37-3 The Lorentz Transformation

37-3 The Lorentz Transformation#

Prompts

  • Write the Galilean transformation equations. Why do they fail at high speeds? What replaces them?

  • Write the Lorentz transformation equations for \((x,t) \to (x',t')\) when frame \(S'\) moves at speed \(v\) along \(+x\). How is time “entangled” with space?

  • From the Lorentz equations, show that two events simultaneous in \(S'\) (\(\Delta t' = 0\)) but spatially separated (\(\Delta x' \neq 0\)) are not simultaneous in \(S\).

  • Derive time dilation \(\Delta t = \gamma\,\Delta t_0\) from the Lorentz transformation. (Hint: two events at same place in \(S'\).)

  • Derive length contraction \(L = L_0/\gamma\) from the Lorentz transformation. (Hint: rod at rest in \(S'\); measure simultaneously in \(S\).)

Lecture Notes#

Overview#

  • The Lorentz transformation equations relate the spacetime coordinates \((x,y,z,t)\) of an event as measured in one inertial frame to the coordinates \((x',y',z',t')\) in another frame moving at speed \(v\) along the common \(x\)-axis.

  • They replace the Galilean transformation (\(x' = x - vt\), \(t' = t\)), which is only approximate for \(v \ll c\).

  • Time and space are entangled: \(x'\) depends on \(t\), and \(t'\) depends on \(x\). As \(c \to \infty\), the Lorentz equations reduce to the Galilean ones.

  • Time dilation and length contraction follow directly from the Lorentz transformation.

Setup and Galilean transformation#

Consider two inertial frames \(S\) and \(S'\) with \(S'\) moving at speed \(v\) in the positive \(x\)-direction relative to \(S\). The axes are parallel; origins coincide at \(t = t' = 0\). An event has coordinates \((x,y,z,t)\) in \(S\) and \((x',y',z',t')\) in \(S'\).

Galilean transformation (pre-Einstein, valid only for \(v \ll c\)):

(323)#\[ x' = x - vt, \quad y' = y, \quad z' = z, \quad t' = t \]

It assumes time is universal (\(t' = t\)) and space is separate. For \(v\) approaching \(c\), these equations are wrong.

Lorentz transformation equations#

The Lorentz transformation equations are correct for any physically possible speed:

(324)#\[ x' = \gamma(x - vt), \quad y' = y, \quad z' = z, \quad t' = \gamma\left(t - \frac{vx}{c^2}\right) \]

where \(\gamma = 1/\sqrt{1 - v^2/c^2}\). Note that \(x'\) involves \(t\), and \(t'\) involves \(x\)—space and time are entangled.

Inverse (from \(S'\) to \(S\)): interchange primed and unprimed quantities and replace \(v\) by \(-v\):

(325)#\[ x = \gamma(x' + vt'), \quad t = \gamma\left(t' + \frac{vx'}{c^2}\right) \]

Limit \(c \to \infty\): \(\gamma \to 1\), and the Lorentz equations reduce to the Galilean ones.

Difference form for pairs of events#

For two events with separations \(\Delta x = x_2 - x_1\), \(\Delta t = t_2 - t_1\) in \(S\), and \(\Delta x'\), \(\Delta t'\) in \(S'\):

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

Use the same frame consistently for both events when substituting values.

Consequences: simultaneity, time dilation, length contraction#

Simultaneity. If two events are simultaneous in \(S'\) (\(\Delta t' = 0\)) but spatially separated (\(\Delta x' \neq 0\)), then from Eq. (327):

(328)#\[ \Delta t = \gamma\left(0 + \frac{v\,\Delta x'}{c^2}\right) = \gamma\,\frac{v\,\Delta x'}{c^2} \neq 0 \]

They are not simultaneous in \(S\). Spatial separation in one frame implies temporal separation in the other.

Time dilation. If two events occur at the same place in \(S'\) (\(\Delta x' = 0\)) but at different times (\(\Delta t' = \Delta t_0\), proper time):

(329)#\[ \Delta t = \gamma(\Delta t' + 0) = \gamma\,\Delta t_0 \]

Length contraction. A rod at rest in \(S'\) has proper length \(\Delta x' = L_0\). In \(S\), we measure its length by marking both ends simultaneously (\(\Delta t = 0\)). From Eq. (326): \(\Delta x' = \gamma(\Delta x - v\cdot 0) = \gamma\,\Delta x\), so

(330)#\[ L = \Delta x = \frac{\Delta x'}{\gamma} = \frac{L_0}{\gamma} \]

Sign of \(v\)

\(v\) is the velocity of \(S'\) relative to \(S\) (positive when \(S'\) moves in \(+x\)). If \(S\) moves relative to \(S'\), use \(-v\) in the inverse transformation.

Summary#

  • Lorentz transformation: \(x' = \gamma(x - vt)\), \(t' = \gamma(t - vx/c^2)\); \(y\) and \(z\) unchanged.

  • Entanglement: \(x'\) depends on \(t\); \(t'\) depends on \(x\). Galilean form recovered as \(c \to \infty\).

  • Difference form for event pairs: use Eq. (326) or Eq. (327); keep both events in the same frame.

  • Derived: relativity of simultaneity, time dilation \(\Delta t = \gamma\,\Delta t_0\), length contraction \(L = L_0/\gamma\).