"The sun creates the curved structure of space-time, and the planets move on geodesics or "straight lines" in curved space-time." This sounds great and is a line you can drop at your next party, but what does it mean? Is it any better than saying, "The sun exerts a force on the planets varying inversely as the square of the distance, and this force, plus the initial conditions leads to the elliptical orbits of the planets," Students, rightly so, say that they cannot visualize accelerated motion without force. After all didn�t we say that for an object to accelerate, a force must act on it? Of course, we defined a force in terms of the acceleration it produces so that the concept of force as something that produces acceleration is a creation of scientists. I would like to try to clarify the concept of motion in curved space-time, along the way, remind you of the manner in which physicists construct theories of natural phenomena.

I do not want to discuss philosophy, but some of the comments may seem philosophical in nature. Most people are limited to a large extent by what they have experienced. As Paul Weiss, a noted philosopher once said, "You don�t have to be a bird to know about flying;" however, it certainly helps to have seen a bird fly to understand what flight is all about. The ancient Greeks observed the moon, sun, and stars rotate around the Earth, and the planets move among the stars. They tried to construct theories in which the heavenly bodies were attached to transparent spheres that were turned by an "unmoved mover." (Is "unmoved mover" any worse a term than "gravity?") As their measurements improved, the complexity of their system increased to explain the data, but they never abandoned the idea of circular motion as the central component of all planetary motion. They also "explained" that atoms fell to Earth owing to the fact that they returned to their "natural" resting place. These ideas may seem ridiculous to you now, but they seemed reasonable at the time. The Greeks were a highly advanced an intelligent culture.

As time went by, new data was accumulated and attitudes changed. With the contributions of Copernicus, Brahe, Kepler, and Galileo, a new view of the Universe was emerging. The sun was moved to the center of the Universe and the planets were found to follow elliptical orbits. Objects were said to fall because the Earth exerted a force on them. Accelerated motion became associated with force. Newton synthesized all the data into his laws of mechanics and law of Universal Gravitation. The planets moved in their orbits because the sun exerted a force on them varying as the inverse square of the distance. Was this the only possible explanation of planetary motion and the attraction of objects? Not necessarily, but it explained the data and allowed one to predict new phenomena that could be tested experimentally. The theory explained all existing data and had predictive attributes. As long as any new data was not in conflict with the theory, the theory can remain as a physically accepted law. Remember that physics is an experimental science and theories are valid only insofar as they agree with experiment. Did Newton�s theory explain why objects attract one another? NO! Newton himself said "he frames no hypotheses" as to the cause of gravity. It is a phenomenon of nature that he attempted (successfully) to quantify.

All fundamental laws of physics eventually lead to some observation that cannot be "explained." It is just the way nature is. If the law of physics is related to your everyday experience, it may seem"reasonable" to you, If not, you may have difficulty in accepting the law as something that is reasonable. Why does the gravitational force fall off inversely as the square of the distance � it just does. Why is the speed of light independent of inertial frame � it just is. You cannot explain fundamental laws of physics, but you can examine the consequences of the laws. The theory of special relativity was developed to explain the experimental laws of electricity and magnetism. Consequences of the postulates of special relativity were far reaching, however. Space and time were no longer independent, length depended on inertial frame, time intervals depended on inertial frame, and simultaneity of spatially separated events depended on inertial frame. A twin who went out and came back to Earth was younger than the twin who stayed on Earth when they are reunited. All these results follow directly from the postulates. Given the postulates, one can explain why the space twin is younger as related to the structure of space-time (the space twin is not younger because of any physical effect related to acceleration). However, the validity of the postulates themselves can only be tested by experiment.

We now return to gravitation. Newton�s theory had already been found to be incorrect for relative speeds approaching the speed of light. Trying to incorporate Newton�s law of Universal Gravitation into special relativity was not obvious at all since the force varies inversely as the distance squared and distance depends on inertial frame. Moreover, there was this strange "accident" in Newtonian theory that inertial mass = gravitational mass. If you are on the space shuttle in free fall, it appears that gravity is absent and you sense no acceleration or force acting on you. Perhaps the concept of force is not all it is cracked up to be. Einstein did not have that much more data than Newton, but he had already developed the special theory. He had a vision of trying to explain gravity in a new way, in which force would be eliminated totally. Clearly, for his theory to be successful, it had to explain all existing phenomena and, to be significant, it had to predict new phenomena.

Without forces, objects move on straight lines. This is a little difficult to prove since we use the fact that they move on straight lines to show that no forces act. The reasoning is circular. Instead we can postulate that objects move on straight lines in the absence of mass. What do we do when mass is present. We can now postulate that objects move on "straight line" paths (geodesics) in curved space-time produced by the masses in the Universe. However, it is not necessary to postulate this result. This is a consequence of the postulates of the General Theory of Relativity. The Einstein field equations that emerge from the postulates of the General Theory describe a world in which gravity as a force is absent. Instead mass (and electromagnetic fields) determine both the geometry of space time and the manner in which objects move in this space time. As in Newtonian theory, the masses of objects and the constant of Universal Gravitation appear, but the speed of light also appears in the General Theory. At distances from the center of an object which are much greater than the Schwarzschild radius of the object, the General theory essentially gives the same motion as that predicted by Newtonian theory (this was a requirement of the theory). However, there are corrections to Newtonian theory of order of the Schwarzschild radius divided by the distance. Moreover, light is found to be deflected as it passes by the sun or any other object. Thus there were predictions of the theory that deviated from Newton�s theory that could be tested experimentally. The experimental tests of Einstein�s exact form of the field equations are not conclusive, but the General Theory is the currently accepted theory of gravitation. Thus objects move on straight line paths in curved space-time follows directly from the postulates of the General Theory, which must be tested experimentally. Since, for masses such as the sun or the earth, the corrections to Newtonian theory are small, the concepts of the Newtonian theory of gravity retains its validity to a high degree of approximation. You may be comfortable with the idea that the gravitational force produces acceleration, but it appears that the more fundamental theory of gravity explains motion in the presence of masses in terms of the geometry of space-time created by the masses.