Warped Passages (18 page)

The resolution of this paradox is that the equivalence principle asserts only that gravity can be replaced by acceleration
locally
. At
different places in space, the acceleration that would replace gravity according to the principle would generally be in different directions. The answer to our problem with Chinese/American relations is that American gravity is equivalent to an acceleration in a different direction from the acceleration that would reproduce Chinese gravity.

Figure 37.
The “Einstein Cross” is formed when multiple images of a bright, distant quasar are formed by light bending in different directions as it passes by a massive foreground galaxy.

This critical insight led Einstein to a complete reformulation of the theory of gravity. He no longer saw gravity as a force that acts directly on an object. Instead, he described it as a distortion of the geometry of spacetime that reflects the different accelerations required to cancel gravity in different places. Spacetime is no longer a parenthetical background to an event—it is an active player. With Einstein’s theory of general relativity, the force of gravity is understood in terms of the curvature of spacetime, which in turn is determined by the matter and energy that are present. Let’s now consider the notion of the curvature of spacetime, on which Einstein’s revolutionary theory rests.

Curved Space and Curved Spacetime

A mathematical theory must be internally consistent but, unlike a scientific theory, it has no obligation to correspond to an external physical reality. True, mathematicians have often drawn inspiration from what they see in the world around them. Mathematical objects such as cubes and natural numbers do have real-world counterparts. But mathematicians extend their assumptions about these familiar concepts to objects whose physical reality is less certain, such as tesseracts (hypercubes in four-dimensional space) and quaternions (an exotic number system).

Euclid wrote his five fundamental postulates of geometry in the third century
BC
. From these assumptions a beautiful logical structure developed, one that you might have had a taste of in high school. But later mathematicians found themselves having trouble with the fifth postulate, the one known as the parallel postulate. This postulate states that, given a line and a point outside that line, there is one and only one line that can be drawn through the point that is parallel to the initial line.

For two millennia after Euclid formulated his postulates, mathematicians argued about whether this fifth postulate was actually independent or merely a logical consequence of the other four. Could there be a system of geometry for which all but the last postulate was true? If no such system of geometry existed, the fifth postulate would not be independent, and would therefore be disposable.

Only in the nineteenth century did mathematicians put the fifth postulate in its proper place. The great German mathematician Carl Friedrich Gauss discovered that Euclid’s fifth assumption was exactly what Euclid had claimed: a postulate that could be replaced by another. He went ahead and replaced it, discovering other systems of geometry and thereby demonstrating that the fifth postulate was independent. With that, non-Euclidean geometry was born.

A Russian mathematician, Nikolai Ivanovich Lobachevsky, also developed non-Euclidean geometry, but when he sent his work to Gauss he was disappointed to learn that the older mathematician had come up with the same idea fifty years before. But neither Lobachevsky
nor anyone else had known about Gauss’s results, which the German had hidden for fear that his colleagues would ridicule him.

Gauss shouldn’t have worried. It is obvious that Euclid’s fifth postulate is not always true, because we all know about alternatives. For example, lines of longitude meet at the North Pole and at the South Pole, even though they are parallel at the equator. Geometry on a sphere is an example of non-Euclidean geometry. Had the ancients written on spheres rather than scrolls, this might have been obvious to them, too.

But there are many examples of non-Euclidean geometry which, unlike the sphere, cannot be realized physically in a three-dimensional world. The original non-Euclidean geometries of Gauss, Lobachevsky, and the Hungarian mathematician János Bolyai
^*
dealt with such undrawable theories, which makes it less surprising that they took so long to discover them.

A few examples illustrate what makes curved geometries different from the flat geometry of this page. Figure 38 shows three two-dimensional surfaces. The first, the surface of a sphere, has constant positive curvature. The second, a section of a flat plane, has zero
curvature. And the third, a hyperbolic paraboloid, has constant negative curvature. Examples of negatively curved surfaces are the shape of a horse’s saddle, the terrain between two mountain peaks, and a Pringles potato chip.

Figure 38.
Surfaces of positive, zero, and negative curvature.

There are many litmus tests that will tell us which of the three possible types of curvature any particular geometric space possesses. For example, you can draw a triangle on each of the three surfaces. On the flat surface the sum of the angles of a triangle is always precisely 180 degrees. But what about a triangle on the surface of the sphere, with one vertex on the North Pole and the remaining two vertices on the equator, a quarter of the way around the equator from each other? Each of the angles of this triangle is a right angle of 90 degrees. Therefore the sum of the angles on the triangle is 270 degrees. This could never happen on a flat surface, but on a surface of positive curvature the sum of the angles of a triangle must exceed 180 degrees because the surface bulges out.

Similarly, the sum of the angles of a triangle drawn on a hyperbolic paraboloid is always less than 180 degrees, a reflection of its negative curvature. This is a bit harder to see. Draw two vertices near the top of the saddle and one down low, along one of the lower parts of the hyperbolic paraboloid, where one of your feet would go if you were sitting on a horse. This last angle is less than it would be if the surface were flat. The angles add up to less than 180 degrees.

