Архив за воскресенье, 6 июля 2025 г.

• Why 𝐸 = 𝑚𝑐²?

  Создано: roberto.c.alfredo в physics 6 июл. 2025 г., 01:54

  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.

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• Why Three Spatial Dimensions Support Stable Complexity

  Создано: roberto.c.alfredo в physics 6 июл. 2025 г., 21:46

  Why does the universe have three large spatial dimensions?

  That question can sound decorative at first, almost philosophical in the loose sense. We are used to treating space as the given background, the silent room in which physics happens. But dimensionality is not neutral scenery. It changes how influences spread, how forces weaken, and what kinds of structures can remain intact through time.

  So the real question is sharper than it first appears:

  Would anything familiar survive if space had fewer dimensions, or more?

  Not just geometry in the abstract. Not just whether equations could still be written down. Would there still be long-lived orbits? Would there still be atoms? Would there still be the kind of persistent, layered complexity that gives you stars, chemistry, planets, and stable environments?

  The striking possibility is that three-dimensional space is not an arbitrary stage. It may be one of the conditions that makes a durable world possible.

  Space changes the law, not just the picture

  A first way to see this is to notice that forces do not merely exist in space. They spread through it.

  Imagine a source sending influence outward equally in all directions. In ordinary three-dimensional space, that influence spreads over the surface of a sphere, whose area grows like \( r^2 \). As the same total influence is smeared over more and more area, the strength falls like

  $$ F(r) \propto \frac{1}{r^2}. $$

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