From crystal color symmetry to quantum spacetime

Abstract

More than one hundred years after the inception of relativistic physics, the concept of time remains incompletely understood. Relativity provides means to perform calculations of geometrical properties of spacetime, such as distances or curvature, and to interpret them in terms of physical observations, for instance as time dilation or gravitational effects. However, an intuitive understanding of spacetime is complicated, not so much because it is four-dimensional (which, after all, can be evaded by visualizing twodimensional cross sections) but mainly because its geometry does not obey Euclid’s axioms even in the absence of curvature. Through a well defined and clever transformation (RBS: renormalized blended spacetime), Venkatraman Gopalan (2021) has demonstrated how the hyperbolic geometry in Minkowski spacetime can be mapped to a circular Euclidean geometry. In particular, Lorentzian boosts become Euclidean rotations which enables new frontiers of exploration in color symmetry and magnetic crystals. His idea of general relativistic spacetime crystals and how to obtain them is both powerful and broad, although the notion of relativistic crystals and lattices in two dimensions has existed for a while (Janner & Ascher, 1969a,b). Finally, the breadth of this new work is underlined by the present opinion piece, written by two co-authors with distinct yet connected areas of expertise.