©Robert DiSalle 2001
 

When the "special" theory of relativity was introduced in 1905, it became known simply as "the theory of relativity"; the idea that this was a "special" theory arose with the the idea that a more general theory was required. And the latter idea arose from some philosophical questions that Einstein immediately asked about the special theory:

If motion is relative, why are we clinging to the idea of an absolute difference between rotating and not rotating?

Why do we maintain a distinction between certain "privileged" coordinate systems-the inertial systems- and every other kind of coordinate system? "What does nature care about our coordinate systems?"

Why shouldn't the laws of physics be the same for all coordinate systems?

If we introduce "rotation relative to an inertial frame" as the cause of centrifugal forces, aren't we introducing an unobservable cause for an observable effect? Doesn't that mean slipping from empirical science into metaphysics?
 

So the idea behind general relativity was that, where the 1905 theory said that the laws of physics would not distinguish between uniform motion and rest, the general theory would express laws that don't distinguish any states of motion; where the 1905 theory said that the laws of physics are the same in any inertial frame, the general theory would express laws that are the same in any frame of reference.
 

This might seem to be an odd idea. If I am on a moving carousel, I become ill; if the carousel is at rest and the surroundings rotate around it, why should I become ill? If the earth rotates, it bulges out at the equator; if the stars are rotating around the earth, why should the earth bulge? Is it really plausible that the distant stars are exerting some strange force by their rotation that is causing the bulge? According to Ernst Mach, however, we can't rule out such possibilities, because we can't take the stars away and check to see whether the bulge is still there. ("The universe is not given to us twice.")

Einstein took this view of Mach's extremely seriously. But he realized that both Newtonian physics and special relativity implied that absolute rotation is independent of relative rotation, and absolute acceleration is independent of relative acceleration. Both theories say that, if a body is alone in the universe, its rotation will be detectable by its centrifugal forces; if a particle is alone in the universe, or if it is immeasurably far away from other masses, its motion will be uniform and rectilinear. The lack of relative motion doesn't matter in either case. This means that one can't simply get rid of the concept of absolute rotation or absolute acceleration, because they are, so to speak, woven into the fabric of the laws of Newtonian and relativistic mechanics. Therefore, to satisfy his philosophical objections to these concepts, he would have to come up with altogether new laws of physics.

Another way of making the same point: The theory that the earth is rotating and the stars are at rest is "kinematically equivalent" to the theory that the stars are rotating and the earth is at rest. Kinematics is the simple geometrical description of motion, without regard to forces, masses, or causal relations of any kind; it doesn't matter from the kinematical point of view which body is at rest. But dynamics is concerned with forces, masses, and causes. True acceleration is caused by forces, and depends on the mass; true rotation, according to Newtonian mechanics and special relativity, is accompanied by a dynamical effect, i.e. centrifugal force. Therefore the two theories, while kinematically equivalent, are not dynamically equivalent. To make them equivalent would require a new theory of dynamics. This is what Einstein set out to create after 1905.

The key to the new theory was gravity. In 1905, it was clear that Newton's theory of gravity could not be maintained in its old form. According to the theory, gravitational influence is propagated instantaneously across empty space-unlike electromagnetic radiation, which is propagated as a "wavelike" disturbance, moving at the speed of light through the electromagnetic field. According to special relativity, however, the speed of light is a limiting velocity for all physical propagation. Therefore, at the very least, the theory of gravity would have to be revised to make it more like the theory of the electromagnetic field. This approach turned out to be impossible, however, because of strange features of the gravitational field.

