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Special Relativity

Syllabus reference

Unit 4, Topic 1 — 16 hours (including practicals)

Einstein's postulates

  1. The laws of physics are the same in all inertial frames of reference.
  2. The speed of light in a vacuum is the same for all observers, regardless of the motion of the source or observer: \(c = 3 \times 10^8\) m s⁻¹.

The Lorentz factor appears in all relativistic equations:

\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Time dilation

A moving clock runs slower relative to a stationary observer.

Key formula

\[ t = \gamma t_0 = \frac{t_0}{\sqrt{1 - v^2/c^2}} \]

where \(t_0\) = proper time (measured in the rest frame of the event), \(t\) = dilated time

Experimental evidence: Muons created in the upper atmosphere by cosmic rays. Their half-life is about 1.5 μs — classically they should not reach the Earth's surface, but due to time dilation (from our frame) they do.

Worked example: Time dilation

Question: A spacecraft travels at 0.80c. A clock on the spacecraft measures 10.0 s for a process. What time does an Earth observer measure?

Solution:

\(\gamma = \frac{1}{\sqrt{1 - 0.80^2}} = \frac{1}{\sqrt{0.36}} = \frac{1}{0.60} = 1.667\)

\(t = 1.667 \times 10.0 = 16.7\) s

Length contraction

A moving object is shorter in the direction of motion as measured by a stationary observer.

Key formula

\[ l = \frac{l_0}{\gamma} = l_0\sqrt{1 - \frac{v^2}{c^2}} \]

where \(l_0\) = proper length (measured in the rest frame of the object), \(l\) = contracted length

From the muon's frame of reference, the muon's lifetime is normal but the distance to Earth's surface is contracted — both frames give consistent results.

Relativistic momentum

At speeds approaching \(c\), classical momentum (\(p = mv\)) is insufficient. Relativistic momentum:

Key formula

\[ p = \gamma mv = \frac{mv}{\sqrt{1 - v^2/c^2}} \]

As \(v \to c\), \(\gamma \to \infty\) and the momentum approaches infinity. This means an infinite force would be required to accelerate an object with mass to the speed of light — which is impossible.

Mass–energy equivalence

Key formula

\[ E = mc^2 \]

This shows that mass and energy are equivalent. The total relativistic energy of an object is:

\[ E_{total} = \gamma mc^2 \]

The rest energy (when \(v = 0\)) is simply \(E_0 = mc^2\).

Why no object with mass can travel at \(c\): As \(v \to c\), the kinetic energy required approaches infinity. Only massless particles (photons) travel at exactly \(c\).

Simulations and videos

Crash Course Physics:

External resources: