Matching conditions at null infinity in the presence of logarithms: the role of advanced and retarded radiation

We provide a new perspective on the general matching conditions between the future of past null infinity and the past of future null infinity, emphasizing the impact of dominant logarithmic terms in the asymptotic expansion of the fields near null infinity. We explicitly consider the cases of a massless scalar field and of electromagnetism. Key in our derivation is the identification of the physical origin of these logarithms, which are associated with advanced and retarded radiation saturating the finite energy flux condition at null infinity (in a space of functions which is made precise). The matching conditions arise then from the requirement of Coulombic (i.e., $1/r$) behaviour at spatial infinity.

💡 Research Summary

The paper revisits the matching conditions that relate the future of past null infinity (𝓘⁻) to the past of future null infinity (𝓘⁺) in four‑dimensional Minkowski space, focusing on the role of logarithmic terms that can appear in the asymptotic expansion of massless fields. Traditional analyses assume that the leading scalar or electromagnetic field behaves as ϕ∼a(u,Ω)/r on 𝓘⁺ and ϕ∼b(v,Ω)/r on 𝓘⁻, with a and b approaching finite limits as u→−∞ and v→+∞. Under these assumptions the matching condition reduces to an antipodal identification A(Ω)=B(−Ω). The authors point out that this assumption is more restrictive than necessary for finite energy flux through null infinity.

For a massless scalar field satisfying □ϕ=0, the authors decompose the solution into a retarded part ϕ_R(u,Ω)/r and an advanced part ϕ_A(v,Ω)/r, plus subleading terms. The total radiated energy through 𝓘⁺ is ΔE|𝓘⁺=−∫du d²Ω √γ (∂_u ϕ_R)², and the analogous expression holds for incoming energy through 𝓘⁻. Requiring these integrals to be finite allows ϕ_R and ϕ_A to contain logarithmic growth: ϕ_R(u)=ϕ_R^± log(±u)+φ_R^±+o(1) as u→±∞, and similarly for ϕ_A(v). The logarithmic pieces arise from “advanced radiation” that saturates the finite‑energy bound at 𝓘⁻ and from “retarded radiation” that saturates the bound at 𝓘⁺.

When the solution is expressed entirely in retarded time (v=u+2r), the advanced component contributes a term proportional to log r/r at 𝓘⁺. This term is independent of u, so its u‑derivative vanishes and it does not affect the energy flux. An analogous log r/r term appears at 𝓘⁻ from the retarded component. Near spatial infinity i⁰, the limit is taken with r→∞ while keeping t=u+r (or t=v−r) finite. Both the retarded and advanced pieces then generate log r/r contributions. To preserve the Coulombic 1/r behavior expected from the elementary solution of Poisson’s equation, these logarithmic terms must cancel. This cancellation imposes the condition ϕ_R^+ = −ϕ_A^+, equivalently Ψ = −Ψ′, where Ψ and Ψ′ are the coefficients of the log r/r terms on 𝓘⁺ and 𝓘⁻ respectively. This is precisely the “mixed matching condition” previously derived in references