These features of the gravitational field suggested to Einstein both a new theory of gravitation, and an answer to the problem of "generalizing" relativity. They are all connected to the "equivalence principle": that inertial mass- the quantity that enters into Newton's second law, and that is a measure of a body's resistance to acceleration- is equivalent to gravitational mass, the quantity that enters into Newton's law of universal gravitation. A more empirical way of expressing it is that all bodies fall with the same acceleration in the same gravitational field, or, the trajectory of a body in a given gravitational field will be independent of its mass and composition. This is essentially the principle that Galileo tested by dropping balls of different weights from a tower, and verifying that they hit the ground at the same time. It means that, if you were to leap from a cliff into the ocean, you could remove the change from your pocket and let go, but the change would fall right alongside you. The most vivid demonstration of it is the phenomenon of "weightlessness" experienced by people in orbiting spacecraft: even though the shuttle and all its contents are being pulled toward the earth by gravity-otherwise they wouldn't be in orbit, but would be flying off into space at a tangent to the orbit- everything inside behaves as if there's no force at all. If you hold a sandwich in front of you and release it, it doesn't fall; your own mass isn't pressed against the floor, but "hovers". In fact, everything happens in the orbiting shuttle as if you were in an inertial frame, i.e. as if the shuttle were moving uniformly in a straight line, and no external forces were acting on it or its contents. This is because the shuttle is actually "falling" toward the earth-not getting any closer, but falling just enough to keep it from flying off on the tangent- and the shuttle, and all of its contents, are falling at exactly the same rate. Therefore your sandwich behaves exactly as if the two of you were leaping from a cliff: gravity is affecting both of you to exactly the same degree, and neither is falling faster than the other. The interesting difference is that, inside the space shuttle, you might have no clue that you were falling at all, rather than simply isolated in space far from any gravitational source.

This has an important implication: it shows that the classical distinction between an inertial frame and an accelerated frame isn't so clear after all. How can we tell whether the frame we are in is moving uniformly in a straight line or being accelerated by a gravitational field? It would seem to be quite a bizarre coincidence if a force were to have exactly the same effect on every body. Electricity and magnetism, for example, don't affect all bodies in the same way; e.g. they don't have the same effects on wood and iron. And this is why you can insulate against electricity and magnetism, putting a rubber shield around conducting wires and so on. But there is no insulating against gravity: if you jump from a tree, nothing you could place on the ground would prevent you from falling. (The antigravity device is still a very remote piece of science fiction.) A classic example of an inertial frame is the center of mass frame of the solar system. Relative to the center of mass, every acceleration of every planet (including the sun) has an equal and opposite acceleration, so that every acceleration of every planet in the system can be said to be caused by the gravitational force of some other planet in the system. It would seem to follow that the whole system is either at rest or uniformly moving in a straight line, and no experiment or observation within the system could tell you one way or the other. But what if all the planets are being accelerated equally, and in parallel directions, by the gravitational force of some distant mass? Then the entire system would behave like the interior of the orbiting shuttle: it would behave as if there were no force acting at all- that is, it would behave exactly as it does now.

Thus, an inertial frame is not empirically distinguishable, within the system, from a freely-falling frame in a gravitational field. For the same reason, inertia is not distinguishable from gravity. Consider a frame (say, a windowless laboratory) sitting on the earth. Every object in the lab exhibits the same acceleration toward the floor, i.e. the acceleration due to the earth's gravity (9.8 meters/second/second). You feel your weight on the floor, and you feel the weight of an object in your hand, and any object you let go of falls to the floor with the same acceleration. Now imagine the same lab, accelerating upwards at 9.8 meters/second/second. In that case you will feel your inertia resisting the upward acceleration: your feet will press against the floor, an object in your hand will pull downward against your hand, and anything you let go of will fall towards the floor with the same acceleration. In other words, inertia will behave exactly like weight. Every experiment in this accelerating frame of reference will happen just as if the frame were at rest in a gravitational field.

So, in a sense, Einstein's philosophical idea turns out to be realized in physics after all. The "special" principle of relativity does need to be generalized in some way, since the inertial frames of reference are indistinguishable, not only from one another, but from an uniformly accelerating frames. Inertial motion is not really distinguishable from free fall in a gravitational field. If two people are falling from a great distance toward a planet, neither one will feel the effects of acceleration. Yet each will think that the other is accelerating and coming closer. The situation is something like that in the Michelson-Morely experiment. The experiment is supposed to test whether you are at rest relative to the ether, but the same test can be passed by laboratories that are in relative motion. Similarly, the test for being in an inertial state of motion can be passed by freely-falling trajectories that are in relative acceleration.
 

From general relativity to curved spacetime
 

These apparent absurdities are explained by the theory that spacetime is curved-that mass and energy, rather than creating a "field of force" in spacetime, cause a curvature of spacetime. Newton held that particles would naturally move uniformly in straight lines, but the gravitational pull of massive objects forces them to deviate; Einstein's theory says that particles naturally follow the "straightest possible paths" of spacetime, but massive objects create curvature in spacetime, so that the straightest possible paths are curved-objects follow the natural "contours" of spacetime, but their paths are curved because spacetime is curved.

