Critical spacetime crystals in continuous dimensions

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.