作成者: roberto.c.alfredo 内 physics 日付
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.
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.
3. Special Relativity in Three Ideas
- Relativity principle: the laws of physics are the same in all inertial frames.
- Speed limit \(c\): no signal travels faster than light in a vacuum.
- 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.
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.