* "In relativity, movement is continuous, causally determinate and well defined, while in quantum mechanics it is discontinuous, not causally determinate and not well defined"- David Bohm *

A cosmos without stars would have no space-time structure relative to which the earth could spin. For there to be gravitational fields capable of bulging a planet's equator and spilling water over the sides of a rotating bucket, there must be stars to create a space-time structure.

In developing the general theory of relativity Einstein found it necessary to adopt a four-dimensional general Riemannian geometry. Instead of a fourth space dimension, however, Einstein made time his fourth dimension. There is nothing mysterious or occult about this concept. It merely means that every event that takes place in the universe is an event occurring in a four-dimensional world of space-time. This can be made clear by considering the following event. You get into a car at 2 pm and drive from your home to a restaurant that is 3 kilometres south and 4 kilometres east of your house. On the two-dimensional plane the actual distance from your house to the restaurant is the hypotenuse of a right triangle with sides of 3 and 4 kilometres. This hypotenuse has a length of 5 kilometres. But it also took you a certain length of time, say, ten minutes, to make this drive. One coordinate of the graph is the distance south in kilometres, another is the distance east in kilometres; the vertical coordinate is the time in minutes. On this three-dimensional graph of spacetime, the "interval" (spacetime distance) between the two events. On leaving your house and arriving at the restaurant is a straight line. This straight line is not a graph of the actual trip. It is simply a measure of the spacetime distance between the two events. A graph of the actual trip would be a complicated curved line. It would be complicated because your car accelerates when it starts, the arrangement of streets may make it impossible to drive to the restaurant on a straight line, perhaps you stopped at traffic lights along the way, and finally, you had to accelerate negatively when you stopped the car.

The complicated wavy graph of the actual trip is, in relativity theory, called the "world line" of the trip. In this case, it is a world line in a spacetime of three dimensions, or (as it is sometimes called) in a Minkowski three-space. Because the trip by car took place on a plane of two dimensions, it was possible to add the one dimension of time and show the trip on a three-dimensional graph. When events occur in three-dimensional space it is not possible to draw an actual graph of four-dimensional spacetime, but mathematicians have ways of handling such graphs without actually drawing them. Try to imagine four-dimensional hyperscientist who can construct four-dimensional graphs as easily as the ordinary scientist can construct graphs with two and three dimensions. Three of the coordinates of his graph are the three dimensions of our space. The fourth coordinate is our time. If a spaceship leaves the earth and lands on Mars, our imaginary hyperscientist will draw the world line of this trip as a curve on his four-dimensional graph. (The line is curved because the ship cannot make such a trip without accelerating.) The spacetime "interval" between take-off and landing will appear as a straight line on the graph. In relativity theory, every object is a four-dimensional structure lying timelessly along its world line in the four-dimensional world of spacetime. If an object is considered at rest with respect to the three space coordinates, it is still traveling through the dimension of time. Its world line will be a straight line that is parallel with the time axis of the graph. If the object moves through space with uniform motion, its world line will still be straight, but no longer parallel with the time axis. If the object moves with nonuniform motion, its world line becomes curved. Strictly speaking, one should not say that an object moves along its world line, because "moves" implies movement in time, whereas time is already represented by the world line.

The world line is no more than a convenient way to graph the motions of an object in three-dimensional space. The fact that a Minkowski graph is, in a sense, a static, timeless picture of the world has nothing whatever to do with the question of whether the future is or is not completely determined by the present. An object moving in a random, unpredictable way can be graphed by a world line just as easily as an object moving in a predictable way. After an event has occurred, its Minkowski graph does indeed freeze the event in a timeless "block universe" but this has no bearing on the question of whether the event had to happen the way it did. We are now in position to look at the Lorentz-FitzGerald contractions of the special theory from a new point of view: the Minkowski point of view, or the viewpoint of our hyperscientist. As we have seen, when two spaceships pass each other in relative motion, observers on each ship see certain changes in the shape of the other ship as well as changes in the rate of the other ship's clock. This is because space and time are not absolutes that exist independently of each other. They are, so to speak, like shadow projections of a four-dimensional spacetime object. If a book is held in front of a light and its shadow projected on a two-dimensional wall, a turn of the book will alter the shape of its shadow. With the book in one position the shadow is a fat rectangle. In another position it is a thin rectangle. The book does not change its shape; only its two-dimensional shadow changes. In a similar way, an observer sees a four-dimensional structure, say, a spaceship, in different three-dimensional projections depending on his motion relative to the structure. In some cases, the projection shows more of space and less of time; in other cases, the reverse is true. The changes that he observes in the space and time dimensions of the other ship can be explained by a kind of "rotation" of the ship in spacetime, causing its shadow projections in space and time to alter. The important point to grasp here is that the spacetime structure, the four-dimensional structure, of the spaceship is just as rigid and unchanging as it is in classical physics.

