The Geometry of Special Relativity

Errata

(Last update: 2/28/19)

\[ \tan\theta' = \frac{u_y'}{u_x'} = \frac{u_y}{u_x-v} = \frac{\tan\theta}{1-\frac{v}{u\cos\theta}} = \frac{u\sin\theta}{u\cos\theta-v} \]

\[\Partial{F^{\mu\nu}}{x^\nu} = \frac{1}{c}\Partial{F^{\mu0}}{t} + \Partial{F^{\mu1}}{x} + \Partial{F^{\mu2}}{y} + \Partial{F^{\mu3}}{z}\]

\[ \frac12 F_{\mu\nu} F^{\mu\nu} = -\frac{1}{c^2}\,{|\EE|}^2 + |\BB|^2 = - \frac12 G_{\mu\nu} G^{\mu\nu} \]

\[d_H(U,V) = \rho\,\cosh^{-1}\left(-\frac{\uu\cdot\vv}{\rho^2}\right)\]

\begin{align*} d_H(U,V) &= \rho\,\cosh^{-1}(-\sinh\alpha\sinh\beta+\cosh\alpha\cosh\beta) \\ &= \rho\,\cosh^{-1}(\cosh(\beta-\alpha)) \\ &= \rho\,|\beta-\alpha| \end{align*}

\[ X + iY = -2\frac{i(x+iy)+\rho}{(x+iy)+\rho\,i} \]

\[ ds^2 = \frac{4\,\rho^4\left(dx^2+dy^2\right)}{(\rho^2-x^2-y^2)^2} = \rho^2\>\frac{dX^2+dY^2}{Y^2} \]