🏷️ タグ: length-contraction

1. Setting the Scene

Imagine two perfectly synchronized clocks. Now place one on a spaceship traveling at high speed and leave the other on Earth. When they reunite, you’ll discover something incredible: the clocks no longer show the same time. How can this be? The answer lies in the relativity of simultaneity.

---

2. Relativity of Simultaneity: What Does “Now” Mean?

Two observers moving relative to each other will disagree on whether two events occur “at the same time.” What is simultaneous for one might not be for the other, thanks to the Lorentz transformations.

Mathematical Reminder (Lorentz Transformations): $$ t = \gamma\bigl(t_0 + \frac{v\,x_0}{c^2}\bigr), \quad x = \gamma\bigl(x_0 + v\,t_0\bigr) \\[4pt] \text{where}\quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $$

If you’re interested in the math behind these transformations, see The Spacetime Invariant: From Minkowski to 𝐸² = 𝑝²𝑐² + 𝑚²𝑐⁴.

---

3. Time Dilation: Time Depends on Speed

Historically, in 1971 Joseph Hafele and Richard Keating flew atomic clocks around the world on airplanes and confirmed Einstein’s predicted dilation in situ. They compared the flown clocks to ones left on the ground and found they recorded different time intervals, just as special relativity predicts.

The duration of a time interval depends on the motion of the observer measuring it. Mathematically: $$ \Delta t = \gamma\,\Delta \tau \quad\text{where}\quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $$

In classical physics, adding velocities is easy. If a Mustang 🐎 travels at 100 km/h and throws a ball forward at 20 km/h (relative to the car), a stationary observer sees the ball at 120 km/h. Simple, right? But when Einstein entered the scene, these comfortable sums ended.

---

Why Don’t Velocities Add Normally?

Special relativity forces us to use a different formula at speeds near the speed of light:

$$ u = \frac{u' + v}{1 + \frac{u'v}{c^2}} $$

• The left-hand \(u\) is the resulting velocity, measured in the stationary frame.
• \(u'\) is the velocity of the object in the moving frame.
• \(v\) is the velocity of the moving frame relative to the stationary one.
• \(c\) is the speed of light (299,792,458 m/s).

This formula ensures that you never exceed the speed of light, preserving the universe’s consistency (phew 😅).

For example, if a spaceship travels at \(0.8c\) and launches a probe forward at \(0.7c\) relative to it, the speed measured from outside is NOT \(1.5c\). It is:

$$ u = \frac{0.7c + 0.8c}{1 + \frac{(0.7c)(0.8c)}{c^2}} = \frac{1.5c}{1 + 0.56} \approx 0.96c $$

Fast? Yes. Faster than light? Never!

---

The Ladder Paradox 🪜🏠

This relativistic logic gives rise to amusing and perplexing situations, like the ladder paradox (also known as the “barn paradox”).

Imagine a ladder too long to fit in your garage. Normally, you can’t squeeze it in, right? But now make the ladder race toward the garage at high speed (yes, a fast ladder—physicists can be crazy).