Special relativity

• energy–momentum relation (or relativistic dispersion relation)
  • $E^2=(pc{)}^2+{\left(m{c}^2\right)}^2$
    • $E$ is the total relativistic energy (or relativistic energy (אנרגייה יחסותית))
      • $E=\gamma m{c}^2$
  • special cases:
    • particle at rest ($p=0$): $E=m{c}^2$
      • It is called rest energy $(E_0$) (or rest mass energy)
    • Massless particles ($m=0$): $E=pc$
      • see also: Planck relation, de Broglie waves
• $m$ is the rest mass (or just mass)
• $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ is the Lorentz factor
• $p=\gamma mv$ is the momentum
• $K=E-E_0=(\gamma -1)m{c}^2$ is the relativistic kinetic energy

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• time dilation $\Delta t=\gamma \Delta t_0$

  • $\Delta t_0$ is the proper time interval
  • $\Delta t$ is the dilated time interval

• length contraction $L=\frac{L_0}{\gamma }$

  • $L_0$ is the proper length (or rest length)
  • $L$ is the contracted length

• atmospheric muons

  • $\Delta t_0$ is the lifetime of the muon in its own rest frame
  • $\Delta t$ is the lifetime of the muon measured by the Earth observer
  • $L_0$ is the height of the atmosphere measured by the Earth observer
  • $L$ is the height of the atmosphere measured by the moving muon

🪼 https://lamda-openu.psik.io/pdfViewer/6088

• relativistic velocity addition (one dimension)
  • $u^{\prime }=\frac{u+v}{1+\frac{uv}{c^2}}$
  • $u$ is the object velocity in one inertial frame
  • $v$ frame velocity relative to another frame along the same line
  • $u^{\prime }$ the object’s velocity in the second frame