A spaceship \((A)\) is moving at speed \(c / 2\) with respect to another spaceship \((B)\). Observers in \(A\) and B set their clocks so that the event at \((x, y, z, t)\) of turning on a laser in spaceship \(B\) has coordinates (0,0,0,0) in \(A\) and also \((0,0,0,0)\) in \(B .\) An observer at the origin of \(B\) turns on the laser at \(t=0\) and turns it off at \(t=\tau\) in his time. What is the time duration between on and off as seen by an observer in \(A?\)

Imagine boarding a spaceship and zipping away into the cosmos at speeds you'd think were only possible in sci-fi movies. This is where the mind-bending concept of special relativity comes in—it's a fundamental part of modern physics that describes how time, space, and motion are interwoven, especially when objects move at significant fractions of the speed of light, known as 'c'.

Introduced by Albert Einstein in 1905, special relativity is grounded on two main ideas: the laws of physics are the same for all observers in uniform motion relative to one another, and the speed of light in a vacuum is constant for all observers, regardless of their motion or the motion of the light source. It’s those principles that lead to fascinating phenomena such as time dilation, where time can tick differently depending on how fast you are moving.

This gets especially interesting when we consider events such as turning on and off a laser aboard a spaceship. If different observers are in motion relative to one another, as in our textbook exercise, they would disagree on measurements like the duration of the event, hinting at the relative nature of time itself.

Zooming through the universe at incredible speeds changes the rules of the game. To understand how, meet the Lorentz factor, denoted by the Greek letter \(\gamma\), a key player in special relativity. The Lorentz factor is the mathematical 'knob' that we turn to adjust for the differences in perception caused by relative motion at high speeds.

In the textbook problem, we crunched numbers to get \(\gamma = \frac{2}{\sqrt{3}}\) which reflects how time and other physical quantities transform when one object moves at a substantial fraction of the speed of light relative to another. The essence of the Lorentz factor is to ensure that even when things seem off-kilter due to high-speed travel, the laws of physics continue to hold up from different viewpoints. This is what keeps everything consistent across different frames of reference, preserving the core principle that the laws of physics are the same for all observers moving uniformly relative to one another.

Picture two spaceships drifting in the vast emptiness of space. How fast one appears to move from the other’s perspective depends on their relative velocity. It's this notion of speed in relation to something else that is pivotal in the realm of special relativity.

When spaceship A observes spaceship B, as in the given exercise, we’re looking at A's viewpoint to understand how B's actions, such as the laser turning on and off, unfold in A's own timeline. The relative velocity \(v = \frac{c}{2}\) is the speed at which spaceship A sees B moving away from or towards it. This rate of motion is significant enough (half the speed of light, no less!) to make certain that everyday intuitions about time and space just don’t apply. Time dilation occurs as a direct consequence of this speedy relative motion, and it's the reason why the observer in spaceship A would perceive the laser being on for longer than the observer in B. Evidently, in the cosmic dance of spacefaring objects, relative velocity doesn't just dictate the rhythm but rewires the fabric of reality as we perceive it.