38.3 Content
38.3.1 Einstein’s Postulates (1905)
Postulate 1 (Principle of Relativity): The laws of physics are the same in all inertial reference frames.
Postulate 2 (Constancy of c): The speed of light in vacuum is the same for all observers, regardless of the motion of the source or observer.
These simple postulates lead to profound consequences: time dilation, length contraction, and the relativity of simultaneity.
38.3.2 The Lorentz Factor
The Lorentz factor appears throughout relativistic physics:
\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]
where: - \(\gamma\) ≥ 1 always - At low speeds (v << c): \(\gamma \approx 1\) - As v → c: \(\gamma \to \infty\)
| Speed (fraction of c) | γ |
|---|---|
| 0.10c | 1.005 |
| 0.50c | 1.155 |
| 0.80c | 1.667 |
| 0.90c | 2.294 |
| 0.99c | 7.089 |
| 0.999c | 22.37 |
38.3.3 Time Dilation
Moving clocks run slower relative to stationary observers:
\[t = \gamma t_0 = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}}\]
where: - \(t_0\) = proper time (measured in the rest frame of the clock) - \(t\) = dilated time (measured by an observer relative to whom the clock moves)
38.3.4 Interactive: Time Dilation
38.3.5 Length Contraction
Moving objects are shorter in the direction of motion:
\[l = \frac{l_0}{\gamma} = l_0\sqrt{1 - \frac{v^2}{c^2}}\]
where: - \(l_0\) = proper length (measured in the rest frame of the object) - \(l\) = contracted length (measured by an observer relative to whom the object moves)
Length contraction only occurs along the direction of motion. Perpendicular dimensions are unchanged.
38.3.6 Evidence for Relativity
Muon decay (cosmic rays): - Muons created in upper atmosphere have half-life of 2.2 μs - At v ≈ 0.998c, they should travel only ~660 m before decaying - Yet they’re detected at Earth’s surface (10+ km away) - Time dilation extends their observed lifetime by factor of ~15
Particle accelerators: - Particles at near-c speeds require increasing force to accelerate - Consistent with relativistic momentum increase - Precisely predicted by γm₀v
Atomic clocks on aircraft: - Hafele-Keating experiment (1971) - Clocks flown around Earth showed time differences - Matched special and general relativity predictions
38.3.7 Relativistic Momentum
At high speeds, momentum is:
\[p = \gamma m_0 v = \frac{m_0 v}{\sqrt{1 - \frac{v^2}{c^2}}}\]
As v → c, momentum → ∞, which is why nothing with mass can reach c.
38.3.8 Mass-Energy Equivalence
Einstein’s most famous equation:
\[E = mc^2\]
Rest energy: \(E_0 = m_0 c^2\)
Total energy: \(E = \gamma m_0 c^2\)
Kinetic energy: \(K = E - E_0 = (\gamma - 1)m_0 c^2\)
Mass is a form of energy. A small amount of mass converts to an enormous amount of energy because c² is so large (~9 × 10¹⁶ m²/s²).