We consider the hyperboloidal initial value problem for the cubic focusing wave equation. Without symmetry assumptions, we prove the existence of a co-dimension 4 Lipschitz manifold of initial data that lead to global solutions in forward time which do not scatter to free waves.
The numerical investigation of wave propagation in the asymptotic domain of Kerr spacetime has only recently been possible thanks to the construction of suitable hyperboloidal coordinates. The asymptotics revealed an apparent puzzle in the decay rate
s of scalar fields: the late-time rates seemed to depend on whether finite distance observers are in the strong field domain or far away from the rotating black hole, an apparent phenomenon dubbed splitting. We discuss far-field splitting in the full field and near-horizon splitting in certain projected modes using horizon-penetrating, hyperboloidal coordinates. For either case we propose an explanation to the cause of the splitting behavior, and we determine uniquely decay rates that previous studies found to be ambiguous or immeasurable. The far-field splitting is explained by competition between projected modes. The near-horizon splitting is due to excitation of lower multipole modes that back excite the multipole mode for which splitting is observed. In both cases splitting is an intermediate effect, such that asymptotically in time strong field rates are valid at all finite distances. At any finite time, however, there are three domains with different decay rates whose boundaries move outwards during evolution. We then propose a formula for the decay rate of tails that takes into account the inter--mode excitation effect that we study.
We present the first numerical construction of the scalar Schwarzschild Green function in the time-domain, which reveals several universal features of wave propagation in black hole spacetimes. We demonstrate the trapping of energy near the photon sp
here and confirm its exponential decay. The trapped wavefront propagates through caustics resulting in echoes that propagate to infinity. The arrival times and the decay rate of these caustic echoes are consistent with propagation along null geodesics and the large l-limit of quasinormal modes. We show that the four-fold singularity structure of the retarded Green function is due to the well-known action of a Hilbert transform on the trapped wavefront at caustics. A two-fold cycle is obtained for degenerate source-observer configurations along the caustic line, where the energy amplification increases with an inverse power of the scale of the source. Finally, we discuss the tail piece of the solution due to propagation within the light cone, up to and including null infinity, and argue that, even with ideal instruments, only a finite number of echoes can be observed. Putting these pieces together, we provide a heuristic expression that approximates the Green function with a few free parameters. Accurate calculations and approximations of the Green function are the most general way of solving for wave propagation in curved spacetimes and should be useful in a variety of studies such as the computation of the self-force on a particle.
We describe the hyperboloidal compactification for Teukolsky equations in Kerr spacetime. We include null infinity on the numerical grid by attaching a hyperboloidal layer to a compact domain surrounding the rotating black hole and the orbit of an in
spiralling point particle. This technique allows us to study, for the first time, gravitational waveforms from large- and extreme-mass-ratio inspirals in Kerr spacetime extracted at null infinity. Tests and comparisons of our results with previous calculations establish the accuracy and efficiency of the hyperboloidal layer method.
Black hole perturbation theory is typically studied on time surfaces that extend between the bifurcation sphere and spatial infinity. From a physical point of view, however, it may be favorable to employ time surfaces that extend between the future e
vent horizon and future null infinity. This framework resolves problems regarding the representation of quasinormal mode eigenfunctions and the construction of short-ranged potentials for the perturbation equations in frequency domain.