Once it was established that non-Euclidean geometries were internally consistent—that is, their premises didn’t result in paradoxes or contradictions—the German mathematician Georg Friedrich Bernhard Riemann developed a rich mathematical structure to describe them. A piece of paper cannot be rolled into a sphere, but it can be rolled into a cylinder. You can’t flatten a saddle without having it crumple or fold back on itself. Building on Gauss’s work, Riemann created a mathematical formalism that encompassed such facts. In 1854 he found a general solution to the problem of how to characterize all geometries through their intrinsic properties. His studies laid the groundwork for the modern mathematical field of differential geometry, a branch of mathematics that studies surfaces and geometry.

Because I will almost always consider space and time together from now on, we will generally find the notion of
spacetime
more useful than the notion of space. Spacetime has one more dimension than space: in addition to “up-down,” “left-right,” and “forwards-backwards,” it includes time. In 1908 the mathematician Hermann Minkowski used geometric notions to develop this idea of an absolute spacetime fabric. Whereas Einstein studied spacetime using time and space coordinates that depended on a frame of reference, Minkowski identified the observer-independent spacetime fabric that can be used to characterize a given physical situation.

In the rest of the book, when I refer to dimensionality I will be giving the number of spacetime dimensions, except where I explicitly state otherwise. For example, when we look around us we see what I will from now on refer to as a four-dimensional universe. Occasionally I will single out time and talk about a “three-plus-one”-dimensional universe, or three spatial dimensions. Bear in mind that all these terms refer to the same setting—one that has three dimensions of space and one of time.

The spacetime fabric is a very important notion. It concisely characterizes the geometry that corresponds to the gravitational field produced by a particular distribution of energy and matter. But Einstein initially disliked the idea, which had seemed to him like an overly fancy way to reformulate the physics that he had already explained. However, he eventually recognized that the spacetime fabric was essential for completely describing general relativity and calculating gravitational fields. (For the record, Minkowski wasn’t overly impressed with Einstein on first acquaintance, either. Based on Einstein’s performance in Minkowski’s calculus class back when Einstein was a student, Minkowski had concluded then that Einstein was a “lazy dog.”)

Einstein wasn’t alone in resisting non-Euclidean geometry. His friend Marcel Grossmann, a Swiss mathematician, also considered it unduly complicated and tried to talk Einstein out of using it. However, they eventually agreed that the only tractable way of explaining gravity was by using non-Euclidean geometry to represent the spacetime fabric. Only then could Einstein interpret and calculate the warping
of spacetime that was equivalent to gravity, which turned out to be the key to completing general relativity. After Grossmann conceded defeat, both he and Einstein struggled through the intricacies of differential geometry to simplify their highly complicated earlier attempts to arrive at a formulation of the theory of gravity. In the end, they completed the theory of general relativity and reached a deeper understanding of gravity itself.

Einstein’s Theory of General Relativity

General relativity presented a radical revision of the concept of gravity. We now understand gravity—the force that keeps your feet on the ground and binds together our galaxy and the universe—not as a force acting directly on objects, but as a consequence of the geometry of spacetime, an idea that took Einstein’s view of the union of space and time to its logical conclusion. General relativity exploits the deep connection between inertial and gravitational mass to formulate the effect of gravity
solely
in terms of the geometry of spacetime. Any distribution of matter or energy curves or warps spacetime. Curved pathways in spacetime determine gravitational motion, and the matter and energy of the universe cause spacetime itself to expand, undulate, or contract.

In flat space the shortest distance between two points, the
geodesic
, is a straight line. In curved space we still can define a geodesic as the shortest path between two points, but that path won’t necessarily look straight. For example, routes of airplanes that follow great circles on the Earth are geodesics. (A great circle is any circle, such as the equator or a line of longitude, that goes around the fattest part of a sphere.) Although these paths are not straight, they are the shortest routes that don’t tunnel through the Earth.

In curved four-dimensional spacetime, we can also define a geodesic. For two events separated in time, a geodesic is the natural path things would take in spacetime to connect one event to the other. Einstein realized that free fall, which is the path of least resistance, is motion along the spacetime geodesic. He concluded that, in the absence of external forces, dropped objects will fall along a geodesic, as with the
path of the person on the falling elevator who doesn’t feel his weight or see a ball drop.

However, even when things are following their geodesics through spacetime and there are no external forces, gravity has noticeable effects. We’ve already seen that the local equivalence between gravity and acceleration was one of the critical insights that led Einstein to develop an entirely new way of thinking about gravity. He deduced that, because the acceleration induced by a gravitational force is locally the same for all masses, gravity must be a property of spacetime itself. That’s because “freely falling” means different things in different places, and it is only
locally
that gravity can be replaced by a single acceleration. My Chinese counterpart and I fall in different directions, even if we are both in our local version of Einstein’s elevator. The fact that the direction of free fall is not the same everywhere is a reflection of the curvature of spacetime. There isn’t a
single
acceleration that can cancel the effects of gravity everywhere. In curved spacetime, the geodesics of different observers will in general be different. So globally, gravity has observable consequences.