An alien spacecraft is flying overhead at a great distance...

An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.150 s. The first officer on the spacecraft measures that the searchlight is on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light $c$?

Key Concepts

Lorentz Factor

The Lorentz factor, denoted by ? (gamma), quantifies the amount of time dilation and length contraction experienced by an object moving at a significant fraction of the speed of light. It is given by the formula ? = 1/?(1 - v²/c²), where v is the relative velocity and c is the speed of light, and it serves as a key multiplier in the equations of special relativity.

Proper Time

Proper time is the time interval between two events as measured by an observer for whom the events occur at the same spatial location. It is the shortest time between events and remains invariant, serving as the baseline measurement for comparing time intervals across different inertial frames.

Time Dilation

Time dilation is a relativistic phenomenon predicted by special relativity where a moving clock appears to tick slower compared to a clock at rest relative to the observer. This concept explains how measurements of time can vary between observers in different inertial frames, hinging on the relative velocity between them.

Relative Velocity

Relative velocity is the measure of the speed of one object as observed from another moving object. In the context of special relativity, it is critical because effects like time dilation and length contraction become significant when this relative speed approaches the speed of light, necessitating the use of the Lorentz transformation.

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