Lorentz factor

Examples of Lorentz factor in the following topics:

• Time Dilation

  • is the Lorentz factor.
  • Note that for speeds below 1/10 the speed of light, Lorentz factor is approximately 1 .
  • Lorentz factor as a function of speed (in natural units where c = 1).

• Problems

  • What is the synchrotron emission from a single electron passing through a magnetic field in terms of the energy density of the magnetic field and the Lorentz factor of the electron?
  • What is the inverse Compton emission from a single electron passing through a gas of photons field in terms of the energy density of the photons and the Lorentz factor of the electron?

• Back to Maxwell's Equations

  • Lorentz proposed that to understand the null result of the experiment objects moving through the aether contract by γ−1 = $\sqrt{1-v^2/c^2}$where γ is the Lorentz factor.
  • Einstein's insight was that if the speed of light was the same for everyone moving uniformly, one would get the apparent "Lorentz" contraction without needing the aether through which light propagates or for the aether to contract objects.

• The Speed of Light

  • The Lorentz factor (γ) is the factor by which length shortens and time dilates as a function of velocity (v):

• Relativistic Energy and Mass

  • For a slower than light particle, a particle with a nonzero rest mass, the formula becomes where is the rest mass and is the Lorentz factor.
  • The Lorentz factor is equal to: $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$, where v is the relative velocity between inertial reference frames and c is the speed of light.

• Inverse Compton Scattering

  • Let's imagine that the electron is traveling along the $x$-axis with Lorentz factor $\gamma$.

• B.6 Chapter 6

  • If the Lorentz factor at , integrating this yields

• Relativistic Momentum

  • Relativistic momentum is given as $\gamma m_{0}v$ where $m_{0}$ is the object's invariant mass and $\gamma$ is Lorentz transformation.
  • As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation.
  • Newton's second law [with mass fixed in the expression for momentum (p=m*v)], is not invariant under a Lorentz transformation.

• Problems

  • One could do a Lorentz transformation but it is easier to use ${U}_{r}^\mu p_{\mu}$ to determine the energy of the particle in the primed frame.
  • Again one could do a Lorentz transformation but it is easier to use ${U'}_{r}^\mu {p'}_{\mu}$ to determine the energy of the particle in the unprimed frame.
  • $\beta \approx 1 - \frac{1}{2\gamma^2}$By what factor does the energy of the particle increase each time it goes back and forth.
  • Write out the Lorentz transformation matrix for a boost in the x−direction to a velocity $\beta _x$.
  • Write out the Lorentz transformation matrix for a boost in the y−direction to a velocity$\beta_y$.

• Lorentz Transformations