Consider a society of two-dimensional beings living on a surface that is almost perfectly flat. In one place the surface contains a bump, which is visible from the perspective of a larger three-dimensional space in which the surface is contained.

From the three-dimensional perspective, imagine a point P at the top of the bump, a circle L at its base, and several lines, R₁, R₂, R₃, ... R_n, running from P to different points on L.

The two-dimensional beings will have no trouble confirming, entirely by means of measurements carried out with their two-dimensional rulers on their two-dimensional surface, that all of the lines R have the same length and that therefore all of the endpoints of the lines R on L are equidistant from P. In other words, it will be easy for them to confirm that P is a circle. The beings can also easily carry out a measurement of L’s circumference. When they do so, however, they will discover that the ratio between the circumference of this circle and its radius (the length of any line R) is smaller than 2π. In this way the beings will be able to discover that the surface inside L is not flat: their world contains an extra dimension that they cannot experience directly themselves.

Until about the middle of the 19th century, no one entertained the slightest suspicion that considerations like these might apply to three-dimensional space—that ordinary space might contain extra dimensions that humans cannot experience directly. The mathematical possibility of spaces of three and higher dimensions that are curved, and whose curvature could in principle be discovered by observers within them, was articulated in magnificent and profoundly illuminating detail by Bernhard Riemann (1826–66), Nicolay Ivanovich Lobachevsky (1792–1856), and others. They developed a powerful and intuitive generalization of the notion of a “straight line” for non-Euclidean geometries: a line that is precisely as straight as the space it traverses will accommodate. These “generalized straight lines” are referred to in the mathematical literature as geodesics.

Einstein was intrigued by the fact that the mass that figures in Newton’s law of motion, F = ma—the mass that measures the resistance of material bodies to being accelerated by an impressed force (inertial mass)—is invariably exactly the same as the mass that determines the extent to which any material body exerts an attractive gravitational force on any other. The two concepts seem to have nothing to do with each other. In the context of Newtonian mechanics, the fact that they are always identical amounts to an astonishing and mysterious coincidence.

It is this equivalence that entails that any given gravitational field will accelerate any two material bodies, whatever their weights or their constitutions, to precisely the same degree. This equivalence also entails that any two material bodies, whatever their weights or their constitutions, will share precisely the same set of physically possible trajectories in the presence of a gravitational field, just as they would in empty space.

The observation that the effects of a gravitational field on the motions of a body are completely independent of the body’s physical properties positively cried out for a geometrical understanding of gravitation. The idea would be that there is only a single, simple law of the motions of material bodies, both in free space and in the presence of gravitational fields. The law would state that the trajectories of material bodies are geodesics rather than Euclidean straight lines and that gravitation is not a force but rather a departure from the laws of Euclidean geometry. Such a law would be out of the question if the geometry involved were that of three-dimensional physical space.

Einstein began by proposing a principle of the local equivalence of inertial and gravitational fields, a more powerful and more general version of the equivalence of inertial and gravitational mass. According to this principle, the laws that govern the time-evolutions of all physical phenomena relative to a frame of reference that is freely falling in a gravitational field are precisely the same as the laws that govern the time-evolutions of those phenomena relative to an inertial frame. After years of prodigious effort, Einstein was able to develop this principle into a fully relativistic and fully geometrical theory of gravitation: the general theory of relativity.

What Einstein produced in the end was a set of differential equations, the so-called Einstein field equations, relating the geometry of space-time to the distribution of mass and energy within it. The general theory of relativity consists of a law to the effect that the four-dimensional geometry of space-time and the four-dimensional distribution of mass and energy within space-time must together amount to a solution to the Einstein field equations.

In this theory, space-time is no longer a fixed backdrop against which the physical history of the world plays itself out but an active participant in its own right, a dynamical entity on a par with all others. Although this may suggest a relationist conception of space-time, it is now widely agreed that such a project is impossible. General relativity is committed to claims regarding which motions are and are not possible for a single particle entirely alone in the universe, and it also allows for the existence of universes whose total angular momentum is other than zero. Moreover, it inherits all of the structural hostility to relationism characteristic of the special theory, as described above.

There are solutions to Einstein’s field equations that result in universes that are finite but have no boundary or “outside”—universes in which, for example, a straight line extended far enough in any direction in space will eventually return to its starting point. Other solutions result in universes in which time travel into the past is possible. This particular example has been the subject of much scientific and philosophical scrutiny in recent years, since it seems to lead to outright logical contradiction—as in the case of the person who travels into the past and murders his parents before his own birth.