Fig. 1: Continuum spectra.
a The hyperboloid (orange) defined by t2 − x2 − y2 = +1 in (2 + 1)-dimensional (x, y, t) Minkowski space is mapped (black rays) by the stereographic projection through the point (0, 0, −1) (black dot) to the unit disk (blue) at t = 0. The geodesics (red) are given by intersections of the hyperboloid with planes passing through the origin (0, 0, 0) (green dot), and are mapped by the projection to circular arcs perpendicular to the boundary of the Poincaré disk. b Comparison of the first few eigenmodes of the Euclidean and hyperbolic drum of radius r0 = 0.94 according to increasing eigenvalues \({\lambda }_{{{{{{{{\rm{g}}}}}}}}}^{n\ell }\). Their spatial profile \({u}_{{{{{{{{\rm{g}}}}}}}}}^{n\ell }\) is shown with yellow (green, blue) denoting maxima (zeros, minima). The number of radial zeros inside the disk, n, and the angular momentum (number of angular zeroes), ℓ, can easily be inferred from the plots. Modes with ℓ = 0 are indicated with a gray background.
