Space-Time Geometry
The Special theory of relativity, which included the concept of space-time, dealt only with relations between measurements made by inertial observers. This restriction to non-accelerated observers was a serious one, and German-American physicist Albert Einstein almost immediately began the search for a more inclusive theory, one that could relate observations made by any observer, regardless of his state of motion.
Any inertial observer perceives a three-dimensional Euclidean space, one in which, for example, the distance between two points satisfies the Pythagorean theorem, and the ratio of the circumference of a circle to its radius is . This is flat space. And, an inertial observer who is at rest with respect to his clocks (no matter where they are located) sees them all run at an identical rate. This is uniform time. In the four-dimensional space-time of special relativity, for any inertial observer space is flat, time is uniform, and the interval between two events satisfies the Pythagorean relation s 2 = x 2 +y2 +z 2 -t2 . Such Minkowskian space-time is the analog of Euclidean space: Einstein soon realized, however, that Minkowskian space-time could not suffice for a theory that allowed accelerated observers.
Consider, Einstein proposed, an observer (whom we'll call Nicholas), sitting on a uniformly rotating disk. Equipped with a standard meter-stick, this accelerated physicist will, by laying out the stick end to end, measure both the disk's radius and circumference. Previously, while the disk was not rotating, another physicist (whom we'll call Jonathan), performed the same set of measurements using the same standard meter-stick. When Nicholas measures the circumference, the rotating meter-stick lies parallel to the direction of the disk's motion; when he measures the radius, it lies perpendicular. Thus the stick is relativistically shortened as the circumference is measured, but when measuring the radius, it is not. Nicholas and Jonathan obtain the same value for the radius, but Nicholas obtains a greater value for the circumference than did Jonathan. (He needs to lay out his shortened meter-stick more times than Jonathan did.) Each observer calculates the circumference/radius ratio: non-accelerated Jonathan finds , while accelerated Nicholas obtains a value greater than . Accordingly, the accelerated observer perceives a non-Euclidean space.
There are two clocks on the disk, as well. One clock is at the center, and the other on the edge. When the disk was stationary, Jonathan found they ran at the same rate. As the disk rotates, however, Jonathan sees the clock at the edge, which is moving with respect to him, lag behind the other. As Einstein said, "It is obvious that the same effect would be noted by an observer [Nicholas] whom we will imagine sitting alongside his clock at the center of the disk. Thus, on our circular disk, a clock will go more quickly or less quickly, according to the position in which the clock is situated." Time, on the disk, is not uniform.
So, for our accelerated observer, space is not flat, and time is not uniform. Such curved space-time is called Riemannian space-time.
Einstein's crucial insight, made in 1907, which he later called "the happiest thought of my life" argued that the effects of acceleration are indistinguishable from those of gravitation. This is the principle of equivalence. The prototypical illustration is an elevator in deep space, free from all gravitational forces, accelerating at a constant rate. There is no experiment a physicist in such an elevator could perform which will determine whether he is indeed in an accelerating elevator, or in an elevator at rest in a uniform gravitational field. All objects in the elevator, for example, fall towards the floor with an acceleration independent of their mass, the hallmark of Newtonian gravitation. A theory, then, which allows accelerated observers must necessarily be a theory of gravity.
The effects of acceleration, we have seen, are a curving of space-time. Those effects are indistinguishable from those of gravity. Thus, gravity is a curvature of space-time. All the gravitational effects we observe, from Newton's apple falling from the tree, to the orbiting of the Moon around Earth, are due to the non-flat, Riemannian geometry of space-time.
Gravity, Newton postulated, was an effect of mass. Gravity, Einstein postulated, is a curving of space-time. Thus mass (or its equivalent, energy) curves space-time. This is the essence of Einstein's masterwork, the General theory of relativity, completed in 1915.
In the presence of a massive object, a body still traverses the shortest possible path between points, but, since the space-time has been warped, that path is no longer a straight line. It is, rather, a geodesic, and its precise form depends upon the space-time geometry, the mathematical description of the curvature. An easily visualized example is a two-dimensional rubber sheet. When it is flat, its geodesic is a straight line. But add a mass and the sheet distorts, its geodesic will be a curved path. In this model there is no gravitational force. Bodies always take the shortest possible path, but the particular mass distribution present will determine what that path will be. Gravity is a manifestation of space-time geometry.
This is the complete article, containing 837 words
(approx. 3 pages at 300 words per page).
