Critical spacetime crystals in continuous dimensions

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.

💡 Research Summary

This paper investigates the phenomenon of critical gravitational collapse in a novel setting where the spacetime dimension D is treated as a continuous real parameter greater than three. Building on the classic Choptuik discovery of discretely self‑similar (DSS) solutions in four‑dimensional Einstein–Klein‑Gordon (EKG) collapse, the authors generalize the construction to arbitrary D > 3 and refer to the resulting solutions as critical spacetime crystals (CSCs). A CSC is a DSS spacetime whose “crystal vector” ∂τ can change its causal character (timelike, null, spacelike) across a self‑similar horizon (SSH), a feature absent in ordinary crystals but natural in a relativistic context.

The theoretical framework starts by reducing the D‑dimensional spherically symmetric EKG system to an effective two‑dimensional dilaton‑gravity model. The dimensional parameter κ = D − 3 appears as a coupling constant in this reduced theory, allowing analytic continuation of D to non‑integer values. In adapted coordinates (τ, x) the DSS condition reads g(τ+Δ, x)=e^{-2Δ}g(τ, x), where Δ is the echoing period. The authors adopt Gundlach’s method of directly constructing the self‑similar solution rather than scanning a one‑parameter family of initial data. They prescribe regularity conditions at the centre (x=0) and at the SSH (x=1), make an initial guess for Δ, and solve the resulting first‑order nonlinear PDE system using a spectral (Fourier) discretization combined with Newton‑type iteration. This approach yields a single, highly accurate CSC for any chosen D, and the solution for one D can be used as an initial ansatz for a nearby D ± ε, facilitating a fine scan of the dimension space.

Numerically, the authors compute Δ(D) and the Choptuik scaling exponent γ(D) for D ranging from 3.05 to 5.5 with step sizes as small as 0.01. Their main findings are:

• The echoing period Δ exhibits a pronounced maximum Δ≈3.58 at D≈3.76. For larger D, Δ decreases and follows a 1/D scaling consistent with a large‑D expansion: Δ∼c D^{-1/2}+… .
• The Choptuik exponent γ grows with D, approaching γ≈0.41 for D→∞, but tends to zero linearly as D→3⁺. This suggests that the single unstable mode becomes marginal in the three‑dimensional limit.
• Both Δ and γ vanish as D→3⁺, indicating that κ = D‑3 can serve as a small expansion parameter. The authors perform a systematic κ‑expansion, obtaining Δ≈α κ^{1/2}+β κ^{3/2}+… and γ≈δ κ+ε κ^{2}+…, which match the numerical data in the near‑three regime.
• The null‑energy‑condition (NEC) saturation lines, characterized by an opening angle α, also scale linearly with κ, confirming the consistency of the dilaton‑gravity reduction.

In addition to the D‑dimensional analysis, the paper extends the results to two‑dimensional dilaton gravity models such as Jackiw–Teitelboim and the Witten black hole. The same CSCs exist in these theories, and the dependence of Δ, γ, and the NEC angle on κ mirrors the higher‑dimensional case, reinforcing the physical relevance of analytic continuation in D.

The authors discuss the broader implications of their work. Treating D as a continuous parameter provides a new small‑parameter expansion (κ → 0) that complements the well‑established large‑D expansion of general relativity. This opens the possibility of analytic control over critical collapse, a problem traditionally accessible only via numerics. Moreover, the concept of a spacetime crystal bridges general relativity, condensed‑matter physics, and quantum field theory: the discrete self‑similarity is analogous to time‑crystals, while the single unstable mode plays the role of a non‑thermal fixed point. The paper suggests several future directions, including extensions to non‑spherical symmetry, inclusion of additional matter fields, quantum corrections, and exploration of CSCs in asymptotically anti‑de Sitter spacetimes.

In summary, the study delivers a high‑precision numerical map of the echoing period and Choptuik exponent across continuous dimensions, validates these results with complementary analytic expansions, and establishes a versatile framework for investigating critical phenomena in gravity beyond integer dimensions. This work significantly advances our theoretical understanding of gravitational criticality and introduces the intriguing notion of spacetime crystals as universal attractors in the dynamics of collapse.