Sections#

Review & Summary#

Simultaneity and Time Dilation#

Events that are simultaneous in one inertial frame need not be simultaneous in another. Time dilation: If a clock at rest in a frame measures a time interval \(\Delta t_0\) (the proper time), then an observer in a frame moving at speed \(v\) relative to that clock measures a longer interval:

(317)#\[ \Delta t = \gamma \Delta t_0 \]

where \(\gamma = 1/\sqrt{1 - v^2/c^2}\) is the Lorentz factor.

The Relativity of Length#

Length contraction: If an object has length \(L_0\) in its rest frame (the proper length), then an observer moving parallel to that length measures

(318)#\[ L = \frac{L_0}{\gamma} \]

The Lorentz Transformation#

For motion along the \(x\) axis with relative speed \(v\), the coordinates \((x,t)\) and \((x',t')\) in two inertial frames are related by \(x' = \gamma(x - vt)\), \(t' = \gamma(t - vx/c^2)\), with inverse \(x = \gamma(x' + vt')\), \(t = \gamma(t' + vx'/c^2)\).

The Relativity of Velocities#

If a particle has velocity \(u\) in one frame and the frame moves at \(v\) relative to another, the velocity \(u'\) in the second frame is \(u' = (u - v)/(1 - uv/c^2)\) for motion along the same line. Velocities do not add simply; \(c\) is the invariant speed limit.

Doppler Effect for Light#

The observed frequency \(f'\) differs from the source frequency \(f\) when source and detector move relative to each other. For a source approaching the detector: \(f' = f\sqrt{(1 + \beta)/(1 - \beta)}\) with \(\beta = v/c\). For receding, \(\beta \to -\beta\). Transverse Doppler (source moving perpendicular to line of sight): \(f' = f/\gamma\).

Momentum and Energy#

Relativistic momentum: \(\vec{p} = \gamma m \vec{v}\). Total energy: \(E = \gamma mc^2\). Rest energy: \(E_0 = mc^2\). Kinetic energy: \(K = E - E_0 = (\gamma - 1)mc^2\). The energy and momentum satisfy the invariant relation

(319)#\[ E^2 = (pc)^2 + (mc^2)^2 \]