📂Physics

Features of Special Relativity due to Lorentz Transformation: Time Dilation

Characteristics of Lorentz Transformation

In special relativity, the transformation between two coordinate systems differs from the classical transformation. This is because ’the speed of light is the same for all observers’. Taking this condition into account leads to the derivation of the Lorentz transformation. The Lorentz transformation introduces three new phenomena that do not appear in classical physics.

• Loss of simultaneity
• Time dilation
• Length contraction

Time Dilation

Simply put, time dilation means that the closer an object moves to the speed of light, the slower (more slowly) time flows for it. This is also referred to as time expansion in the sense that the time axis becomes longer.

Let’s assume there is a system $A$ and a system $A$ moving at a constant velocity $v_{0}$ in the $x$ direction relative to $A^{\prime}$. The world line of an object at rest in the $A$ system is as follows.

When an event starts and ends, it takes longer from the perspective of the moving system $(A^{\prime}$$)$. That is, time flows more slowly. In comparison to the stationary system, time is extended by a factor of $\gamma_{0}$, and $\gamma_{0}$ depends on the speed $v_{0}$ of the $A^{\prime}$ system. The faster you move, the larger the value of $\gamma_{0}$ becomes, resulting in increasingly slower passage of time.

This is exactly the phenomenon where people on Earth and aboard a spaceship age differently, as seen in movies like The Martian or Interstellar. Similar to length contraction, time dilation does not occur in the direction perpendicular (orthogonal) to the movement of the $A^{\prime}$ system. If you’re curious, try calculating with different coordinates. Simultaneity breaks down, and time dilation and length contraction occur only in the direction parallel to the movement.

Thought Experiment

This is the most representative experiment to confirm the phenomenon of time dilation. Since the speed of light is constant, let’s denote it as a constant $c$.

Let’s say there is a rocket as shown above. The rocket moves to the right and shoots light vertically from the floor to the ceiling. The light reflected from the ceiling returns to the floor. Let $L$ be the distance from the floor to the ceiling inside the rocket. The time it takes for the light to make a round trip inside the rocket is $t_{r} = \dfrac{2L}{c}$. This result is when the observer is inside the rocket. Then, what happens if we observe this situation from outside the rocket?

From outside the rocket, the light making a round trip between the floor and ceiling will appear to follow the red line path. Let’s denote the speed of the rocket as $v_{0}$. Using Pythagoras’ theorem, we can calculate the time outside the rocket.

$$\begin{align*} && {\left( \frac { ct }{2} \right)} ^{2} &= {\left( \frac { v_{0}t }{2} \right)} ^{2}+L^{2} \\ \implies && t^{2}\frac { c^{2}-{v_{0}} ^{2} }{4} &= L^{2} \\ \implies && t &= \frac { 2L }{ \sqrt { c^{2}-{v_{0}} ^{2} } }=\frac { 2L }{ c }\frac { 1 }{ \sqrt { 1-\frac { v_{0}}{ c^{2}}}} =\gamma _{0}\frac { 2L }{ c } \end{align*}$$

Here, $\gamma _{0} = \dfrac { 1 }{ \sqrt { 1-\frac {{v_{0}}^{2} }{ c^{2} } } }$ is called the Lorentz factor. Now, comparing $t$ and $t_{r}$ gives the following:

$$t_{r}=\frac { 2L }{ c },\quad t=\gamma _{0}\frac { 2L }{ c}$$

The time inside and outside the rocket differs by a factor of $\gamma_{0}$. Since the speed of the rocket $v_{0}$ is less than the speed of light $c$, $\gamma_{0}>1$ and $t=\gamma_{0}\frac{2L}{c}=\gamma_{0}t_{r}$, hence $t>t_{r}$. Thus, despite the same situation, the flow of time inside and outside the rocket is different. Time flows slower inside the rocket than outside.