37-4 The Relativity of Velocities

37-4 The Relativity of Velocities#

Prompts

  • In Galilean relativity, if a particle has speed \(u'\) in frame \(S'\) and \(S'\) moves at \(v\) relative to \(S\), the speed in \(S\) is \(u = u' + v\). Why does this fail when \(u'\) or \(v\) is close to \(c\)?

  • Derive the relativistic velocity addition formula \(u = (u' + v)/(1 + u'v/c^2)\) from the Lorentz transformation. (Hint: \(u = \Delta x/\Delta t\), \(u' = \Delta x'/\Delta t'\).)

  • Show that if \(u' = c\) (e.g., a light beam in \(S'\)), then \(u = c\) in \(S\)—the speed of light is invariant.

  • A spaceship moves at \(0.9c\) relative to Earth. It fires a probe at \(0.9c\) relative to the ship (in the same direction). What is the probe’s speed relative to Earth? Compare with the naive \(1.8c\).

  • When is the relativistic formula approximately equal to \(u = u' + v\)?

Lecture Notes#

Overview#

  • Velocities do not add simply in relativity. The Galilean rule \(u = u' + v\) fails when speeds approach \(c\).

  • The relativistic velocity transformation gives the correct relation between a particle’s velocity as measured in two inertial frames in relative motion.

  • A key consequence: if a particle (or light) has speed \(c\) in one frame, it has speed \(c\) in all frames—the speed of light is invariant.

  • The formula follows directly from the Lorentz transformation.

Relativistic velocity addition#

Consider frames \(S\) and \(S'\) with \(S'\) moving at speed \(v\) in the \(+x\) direction relative to \(S\). A particle moves with constant velocity parallel to the \(x\)-axis. In \(S'\) its speed is \(u' = \Delta x'/\Delta t'\); in \(S\) its speed is \(u = \Delta x/\Delta t\).

From the Lorentz transformation (difference form):

\[ \Delta x = \gamma(\Delta x' + v\,\Delta t'), \quad \Delta t = \gamma\left(\Delta t' + \frac{v\,\Delta x'}{c^2}\right) \]

Dividing \(\Delta x\) by \(\Delta t\):

(331)#\[ u = \frac{\Delta x}{\Delta t} = \frac{\Delta x' + v\,\Delta t'}{\Delta t' + v\,\Delta x'/c^2} = \frac{\Delta x'/\Delta t' + v}{1 + v(\Delta x'/\Delta t')/c^2} = \frac{u' + v}{1 + u'v/c^2} \]

Inverse (from \(S\) to \(S'\)): interchange \(u\) and \(u'\), replace \(v\) by \(-v\):

(332)#\[ u' = \frac{u - v}{1 - uv/c^2} \]

Key properties#

Invariance of \(c\). If \(u' = c\) (e.g., a light signal in \(S'\)):

(333)#\[ u = \frac{c + v}{1 + cv/c^2} = \frac{c + v}{1 + v/c} = \frac{c(c + v)}{c + v} = c \]

The speed of light is the same in all inertial frames—consistent with the second postulate.

Galilean limit. For \(u' \ll c\) and \(v \ll c\), the denominator \(1 + u'v/c^2 \approx 1\), so \(u \approx u' + v\). The relativistic formula reduces to the familiar Galilean addition.

No speed exceeds \(c\). For any \(u' < c\) and \(v < c\), one can show \(u < c\). Velocities “add” in such a way that \(c\) is never exceeded.

Sign convention

\(v\) is the velocity of \(S'\) relative to \(S\) (positive when \(S'\) moves in \(+x\)). For motion in the negative \(x\) direction, use negative values of \(u'\), \(u\), or \(v\) as appropriate.

Example: ship and probe

Ship moves at \(v = 0.9c\) relative to Earth. Probe fired at \(u' = 0.9c\) relative to ship (same direction). Speed relative to Earth: \(u = (0.9c + 0.9c)/(1 + 0.81) = 1.8c/1.81 \approx 0.994c\). Galilean addition would give \(1.8c\) (impossible); relativity keeps \(u < c\).

Summary#

  • Velocity transformation: \(u = (u' + v)/(1 + u'v/c^2)\); inverse \(u' = (u - v)/(1 - uv/c^2)\).

  • Derived from the Lorentz transformation; valid for motion along the common \(x\)-axis.

  • \(c\) is invariant: if \(u' = c\) then \(u = c\). No velocity exceeds \(c\).

  • Galilean limit: \(u \approx u' + v\) when \(u', v \ll c\).