Twist and higher modes of a complex scalar field at the threshold of collapse

TL;DR

This work investigates Type II critical phenomena in axisymmetric gravitational collapse of a massless complex scalar field with angular momentum, extending Choptuik-type results to higher angular modes using an $m$-cartoon symmetry in a pseudospectral code. The authors establish universality within each fixed angular mode $m$ while revealing $m$-dependent discrete self-similarity periods $\Delta$ and scaling exponents $\gamma$, with $m=1$ yielding $\Delta\approx0.42$, $\gamma\approx0.11$ and $m=2$ yielding $\Delta\approx0.09$, $\gamma\approx0.035$. They find that angular momentum on the apparent horizon vanishes at threshold ($\chi_{\mathrm{AH}} \to 0$) for both $m=1$ and $m=2$ families, indicating that extremality is not approached in these setups, and show that gravitational-wave content remains substantial but does not dominate threshold dynamics. The results confirm the robustness of the DSS-critical picture within each angular sector and motivate future exploration of twisting vacuum thresholds and more general initial data to probe extremal limits and potential threshold competition with gravitational radiation.

Abstract

We investigate the threshold of collapse of a massless complex scalar field in axisymmetric spacetimes under the ansatz of Choptuik et al. 2004, in which a symmetry depending on the azimuthal parameter $m$ is imposed on the scalar field. This allows for both non-vanishing twist and angular momentum. We extend earlier work to include higher angular modes. Using the pseudospectral code bamps with a new adapted symmetry reduction method, which we call $m$-cartoon, and a generalized twist-compatible apparent horizon finder, we evolve near-critical initial data to the verge of black hole formation for the lowest nontrivial modes, $m=1$ and $m=2$. For $m=1$ we recover discrete self-similarity with echoing period $Δ\simeq0.42$ and power-law scaling with exponent $γ\simeq0.11$, consistent with earlier work. For $m=2$ we find that universality is maintained within this nonzero fixed-$m$ symmetry class but with smaller period and critical exponents, $Δ\simeq0.09$ and $γ\simeq0.035$, establishing an explicit dependence of the critical solution on the angular mode. Analysis of the relation between the angular momentum and the mass of apparent horizons at the instant of formation, $J_{\mathrm{AH}}{-}M_{\mathrm{AH}}$, shows that the effect of angular momentum is minimal at the threshold, with $χ_{\mathrm{AH}}=J_{\mathrm{AH}}/M_{\mathrm{AH}}^2\to0$, and, therefore, excludes extremal black holes for the families under consideration. Our results demonstrate that while universality and DSS hold within each $m$-sector, the critical universal values vary with $m$, and neither extremality nor bifurcation occur in the complex scalar field model within the families considered here.

Figures (10)

• Figure 1: Scaled derivatives of the scalar field at the center for the best tuned subcritical evolutions of Family I (top) and Family II (bottom). The quantity $(\tau_*-\tau)\,\partial_{\rho_c} \psi$ is plotted against slow time $-\ln(\tau_*-\tau)$. Both the real part (red dashed) and the imaginary part (blue dash-dotted) display oscillatory behavior, with the real component showing larger amplitude oscillations. While the oscillation period seems to be universal across families, the curves themselves are not. Looking at curvature scalars the spacetime metric itself is universal at the threshold.
• Figure 2: Color maps of the normalized Kretschmann invariant $(\tau_*-\tau) \left|R_{abcd}R^{abcd}\right|^{\frac{1}{4}}$ for our best-tuned subcritical runs of Family I and II initial data setups, all along the symmetry axis in single-null similarity coordinates.
• Figure 3: Global maximum of the Ricci scalar, $R_{\text{max}}$, as a function of the inverse of the phase space distance to criticality, $|a-a_*|^{-1}$, for the $m=1$ family I. The data exhibit a clear power-law scaling, with slope $2\gamma_{\textrm{sub}}$. Fitting yields a universal critical exponent $\gamma\simeq 0.11$, in agreement with the results of choptuik2004critical. The regression parameter $r^2$ quantifies the level of scattering of the data.
• Figure 4: Mass of the first apparent horizon that appears in each simulation, $M_{\text{AH}}$, against the phase space distance to the threshold solution, $|a-a_*|$, for the $m=1$ family I. The mass scales with an exponent $\gamma_\textrm{sup}\simeq0.1$ the same universal number found on the subcritical side.
• Figure 5: Angular momentum versus mass at horizon formation in near-threshold $m=1$ evolutions of family I. Each data point shows the angular momentum $J_{\mathrm{AH}}$ against the mass $M_{\mathrm{AH}}$ measured at the first apparent horizon of each evolution. Although both $J_{\mathrm{AH}}$ and $M_{\mathrm{AH}}$ are sensitive to gauge and become very small near the threshold — leading to increased scatter — a clear overall trend is evident. Compare this figure with Fig. 4 in choptuik2004critical. We also find that the dimensionless spin $\chi_{\mathrm{AH}} = J_{\mathrm{AH}}/M_{\mathrm{AH}}^{2}$ shows a vanishing trend, decreasing to values $\mathcal{O}\left(10^{-4}\right)$ at our best tuning, revealing that angular momentum is irrelevant at threshold.