Seeking the Geometry of the Universe
Seeking the Geometry of the Universe
The German mathematician Georg Friedrich Riemann died shortly before his fortieth birthday and long before the importance of his work was truly recognized. He left the equivalent of only one volume of writings. Yet these provided the tools with which, 50 years after Riemann's death, Albert Einstein formulated his general theory of relativity.
In traditional Euclidean geometry, codified in about 300 b.c., a straight line is the shortest distance between two points. Euclidean geometry is easily visualized on a two-dimensional flat surface, or plane (like a piece of paper). Spaces of any number of dimensions in which Euclidean geometry holds are called Euclidean or flat spaces.
On a curved surface, however, the shortest distance between two points is a curve called a geodesic. If the surface is spherical, the geodesic falls on an arc of a great circle, the intersection of the surface of the sphere with a plane passing through its center. The usefulness of curved-surface geometry for such important pursuits as navigation seems obvious. However, progress was delayed for centuries by religious proscriptions against admitting that Earth was round, even in the face of observations by astronomers and mariners. The classic work on curved-surface geometry, General Investigations of Curved Surfaces, was published by Carl Friedrich Gauss (1777-1855) in 1827. Georg Friedrich Riemann (1826-1866) was among his students.
Riemann generalized geometry so that it applied to spaces of any curvature and in any number of dimensions. He began with the concept of coordinates, invented by the French mathematician and philosopher René Descartes (1596-1650). Coordinates are ordered sets of numbers that describe the position of a point. On a plane, which has two dimensions, the point is described by an ordered pair (x1, x2). In three-dimensional space, an ordered set of three numbers (x1, x2, x3) is required.
This was as far as Descartes deemed it necessary to go, since three dimensions defined his view of the physical world. Riemann extended Descartes' invention to accommodate any number of dimensions n, with n-tuples, coordinates defined as (x1, x2, . . . xn). Two n-tuples (x1, x2, . . . xn) and (y1, y2, . . . yn) are equal if all their elements are equal; that is, x1 = y1, x2 = y2, . . . xn = yn
If we let all the elements in an n-tuple vary over some range of real numbers, the set of all the distinct n-tuples we obtain is called a manifold of n dimensions. If the elements can only take discrete values, such as integers, we have a discrete manifold; otherwise, it is a continuous manifold. The intermediate possibility, in which some elements are continuous and some discrete, can always be handled as the sum of a continuous and a discrete manifold, so it need not be considered separately.
Just as the Earth has mountains and valleys, vast featureless plains and the Grand Canyon, the geometry of space must allow for the possibility that the curvature is not a constant. Distance must therefore be computed piecewise, as differentials, and added up. Such techniques of the calculus as applied to manifolds are called differential geometry.
For any Riemannian manifold, distances are computed in quadratic form. This means that the differential terms are squared, just like the terms in the familiar Pythagorean theorem for computing distances in flat space. The coefficients of the differentials define the metric, or theory of measurement, for that manifold. These coefficients may be functions of any type, so an infinite number of Riemannian geometries are mathematically possible.
Riemann developed formulas describing the curvature of manifolds in terms of the coefficients of the differentials. These were extended by his fellow German Hermann von Helmholtz (1821-1894) and Englishman William Clifford (1845-1879). There are three basic possibilities for the curvature of a space or a region of space. In positive or spherical curvature, straight lines eventually return back upon themselves. Such a space would have no end but would still not be infinitely large. Negatively curved or flat (zero curvature) space both have straight lines running out to infinity, but they differ in the behavior of parallel lines.
The mathematics of curved space were to prove useful to Albert Einstein (1879-1955) decades later. Riemann had perhaps been ahead of his time, but finally a physicist came along with the genius to apply his mathematics to the universe.
Relativity theory is based on four-dimensional space-time, consisting of the familiar three spatial dimensions with time as the fourth. The "distance" between two events is thus a spacetime interval, and Einstein hypothesized that this interval is invariant for continuous transformations of the four coordinates, referred to as a change in frame of reference. Any observer, in other words, should come up with the same answer. Einstein's first theory of relativity was "special" as opposed to "general"; it was restricted to flat space-times and observers moving with constant velocity with respect to each other.
An accelerated reference frame presented Einstein with a more difficult problem, because acceleration implies a force, and classical mechanics demands that a force have an effect. How could the space-time interval between events thus remain invariant? Fortunately, Riemann had conveniently provided Einstein with another tool, the tensor calculus. He had developed it for a problem on heat conduction he addressed for a French Academy of Sciences competition. Later it was elaborated by Matteo Ricci (1835-1925) and Tullio Levi-Civita (1873-1941).
A tensor is a general way of expressing a physical or mathematical quantity and its transformation properties. A scalar, or constant, which is invariant under all transformations, can be thought of as a tensor of "rank 0." A vector quantity, which is invariant under coordinate transformations, is a tensor of first rank. The quantities commonly called "tensors" are those with rank of two or greater. Tensor analysis helps in identifying differential equations that are invariant for all transformations of their variables, which in the case of relativity are the 4-tuple coordinates of a point in space-time. If the system of equations results in a tensor that goes to zero, the system is invariant under transformations of its variables.
Einstein postulated that the square of the space-time interval is invariant, and zero in the case of the path of light. The path of a free particle was assumed to be a geodesic in space-time. Next came Einstein's famous thought-experiment of the man in the elevator. His accelerating reference frame produces effects that are to him indistinguishable from changes in the force of gravity. Similarly, Einstein reasoned, gravity is equivalent to a transformation of coordinates in a local region. The curvature of space-time in a local area, which is perceived as gravity, is a result of a mass in that area. This theory of general relativity has been supported by evidence of light's path curving in the vicinity of a large mass such as a star, and other observations.
While the Riemannian geometries were more general than anything that had been developed before, they did not cover all possibilities. The advent of general relativity, however, concentrated work in this area. Still, it is conceivable to envision geometries in which the differentials are not quadratic, or the transformations work differently. Hermann Weyl (1885-1955) was an important contributor to the discipline of non-Riemannian geometry.
SHERRI CHASIN CALVO
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