Bleach Shinigami

The Fourth Dimension

jordynicki — Tue, 08 Sep 2009 13:56:28 +0000

Einstein did not quite finish the job, however. Contrary to popular belief, he did not draw the conclusion that space and time could be seen as components of a single four-dimensional spacetime fabric. That insight came from Hermann Minkowski (1864-1909), who announced it in a 1908 colloquium with the dramatic words: “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”.

Four-dimensional Minkowski spacetime is often pictured in the form of a two-dimensional lightcone diagram, with the horizontal axes representing “space” (x) and the vertical axis “time” (ct). The walls of the cone are defined by the evolution of a flash of light passing from the past (lower cone) to the future (upper cone) through the present (origin). All of physical reality is contained within this cone; the region outside (“elsewhere”) is inaccessible because one would have to travel faster than light to reach it. The trajectories of all real objects lie along “worldlines” inside the cone (like the one shown here in red). The apparently static nature of this picture, in which history does not seem to “happen” but is rather “already there”, has given writers and philosophers a new way to think about old issues involving determinism and free will.

Einstein initially dismissed Minkowski’s four-dimensional interpretation of his theory as “superfluous learnedness” (Abraham Pais, Subtle is the Lord…, 1982). To his credit, however, he changed his mind quickly. The language of spacetime (known technically as tensor mathematics) proved to be essential in deriving his theory of general relativity.

General Relativity

jordynicki — Tue, 08 Sep 2009 13:48:27 +0000

General relativity is based physically on the equivalence principle, but the theory also has a second, more mathematical foundation. Known as the principle of general covariance, it is the requirement that the law of gravitation be the same for all observers — even accelerating ones — regardless of the coordinates in which it is described. (It is for this reason that Einstein named his new theory “general”, as opposed to “special” relativity — he dropped the earlier restriction to uniformly moving observers.) This proved to be the most difficult challenge that Einstein ever faced. As he later said, to express physical laws without coordinates is like “describing thoughts without words”. Einstein was obliged to master the abstract mathematics of surfaces and their description in terms of tensors, a field pioneered by Carl Friedrich Gauss (1777-1855) and generalized to higher dimensions and more abstract spaces by Georg Friedrich Bernard Riemann (1826-1866). In this labor he was aided above all by his friend the mathematician Marcel Grossmann (1878-1936). Another mathematician named David Hilbert (1862-1943) nearly beat him to his final equations.

But general relativity is above all Einstein’s achievement, and the phrase “Einstein’s spacetime” is entirely appropriate. No theory of comparable significance before or since is more nearly due to the struggle of a single scientist. At the end of 1915 Einstein wrote to a friend that he had succeeded at last, and that he was “content but rather worn out”. He later described this period as follows: “The years of searching in the dark for a truth that one feels but cannot express, the intense desire and the alternations of confidence and misgiving until one breaks through to clarity and understanding, are known only to those who have themselves experienced them”.

Special Relativity

jordynicki — Tue, 08 Sep 2009 13:30:26 +0000

Physics at the end of the nineteenth century found itself in crisis: there were perfectly good theories of mechanics (Newton) and electromagnetism (Maxwell), but they did not seem to agree. Light was known to be an electromagnetic phenomenon, but it did not obey the same laws of mechanics as matter. Experiments by Albert A. Michelson (1852-1931) and others in the 1880s showed that it always traveled with the same velocity, regardless of the speed of its source. Older physicists struggled with this contradiction in various ways. In 1892 George F. FitzGerald (1851-1901) and Hendrik A. Lorentz (1853-1928) independently found that they could reconcile theory and experiment if they postulated that the detector apparatus was changing its size and shape in a characteristic way that depended on its state of motion. In 1898, J. Henri Poincaré (1854-1912) suggested that intervals of time, as well as length, might be observer-dependent, and he even speculated (in 1904) that the speed of light might be an “unsurpassable limit”.

None of these eminent physicists, however, put the whole story together. That was left to the young Albert Einstein (1879-1955), who already began approaching the problem in a new way at the age of sixteen (1895-6) when he wondered what it would be like to travel along with a light ray. By 1905 he had shown that FitzGerald and Lorentz’s results followed from one simple but radical assumption: the laws of physics and the speed of light must be the same for all uniformly moving observers, regardless of their state of relative motion. For this to be true, space and time can no longer be independent. Rather, they are “converted” into each other in such a way as to keep the speed of light constant for all observers. (This is why moving objects appear to shrink, as suspected by FitzGerald and Lorentz, and why moving observers may measure time differently, as speculated by Poincaré.) Space and time are relative (i.e., they depend on the motion of the observer who measures them) — and light is more fundamental than either. This is the basis of Einstein’s theory of special relativity (“special” refers to the restriction to uniform motion).

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jordynicki — Tue, 08 Sep 2009 05:33:33 +0000

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