The ďAlembert solution in hyperboloidal foliations

Abstract

We explicitly construct the analogue of the ďAlembert solution to the 1+1 wave equation in an hyperboloidal setting. This hyperboloidal ďAlembert solution is used, in turn, to gain intuition into the behaviour of solutions to the wave equation in a hyperboloidal foliation and to explain some apparently anomalous behaviour observed in numerically constructed solutions discussed in the literature.

21 pages, 10 figures; Amended to accommodate for referees requests and published version: important typo in Eqs.(12,13) corrected, and Appendix added with further examples in different hyperboloidal coordinate charts; Part of the topical collection "Hyperboloidal foliations in the era of gravitational-wave astronomy: from mathematical relativity to astrophysics"