Space-Time
In 1905 Albert Einstein published his seminal paper "On the Electrodynamics of Moving Bodies." In it he put forth his "Principle of Relativity," which soon gave the work the name by which it has been known since. All inertial (i.e., unaccelerated) observers, he said, regardless of their state of relative motion, must find the identical mathematical form for all laws of physics. (This is an example of what is today known as an "invariance principle"; in this case the laws are invariant--unchanging in form--when the space and time coordinates used by one observer are transformed into those used by another.) The laws known at that time were those of Isaac Newton's mechanics and James Clerk Maxwell's electrodynamics. Newton's laws of motion obeyed Einstein's principle, but Maxwell's equations did not.
The problem, Einstein realized, lay with the set of transformation equations relating position and time measurements made by observers in uniform relative motion. Galileo had been the first to consider these, and the relationships between the coordinates used by two such observers were known as the Galilean transformations. Newton's laws, but not those of Maxwell, were "Galilean invariant." A new set of transformations was needed, under which all laws of physics remained unchanged. The Dutch physicist Hendrik Antoon Lorentz had discovered the proper set of transformations in 1904 (although for a different reason). Maxwell's equations are Lorentz invariant, while Newton's laws must be modified somewhat to bring them in line. These modifications, which change "classical" mechanics into "relativistic" mechanics, lead to one of the most well known predictions of relativity theory, the equivalence of mass and energy. The mathematical statement of Einstein's Principle of Relativity is simply this: that the laws of physics are Lorentz invariant.
The Lorentz transformations do something quite strange; they mix the space and time coordinates in a way that the Galilean transformations do not. Consider two observers, Nicholas at rest, and Jonathan in uniform motion with respect to him. Nicholas measures an object's position. It is no surprise that the position coordinate, as measured by Jonathan, depends upon both the position measured by Nicholas, and the time at which Nicholas made that measurement (since Jonathan is moving). But it is quite surprising that the two observers do not agree upon the time at which the position measurement occurred. It had always been assumed that there existed a single "universal time," measured by every observer. Of this unvarying, absolute time Newton had written, "Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external." The root of all of the disturbing and counterintuitive results of relativity (e.g., length contraction and time dilation) is the error of that assumption. Time is observer-dependent.
There is a beautiful symmetry, though, to this mixing of space and time. The structure of the Lorentz transformations is such that the space coordinates and the time behave in an identical manner. Thus, we should not treat the world as three-dimensional, requiring three coordinates for its description, with time a "different thing," a universal parameter, agreed upon by all observers. Rather, the world is four dimensional, time being merely another coordinate needed to localize an event.
In 1908 the mathematician Hermann Minkowski showed that the equations of relativity simplify enormously if the three spatial and one time coordinate needed to define an event (a particular place at a particular time) are treated as the four coordinates of a higher dimensional "space"--a four-dimensional "space-time." Speaking of this new view, Minkowski wrote, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
That "independent reality" is made manifest by the fact that it is quantities in space-time, not quantities in space, such as lengths, or time intervals independently, that are Lorentz invariant; i.e., that have the same values for all inertial observers. According to Einstein, "Pure 'space distance' of two events with respect to [one observer] results in 'time-distance' of the same events with respect to [another]." Minkowski showed that the Lorentz transformations have the mathematical form of rotations in space-time. Just as a three-dimensional vector (for example, the displacement between two points in space) has a length, which is invariant under rotations in space, a "four-vector" (for example, the "interval" between two events in space-time) has a magnitude that is invariant under the Lorentz transformations. The world as we observe it at any particular time, described by physicists in terms of three-dimensional vectors such as momentum, is simply a three-dimensional "slice" of four-dimensional space-time.
As Einstein wrote in 1916, "The non-mathematician is seized by a mysterious shuddering when he hears of "four-dimensional" things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more commonplace statement than that the world in which we live is a four-dimensional space-time.... That we have not been accustomed to regard the world in this sense ... is due to the fact, that in physics, before the advent of the theory of relativity, time played a different and more independent role, as compared with the space coordinates.... The four-dimensional mode of consideration of the 'world' is natural in the theory of relativity, since according to this theory time is robbed of its independence."
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