Fine-Structure of Choptuik's Mass-Scaling Relation

TL;DR

The paper addresses the critical gravitational collapse of a spherical massless scalar field (Choptuik problem) and the well-known mass scaling $M_{bh} \propto (p-p^*)^{\beta}$ with discrete self-similarity of period $\Delta$. It proposes a fine-structure correction $\Psi[\ln(p-p^*)]$ with a universal period $\varpi = \Delta/\beta$, based on an analytical argument about the final non-self-similar evolution and the echoing dynamics. The exponent relation $\beta = 1/\alpha$ links the mass scaling to perturbation growth, and the final mass is modulated by a universal function $G[\ln(p-p^*)]$ with period $\varpi$ leading to $\ln(M_{bh}) = \beta \ln(p-p^*) + c_k + \Psi[\ln(p-p^*)]$. Numerical simulations across four initial-data families confirm the predicted universal period $\varpi \approx 4.6$ and $\beta \approx 0.37$, and robustness is demonstrated via grid-convergence tests; the work also discusses implications for axisymmetric collapse, predicting a similar fine-structure with $\varpi \approx 0.8$ in that context.

Abstract

We conjecture (analytically) and demonstrate (numerically) the existence of a fine-structure above the power-law behavior of the mass of black-holes that form in gravitational collapse of spherical massless scalar field. The fine-structure is a periodic function of the critical-separation $(p-p^*)$. We predict that the period $\varpi$ is {\it universal} and that it depends on the previous universal parameters, the critical exponent, $β$, and the echoing period $Δ$ as $\varpi = Δ/β$.

Figures (3)

• Figure 1: Illustration of the conjectured universal periodic fine-structure of $-T_{bh}$. The quantity $[-T_{bh}- \langle -T_{bh} \rangle]$ is plotted as a function of $\ln(a)$ , where $a\equiv (p-p^*)/p^*$ , for the four families. The curves were shifted horizontally (but not vertically) in order to overlap the first oscilation of each family with the first one of family $(a)$. $\langle T_{bh} \rangle$ is the value of $T_{bh}$ determined from a straight line approximation, i.e. $\langle T_{bh} \rangle =Const+ \beta \ln(p-p^*)$. The numerical results agree with the predicted relation $\varpi= \alpha \Delta \approx 4.6$.
• Figure 2: Illustration of the conjectured universal periodic fine-structure generalization of Choptuik's mass-scaling relation. $\ln(m)-\langle \ln(m) \rangle$ is plotted as a function of $\ln(a)$ for the four families, where $m \equiv M_{bh}/M_{bh,c}$ is the normalized black hole mass in units of the initial mass in the critical solution $M_{bh,c}$. $\langle \ln(m) \rangle$ is the value of $\ln(m)$ determined from a straight line approximation. The curves were shifted horizontally (but not vertically) in order to overlap the first oscilation of each familiy with the first one of family $(a)$. The numerical results agree with the it predected relation $\varpi= \Delta /\beta \approx 4.6$.
• Figure 3: $\ln(m)-\langle \ln(m) \rangle$ is plotted as a function of $\ln(a)$ for family $(a)$ and for it five different resolution grids with 100,200,400, 800 and 1600 gridpoints. The five curves overlap and all show the same periodic behavior.