Modern Physics: Special Relativity Class Notes

Galilean Transformation

The classical transformation equations used when coordinates $(x, y, z, t)$ are available in frame $S$ to estimate $(x', y', z', t')$ in frame $S'$, given by $x' = x - v t$, $y' = y$, $z' = z$, and $t' = t$.

Inverse Galilean Transformation (IGT)

The transformation equations used when coordinates $(x', y', z', t')$ are available in frame $S'$ to estimate $(x, y, z, t)$ in frame $S$, given by $x = x' + v t'$, $y = y'$, $z = z'$, and $t = t'$.

Lorentz Transformation (L.T.)

The relativistic coordinate transformation equations used to estimate $(x', y', z', t')$ from $(x, y, z, t)$, given by $x' = \gamma(x - v t)$, $y' = y$, $z' = z$, and $t' = \gamma\left(t - \frac{v x}{c^2}\right)$.

Lorentz Factor ($\gamma$)

The relativistic factor that scales time dilation, length contraction, and Lorentz transformations, defined as $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$.

Inverse Lorentz Transformation (ILT)

The relativistic transformation equations used to estimate $(x, y, z, t)$ from $(x', y', z', t')$, given by $x = \gamma(x' + v t')$, $y = y'$, $z = z'$, and $t = \gamma\left(t' + \frac{v x'}{c^2}\right)$.

Proper Length ($L_0$)

The length of an object measured in its own rest frame, represented by the difference between coordinates $L_0 = x_2' - x_1'$.

Length Contraction

The relativistic shortening of an object along its direction of motion as observed from a moving frame, given by the relation $L_0 = \gamma L$ or $L = L_0 \sqrt{1 - \frac{v^2}{c^2}}$.

Proper Time ($\tau_0$)

The time interval measured by a clock at rest relative to the events, denoted as $\tau_0 = t_2' - t_1'$.

Time Dilation

The relativistic effect where the elapsed time interval $\tau$ between events is measured to be longer by an observer in relative motion compared to proper time $\tau_0$, expressed as $\tau = \frac{\tau_0}{\sqrt{1 - \frac{v^2}{c^2}}}$.

Principle of Relativity of Simultaneity

The relativistic principle showing that events occurring at different spatial locations that are simultaneous in one reference frame are not simultaneous in another, described by $t_2' - t_1' = \gamma(t_2 - t_1) - \frac{\gamma v}{c^2}(x_2 - x_1)$.

Relativistic Velocity Addition ($x$-component)

The transformation relation for the component of velocity parallel to the relative motion between frames, given by $u_x' = \frac{u_x - v}{1 - \frac{u_x v}{c^2}}$.

Inverse Relativistic Velocity Addition ($x$-component)

The inverse relation used to calculate the parallel velocity component $u_x$ in frame $S$ from $u_x'$ in moving frame $S'$, given by $u_x = \frac{u_x' + v}{1 + \frac{u_x' v}{c^2}}$.

Perpendicular Relativistic Velocity Transformation ($y$ and $z$ components)

The velocity transformations perpendicular to the direction of motion, given by $u_y' = \frac{u_y}{\gamma\left(1 - \frac{u_x v}{c^2}\right)}$ and $u_z' = \frac{u_z}{\gamma\left(1 - \frac{u_x v}{c^2}\right)}$.