Once Before Time - Martin Bojowald [14]
CURVED SPACE-TIME: A STAGE SHAKING UNDER THE WEIGHT OF THE ACTORS
Ay!
There are times when the great universe
Like cloth in some unskilful dyer’s vat
Shrivels into a hand’s-breadth, and perchance
That time is now! Well! Let that time be now.
—OSCAR WILDE, A Florentine Tragedy
In general relativity, the form of space-time is determined by the matter it contains. Here the gravitational force finds its very origin, intimately connected with the structure of space and time in a way not realized for any of the other known forces in physics. Mathematically, all this is described by means of a curved space-time, a space-time whose degree of transformation between space and time, in the relativistic sense, depends not only on an observer’s motion but also on the position in space and time. By this dependence on the observer’s position, the concepts of special relativity are generalized. The theory is no longer constrained to what (inertial) observers moving at different but constant velocities along straight lines see. This assumption was a simplification employed in special relativity to understand the effects of different velocities, but it is not realistic: When we make observations, we are moving along complicated trajectories in space and time. We may be standing more or less still in a lab, but we are standing on the earth, which is rotating and orbiting the sun. The sun is moving, too, and so is the Milky Way. A general theory must be able to describe observers moving along arbitrary curves, possibly accelerating when forces are acting on them. General relativity does so by allowing for position-dependent transformations of space and time, thereby endowing space-time with a curved form.
The prime example of a curved space is the two-dimensional surface of a ball, or a sphere. It is a curved surface, and also closed in on itself, although the latter property is not shared by all curved spaces. What is illustrated by the sphere is the fact that lines on its surface must be curved, as seen from the surrounding space, in order to stay on the surface. Every straight line in space starting on its surface would immediately leave it. This behavior can be seen as a general consequence of curvature, even though abstract curved spaces do not need a surrounding space such as the three-dimensional one around the sphere. Space-time itself, for instance, is four-dimensional and would require an even higher dimensional ambient space. All consequences of curvature can mathematically be described without referring to such surrounding spaces—a convenient fact crucially exploited by Einstein in his formulation of general relativity. The relevant branch of mathematics, differential geometry, was founded on work by Bernhard Riemann in the nineteenth century.
Returning to the example of the sphere in its ambient three-dimensional space, we can see a further important consequence of curvature. When we move and change our position on a sphere, as we regularly do on the earth’s surface, we are, as seen from the ambient space, forced to rotate. We usually do not notice this because, for one thing, Earth is very large, and moreover, we can rarely take this view from outer space. But the forced rotation can easily be visualized on a globe: The head of a person in Europe points in a different direction in space than does that of a person in America, even if both people are standing up ramrod-straight.6 No such rotation would occur if one were moving on a planar surface such as the level floor of a room; it must be a consequence of curvature.
Space-time is curved by the matter it contains and should show effects comparable to those on a sphere. This is more difficult to visualize, for we now have a four-dimensional situation involving time as well. Our earlier analogy shows the most important