Question

A problem from relativity theory Suppose a spaceship of length \(L_{0}\) travels at a high speed \(v\) relative to an observer. To the observer, the ship appears to have a smaller length given by the Lorentz contraction formula $$L=L_{0} \sqrt{1-\frac{v^{2}}{c^{2}}}$$ where \(c\) is the speed of light. a. What is the observed length \(L\) of the ship if it is traveling at \(50 \%\) of the speed of light? b. What is the observed length \(L\) of the ship if it is traveling at \(75 \%\) of the speed of light? c. In parts (a) and (b), what happens to \(L\) as the speed of the ship increases? d. Find \(\lim _{v \rightarrow c^{-}} L_{0} \sqrt{1-\frac{v^{2}}{c^{2}}}\) and explain the significance of this limit.

Short answer

Short Answer: The observed length L of the spaceship decreases as its speed relative to an observer increases. Specifically, for a spaceship traveling at 50% and 75% of the speed of light, the observed lengths are \(\sqrt{0.75}L_0\) and \(\sqrt{0.4375}L_0\), respectively. As the spaceship's speed approaches the speed of light, its observed length contracts to nearly 0, meaning it will appear almost non-existent in length from an observer's perspective.

Step-by-step solution

Calculate observed length L at 50% of the speed of light

Using the Lorentz contraction formula, we will calculate the observed length L of the spaceship when it is traveling at 50% of the speed of light (0.5c). $$ L = L_{0}\sqrt{1 - \frac{(0.5c)^2}{c^2}} $$ $$ L = L_{0}\sqrt{1 - \frac{0.25c^2}{c^2}} $$ $$ L = L_{0}\sqrt{1 - 0.25} $$ $$ L = L_{0}\sqrt{0.75} $$

Calculate observed length L at 75% of the speed of light

Using the Lorentz contraction formula, we will calculate the observed length L of the spaceship when it is traveling at 75% of the speed of light (0.75c). $$ L = L_{0}\sqrt{1 - \frac{(0.75c)^2}{c^2}} $$ $$ L = L_{0}\sqrt{1 - \frac{0.5625c^2}{c^2}} $$ $$ L = L_{0}\sqrt{1 - 0.5625} $$ $$ L = L_{0}\sqrt{0.4375} $$

Discuss the change in L as the speed of the ship increases

Based on the results from Step 1 and Step 2, we can observe that as the speed of the ship increases from 50% to 75% of the speed of light, the observed length L of the spaceship gets smaller (from \(\sqrt{0.75}L_0\) to \(\sqrt{0.4375}L_0\)) due to the Lorentz contraction effect.

Find the limit as v approaches the speed of light and explain its significance

We have to find the limit of the Lorentz contraction formula as the speed of the spaceship (v) approaches the speed of light (c). $$ \lim_{v\to c^{-}} L_0\sqrt{1 - \frac{v^2}{c^2}} $$ As v approaches c, the term inside the square root approaches 0, which means that the observed length L approaches 0. The significance of this limit is that as the spaceship's speed approaches the speed of light, its observed length contracts to nearly 0. This means that, from an observer's perspective, the spaceship will appear almost non-existent in length as it gets closer to the speed of light.

Key concepts

Relativity Theory

Relativity theory, introduced by Albert Einstein, revolutionized our understanding of space, time, and gravity. This theory is divided into two parts: special relativity and general relativity. Special relativity focuses on the physics of moving objects in the absence of gravity and introduces several counterintuitive concepts, one of which is length contraction. According to this theory, the length of an object as measured by an observer in relative motion to the object is shorter than its length in its own rest frame. The formula used to calculate this so-called Lorentz contraction, named after physicist Hendrik Lorentz, is \[ L = L_{0} \sqrt{1 - \frac{v^2}{c^2}} \].In this context, \( L_0 \) is the proper length (the object's length in its rest frame), \( v \) is the relative velocity between the observer and the moving object, and \( c \) represents the speed of light. As shown with the spaceship example, the faster the ship moves compared to the observer, up to a certain limit, the more its length appears to contract.

Understanding these concepts helps grasp the fundamentally different behavior of objects moving at speeds comparable to the speed of light, which is crucial for any advanced study in physics or for anyone keen on understanding the fabric of our universe.

Speed of Light

The speed of light, denoted as \( c \), is a fundamental constant in physics with a value of approximately \( 299,792,458 \) meters per second. It is not only the speed at which light propagates through a vacuum but also represents the maximum speed at which all energy, matter, and information in the universe can travel. Special relativity posits that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer. This has profound implications on how we experience time and distance.

In the context of the Lorentz contraction, the speed of light acts as an upper boundary for the velocity variable within the formula. As a spaceship approaches the speed of light, the length contraction becomes more and more pronounced, leading to significant shrinking as perceived by a stationary observer. This concept is essential to understand how the universe would appear differently if we could travel close to the speed of light. For example, space travel at such speeds would make the vast distances between stars appear much shorter, potentially making interstellar travel more feasible – at least theoretically.

Limits in Calculus

Limits in calculus are fundamental to understanding the behavior of functions as they approach a specific input value, and they play a crucial role in many fields, including physics and engineering. The limit concept is used to describe the value that a function or sequence 'approaches' as the input (or index) approaches some value. In the exercise provided, the limit as the velocity \( v \) approaches the speed of light is essential to understand what happens to the length of the spaceship.

The calculation of the limit is written as \[ \lim_{v \rightarrow c^{-}} L_0 \sqrt{1 - \frac{v^2}{c^2}} \], and it describes what happens to the Lorentz contraction as the speed \( v \) gets infinitesimally close to, but does not reach, the speed of light \( c \). In our case, the limit indicates that the observed length of the spaceship tends to zero. This mathematical conclusion reinforces the idea from relativity theory that an object would appear infinitely contracted in the direction of motion as its speed approaches the speed of light, suggesting that at the speed of light itself, normal spatial dimensions cease to have the meaning we ascribe to them at lower speeds.