Theory of relativity
Lorentz Transformation¶
Let \(S'\) be a frame of a moving observer with velocity \(v\) in the x-direction with respect to the frame \(S\). Then the space-time coordinates in \(S\) and \(S'\) are related by the Lorentz Transformation. For our convenience, we express the velocity as \(v=\beta c\) and the Lorentz factor is \(\gamma = (1-\beta^2)^{-1/2}\).
\[
ct'=\gamma(ct-\beta x),x'=\gamma(x-\beta ct),y'=y,z'=z
\]
\[
p_x'=\gamma(p_x-mv),m'=\gamma(m-p_xv/c^2)
\]
Velocity Addition¶
\[
u_x=\dfrac{u_x'+v}{1+u_x'v/c^2},u_y=\dfrac{u_y'}{\gamma(1+u_x'v/c^2)}
\]
Doppler effect¶
\[
\nu'=\nu_0\sqrt{(1-\beta)/(1+\beta)}
\]
4-vector¶
\[
s^2=(ct)^2-x^2-y^2-z^2\\
m_0^2c^2=m^2c^2-p_x^2-p_y^2-p_z^2
\]