📘 🛰️ Special Relativity: Einstein, Time, and the Speed of Light

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A didactic dive into the theory of Special Relativity, with graphs, examples, and historical context.
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Blog do Marcellini

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September 24, 2025

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1 📘 🛰️ Special Relativity: Einstein, Time, and the Speed of Light

A didactic dive into the theory of Special Relativity, with graphs, examples, and historical context.

1.1 🌟 What Was Einstein’s Annus Mirabilis?

ℹ️ In 1905, Albert Einstein published five revolutionary papers. Among them, “On the Electrodynamics of Moving Bodies” gave rise to the theory of Special Relativity1. The theory completely reshaped what we understand about time, space, and motion.

How can time not be absolute? And if two clocks are moving relative to each other, which one shows the “true” time?

1.2 📜 Postulates of Special Relativity

✅ The theory is based on two simple — yet revolutionary — postulates:

  1. The laws of physics are the same in all inertial frames (without acceleration).
  2. The speed of light in a vacuum is constant for all observers, regardless of the velocity of the source or the observer.

1.3 ⏱️ Time Dilation: What It Is and How It Works

When an object moves very fast (close to the speed of light), time passes more slowly for it relative to a stationary observer.

library(plotly)
velocity <- seq(0, 0.99, by = 0.01)
time <- 1 / sqrt(1 - velocity^2)
plot_ly(x = ~velocity, y = ~time, type = "scatter", mode = "lines") %>%
  layout(title = "Time Dilation as a Function of Velocity",
         xaxis = list(title = "Relative velocity (v/c)"),
         yaxis = list(title = "Time dilation factor"))

\[ \Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]

1.4 📏 Length Contraction: Does Space Shrink Too?

⚠️ Surprisingly, moving objects also appear to shrink in length, for an observer at rest:

\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]

velocity <- seq(0, 0.99, by = 0.01)
L <- sqrt(1 - velocity^2)
plot_ly(x = ~velocity, y = ~L, type = "scatter", mode = "lines") %>%
  layout(title = "Length Contraction as a Function of Velocity",
         xaxis = list(title = "Relative velocity (v/c)"),
         yaxis = list(title = "Apparent length (L/L₀)"))

1.5 🔁 Lorentz Transformations

These transformations describe how space and time relate between observers moving at different velocities:

\[ \begin{aligned} x' &= \gamma (x - vt) \\ t' &= \gamma \left(t - \frac{vx}{c^2} \right) \end{aligned} \quad \text{with} \quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \]

1.6 👯 The Twin Paradox

Imagine two twins: one stays on Earth and the other travels on a spaceship near the speed of light. When the traveler returns, he is younger than the sibling who stayed behind. This happens because time “ran more slowly” for the one who experienced accelerated motion.

1.7 🧪 Relativity vs. Classical Mechanics

Concept Classical Mechanics Special Relativity
Time Absolute Relative
Addition of velocities A + B Uses relativistic formula
Speed of light Varies with the observer Constant for all
Transformations Galilean Lorentzian

1.8 🔍 Common Myths About Relativity

⚠️ “Einstein said everything is relative.”
❌ Not true. He showed that there are invariants, such as the speed of light and the laws of physics.

⚠️ “Relativity is only about things moving fast.”
❌ Wrong. Relativity also affects everyday technologies, such as GPS!

1.9 ❓ Questions to Reflect On

1.9.1 🔦 Question 1: Turning on a Flashlight Inside the Train

You are traveling on a train at an extremely high speed, close to that of light. In the middle of the carriage, you switch on a flashlight. For you, the light:

    1. Travels faster forward, since the train is also moving forward
    1. Travels faster backward, since the rear is approaching the light
    1. Travels forward and backward at the same speed

✅ Correct Answer: C

📘 Comment: For you, who are inside the train and consider yourself at rest, the light propagates symmetrically. This is the principle of relativity: the laws of physics (including the speed of light) are the same in all inertial frames.

But for someone standing at the station, the light appears to reach the rear of the carriage more quickly, since the carriage is “running” forward — leading us to the idea that simultaneity depends on the frame of reference.

1.9.2 ⏳ Question 2: Time for Astronauts and Farmers

An astronaut is traveling at a significant fraction of the speed of light. A farmer remains on Earth. At the end of the journey:

    1. Both will be the same age, since time is universal
    1. The astronaut will be older, since he moved more
    1. The astronaut will be biologically younger

✅ Correct Answer: C

📘 Comment: This is the famous twin paradox. From the Earth observer’s point of view, time passes more slowly for the astronaut — the clock on board the spaceship runs slower.

This effect has already been experimentally observed in particles such as muons and even in atomic clocks flown on airplanes and satellites. In the case of GPS, relativistic corrections are essential for navigation accuracy.

1.10 📚 Conclusion

Special Relativity forces us to rethink time, space, and the notion of simultaneity. What seems simple at first reveals a world where two events that are simultaneous for one observer may not be for another.

Einstein started with two postulates — and led us to a revolution in physical thought.

2 📚 Main Reference:

Resnick, R. (1968). Introduction to Special Relativity. Wiley.

⚠️ In upcoming posts we will explore these concepts in more detail.

3 🧭 📘 Reading Guide — Special Relativity

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Footnotes

  1. Einstein, A. (1905). “Zur Elektrodynamik bewegter Körper.” Annalen der Physik, 17(10), 891–921.↩︎