1. Setting the Stage

It all began in 1905 when Albert Einstein asked whether light could exert a push (the “recoil” of a photon cannon) and what that implied for conservation of energy and momentum. The radical answer was: mass is stored energy. Here’s why.

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2. Historical Clues and Key Experiments

• Fusion in the Sun: the “missing” mass in the reaction converts into the energy that lights our star.
• Marie Curie’s radioactivity: nuclei “lose” mass by emitting particles and photons.
• Atomic bomb (Trinity, 1945): \(\sim0.7 g\) of mass converted into \(5\times10^{13}\,\text{J}\) of energy.

🎼 Musical fact:   Metastasis by Iannis Xenakis (1954) applies mathematical formulas and architectural structures to music—an art–science fusion.

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3. Special Relativity in Three Ideas

1. Relativity principle: the laws of physics are the same in all inertial frames.
2. Speed limit \(c\): no signal travels faster than light in a vacuum.
3. Spacetime invariant:     $$   s^2 = c^2t^2 - x^2 - y^2 - z^2.   $$

These pillars lead to new expressions for momentum and energy.

Want to dive deeper into this fundamental invariant? See The spacetime invariant: from Minkowski to 𝐸² = 𝑝²𝑐² + 𝑚²𝑐⁴.

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4. Quick Quantitative Derivation

4.1 Relativistic Momentum

$$ \mathbf{p} = \gamma\,m\,\mathbf{v}, \quad \text{where} \quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}. $$

4.2 Total Energy

$$ E^2 = p^2c^2 + m^2c^4. $$

4.3 Rest Energy

If \(v = 0\) ⇒ \(p = 0\), then   $$ E_0 = mc^2. $$   Voilà! The rest energy is bottled up in mass.