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Lorentz Transformations


Albert Einstein didn't create the coordinate transformations needed for special relativity. He didn't have to, because the Lorentz transformations that he needed already existed. Einstein was a master at taking previous work and adapting it to new situations, and he did so with the Lorentz transformations just as he had used Planck's 1900 solution to the ultraviolet catastrophe in black body radiation to craft his solution to the photoelectric effect, and thus develop the photon theory of light.

Origins of Lorentz Transformations

The transformations were actually first published by Joseph Larmor in 1897. A slightly different version had been published a decade earlier by Woldemar Voigt, but his version had a square in the time dilation equation. Still, both versions of the equation were shown to be invariant under Maxwell's equation.

The mathematician and physicist Hendrik Antoon Lorentz proposed the idea of a "local time" to explain relative simultaneity in 1895, though, and began working independently on similar transformations to explain the null result in the Michelson-Morley experiment. He published his coordinate transformations in 1899, apparently still unaware of Larmor's publication, and added time dilation in 1904.

In 1905, Henri Poincare modified the algebraic formulations and attributed them to Lorentz with the name "Lorentz transformations," thus changing Larmor's chance at immortality in this regard. Poincare's formulation of the transformation was, essentially, identical to that which Einstein would use.

The transformations apply to a four-dimensional coordinate system, with three spatial coordinates (x, y, & z) and one time coordinate (t). The new coordinates are denoted with an apostrophe, pronounced "prime," such that x' is pronounced x-prime. In the example below, the velocity is in the xx' direction, with velocity u:

x' = ( x - ut ) / sqrt ( 1 - u2 / c2 )

y' = y

z' = z

t' = { t - ( u / c2 ) x } / sqrt ( 1 - u2 / c2 )

The transformations are provided primarily for demonstration purposes. Specific applications of them will be dealt with separately. The term 1/sqrt (1 - u2/c2) so frequently appears in relativity that it is denoted with the Greek symbol gamma in some representations.

It should be noted that in the cases when u << c, the denominator collapses to essentially the sqrt(1), which is just 1. Gamma just becomes 1 in these cases. Similarly, the u/c2 term also becomes very small. Therefore, both dilation of space and time are non-existent to any significant level at speeds much slower than the speed of light in a vacuum.

Consequences of the Transformations

Special relativity yields several consequences from applying Lorentz transformations at high velocities (near the speed of light). Among them are:
  • Time dilation (including the popular "Twin Paradox")
  • Length contraction
  • Velocity transformation
  • Relativistic velocity addition
  • Relativistic doppler effect
  • Simultaneity & clock synchronization
  • Relativistic momentum
  • Relativistic kinetic energy
  • Relativistic mass
  • Relativistic total energy

Lorentz & Einstein Controversy

Some people point out that most of the actual work for the special relativity had already been done by the time Einstein presented it. The concepts of dilation and simultaneity for moving bodies were already in place and the mathematics had already been developed by Lorentz & Poincare. Some go so far as to call Einstein a plagiarist.

There is some validity to these charges. Certainly, the "revolution" of Einstein was built on the shoulders of a lot of other work, and Einstein got far more credit for his role than those who did the grunt work.

At the same time, it must be considered that Einstein took these basic concepts and mounted them on a theoretical framework which made them not merely mathematical tricks to save a dying theory (i.e. the ether), but rather fundamental aspects of nature in their own right. It is unclear that Larmor, Lorentz, or Poincare intended so bold a move, and history has rewarded Einstein for this insight & boldness.

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