Lorentz Transformations

• Lorentz Transformations relate space and time measurements of an event made by different observers.
• A transformation is required to relate measurements made in one inertial frame to measurements made in another.

Motivation

• The classical Galilean transformation is defined as $x' = x - vt$ and $t' = t$.
• Galilean transformation is not consistent with the observed constancy of the speed of light.
• The Lorentz transformation is needed as a generalization of the Galilean transformation that works for any relative speed $v < c$.

Setup

• Two inertial reference frames are defined as S: $(x, y, z, t)$ and S': $(x', y', z', t')$.
• S' moves with velocity $v$ in the $+x$ direction relative to S.

The Lorentz Transformation

• The Lorentz transformation equations are:
  • $x^{\prime} = \gamma(x - vt)$
  • $y^{\prime} = y$
  • $z^{\prime} = z$
  • $t^{\prime} = \gamma(t - \frac{v}{c^{2}} x)$
• Here, $\gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$

Remarks

• The Lorentz transformation is linear.
• The Lorentz transformation preserves the interval $\Delta s^{2}=c^{2} \Delta t^{2}-\Delta x^{2}-\Delta y^{2}-\Delta z^{2}$.
• For $v \ll c$, $\gamma \approx 1$ and the Lorentz transformation reduces to the Galilean transformation.

Inverse Lorentz Transformation

• The inverse Lorentz transformation equations are:
  • $x = \gamma(x^{\prime} + vt^{\prime})$
  • $y = y^{\prime}$
  • $z = z^{\prime}$
  • $t = \gamma(t^{\prime} + \frac{v}{c^{2}} x^{\prime})$
• S moves with velocity $v$ in the $-x'$ direction relative to S', obtained by flipping the sign of $v$.

Velocity Transformation

• If an object has velocity $\vec{u} = (u_x, u_y, u_z)$ in S and $\vec{u'} = (u'_x, u'_y, u'_z)$ in S', the velocities are related by:
  • $u_{x}^{\prime} = \frac{u_{x} - v}{1 - \frac{v u_{x}}{c^{2}}}$
  • $u_{y}^{\prime} = \frac{u_{y}}{\gamma(1 - \frac{v u_{x}}{c^{2}})}$
  • $u_{z}^{\prime} = \frac{u_{z}}{\gamma(1 - \frac{v u_{x}}{c^{2}})}$

Remarks

• Even if $u_x < c$ and $v < c$, it is not necessarily true that $u'_x < c$.
• If $u_x = c$, then $u'_x = c$, indicating that the speed of light is invariant across all frames.
• The inverse transformation is obtained by flipping the sign of $v$.

Example

• A spaceship moves at $0.6c$ relative to Earth.
• An astronaut shines a laser pointer in the spaceship’s direction of motion.

Solution

• In the spaceship frame, the laser beam has speed $u' = c$.
• Using the inverse velocity transformation: $u=\frac{u^{\prime}+v}{1+\frac{v u^{\prime}}{c^{2}}}=\frac{c+0.6 c}{1+\frac{(0.6 c)(c)}{c^{2}}}=\frac{1.6 c}{1.6}=c$
• The observer on Earth measures the laser beam to travel at the speed of light.