The Boost Operator: Properties, Computation and Applications

Abstract: The transformation of radiation signals (e.g., photon occupation number and integrated intensity) between moving frames is a common task is physics, astrophysics and cosmology. Here we show that the required boost operator, relating the frequency-dependent spin-weighted spherical harmonic coefficients of the considered observable between the frames, is directly given by the aberration kernel with the Doppler weight parameter being replaced by a differential operator. The aberration kernel has been previously studied in great detail, meaning that this simplification allows us to directly compute the boost operator using the expressions of the aberration kernel. As a preparatory step, we generalize the differential equation that determines the aberration kernel to general Doppler weight. This avoids the intermediate step of Doppler-weight raising and lowering operations in computations of the boost operator. We then clarify all the properties of the boost operator (e.g., raising and lowering operations, symmetries and commutation relations) and derive a formal operator differential equation for the boost operator. This differential equation allows us to quickly generate the boost operator for which we give exact expressions up to second order in v/c. For illustration, we then apply the boost operator to transformations of the cosmic microwave background (CMB), validating that measurements of the lowest CMB multipoles do not allow determining the amplitude of the primordial CMB dipole. We also derive the kinematic corrections to the Thomson scattering process (to all orders in v/c), giving explicit expressions up to second order in v/c, showcasing an application of the boost operator in radiative transfer problems.