Relativistic spacetime crystals

Periodic space crystals are well established and widely used in physical sciences. Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other hand need to account for the mixing of space and time in special relativity through Lorentz transformation, and have been listed only in 2D. This work shows that there exists a transformation between the conventional Minkowski spacetime (MS) and what is referred to here as renormalized blended spacetime (RBS); they are shown to be equivalent descriptions of relativistic physics in flat spacetime. There are two elements to this reformulation of MS, namely, blending and renormalization. When observers in two inertial frames adopt each other's clocks as their own, while retaining their original space coordinates, the observers become blended. This process reformulates the Lorentz boosts into Euclidean rotations while retaining the original spacetime hyperbola describing worldlines of constant spacetime length from the origin. By renormalizing the blended coordinates with an appropriate factor that is a function of the relative velocities between the various frames, the hyperbola is transformed into a Euclidean circle. With these two steps, one obtains the RBS coordinates complete with new light lines, but now with a Euclidean construction. One can now enumerate the RBS point and space groups in various dimensions with their mapping to the well known space crystal groups. The RBS point group for flat isotropic RBS spacetime is identified to be that of cylinders in various dimensions: mm2 which is that of a rectangle in 2D, (∞/ m ) m which is that of a cylinder in 3D, and that of a hypercylinder in 4D. An antisymmetry operation is introduced that can swap between space-like and time-like directions, leading to color spacetime groups. The formalism reveals RBS symmetries that are not readily apparent in the conventional MS formulation. Mathematica script is provided for plotting the MS and RBS geometries discussed in the work.

Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space

When a pulsed, few-cycle electromagnetic wave is focused by optics with f-number smaller than two, the frequency components it contains are focused to different regions of space, building up a complex electromagnetic field structure. Accurate numerical computation of this structure is essential for many applications such as the analysis, diagnostics, and control of high-intensity laser-matter interactions. However, straightforward use of finite-difference methods can impose unacceptably high demands on computational resources, owing to the necessity of resolving far-field and near-field zones at sufficiently high resolution to overcome numerical dispersion effects. Here, we present a procedure for fast computation of tight focusing by mapping a spherically curved far-field region to periodic space, where the field can be advanced by a dispersion-free spectral solver. In many cases of interest, the mapping reduces both run time and memory requirements by a factor of order 10, making it possible to carry out simulations on a desktop machine or a single node of a supercomputer. We provide an open-source C++ implementation with Python bindings and demonstrate its use for a desktop machine, where the routine provides the opportunity to use the resolution sufficient for handling the pulses with spectra spanning over several octaves. The described approach can facilitate the stability analysis of theoretical proposals, the studies based on statistical inferences, as well as the overall development and analysis of experiments with tightly-focused short laser pulses.

Asymptotic structure in substitution tiling spaces

Every sufficiently regular non-periodic space of tilings of $\mathbb {R}^d$ has at least one pair of distinct tilings that are asymptotic under translation in all the directions of some open $(d-1)$-dimensional hemisphere. If the tiling space comes from a substitution, there is a way of defining a location on such tilings at which asymptoticity ‘starts’. This leads to the definition of the branch locus of the tiling space: this is a subspace of the tiling space, of dimension at most $d-1$, that summarizes the ‘asymptotic in at least a half-space’ behavior in the tiling space. We prove that if a $d$-dimensional self-similar substitution tiling space has a pair of distinct tilings that are asymptotic in a set of directions that contains a closed $(d-1)$-hemisphere in its interior, then the branch locus is a topological invariant of the tiling space. If the tiling space is a two-dimensional self-similar Pisot substitution tiling space, the branch locus has a description as an inverse limit of an expanding Markov map on a zero- or one-dimensional simplicial complex.