Lorentz invariance

(noun)

First introduced by Lorentz in an effort to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism.

Related Terms

  • absolute space

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

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