Time Dilation from Lorentz Transformations

The Concept of Proper Time

Before we begin the derivation, we need to define a crucial concept: proper time.

The proper time interval, often denoted as $\Delta t_0$, is the time interval between two events as measured by a single clock that is present at both events. This means that in the reference frame of this clock, the two events occur at the same spatial location.

Imagine an astronaut on a spaceship checking their watch. The time elapsed on their watch between two glances is a proper time interval because the watch is at rest relative to the astronaut. For any other observer moving relative to the spaceship, the two events (the two glances at the watch) occur at different spatial locations. As we will now show, that observer will measure a longer time interval. The proper time is the shortest possible time interval measured between two events.

Deriving Time Dilation

Let's use the machinery of the Lorentz transformations to derive the relationship between the proper time interval $\Delta t_0$ and the time interval $\Delta t$ measured by an observer in relative motion.

1. Set up the frames: We have our standard two inertial frames. Frame S is the "stationary" frame (e.g., Earth), and frame S' is the "moving" frame (e.g., a spaceship) traveling at a constant velocity $v$ along the x-axis relative to S.

2. Define the events: A clock is at rest in the moving frame, S'. It ticks twice.

   • Event 1: The first tick occurs at coordinates $(t_1^{\prime },x_1^{\prime })$ in frame S'.
   • Event 2: The second tick occurs at coordinates $(t_2^{\prime },x_2^{\prime })$ in frame S'.

3. Apply the definition of proper time: Since the clock is stationary in frame S', both ticks happen at the same location in that frame. Therefore, $x_1^{\prime }=x_2^{\prime }$, which means the spatial separation is $\Delta x^{\prime }=x_2^{\prime }-x_1^{\prime }=0$. The time interval measured in S', $\Delta t^{\prime }=t_2^{\prime }-t_1^{\prime }$, is the proper time interval, so we can write $\Delta t^{\prime }=\Delta t_0$.

4. Transform to the stationary frame: We want to find the time interval $\Delta t=t_2-t_1$ as measured by an observer in frame S. To do this, we use the inverse Lorentz transformation for time, which you'll recall from our last lesson:
   $t=\gamma \left(t^{\prime }+\frac{v{x}^{\prime }}{c^2}\right)$

5. Calculate the interval: We apply this equation to both events:
   $t_1=\gamma \left(t_1^{\prime }+\frac{v{x}_1^{\prime }}{c^2}\right)$
   $t_2=\gamma \left(t_2^{\prime }+\frac{v{x}_2^{\prime }}{c^2}\right)$
   Now, we subtract the first equation from the second to find the time interval $\Delta t$ in frame S:
   $\Delta t=t_2-t_1=\gamma \left((t_2^{\prime }-t_1^{\prime })+\frac{v(x_2^{\prime }-x_1^{\prime })}{c^2}\right)$
   This simplifies to:
   $\Delta t=\gamma \left(\Delta t^{\prime }+\frac{v\Delta x^{\prime }}{c^2}\right)$

6. The Result: Finally, we substitute our condition from step 3, namely $\Delta x^{\prime }=0$ and $\Delta t^{\prime }=\Delta t_0$:
   $\Delta t=\gamma (\Delta t_0+0)$
   $\Delta t=\gamma \Delta t_0$

Since the Lorentz factor $\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$ is always greater than or equal to 1, this equation tells us that the time interval $\Delta t$ measured in the stationary frame S is always longer than (or equal to, if $v=0$) the proper time interval $\Delta t_0$. A moving clock is observed to run slow.

The following image provides a compact visual summary of this derivation on a spacetime diagram.

To see this derivation in motion, let's watch the concluding segment of the video from our last lesson.

For a more detailed explanation in a text format, which also provides a nice geometric derivation using a "light clock", please review the following resource.

Experimental Evidence: The Muon Experiment

Time dilation is not just a mathematical abstraction; it's a real, measurable physical phenomenon. One of the most classic and compelling pieces of evidence comes from the study of cosmic-ray muons, a topic highly relevant to your interest in particle physics.

• The Puzzle: Muons are unstable elementary particles, similar to electrons but much heavier. They are created when high-energy cosmic rays strike the upper atmosphere, about 15 km above the Earth's surface. A muon at rest has a very short mean lifetime of about ${\tau }_0=2.2\times 10^{-6}$ seconds (2.2 microseconds). Even traveling near the speed of light ($c$), classical physics would predict a muon could only travel an average distance of $d=c\times {\tau }_0\approx (3\times 10^8\text{m/s})\times (2.2\times 10^{-6}\text{s})\approx 660$ meters before decaying. So, how do they reach detectors at sea level in large numbers?

• The Solution: Time dilation. The muon's short lifetime of 2.2 µs is its proper lifetime, measured in its own rest frame. From our perspective on Earth, the muon is a moving clock. These muons travel at about $0.995c$, which gives a Lorentz factor of $\gamma \approx 10$. According to our clocks on Earth, the muon's average lifetime is dilated:
  $\Delta t=\gamma {\tau }_0\approx 10\times 2.2\mu \text{s}=22\mu \text{s}$
  With this dilated lifetime, the average distance a muon can travel before decaying becomes $d\approx c\times \Delta t\approx (3\times 10^8\text{m/s})\times (22\times 10^{-6}\text{s})\approx 6600$ meters, or 6.6 km. This is much more consistent with the fact that many muons survive the journey from the upper atmosphere to the ground.

The following video provides an excellent explanation of this cornerstone experiment.

This remarkable agreement between theory and observation was one of the first major triumphs of special relativity and serves as undeniable proof that time is, indeed, relative.

Conclusion

In this lesson, we have seen how the seemingly abstract Lorentz transformations lead to the very real and measurable effect of time dilation.

Key Takeaways:

• Proper time ($\Delta t_0$) is the time interval between two events measured in a reference frame where the events occur at the same location.
• Time dilation is derived directly from the Lorentz transformation for time, resulting in the formula $\Delta t=\gamma \Delta t_0$.
• An observer measures a clock moving relative to them to run slower by a factor of $\gamma$.
• This is not a mechanical or illusory effect; it is a fundamental property of time itself.
• The survival of cosmic-ray muons at sea level provides powerful experimental evidence for time dilation.

In our next lesson, we will explore the other side of this coin: length contraction. Just as observers disagree on time intervals, they also disagree on measured lengths. We will see that from the muon's perspective, its lifetime isn't dilated, but rather the distance from the atmosphere to the Earth's surface is contracted, providing another beautiful example of the symmetry at the heart of relativity.