Lorentz invariance

Examples of Lorentz invariance in the following topics:

• Transformation of Radiative Transfer

  • where we have used $v'=p'_1/p'_t$ and the inverse Lorentz transformation, so we find
  • Therefore, $d^3 {\bf x} d^3 {\bf p}$ is Lorentz invariant and
  • Because the left-hand side is a bunch of Lorentz invariants we find that $\frac{I_\nu}{\nu^3} = $Lorentz invariant.
  • $\displaystyle \frac{I_\nu}{\nu^3} = \frac{B_\nu(T)}{\nu^3} = \frac{2 h}{c^2} \frac{1}{\exp ( h \nu / k T) - 1} = \mbox{Lorentz invariant}.$
  • $\displaystyle \tau = \frac{l \alpha_\nu}{\sin \theta} = \frac{l}{\nu \sin \theta} \nu \alpha_\nu = \frac{l c}{k_y} \nu \alpha_\nu = \mbox{Lorentz invariant}$

• Gallilean-Newtonian Relativity

  • The puzzle lied in the fact that the Galilean invariance didn't work in Maxwell's equations.
  • Albert Einstein's central insight in formulating special relativity was that, for full consistency with electromagnetism, mechanics must also be revised, such that Lorentz invariance (introduced later) replaces Galilean invariance.
  • At the low relative velocities characteristic of everyday life, Lorentz invariance and Galilean invariance are nearly the same, but for relative velocities close to that of light they are very different.
  • Newtonian mechanics is invariant under a Galilean transformation between observation frames (shown).
  • This is called Galilean invariance.

• Lorenz Gauge

  • This is the Lorenz gauge (which happens to be Lorentz invariant).

• Inverse Compton Spectra - Single Scattering

  • What does the intensity look like in the rest frame of the electrons.Remember that $I_\nu/\nu^3$ was a Lorentz invariant so we have
  • Now we can transform into the lab frame, using the fact that $j_\nu/\nu^2$ is a Lorentz invariant.

• 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.
  • Conservation laws in physics, such as the law of conservation of momentum, must be invariant.
  • Newton's second law [with mass fixed in the expression for momentum (p=m*v)], is not invariant under a Lorentz transformation.
  • However, it can be made invariant by making the inertial mass m of an object a function of velocity:

• Inverse Compton Power - Single Scattering

  • The number of photons in a box over the energy range is a Lorentz invariant
  • $\displaystyle \frac{v dE}{E} = \frac{v' dE'}{E'} = \text{ Lorentz Invariant} $
  • The first equality holds because the emitted power is a Lorentz invariant.Why is this true?

• Tensors

  • Let's use the Lorentz matrix to transform to a new frame
  • This just means a Lorentz invariant number at each point and time.
  • Let's look first at the Lorentz force equation,

• Four-Dimensional Space-Time

  • Due to the invariance of the speed of light both observers will agree on:
  • The set of all coordinate transformations that leave the above quantity invariant are known as Lorentz Transformations.
  • It follows that the coordinate systems of all physical observers are related to each other by Lorentz Transformations.
  • (The set of all Lorentz transformations form what mathematicians call a group, and the study of group theory has revolutionized physics).

• The Speed of Light

  • In the theory of relativity, c interrelates space and time in the Lorentz transformation; it also appears in the famous equation of mass-energy equivalence: E = mc2.
  • This invariance of the speed of light was postulated by Einstein in 1905 after being motivated by Maxwell's theory of electromagnetism and the lack of evidence for "luminiferous aether"; it has since been consistently confirmed by many experiments.
  • Discuss the invariance of the speed of light and identify the value of that speed in vacuum

• Lorentz Transformations