Critical Collapse of a Complex Scalar Field with Angular Momentum

We report a new critical solution found at the threshold of axisymmetric gravitational collapse of a complex scalar field with angular momentum. To carry angular momentum the scalar field cannot be axisymmetric; however, its azimuthal dependence is defined so that the resulting stress-energy tensor and spacetime metric are axisymmetric. The critical solution found is nonspherical, discretely self-similar with an echoing exponent $\Delta =0.42(\pm 4\%)$, and exhibits a scaling exponent $\gamma =0.11(\pm 10\%)$ in near-critical collapse. Our simulations suggest that the solution is universal (within the imposed symmetry class), modulo a family-dependent constant, complex phase.

Figure 1

A surface plot of the real part (${\Phi }_r\rho$) of the complex field after several echoes of a near-critical evolution. The origin is at the bottom center of the figure, the $\rho$ axis runs vertically through the middle, and the $z$ axis runs horizontally. Only a single echo (roughly) at the origin corresponds to the self-similar part of the spacetime; the other “waves” visible were radiated during earlier echoes of the field. Note also that the solution is not spherically symmetric.

Figure 2

The real ($\rho {\Phi }_r$) and imaginary ($\rho {\Phi }_i$) components of the central value of $\Phi$ (9) multiplied by proper time $\tau$ and divided by proper radius ${\rho }_c\equiv \rho {\psi }^2{e}^{\rho \overline{\sigma }}$ versus $-\ln \tau$ for the near-critical collapse solutions of families A, C, and D (the phase information for family B is trivial as ${\Phi }_r={\Phi }_i$ then, and so for brevity we do not show it). The family D solution shown here is supercritical, and the simulation is stopped soon after an apparent horizon is detected.

Figure 3

$R_m$, the maximum value of the Ricci scalar on axis ($\rho =0$) attained during subcritical evolution, versus distance $|\overline{p}|=p^{\star }-p$ from threshold, for family A. Each point represents a single simulation. The line is a least-squares fit to the data; $R_m$ has dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-2}$, and hence the slope of the fit is expected to be $-2\gamma$. For this case, we infer $\gamma \approx 0.11$.

Figure 4

Estimated black hole mass ($M_{\mathrm{A}\mathrm{H}}$) versus angular momentum ($J_{\mathrm{A}\mathrm{H}}$) in supercritical collapse of family A initial data. Points represent individual simulations, while the two lines are separate linear regression fits to the set of points to the left and right of the “knee” in the curve at $\ln (M_{\mathrm{A}\mathrm{H}})\approx -3.2$, with slopes $\approx 6.0$ and $\approx 2.2$, respectively. In $\ln \overline{p}$, the horizontal scale ranges from $-22$ to $-2$ (compare to Fig. 3).