91Ó°ÊÓ

In the realm of special relativity, the Lorentz factor is crucial for understanding how time and space behave at high speeds. It's defined by the formula:

• \( \gamma = \dfrac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \)

This factor comes into play when objects travel at velocities close to the speed of light \((c)\), causing their time, length, and mass to transform relative to an observer at rest.
In our rocket problem, the speed \(v = 0.866c\) results in a Lorentz factor of 2. This means that from the perspective of an outside observer, time appears to slow down, and lengths contract in the direction of the motion. Understanding this factor helps us calculate how these quantities change, revealing the hidden dynamics of objects moving at relativistic speeds.

Length contraction is a fascinating consequence of special relativity. When an object moves at a significant fraction of the speed of light, its length in the direction of travel seems shorter to a stationary observer. This is given by the formula:

• \( L' = \dfrac{L}{\gamma} \)

Here, \(L'\) is the contracted length, \(L\) is the original length, and \(\gamma\) is the Lorentz factor.

In the exercise, the rocket's pilot sees no contraction of the rocket because they are moving with it, so its length remains \(L\). However, the garage appears shorter to the pilot due to its relative motion. Calculating with the Lorentz factor of 2, the garage's length contracts to \(L/4\) from the pilot's viewpoint.

This concept highlights how observers in different frames perceive different realities, each equally valid in their own context.

The rocket paradox elegantly demonstrates relativity's counterintuitive nature. From the stationary observer's frame, the rocket fits into the garage due to length contraction, giving the illusion of it being shorter.
However, from the rocket pilot's perspective, the garage is the object in motion, and thus it experiences length contraction, appearing only \(L/4\) long.

This paradox illustrates the core tenets of special relativity, where an observer in motion perceives the dimensions of stationary objects differently. It showcases how reality can differ substantively for observers in different frames, challenging our everyday intuitions about space and movement.

By exploring this paradox, we gain insight into the flexible and interconnected nature of time and space, where both perspectives coexist without contradiction in the relativistic world.