Critical spacetime crystals in continuous dimensions

We numerically construct a one-parameter family of critical spacetimes in arbitrary continuous dimensions D;gt3 D > 3 . 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