Question

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?

Short answer

The volume is \(a^3 \sqrt{1 - \frac{u^2}{c^2}}\)."

Step-by-step solution

Understanding Relativity's Effect on Dimensions

When an object moves at a significant fraction of the speed of light, its length along the direction of motion contracts due to the phenomenon known as length contraction. This contraction does not affect dimensions perpendicular to the direction of motion. Here, the cube is moving along the x-axis in frame \(S'\), so only the length along the x-axis is affected.

Apply Length Contraction Formula

The length contraction formula is \(L' = L \sqrt{1 - \frac{u^2}{c^2}}\), where \(L'\) is the contracted length, \(L\) is the rest length, \(u\) is the relative velocity, and \(c\) is the speed of light. Since the cube's side \(a\) is parallel to the x-axis, it becomes \(a' = a \sqrt{1 - \frac{u^2}{c^2}}\).

Calculate the Transverse Dimensions

Because the movement does not affect the other dimensions of the cube transverse to the direction of motion, the sides parallel to the y and z axes remain the same, \(a\).

Calculate the Volume in Frame S'

The volume of the cube in frame \(S'\) is given by the product of its dimensions: \(V' = a' \cdot a \cdot a = (a \sqrt{1 - \frac{u^2}{c^2}}) \cdot a \cdot a = a^3 \sqrt{1 - \frac{u^2}{c^2}}\). This is the volume as observed in frame \(S'\).

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics introduced by Albert Einstein in 1905. It provided a new way to understand the relationship between space and time, drastically changing how we perceive motion. The core idea is that the laws of physics are the same for all observers, regardless of their relative motion. One of the groundbreaking concepts introduced by special relativity is that the speed of light in a vacuum is constant and will be the same for all observers, regardless of their motion relative to the light source.
This invariance of the speed of light leads to some counterintuitive consequences. For example, time dilation, where a moving clock ticks slower than one at rest, and length contraction, which we will delve into more in upcoming sections. The theory of special relativity highlights how measurements of time and space are not absolute but depend on the observer's state of motion. This revelation paved the way for modern physics to explain phenomena occurring at very high speeds, close to the speed of light.

Lorentz Transformation

The Lorentz transformation equations are mathematical formulas used to switch between two reference frames that are moving at a constant velocity relative to each other. These transformations explain how the coordinates of time and space are related in different inertial frames. Named after the physicist Hendrik Lorentz, these equations incorporate the factor \( \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} \), where \(c\) is the speed of light and \(u\) is the relative velocity.
The main use of the Lorentz transformation is to ensure that the laws of physics, especially those involving electromagnetic phenomena, remain consistent for different observers, regardless of their motion. In particular, they account for changes in time, length, and simultaneity that occur when objects move at velocities approaching the speed of light. When applied to the case of length contraction, these transformations show that an object moving in regard to an observer will appear shorter along the direction of motion by the factor of \(\sqrt{1 - \frac{u^2}{c^2}}\). This is a crucial part in understanding how and why the volume of objects changes in different inertial frames.

Relativistic Effects

Relativistic effects become noticeable when objects move at speeds close to the speed of light. These include time dilation and length contraction, both of which are direct consequences of special relativity. Length contraction, the focus of our original exercise, occurs because the length of an object along the direction of motion decreases as its speed approaches the speed of light. This effect only impacts dimensions parallel to the direction of travel. For instance, in the scenario of the metal cube, only the side aligned with the motion direction is contracted. By contrast, dimensions perpendicular to the motion remain unchanged.
These relativistic effects are not just theoretical concepts. They have practical implications in modern technology and experiments. For example, the Global Positioning System (GPS) accounts for these effects to provide accurate positioning data, as the satellites move at high speeds relative to the Earth. Understanding these effects is essential in fields like astrophysics and particle physics, where high-velocity particles are more commonly encountered. Thus, mastering the concept of length contraction and other relativistic effects is crucial for comprehending the universe's behavior at its most fundamental levels.