At the 10.0 -km-long Stanford Linear Accelerator, electrons...

At the $10.0$ -km-long Stanford Linear Accelerator, electrons with rest energy of $0.511 \mathrm{MeV}$ have been accelerated to a total energy of $46 \mathrm{GeV}$. How long is the accelerator as measured in the reference frame of the electrons?

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Key Concepts

Length Contraction

Length contraction is a phenomenon predicted by special relativity, whereby objects moving at relativistic speeds are measured to be shorter in the direction of motion compared to when they are at rest. This contraction is only apparent to stationary observers; in the moving object's own rest frame, its length remains unchanged. The contraction is directly related to the Lorentz factor, demonstrating the relative nature of spatial measurements.

Lorentz Factor

The Lorentz factor, often denoted by ? (gamma), is a dimensionless quantity that arises in special relativity. It quantifies the amount by which time, length, and relativistic mass change for an object while moving relative to an observer. Its value increases dramatically as an object's speed approaches the speed of light, leading to pronounced effects such as time dilation and length contraction.

Special Relativity

Special relativity is a theory developed by Einstein that describes how the laws of physics are the same in all inertial reference frames. It introduces concepts like time dilation and length contraction, which become significant at velocities approaching the speed of light. This theory challenges classical notions of space and time, showing that measurements of time intervals and distances depend on the relative motion between observers.

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