This is the essential difference between the discarded Lorentz contraction theory and the Einstein contraction theory. For Lorentz, the contraction was a real contraction of a three-dimensional object. For Einstein, the "real" object is a four-dimensional object that does not change at all. It is simply seen, so to speak, from different angles. Its three-dimensional projection in space and its one-dimensional projection in time may change, but the four-dimensional ship of space-time remains rigid. Here is another instance of how the theory of relativity introduces new absolutes. The four-dimensional shape of a rigid body is an absolute, unchanging shape. We can slice space-time so the shape of a spaceship depends on the motion of the frame of reference from which we make the slice, but the fact that we can take slices at different angles through a sausage does not force us to give up an absolute theory of sausages. The four-dimensional interval between any two events in space-time is an absolute interval. Observers moving at great speeds and with different relative motions may disagree on how far apart they judge two events to be in space, and on how far apart they judge two events to be in time, but all observers, regardless of their motions, will agree on how far apart they judge two events to be in space-time. Space is different for different observers. Time is different for different observers. Space-time is the same for everyone. In classical physics an object moves through space in a straight line, with uniform velocity, unless acted upon by a force. A planet, for example, would move off in a straight line were it not held by the force of the sun's gravity. From this point of view, the sun is said to "pull" the planet into an elliptical orbit. In relativity physics an object also moves in a straight line, with uniform velocity, unless acted upon by a force, but the straight line must be thought of as a line in space-time instead of space.

This is true even in the presence of gravity. The reason for this is that gravity, according to Einstein, is not a force at all! The sun does not "pull" on the planets. The earth does not "pull" down the falling apple. What happens is that a large body of matter, such as the sun, causes space-time to curve in the area surrounding it. The closer to the sun, the greater the curvature. In other words, the structure of space-time in the neighbourhood of large bodies of matter becomes non-Euclidian. In this non-Euclidian space, objects continue to take the straightest possible paths, but what is straight in space-time is seen as curved when projected onto space. Our imaginary hyperscientist, if he plots the orbit of the earth on his four-dimensional graph, will plot it as a "straight" line. We who are three-dimensional creatures see the space path as an ellipse. Writers on relativity theory often explain it in the following way. Imagine a rubber sheet stretched out flat like a trampoline. A grapefruit placed on this sheet will make a depression. A marble placed near the grapefruit will roll toward it. The grapefruit is not "pulling" the marble. Rather, it has created a field of such a structure that the marble, taking the path of least resistance, rolls toward the grapefruit. In a roughly similar way, space-time is curved or warped by the presence of large masses like the sun. This warping is the gravitational field. A planet moving around the sun is not moving in an ellipse because the sun pulls on it, but because the field is such that the ellipse is the "straightest" possible path the planet can take in space-time. Such a path is called a geodesic On a Euclidian plane, such as a flat sheet of paper, the straightest distance between two points is a straight line. It is also the shortest distance. On the surface of a globe, a geodesic between two points is the arc of a great circle. If a string is stretched as tautly as possible from point to point, it will mark out the geodesic. This, too, is both the "straightest and the shortest distance connecting the two points. In a four-dimensional Euclidian geometry, where all the dimensions are space dimensions, a geodesic also is the shortest and straightest line between two points.

But in Einstein's non-Euclidian geometry of spacetime, it is not so simple. There are three space dimensions and one time dimension, united in a way that is specified by the equations of relativity. This structure is such that a geodesic, although still the straightest possible path in spacetime, is the longest instead of the shortest distance. This concept is impossible to explain without going into complicated mathematics, but it has this curious result: A body moving under the influence of gravity alone always finds the path along which it takes the longest proper time to travel; that is, the longest when measured by its own clock. Bertrand Russell has called this the "law of cosmic laziness." The apple falls straight down, the missile moves in a parabola, the earth moves in an ellipse because they are too lazy to take other routes. It is this law of cosmic laziness that causes objects to move through space in ways sometimes attributed to inertia, sometimes to gravity. If you tie a string to an apple and swing it in circles, the string keeps the apple from moving in a straight line. We say that the apple's inertia pulls on the string. If the string breaks, the apple takes off in a straight line. Something like this happens when an apple falls off a tree. Before it falls, the branch prevents it from moving through space. The apple on the branch is at rest (relative to the earth) but speeding along its time coordinate because it is constantly getting older. If there were no gravitational field, this travel along the time coordinate would be graphed as a straight line on a four-dimensional graph. But the earth's gravity is curving space-time in the neighbourhood of the apple.

This forces the apple's world line to become a curve. When the apple breaks away from the branch, it continues to move through space-time, but (being a lazy apple) it now "straightens" its path and takes a geodesic. We see this geodesic as the apple's fall and attribute the fall to gravity. If we like, however, we can say that the apple's inertia, after the apple is suddenly released from its curved path, carries it to the ground. After the apple falls, suppose a boy comes along and kicks it with his bare foot. He shouts in pain because the kick hurts his toes. A Newtonian would say that the apple's inertia resisted his kick. An Einsteinian can say the same thing, but he can also say, if he prefers, that the boy's toes caused the entire cosmos (including the toes) to accelerate backward, setting up a gravitational field that pulled the apple with great force against his toes. It is all a matter of words. Mathematically the situation is described by one set of space-time field equations, but it can be talked about informally (thanks to the principle of equivalence) in either of two sets of Newtonian phrases.

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