 0.47$, consistent with Fig. 7.

Figure 5

Residuals of the linear fit to Fig. 2 for the Ricci scalar $R_{\mathrm{\Psi }}$ for $q=1-ϵ$. The scaling exponent is $\gamma =0.133$, and the fitted period of the residuals is ${\mathrm{\Delta }}_{\mathrm{res}}=1.25$.

Figure 6

The scalar field $\mathrm{\Psi }(\xi ,T)e^{-{\kappa }_{\mathrm{\Psi }}T}$ for optimal fine-tuning with $q=1-ϵ$, ${\kappa }_{\mathrm{\Psi }}=-0.175$. A black line represents the extrapolation to the regular center $R=0$.

Figure 7

The quantity $B/{\xi }^2$ for optimal fine-tuning with $q=1-ϵ$. A black line represents the extrapolation to the regular center $R=0$.

Figure 8

The compactness $M/R^2$ for optimal fine-tuning with $q=1-ϵ$.

Figure 9

Scaling laws for the radius $R_{\mathrm{ah}}$ of apparent horizon formation, and the global maximum of the Ricci scalar $R_{\mathrm{\Psi }}$ for the case $q=ϵ$ and Gaussian initial data. The latter is rescaled by $-1/2$ in the log plot to account for its dimension ${\text{length}}^{-2}$. The black lines represent the linear fits to each curve. The slope of the lines fitted against $R_{\mathrm{ah}}$ and $(R_{\mathrm{\Psi }}{)}^{-1/2}$ was 0.4131.

Figure 10

Residuals of the linear fit to Fig. 9 for the for the radius $R_{\mathrm{ah}}$ of apparent horizon formation for $q=ϵ$. The scaling exponent is $\gamma =0.4131$, and the fitted period of the residuals is ${\mathrm{\Delta }}_{\mathrm{res}}=1.7$, which is related to the echoing period of the critical solution by ${\mathrm{\Delta }}_{\mathrm{res}}={\mathrm{\Delta }}_{\mathrm{\Psi }}/(\ln (10)\gamma )$, resulting in ${\mathrm{\Delta }}_{\mathrm{\Psi }}\simeq 1.6$, consistent with Fig. 14.

Figure 11

Residuals of the linear fit to Fig. 9 for the Ricci scalar $R_{\mathrm{\Psi }}$ for $q=ϵ$. The scaling exponent is $\gamma =0.4131$, and the fitted period of the residuals is ${\mathrm{\Delta }}_{\mathrm{res}}=1.7$, which is related to the echoing period of the critical solution by ${\mathrm{\Delta }}_{\mathrm{res}}={\mathrm{\Delta }}_{\mathrm{\Psi }}/(\ln (10)\gamma )$, resulting in ${\mathrm{\Delta }}_{\mathrm{\Psi }}\simeq 1.6$, consistent with Fig. 12.

Figure 12

The scalar field $\mathrm{\Psi }(\xi ,T)$ for optimal fine-tuning with $q=ϵ$. A black line represents the extrapolation to the regular center $R=0$.

Figure 13

The quantity $B/{\xi }^2{e}^{-{\kappa }_{B}T}$ for optimal fine-tuning with $q=ϵ$, ${\kappa }_B=-1.5$. A black line represents the extrapolation to the regular center $R=0$. $T$ is restricted to [2, 17] so as to visualize the exponential correction to $B$ after the dominant scalar field starts to approximate the critical solution.

Figure 14

The compactness $M/R^2$ for optimal fine-tuning with $q=ϵ$.

Figure 15

The maxima and minima (over $x$) of the quantities $R^2{R}_B$ (orange) and $R^2{R}_{\mathrm{\Psi }}$ (purple), plotted against $T$ for different values of $q$, with $q_a<q_b<q_c$, extracted from the respective best subcritical evolutions. For reference, the same quantities for the two pure critical solutions are plotted together in Fig. 15.

Figure 16

The scalar field $\mathrm{\Psi }(\xi ,T)$ for optimal fine-tuning with $q=q_b$. A black line represents the extrapolation to the regular center $R=0$.

Figure 17

The field $B$ for optimal fine-tuning with $q=q_b$. It is zero at the origin $R=0\iff \xi =0$ due to Eq. (8).

Figure 18

The compactness $M/R^2$ for optimal fine-tuning with $q=q_b$.

Figure 19

Plot of the critical exponent $\gamma$ estimated from the scaling law for radius of apparent horizon formation $R_{\mathrm{ah}}$. The points in black correspond to initial data with a Gaussian profile, which are plotted against $q$. The points in blue correspond to initial data with the profile of a Gaussian derivative, which are plotted against $sq/(1-(1-s)q)$ for $s=0.5$.

Figure 20

Schematic conjectured phase space picture, with the infinite-dimensional phase space represented in three dimensions. The framed plane represents the black-hole threshold (in reality a hypersurface). All arrow lines represent trajectories (spacetimes). The filled dots represent fixed points (DSS spacetimes); the scalar field critical solution, on the left; the gravitational wave critical solution, on the right; and the codimension-2 critical solution in between. Here, the middle fixed point has two unstable modes, while the left and right ones have one each. An infinite number of phase space dimensions of the black-hole threshold are suppressed, and with them an infinite number of stable modes of each fixed point within the black-hole threshold. The two dashed lines represent three families of initial data with $q=0$ (left) and $q=1$ (right). Hollow dots represent initial data with $p<p_{*}$, $p=p_{*}$ and $p>p_{*}$ for each family. Figure taken from Ref. [7].

Figure 21

The scalar field $\mathrm{\Psi }(x,T)$ in the best near-critical evolution in $8+1$ dimensions. A black line represents the extrapolation to the regular center $R=0$.

Figure 22

The field $h(x,T)$ in the best near-critical evolution in $8+1$ dimensions. A black line represents the extrapolation to the regular center $R=0$.

Figure 23

The compactness $M/R^2$ in the best near-critical evolution in $8+1$ dimensions. A black line represents the extrapolation to the regular center $R=0$.