Critical spacetime crystals in continuous dimensions

TL;DR

This work extends the study of critical gravitational collapse by constructing discretely self-similar CSCs in continuous spacetime dimensions $D>3$, thereby generalizing Choptuik’s $D=4$ solution. The authors develop a direct CSC construction framework that enforces DSS from the outset and extract key observables, notably the echoing period $oldsymbol{igtriangleup(D)}$ and the Choptuik exponent $oldsymbol{gamma(D)}$, across $3.05\le D\le 5.5$ and via analytic expansions in $1/D$ and $D-3$. A central finding is that both $oldsymbol{igtriangleup}$ and $oldsymbol{gamma}$ vary smoothly with $D$, featuring a maximum of $oldsymbol{igtriangleup}$ near $oldsymbol{D\approx 3.76}$ and indicating they vanish as $D\to 3^+$, with complementary large-$D$ behavior consistent with a $gamma\to 1/2$ limit. The paper also generalizes the framework to 2D dilaton gravity, derives several scaling results in the large-$D$ and small-$D$ regimes, and discusses conjectures about the asymptotic vanishing of $oldsymbol{igtriangleup}$ and the fate of criticality in these limits, providing a versatile map between higher-dimensional GR, 2D dilaton models, and potential holographic connections.

Abstract

We numerically construct a one-parameter family of critical spacetimes in arbitrary continuous dimensions D>3. This generalizes Choptuik's D=4 solution to spherically symmetric massless scalar-field collapse at the threshold of D-dimensional Schwarzschild-Tangherlini black hole formation. We refer to these solutions, which share the discrete self-similarity of their four-dimensional counterpart, as critical spacetime crystals. Our main results are the echoing period and Choptuik exponent of the crystals as continuous functions of D, with detailed data for the interval 3.05<D<5.5. Notably, the echoing period has a maximum near D=3.76. As a by-product, we recover the echoing periods and Choptuik exponents in D=4 (5): Delta=3.445453 (3.22176) and gamma=0.373961 (0.41322). We support these numerical results with analytical expansions in 1/D and D-3. They suggest that both the echoing period and Choptuik exponent vanish as D approaches 3 from above. This paves the way for a small-(D-3) expansion, paralleling the large-$D$ expansion of general relativity. We also extend our results to two-dimensional dilaton gravity.

Figures (22)

• Figure 1: Illustration of spacetime crystal for critical solutions in spherically symmetric collapse. The past lightcone of the naked singularity is the patch we solve, but the spacetime may be extended beyond the selfsimilar horizon (SSH) until the Cauchy horizon (CH). The crystal vector $\partial _\tau$ (in green) is timelike, null, or spacelike depending on whether one is inside, on, or outside the SSH.
• Figure 2: Penrose diagram of a CSC taken from Ecker:2024haw. Dark gray shaded: time crystal region. Light gray shaded: (space) crystal region. Dashed line: SSH. Black circle: Naked singularity. Zig-zag line: Cauchy horizon. Vertical line: Center. Solid 45-degree lines: ${\mathscr{I}}^\pm$. Dotted 45-degree lines: Null lines in fundamental domain. CSC metric identified along bold $\tau=\rm const.$ lines. Inset: Fundamental domain with NEC lines (black), positive (red) and negative (blue) Ricci regions.
• Figure 3: Decomposition of $x$-domain into near-boundary regions (solid lines; solution approximated by Taylor series) and interior region (dashed lines; equations solved numerically from cutoffs $x_{\textrm{\tiny L}}$ and $x_{\textrm{\tiny R}}$ toward matching surface $x_{\textrm{\tiny match }}$).
• Figure 4: Comparison of various fields (top to bottom) constituting CSCs in $D = 3.05$, $4$, $5.5$ (left to right) on fundamental domain $\tau/\Delta\in[0,1]$, $x\in[0,1]$.
• Figure 5: Converged boundary values $f_c$ (left), $\Psi_c$ (middle), and $\psi_{-p}$ for approximately equidistant $D\in[3.05,5.5]$, with $\Psi_c$ normalized by ${\rm max}\,(\Psi_c)$.