You and your friend carry identical clocks. Your friend passes by in a rapidly moving train. As your clock ticks off 1.00 s, you see your friend's clock tick off 0.50 s. How much time would your friend see your clock tick off in the time it takes your friends clock to tick off 1.00 s?

Special relativity is a theory in physics that explains how time and space are interwoven. It was developed by Albert Einstein in the early 20th century.
This theory plays a crucial role in understanding how objects moving at high speeds experience time and space.
One of the core ideas of special relativity is that the speed of light is constant for all observers, regardless of their motion.
This leads to several interesting phenomena, one of which is time dilation.
Time dilation means that time is observed to pass differently for observers in different states of motion.
In our exercise, we see this when a clock on a rapidly moving train ticks slower compared to a stationary clock.

The Lorentz factor is a key part of the mathematics behind special relativity.
It quantifies the amount of time dilation experienced by an object moving relative to an observer. The Lorentz factor, represented by \( \beta \), is calculated using the formula:
\[ \beta = \frac{1}{ \sqrt{1 - \frac{v^2}{c^2}} } \]
Here, \( v \) is the relative velocity between the observer and the moving object, and \( c \) is the speed of light.
In the given exercise, we simplified the calculation based on observed times, which gave us another form of the Lorentz factor.
We used \( \beta = \frac{1.00}{0.50} = 2 \) as the ratio between the proper time and the observed time.
This simplification helps to understand that the closer the speed is to the speed of light, the more significant the time dilation becomes.

Observed time refers to the time measured by a stationary observer for an event taking place in a moving reference frame.
In our exercise, you, the observer, see your friend's clock on a moving train ticking slower.
The formula used to express the relationship between the proper time (time in the moving frame) and observed time is:
\[ \tau' =\frac{ \tau}{\beta} \]
Where \( \tau \) is the proper time and \( \tau' \) is the observed time.
We calculated the proper time based on the relationship:
\[ \tau = \beta \times \tau' \]
From the calculations, the observed time dilation tells us that for every 1.00 second on your friend's moving clock, you observe only 0.50 seconds.
Conversely, your friend will see your clock ticking twice as fast.
So, while their clock ticks 1.00 second, they will see your clock tick 2.00 seconds, illustrating the reciprocal nature of time dilation in special relativity.