Key Equations

Key Equations

Time dilation	$\text{Δ}t=\frac{\text{Δ}\tau }{\sqrt{1-\frac{v^2}{c^2}}}=\gamma \tau$
Lorentz factor	$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
Length contraction	$L=L_0\sqrt{1-\frac{v^2}{c^2}}=\frac{L_0}{\gamma }$
Galilean transformation	$x=x\prime +vt,y=y\prime ,z=z\prime ,t=t\prime$
Lorentz transformation	$t=\frac{t\prime +vx\prime \text{/}{c}^2}{\sqrt{1-v^2\text{/}{c}^2}}$
	$x=\frac{x\prime +vt\prime }{\sqrt{1-v^2\text{/}{c}^2}}$
	$y=y\prime$
	$z=z\prime$
Inverse Lorentz transformation	$t\prime =\frac{t-vx\text{/}{c}^2}{\sqrt{1-v^2\text{/}{c}^2}}$
	$x\prime =\frac{x-vt}{\sqrt{1-v^2\text{/}{c}^2}}$
	$y\prime =y$
	$z\prime =z$
Space-time invariants	${\left(\text{Δ}s\right)}^2={\left(\text{Δ}x\right)}^2+{\left(\text{Δ}y\right)}^2+{\left(\text{Δ}z\right)}^2-c^2{\left(\text{Δ}t\right)}^2$
	${\left(\text{Δ}\tau \right)}^2=-{\left(\text{Δ}s\right)}^2\text{/}{c}^2={\left(\text{Δ}t\right)}^2-\frac{\left[{\left(\text{Δ}x\right)}^2+{\left(\text{Δ}y\right)}^2+{\left(\text{Δ}z\right)}^2\right]}{c^2}$
Relativistic velocity addition	$u_x=\left(\frac{u_x^{\prime }+v}{1+v{u}_x^{\prime }\text{/}{c}^2}\right),u_y=\left(\frac{u_y^{\prime }\text{/}\gamma }{1+v{u}_x^{\prime }\text{/}{c}^2}\right),u_z=\left(\frac{u_z^{\prime }\text{/}\gamma }{1+v{u}_x^{\prime }\text{/}{c}^2}\right)$
Relativistic Doppler effect for wavelength	${\lambda }_{\text{obs}}={\lambda }_s\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$
Relativistic Doppler effect for frequency	$f_{\text{obs}}=f_s\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}$
Relativistic momentum	$\overrightarrow{p}=\gamma m\overrightarrow{u}=\frac{m\overrightarrow{u}}{\sqrt{1-\frac{u^2}{c^2}}}$
Relativistic total energy	$E=\gamma m{c}^2,\text{where}\gamma =\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$
Relativistic kinetic energy	$K_{\text{rel}}=(\gamma -1)m{c}^2,\text{where}\gamma =\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$