Lorentz transformation

Examples of Lorentz transformation in the following topics:

• Lorentz Transformations

• 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.
  • The Galilean transformation gives the coordinates of the moving frame as
  • 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.
  • 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$\beta_y$.
  • Write out the Lorentz transformation matrix for a boost in the x−direction to velocity $\beta_x$ followed by boost to a velocity $\beta_y$ in the y−direction.

• Four-Dimensional Space-Time

  • 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).
  • For the coordinate transformation in , the transformations are:
  • For example, for a space-like separation you can always find a coordinate transformation that reverses the time-ordering of the events (try to prove this for the example in ).

• Phase-Space Density

  • We would like to define some quantities that are integrals over momentum space that transform simply under Lorentz transformations.
  • that transforms as a scalar where $n(x^\alpha)$ is the number density.

• 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.

• 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.
  • $\displaystyle \frac{I_\nu}{\nu^3} = \frac{B_\nu(T)}{\nu^3} = \frac{2 h}{c^2} \frac{1}{\exp ( h \nu / k T) - 1} = \mbox{Lorentz invariant}.$
  • Because the source function $S_\nu$ appears in the equations of radiative transfer as , must have the same transformation properties as , i.e.

• 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.
  • Because transforms as a contravariant vector and doesn't transform, must transform as a covariant vector.
  • Let's look first at the Lorentz force equation,
  • This is called a duality transformation.

• Gallilean-Newtonian Relativity

  • This transformation of variables between two inertial frames is called Galilean transformation .
  • That is, unlike Newtonian mechanics, Maxwell's equations are not invariant under a Galilean transformation.
  • 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).

• Inverse Compton Power - Single Scattering

  • We will first make use on some of the transformation rules that we derived for the phase-space density of photons.
  • 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?