The Lorentz transformations are a group of mathematical functions that are central to the special theory of relativity. They express the constancy of the speed of light in all frames of reference and the dependence of space and time measurements on the relative motion of the observer and the system observed.
The transformations describe how the coordinates of an event in one frame of reference compare with the coordinates as measured in another reference frame that is in motion at constant velocity relative to the first frame. Such transformations can be used to test the invariance of physical laws, showing whether the laws apply universally, regardless of the state of motion of the observer. In the Galilean transformations, which predate the Lorentz transformations, it is implicitly assumed that measurements of lengths and time intervals are not affected by the relative motion of the observers, and that the speed of light may have different values depending on the observer's motion. In the equations below, the primed coordinates are those of one system, and the unprimed are in the other system. For simplicity, we assume that the origins of the coordinates overlap, and the motion is only along the x axis:
x = x-vt
y = y
z = z
t = t
Toward the end of the nineteenth century, electromagnetic theory began to suggest that some modification of the Galilean transformations was necessary. Following the Michelson-Morley experiment of 1887, there were many attempts to explain why it failed to detect changes in the velocity of light as a result of Earth's motion through the ether, as was expected. George Francis FitzGerald (1851-1901) was the first to suggest that motion through the ether caused a compression of the measuring apparatus that negated any interference effects in the light beams used in the experiment. A contraction formula was later advanced independently by Hendrik Lorentz. The length change, called Lorentz-FitzGerald length contraction, occurs only along the direction of motion, and causes a dramatic shortening of objects at speeds near the speed of light. Sir Joseph Larmor (1857-1942) and Lorentz independently developed what became known as the Lorentz transformations as part of their efforts to replace the Galilean transformation with equations consistent with electromagnetic theory. It was shown that the Lorentz-FitzGerald contraction could be derived from the Lorentz transformation equations. Using the same simplifications employed with the Galilean transformations, the Lorentz transformations are given by:
x = (x-vt)
y = y
z = z
t = (t-vx/c2 )
m = m
= (1-v2 /c2 )-½
The factor introduces a velocity dependence that becomes significant near c, the speed of light. Unlike the Galilean transformations, the time coordinate is different in the two reference frames.
According to the Lorentz transformations, as a body's relative motion approaches the speed of light, its length in the direction of motion will decrease, its mass will increase, and time dilation will occur. The formulas prohibit travel at the speed of light. Simultaneity is redefined by the transformations such that two events that are simultaneous in one frame of reference may be observed to occur at different times in another reference frame. The twin paradox, in which one of a pair of twins travels at high speed in a spacecraft and returns to Earth younger than the twin remaining behind, is a consequence of the Lorentz transformations.
In 1905 Albert Einstein derived the Lorentz transformations in an entirely new way with his special theory of relativity. Previously, the results of the Lorentz transformations were seen as effects of the ether on bodies, and were essentially an arbitrary device used to preserve the role of the ether in classical mechanics by explaining the anomalous results of the Michelson-Morley experiment. Einstein dismissed ether theory, and instead made two simple assumptions about nature and determined the logical consequences. He postulated that the speed of light should be constant for all observers, and that physical laws are the same in all inertial reference frames, that is, reference frames that are unaccelerated. From these principles he deduced the Lorentz transformations, along with a new form of the law of addition of velocities that accommodated the constancy of the speed of light. He interpreted the transformations to mean that there is no ether or standard of absolute rest with which to define the "real" motion of bodies; only relative motion exists. Measurements of length, time, and mass are not absolute, but depend on the relative motion of the observer and the system being observed. For example, if two observers have identical measuring rods, masses, and synchronized clocks when at rest with respect to each other, when moving apart at a given speed they will each see the other's masses as greater, clocks as running slower, and measuring rods as shortened in the direction of travel. What distinguished Einstein's use of the Lorentz transformations from the work of contemporaries such as Lorentz and Henri Poincaré was Einstein's abandonment of absolute space, absolute time, and the ether, as well as the fact that he built his theory using only two basic principles and a system of light signals between observers.
The validity of the Lorentz transformations has been successfully tested by a variety of means. Unstable subatomic particles traveling at high speeds in accelerators or as cosmic rays are known to have much longer life spans than the same particles moving at low speeds, indicating that we observe time running more slowly for the particles when they are moving at high relative speeds. Time dilation has also been tested by comparing a ground-based atomic clock with one that has been travelling at high speed in an aircraft, and the predicted effect was confirmed to a high degree of accuracy.
This is the complete article, containing 935 words
(approx. 3 pages at 300 words per page).