What does it mean to say that spacetime is curved? To answer this, it helps to consider the question, what does it mean to say that space is curved? We can say that the surface of the earth is curved, by comparing it to the space around it: when you see a ship disappear over the horizon, you are comparing the curvature of the earth's surface to a straight line of sight of the surrounding space. Or, you could consider the fact that a line perpendicular to the surface in London, Ontario is not parallel to a line perpendicular to the surface in Toronto (e.g. by comparing the positions of a star that is directly overhead in London, but not in Toronto). This kind of comparison tells us the "extrinsic" curvature.

But what if you couldn't make such observations, if, for example, the earth was permanently clouded over? Then you would have to discover the curvature of the surface by its "intrinsic" properties. As we saw in Part I, this would mean determining the "straight lines" ("geodesics") of the surface, and determining whether they behave like straight lines in a plane or straight lines on a curved surface. Again, straight lines on a plane that are perpendicular to a given line will not converge; straight lines that intersect once will not intersect again; a triangle with straight sides will have internal angles that sum to 180 degrees. But on the earth, the straightest lines (geodesics) you can draw on the surface are "great circles," or lines of longitude. And these certainly do not behave like straight lines on the plane. Thus, by measurements done on the surface, without comparison to the "flat" three-dimensional space surrounding it, the earth's surface can be determined to be spherical rather than flat- its intrinsic curvature can be measured.

This is very important for understanding the curvature of space or spacetime. For, obviously, we have no "surrounding" space to which we can compare them; the intrinsic measure of curvature is all we could possibly have. Well, what could that be? The answer is not very complicated or surprising, in light of what you already know. First, consider the question, what are the straight lines of space? Then ask, do the straight lines of space behave like straight lines of a flat space, or not? Do they obey Euclid's parallel postulate? Do the internal angles of triangles sum to two right angles? The usual answer to the first question is that the straight lines of space are the paths of light rays. So the answer to the second question can then be arrived at by straightforward measurement. If space were non-Euclidean, for example, the angles of a large triangle whose sides were determined by light signals (e.g. laser beams) would not sum to 180 degrees. Thus the idea of spatial curvature is not particularly strange. If you can understand how space could be Euclidean, you can understand how it could be non-Euclidean. A basic consequence of general relativity was that space would be curved near a massive object like a star; this was confirmed in 1919 by a measurement of light passing by the sun, and more recently by light and radio signals. (See Clifford Will's Was Einstein Right?.)

But what about spacetime curvature? To understand this, first ask, what is a straight line of spacetime? Remember that a line of spacetime is a trajectory, or "worldline," i.e. the history of some particle as it moves in space and time. A straight line of spacetime is a uniform rectilinear motion, i.e. an inertial trajectory. So Newton's first law can be expressed in spacetime terms as follows: A particle not subject to forces follows a straight line (geodesic) of spacetime. It is easy to see why this must be the case- why any accelerated motion, even if it involves changing speed but continuing in the same direction in space, must be a curved trajectory in spacetime.

Once you understand that inertial trajectories are the straight lines of spacetime, to understand spacetime curvature is straightforward. You simply have to consider the question, can inertial trajectories behave like straight lines in a non-Euclidean space? Before answering this, however, remember that, according to the equivalence principle, an inertial motion is indisinguishable from a freely-falling trajectory. Therefore the paths of bodies falling in a gravitational field have every right to be considered inertial trajectories; this means that they have every right to represent the straight lines of spacetime. And the paths of falling bodies do in fact behave like non-Euclidean geodesics. In particular, they can be, as we saw, relatively accelerated; they can begin perpendicular to the same line, yet converge. In short, spacetime is curved because the paths of falling bodies are the straight lines of spacetime, and therefore the straight lines of spacetime behave like the straight lines of a curved space. Newton held that a falling body is being forced out of a spacetime geodesic by the gravitational field; general relativity holds that a falling body is simply following a spacetime geodesic, in a curved region of spacetime. Newton's theory leads to a "field equation" that explains how the relative accelerations of falling particles are related to the distribution of mass; Einstein's theory has a field equation that explains how the curvature of spacetime--as revealed in the convergence and divergence of its geodesics or straightest lines--is related to the distribution of mass and energy.