🏷️ タグ: special-relativity

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.

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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.

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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.

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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!

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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).

1. Setting the Stage

When Albert Einstein published his “miracle year” in 1905, he still treated space and time as separate entities. It was Hermann Minkowski (1908) who famously declared:

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

This “union” is the interval \(s\). We’ll see why it’s invariant—identical for all inertial observers—and how this forces us to redefine energy and momentum.

This interval also underpins fundamental concepts like mass–energy equivalence, explained in detail in Why 𝐸 = 𝑚𝑐².

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2. Brief History of the Interval

Cultural tidbit 🎸: while Minkowski revolutionized physics in 1908, the tango “El choclo” was sweeping Buenos Aires—another example of Latin American vanguard art and science.

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3. Lorentz Transformations in a Nutshell

For two inertial frames \(\mathcal{S}\) and \(\mathcal{S}'\) with relative velocity \(v\) along the \(x\)-axis:

$$ \begin{aligned} x' &= \gamma\,\bigl(x - vt\bigr) \\[4pt] t' &= \gamma\!\left(t - \tfrac{v\,x}{c^2}\right) \\[4pt] \gamma &= \frac{1}{\sqrt{1 - v^2/c^2}} \end{aligned} $$

Rewriting the interval in \(\mathcal{S}'\) gives:

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.

1. From the Siren to the Cosmos: What Does Doppler Measure?

In acoustics we hear the pitch rise as an ambulance approaches and fall as it moves away. In relativity, the Doppler effect applies to light, not sound—and the classical formula no longer suffices.

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2. Longitudinal Doppler: Frequency and Wavelength

For a source and observer receding from or approaching each other along the line of sight:   $$ \frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{em}}} = \sqrt{\frac{1 + \beta}{1 - \beta}}, \quad \beta = \frac{v}{c}. $$   In terms of frequencies:   $$ \nu_{\mathrm{obs}} = \nu_{\mathrm{em}}\, \sqrt{\frac{1 - \beta}{1 + \beta}}. $$

• If \(v > 0\) (receding): redshift (\(\lambda_{\mathrm{obs}} > \lambda_{\mathrm{em}}\)).
• If \(v < 0\) (approaching): blueshift (\(\lambda_{\mathrm{obs}} < \lambda_{\mathrm{em}}\)).

Astronomical example:   A planet orbiting its star at 30 km/s induces a shift of about 0.01 nm in an iron absorption line—detectable with modern spectrographs.

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3. Light Aberration: Apparent Angle Change

Aberration changes the apparent angle \(\theta\) between a light ray and the direction of motion. If \(\theta\) is the source’s emission angle and \(\theta'\) the observer’s reception angle:   $$ \cos\theta' = \frac{\cos\theta - \beta}       {1 - \beta\,\cos\theta}. $$

• For \(\theta = 90^\circ\): stars appear “dragged” toward the direction of Earth’s motion, by up to \(\approx 20.5''\).

Historical note:   In 1727, James Bradley, while searching for stellar parallax, discovered aberration—first proof that Earth moves.

1. What Exactly Are Light Cones?

When you turn on a flashlight in a dark room, the light spreads out in a cone of illumination. Now imagine that effect on a cosmic scale with the universal speed limit: light itself. In relativity, these cones mark the boundaries of the causal past and causal future of every event.

Formally, an event is just a point in spacetime with coordinates \((t, x, y, z)\). The light emitted from that point travels along surfaces that form cones, clearly separating which events can influence it or be influenced by it.

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2. Simple Mathematical Structure of the Light Cone

Recall our spacetime interval:   $$ s^2 = c^2 t^2 - x^2 - y^2 - z^2 $$   The light cones are the regions satisfying:

• \(s^2 = 0\): lightlike trajectories.
• \(s^2 > 0\): inside the cone, timelike trajectories.
• \(s^2 < 0\): outside the cone, spacelike trajectories.

Visually, these cones are the surfaces traced out by light emitted at the central event, neatly partitioning spacetime into causal regions.

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3. Past, Future, and Causality: What’s Causally Possible?

• Causal past (lower cone): all events that could have influenced the present.
• Causal future (upper cone): all events that can be influenced by the present.
• Non-causal region (outside the cone): events that can neither influence nor be influenced, since doing so would require exceeding \(c\).

This is the physical guarantee of cause-and-effect order in the universe. No observer will ever see events flipped in time, avoiding classic time-travel paradoxes.