A cube of metal with sides of length a sits at rest in a...

A cube of metal with sides of length $a$ sits at rest in a frame $S$ with one edge parallel to the $x$-axis. Therefore, in $S$ the cube has volume $a^3$. Frame $S'$ moves along the $x$-axis with a speed $u$. As measured by an observer in frame $S'$, what is the volume of the metal cube?

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Key Concepts

Length Contraction

Length contraction is a relativistic effect predicted by special relativity, stating that an object in motion relative to an observer will appear shortened along the direction of its motion. This effect is quantified using the Lorentz factor, where the contracted length is equal to the proper length divided by ?, emphasizing that only dimensions parallel to the velocity are affected while perpendicular dimensions remain unchanged.

Lorentz Factor

The Lorentz factor, commonly denoted by gamma (?), is a key quantity in special relativity. It quantifies how much time, length, and relativistic mass change for an object moving relative to an observer. The factor is defined as ? = 1/?(1 - u²/c²), where u is the relative velocity and c is the speed of light, and it serves as the multiplier in the formula for length contraction.

Special Relativity

Special relativity is a fundamental theory in physics that describes how measurements of space and time change for observers in different inertial frames of reference, particularly when those frames move at speeds comparable to the speed of light. It provides the framework for understanding phenomena such as time dilation, length contraction, and the relativity of simultaneity.

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