Are the Muonic Hydrogen and Electron Scattering Experiments Measuring the Same Observable?

Elastic scattering of relativistic electrons from the nucleon yields Lorentz invariant form factors that describe the fundamental distribution of charge and magnetism. The spatial dependence of the nucleon's charge and magnetism is typically interpreted in the Breit reference frame which is related by a Lorentz boost from the laboratory frame, where the nucleon is at rest. We construct a model to estimate how the Sachs electric and magnetic form factors can be corrected for the effects of relativistic recoil. When the corrections are applied, the ratio of the proton's Sachs form factors is approximately flat with $Q^2$, i.e. the spatial distributions of the proton's intrinsic charge and magnetization are similar. Further, we estimate the correction due to recoil that must be applied to the determination of the proton charge radius from elastic electron scattering before it can be compared to the value determined using the Lamb shift in hydrogen. Application of the correction brings the two values of the proton charge radius into significantly closer agreement. Predicted corrections based on the model are provided for the rms charge radii of the deuteron, the triton, and the helium isotopes.