Generating fields from data onH ? ?H +andH ? ?I +

Abstract

There are series solutions for characteristic boundary value problems for fields on black hole backgrounds that converge when the data are given on ℋ=ℋ− ∪ ℋ+, or on ℐ=ℐ− ∪ ℐ+, but may not converge when the data are given on ℋ− ∪ ℐ−, or on ℐ+ ∪ ℐ+. We specialize to oscillatory data of frequency ω and calculate approximate reflection and transmission coefficientsR(ω) andT(ω), using a field generated by data on ℋ=ℋ− ∪ ℋ+, and again, using a field generated by data on ℋ− ∪ ℐ−. The first calculation gives qualitatively good results at all frequencies at each order of approximation, and quantitatively better results at higher orders of approximation; the second calculation, using the series which may not converge, gives bad results except at very high frequencies. Thus for the physically unnatural case of a field that vanishes on ℋ− and goes toe iωv on ℋ+ we have a series that is convergent, and uniformly so with respect to frequency, while for the natural case of a field that vanishes on ℋ− and goes toe iωv on ℐ− we are limited to high frequencies. It is argued that a frequency-dependent renormalization of a series of the first type provides an approximation scheme that is convergent, and uniformly so with respect to frequency, for the physically important problems of the second type. The difficulties posed by the ω-dependent renormalization for the study of incident pulses are discussed.