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The theory of relativity, proposed by Albert Einstein, revolutionized our understanding of space and time. It consists of the special and general theories of relativity. Special relativity, in particular, introduces the idea that the laws of physics are the same for all non-accelerating observers. It shows how space and time are interconnected, forming what we call "spacetime."
This theory proposes that the speed of light is constant for all observers, regardless of their relative speed. In practical terms, this means that time and space are not absolute. They are malleable and change depending on the observer's velocity relative to the speed of light.
One fascinating consequence of this is time dilation, where time is observed to pass at different rates in different inertial frames. When an object moves at speeds near the speed of light, time will appear to slow down relative to a stationary observer. This is a key concept when dealing with time experienced on fast-moving spaceships versus stationary observers.

The Lorentz transformation formula is an essential mathematical tool used in the theory of relativity. It provides the means to convert physical measurements from one inertial frame of reference to another.
The formula is particularly important for understanding time dilation and length contraction. It mathematically describes how, at relativistic speeds, time and space are transformed between two observers moving at a constant velocity relative to each other.
For time dilation, the relevant formula is:\[ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \]Here, \( t \) is the proper time measured in the moving reference frame, \( t' \) is the dilated time observed in the stationary reference frame, \( v \) is the relative velocity, and \( c \) is the speed of light.
This equation shows that as an object's velocity \( v \) approaches the speed of light \( c \), time \( t' \) for that object in the stationary frame will appear to increase. This is because of the term \( \sqrt{1 - \frac{v^2}{c^2}} \), which becomes smaller, thus increasing the time \( t' \).

Proper time is a crucial concept in understanding time dilation. It represents the time measured by an observer who is at rest relative to the event being timed. In simpler terms, it's the time interval measured by someone who is experiencing the events without moving relative to them.
In the context of our pendulum problem, the proper time is the 1.50 seconds measured by the astronaut inside the spaceship. This is because the astronaut is at rest relative to the pendulum he/she is observing. To them, the pendulum swings naturally, without any perceived differences.
Proper time differs from dilated time, which is the time interval measured by an observer in a different inertial frame moving relative to the proper time observer. Understanding the distinction is vital when applying the Lorentz transformation to calculate how time behaves at relativistic speeds.

Reference frames are essential for understanding and applying the concepts of relativity. A reference frame is simply a perspective or a point of view from which measurements like time and space are made.
There are two types of reference frames: inertial, where objects move at a constant velocity, and non-inertial, where objects are accelerating. Inertial frames are the ones often discussed in special relativity because they apply constant velocities which are crucial for using the Lorentz transformations.
In our problem scenario, the spaceship and Earth act as two distinct inertial reference frames. The astronaut inside the spaceship has a different frame compared to someone on Earth. Because the spaceship is moving at three-fourths the speed of light relative to Earth, each observer measures time (and space) differently. The key takeaway is that no single reference frame is more correct than the other—they're just different perspectives of the same events.