Defining the proton radius: A unified treatment.

Background: There is significant current interest in knowing the value of the proton radius and also its proper definition. Purpose: Combine the disparate literatures of hydrogen spectroscopy and diverse modern parton distributions to understand the meaning of the proton radius in a manner consistent with the separate bodies of work. Methods: Use perturbation theory, light-front dynamics, and elementary techniques to find relativistically correct definitions of the proton radius and charge density. Results: It is found that the very same proton radius is accessed by measurements of hydrogen spectroscopy and elastic lepton scattering. The derivation of the mean-square radius as a moment of a spherically symmetric three-dimensional density is shown to be incorrect. A relativistically correct, two-dimensional charge density is related to the diverse modern literature of various parton distributions. Relativistically invariant moments thereof are derived in a new relativistic moment expansion, the RME. Conclusion: The equation rp2p =-6G'E(0) is the definition of the proton radius. [ABSTRACT FROM AUTHOR]