Unified formalism for electromagnetic and gravitational probes: Densities
Adam Freese, Gerald A. Miller
Abstract
The use of light front coordinates allows a fully relativistic description of a hadron's spatial densities to be obtained. These densities must be two-dimensional and transverse to a chosen spatial direction. We explore their relationship to the three-dimensional, nonrelativistic densities, with a focus on densities associated with the energy-momentum tensor. The two-dimensional nonrelativistic densities can be obtained from the light front densities through a nonrelativistic limit, and can subsequently be transformed into three-dimensional nonrelativistic densities through an inverse Abel transform. However, this operation is not invertible, and moreover the application of the inverse Abel transform to the light front densities does not produce a physically meaningful result. We additionally find that the Abel transforms of so-called Breit-frame densities generally differ significantly from the light front densities. Numerical examples are provided to illustrate the various differences between the light front, Breit frame, and nonrelativistic treatment of densities.