Question

A person in a spaceship holds a meter stick as the ship moves parallel to the Earth's surface with \(\gamma=2 .\) What does an observer on the Earth notice as the person in the spaceship rotates the stick from parallel to perpendicular to the ship's motion? a) The stick becomes shorter. b) The stick becomes longer. c) The length of the stick stays the same.

Short answer

a) The stick becomes shorter. b) The stick becomes longer. c) The stick stays the same length. Answer: b) The stick becomes longer.

Step-by-step solution

Understand the motion of the spaceship and the gamma factor

The ship moves parallel to the Earth's surface with a gamma factor (\(\gamma)=2\). The gamma factor is a dimensionless quantity that helps us compare the length between the observer in the Earth and the spaceship. It is given by the following equation: $$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$ Here, \(v\) is the velocity of the spaceship relative to the Earth, and \(c\) is the speed of light. A gamma factor value greater than 1 indicates relativistic speeds.

Determine the initial length observed by the Earth

When the meter stick is parallel to the motion of the spaceship, the length contraction is at its maximum. The relation between the proper length (\(L_0\)) and the observed length (\(L\)) is: $$L = \frac{L_0}{\gamma}$$ Here, \(L_0=1\,\text{m}\) (the length of the meter stick), and \(\gamma =2\). So the observed length on Earth is: $$L = \frac{1\,\text{m}}{2} = 0.5\,\text{m}$$

Analyze the rotation of the meter stick

As the person in the spaceship rotates the stick from parallel to perpendicular to the ship's motion, the component of the length contracted in the direction of motion decreases. When the stick is perpendicular to the motion, there is no length contraction observed in the direction of motion. At this position, the length of the stick observed on Earth will be the same as the length in the spaceship.

Determine the final length observed by the Earth

Since the stick is perpendicular to the ship's motion and there is no length contraction, the observer on Earth will see the stick's length equal to the proper length, which is 1 m. That is: $$L_{final} = L_0 = 1\,\text{m}$$

Compare the lengths and choose the correct answer

Comparing the initial length observed on Earth (\(0.5\,\text{m}\)) with the final length observed on Earth (\(1\,\text{m}\)), we see that the stick becomes longer as it is rotated from parallel to perpendicular to the ship's motion. So, the correct answer is: b) The stick becomes longer.

Key concepts

Relativistic Physics

Relativistic physics is an area of physics that studies the behavior of objects moving at speeds close to the speed of light. At these high velocities, classical mechanics as laid out by Newton begins to break down, and we need Einstein's theory of relativity to describe what happens.

In our everyday experience, objects seem to behave independently of their state of motion, at least visually. However, this is not the case when an object moves at a significant fraction of the speed of light. An observer in a different frame of reference might measure different lengths and time intervals for events involving these fast-moving objects. It might sound like science fiction, but it's a very real aspect of our universe, confirmed by numerous experiments.

The concept of length contraction, which we encounter in the given problem, is one of these bizarre effects. It indicates that the length of an object moving at relativistic speeds will appear shorter to a stationary observer, compared to its length when at rest. This is not an illusion; it results from the very nature of space and time as described by the special theory of relativity.

Lorentz Factor

The Lorentz factor, often denoted by the Greek letter gamma ((\gamma)), is crucial for calculating changes in measurements between different inertial frames moving at relativistic speeds relative to each other. It emerges from the Lorentz transformations, which are equations that relate space and time coordinates of events as measured in different inertial frames.

(\gamma) is defined by the equation:
\[\gamma = \frac{1}{{\sqrt{1 - v^2/c^2}}}\]
Here, (v) is the relative velocity between the observer and the moving object, and (c) is the speed of light in a vacuum, the cosmic speed limit. As velocity approaches the speed of light, (\gamma) increases significantly, indicating stronger relativistic effects. For lengths, times, and masses, we use the Lorentz factor to calculate how these quantities appear altered from the perspective of different observers. In our space travel scenario, when the spaceship moves parallel to Earth with a Lorentz factor of 2, any object aligned with the direction of motion would appear half as long to the Earth-bound observer. This is a quantifiable demonstration of length contraction.

Special Relativity

Special relativity is a fundamental theory in physics formulated by Albert Einstein in 1905. It brought about a radical change in our understanding of space, time, and energy. The theory's two postulates are deceptively simple: the laws of physics are invariant in all inertial frames of reference, and the speed of light in a vacuum is the same for all observers, regardless of their relative motion or the motion of the light source.

These postulates lead to surprising conclusions, such as the length contraction mentioned in our exercise. Another implication is time dilation, where a moving clock ticks slower than a stationary one from the perspective of an observer at rest. Mass-energy equivalence, encapsulated by the famous equation (E=mc^2), is also a result of special relativity, indicating that mass can be converted to energy and vice versa.

Applying special relativity to the rotating meter stick in the spaceship, we see that length contraction only applies along the direction of motion. Once the stick is perpendicular, this effect vanishes, leading to the appearance that the stick becomes longer from the Earth observer's perspective. Thus, special relativity is critical for understanding why the length of an object in motion can appear differently based on its orientation relative to the direction of movement.