An Introduction to Modern Astrophysics · Problem 15.20

The Lorentz factor $\gamma$ tells you how much a particle's time, mass, and energy are "stretched" by relativistic motion. Since you're given total energy directly, the key insight is to use the energy-momentum relation $E = \gamma m c^2$ rather than measuring velocity—this avoids the messy conversion and lets you find $\gamma$ by simple division. Ultra-high-energy cosmic rays like this proton are so relativistic that $\gamma$ becomes enormous, which is why they're so interesting to astrophysicists.

The Lorentz factor $\gamma$ tells you how much a particle's time, mass, and energy are "stretched" by relativistic motion. Since you're given total energy directly, the key insight is to use the energy-momentum relation $E = \gamma m c^2$ rather than measuring velocity—this avoids the messy conversion and lets you find $\gamma$ by simple division. Ultra-high-energy cosmic rays like this proton are so relativistic that $\gamma$ becomes enormous, which is why they're so interesting to astrophysicists.