Critical spacetime crystals in continuous dimensions

We numerically construct a one-parameter family of critical spacetimes in arbitrary continuous dimensions

. This generalizes Choptuik’s

D=4 D = 4

solution to spherically symmetric massless scalar-field collapse at the threshold of

D 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 D

, with detailed data for the interval

3.05≤ D≤ 5.5 3.05 ≤ D ≤ 5.5

. Notably, the echoing period has a maximum near

D≈ 3.76 D ≈ 3.76

. As a by-product, we recover the echoing periods and Choptuik exponents in

D=4\,(5) D = 4 ( 5 )

:

\Delta = 3.445453\,(3.22176) Δ = 3.445453 ( 3.22176 )

and

\gamma = 0.373961\,(0.41322) γ = 0.373961 ( 0.41322 )

. We support these numerical results with analytical expansions in

1/D 1 / D

and

D-3 D − 3

. They suggest that both the echoing period and Choptuik exponent vanish as

D\to 3^+ D → 3 +

. This paves the way for a small-

(D-3) ( D − 3 )

expansion, paralleling the large-

D D

expansion of general relativity. We also extend our results to two-dimensional dilaton gravity.