Lorentz Transformations

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Lorentz Transformations

The Lorentz transformations are a group of equations that yield the geometry of special relativity. They are the basis of Einstein's special theory of relativity formulated in 1905. The Lorentz transformations are a form of the Galilean transformations that can be obtained by substituting an infinitely large value for the velocity of light in the Lorentz transformations. The main objective in formulating the equations that comprise the Lorentz transformations was to preserve the velocity of light and hence the invariance of the space-time interval under all reference frames. Using these transformations it is possible to correlate the space-time coordinates of one moving system with the space-time coordinates of any other system.

The Lorentz transformations are a general group of equations that comprise a four-dimensional transformation, a transformation that involves the three-dimensions of space and time as the fourth dimension, that is satisfied by all four-vectors, a four-element vector. The Galilean transformations are a set of equations that involve a transformation from one frame of reference to another frame of reference that is moving with constant velocity with respect to the first. In Einstein's special theory of relativity the Lorentz transformations replace the Galilean transformations to describe relativistic motion and are formulated as: x' = (x - vt)/((1-(v²/c²))) y' = y, z' = z, t' = (t-(vx)/c²)/((1-(v²/c²))), where v is the velocity, c is the speed of light, x', y', z', t' refer to one frame of reference and x, y, z, t refer to the other frame of reference. It is clear from these equations that it is impossible to travel faster than the speed of light since if v is greater than c then there would be imaginary numbers in the denominators of the x' and t' transformations. These transformations make the speed of light independent of the motion of either of the frames of reference. They combine time dilation and length contraction, two ideas principle to the special theory of relativity, into a single transformation.

The Lorentz transformations, considered the mathematical tool of relativity, gives a method that adjusts to the differences in the classical predictions of motion relating to time and the Michelson-Morley null experiment conducted for the first time in 1881. The Michelson-Morley null experiment was one whose results pointed to the conclusion that the velocity of light must be constant, contrary to the universal belief at the time.

It was this experiment that led to the formulation of Einstein's special theory of relativity several years later. After Michelson's first attempt at the null experiment in 1881 he repeated the experiment in 1887 obtaining the same result that the velocity of light was independent of the velocity of the observer. In 1887 Voigt, while studying the Doppler shift, wrote down a set of transformations and showed that certain equations were invariant under them. These transformations were again written down by Larmor in 1898 in an article Ether and matter. Finally in 1899 Lorentz wrote the same transformations, which with a different scale factor (the relativity factor), were named in his honor by Poincaré in 1905. To formulate the transformations Lorentz made two assumptions: that the speed of light was a constant for any observers regardless of their respective motions, and that the transformation between observers in different inertial frames of reference is linear. Working from these two assumptions Lorentz formulated his transformations that are capable correlating the space and time coordinates of one moving system with the known space and time coordinates of any other system. In 1905 Einstein first published his special theory of relativity in which he adopted the Lorentz transformation equations. Einstein gave the transformations and entirely new interpretation. The transformations describe the increase of mass, the shortening of length, and the time dilation of a body moving at speeds approaching that of the velocity of light. In 1912 Lorentz and Einstein were jointly proposed for a Nobel prize for their work concerning special relativity. It is Lorentz who is considered to have been the first to find the mathematical content of the relativity principle.