Let $\mathcal Rg(s,\omega)$ be the <Radon transform>
$$
\mathcal Rg(s,\omega)=\int_{y\cdot\omega=s}g(y)\,dA_y.
$$
Taking the large-radius limit in the Kirchhoff formula, with $\xi=t-r$ fixed for the outgoing limit and $\eta=t+r$ fixed for the incoming limit, gives the <radiation field>[radiation fields]
$$
\psi_+(\xi,\omega)
=\frac1{4\pi}
\left[
\mathcal Ru_1(-\xi,\omega)
-\partial_s\mathcal Ru_0(-\xi,\omega)
\right],
$$
$$
\psi_-(\eta,\omega)
=-\frac1{4\pi}
\left[
\mathcal Ru_1(\eta,\omega)
+\partial_s\mathcal Ru_0(\eta,\omega)
\right].
$$
Indeed, the expanding spheres converge after multiplication by $r/t$ to the planes $y\cdot\omega=-\xi$ and $y\cdot\omega=\eta$, respectively.

The Radon transforms of smooth compactly supported functions are smooth. If the data are supported in a ball of radius $R$, these transforms vanish for $|s|>R$. Hence both radiation fields are well-defined smooth functions of compact support in the null-time variable.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 3 / c / Solution

A radiation field is the rescaled limit of a wave along null infinity. For a three-dimensional wave, the outgoing field is the limit of $ru(t,r\omega)$ with $t-r$ fixed, while the incoming field fixes $t+r$.


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