# Relativity

Invariance of physical laws, simultaneity:  All of the fundamental laws of physics have the same form in all inertial frames of reference.  The speed of light in vacuum is the same in all inertial frames and is independent of the motion of the source.  Simultaneity is not an absolute concept; events that are simultaneous in one frame are not necessarily simultaneous in a second frame moving relative to the first.

Time dilation:  If two events occur at the same space point in a particular frame of reference, the time interval $\Delta t_0$ between the events as measured in that frame is called a proper time interval.  If this frame moves with constant velocity $u$ relative to a second frame, the time interval $\Delta t$ between the events as observed in the second frame is longer than $\Delta t_0$: $\Delta t=\frac{\Delta t_0}{\sqrt{1-\frac{u^2}{c^2}}}=\gamma\Delta t_0$, $\gamma=\frac1{\sqrt{1-u^2/c^2}}$.

Length contraction:  If two points are at rest in a particular frame of reference, the distance $l_0$ between the points as measured in that frame is called a proper length. If this frame moves with constant velocity $u$ relative to a second frame and the distances are measured parallel to the motion, the distance $l$ between the points as measured in the second frame is shorter than $l_0$.  $l=l_0\sqrt{1-\frac{u^2}{c^2}}=\frac{l_0}{\gamma}$.

The Lorentz transformation:  The Lorentz coordinate transformations relate the coordinates and time of an event in an inertial frame $S$ to the coordinates and time of the same event as observed in a second inertial frame $S'$ moving at velocity $u$ relative to the first.  For one-dimensional motion, a particle’s velocities $v_x$ in $S$ and $v_x'$ in $S'$ are related by the Lorentz velocity transformation.  $x'=\frac{x-ut}{\sqrt{1-u^2/c^2}}=\gamma(x-ut)$, $y'=y$, $z'=z$, $t'=\frac{t-ux/c^2}{\sqrt{1-u^2/c^2}}=\gamma(t-ux/c^2)$, $v_x'=\frac{v_x-u}{1-uv_x/c^2}$, $v_x=\frac{v_x'+u}{1+uv_x'/c^2}$.

The Doppler effect for electromagnetic waves:  The Doppler effect is the frequency shift in light from a source due to the relative motion of source and observer.  For a source moving toward the observer with speed $u$, the received frequency $f$ in terms of the emitted frequency $f_0$ is $f=\sqrt{\frac{c+u}{c-u}}f_0$.

Relativistic momentum and energy:  For a particle of rest mass $m$ moving with velocity $\vec{v}$, the relativistic momentum is $\vec{p}=\frac{m\vec{v}}{\sqrt{1-v^2/c^2}}=\gamma m\vec{v}$, the relativistic kinetic energy is $K=\frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2=(\gamma-1)mc^2$.  The total energy $E$ is the sum of the kinetic energy and the rest energy $mc^2$: $E=K+mc^2=\frac{mc^2}{\sqrt{1-v^2/c^2}}=\gamma mc^2$.  Also, $E^2=(mc^2)^2+(pc)^2$.