The central theme of the General Theory of Relativity is the equivalence of inertial to gravitational mass-that a body experiencing an acceleration has an identical relationship to one experiencing a gravitational field. In essence, an accelerating body, in an area of space where gravitating bodies are not acting, experiences, in effect, a gravitational field.

Einstein used a man in an elevator to show how the General Theory works. He supposed that a man in an elevator which is experiencing an accelerating force equivalent to what will give him his natural weight in the gravitational field of Earth will not be able to know whether he is, indeed, standing in an elevator on Earth, or being accelerated at one "g" (the unit of the acceleration due to the earth's gravitational field) in space.

In a second case, if the man is in "free-fall," he will not be able to know whether he is in a freely falling elevator near the surface of the earth, or simply floating in space.

Einstein argued that, for all practical (and theoretical) purposes, in both cases, it makes no difference-and he proved this both analytically and mathematically.

Next, he wanted to show what would happen if the man in the elevator were able to measure the "bending" of a beam of light in a moving elevator. Suppose that there is a pinhole in the side of the elevator, through which a laser beam can be projected, and that the man inside is able to measure (very accurately) how much that beam will bend due to the elevator's motion.

Figure 4.1 Man in the elevator undergoing simple motion (or stationary) sees the laser beam come through in a straight line.

Suppose, now, that the elevator (with the man inside) is in space, far away from gravitating bodies (bodies which exert a gravitational force), but that the elevator, and the man inside, are experiencing simple upward (from the frame of reference of the man in the elevator) motion. The elevator will be doing a "fly-by," since we do not want the laser (or the beam) to be moving "with" the elevator-so the aim and the timing will be very precise. The man inside the elevator knows of the experiment, so he will be awaiting the "arrival" of the laser beam at the appropriate instant and will measure its bending at the precise moment that the direction of his motion is exactly perpendicular to the to the line connecting him to the source of the laser beam. This is shown in Figure 4.1.

Figure 4.2 From the stationary observer's view, to make the angle straight for the passenger, he must aim it at an angle.

In this case, the man inside the elevator is experiencing free-fall, even though he is moving, and the laser beam does not bend (from his perspective). The beam does, however, have to angled, from the stationary observer's point of view, so that it will not appear angled to the man in the elevator. This is done to compensate for the motion of the elevator with respect to the observer. To the stationary observer operating the laser beam, things would look like Figure 4.2. Unless, however, the elevator is moving extremely fast, the angle in this drawing is highly exaggerated.

So the motion of the elevator is compensated for by adjusting the angle of the beam at the site of the laser source. Now, to understand where Einstein got his ideas about general relativity, the elevator must be accelerated, and the man inside will suspect that he is either in a gravitational field or being pulled with a constant acceleration by some external force. Before I go on, however, I should explain why the beam must be angled to make it appear straight for the man in the elevator (this point is actually crucial to the argument).

Figure 4.3 The girl throwing the ball to the boy must lead the motion of the truck in her throw, otherwise, the ball will fall behind the truck after it is gone.

Suppose a boy is sitting in the back of a moving pick-up truck and a girl on the side of the road wants to throw a ball to him. If she wants the boy to catch the ball, then she must "lead" the truck in her throw, as shown in Figure 4.3.

Light, of course, acts the very same way, and this is why the light ray must lead the elevator.

The next step in this (thought) experiment is to give the man inside the elevator the impression that he is in a gravitational field, say, the same as on the surface of the Earth. This is done by applying a constant force which will give the elevator a constant acceleration of 32 feet per-second, per-second (32 ft/s²) or, in the metric system, 9.8 meters per-second squared (m/s²).

Figure 4.4 In an accelerated elevator, the laser beam bends because of the elevator's increasing velocity.

Now the man and the elevator are accelerated past the laser, and the beam is timed so that the man can measure it at the appropriate instant. But this time, he will measure the angle of the laser beam in several different places along its path. Because of the acceleration of the elevator, however, he finds that the beam is bending downward, as shown in Figure 4.4.

Of course, in this example, the bending of the light beam is tremendously exaggerated (which must be done in order to understand this concept). In real life, there are no instruments available to us which can detect such a minute shift of direction like this.

The reason for the bending of the light beam in the elevator is this; In the experiment with the girl on the side of the road and the boy in the truck, the girl would have had to throw the ball at a particular angle depending on the speed of the truck. If the truck were moving faster, the angle of her throw would have to be greater.

For the same reason, if the elevator and the man inside are accelerating, then the speed of the elevator is changing with respect to the observer and the source of the laser beam. As a result, when the laser beam enters the elevator at the pinhole in its side, it is assumed to be parallel to the floor of the elevator. As it moves across the elevator, say to a quarter of the way, the speed of the elevator, due to its acceleration, has changed. As a result, the laser beam is no longer angled correctly for the new velocity, and the beam will be angled slightly downward.

As it moves across the elevator to the half-way point, the new, increased velocity is well removed from the original velocity and the angle of the beam is even steeper. By the time the beam has reached all the way across the elevator to the far wall, the man inside the elevator has observed a curvature in the light beam as shown in Figure 4.4.

This was one of Einstein's arguments, and since he had already shown that it did not make any difference whether the man in the elevator was experiencing a force due to acceleration or a force due to gravity, it followed that light should actually bend in a gravitational field.

Figure 4.5 The precession of Mercury's orbit is a rotation of the ellipse that describes its orbit.

From this idea, he was able to predict two very important things. First, he predicted an orbital precession of 43 seconds of arc per century of the planet Mercury (closest to the Sun) as a result of the sun's gravity. This is depicted in Figure 4.5.

Second, he predicted that light coming from far away stars would bend around the Sun as it traveled through its large gravitational field, as seen in Figure 4.6.

Figure 4.6 Light from distant stars bends around the Sun's gravitational field, but this can only be observed during a solar eclipse.

The precession of Mercury's orbit was known about prior to Einstein's statements and had been measured to be off, failing to agree with Newton's laws of motion, by a certain amount since 1859. Einstein's calculations explained why the measurements were off and verified how closely those measurements had been made.

The second prediction was verified by the British astronomer Arthur Eddington in 1919, who led one of two expeditions during World War I to observe a total eclipse of the Sun. Einstein's theories gained greater acceptance in the scientific community and world-wide critical acclaim.

It was Herman Minkowski's notion that time and space were inextricably linked to one another in some sort of four-dimensional continuum in which space and time were woven into a "fabric" that would bend and twist as a result of being acted upon by matter. He called this the space-time continuum.

Einstein did not like the idea of the space-time continuum at first, but he later accepted it since there really was no other explanation to many of the ideas he, himself, had introduced. The notion that space should "bend" around a gravitational body was repugnant to him. He suggested that Minkowski's idea of the four-dimensional continuum bore an uncanny resemblance to the aether suggested by Descartes and, perhaps, was not a reasonable solution to the problem at hand.

According to his own theories, space and time were interlocked in a continuous existence of reality-a continuum. Much as he wanted to, he was not able to reduce light or matter to more basic constituents and finally decided that these were, indeed, irreducible causalities. Matter had extension in space and time and therefore, the idea of the non-existence of space was meaningless. The gravitational field, likewise, was irreducible, and therefore also had extension in space and time (i.e. space-time).

Once it was decided that the aether did not exist to "carry" electromagnetic waves, space itself became an enigma. Suddenly, time became a part of the "light" problem-space could no longer "act" alone. The idea held and has influenced our thinking about this subject right up to this day.

Figure 4.7 The conic view of space-time. Events affecting the present moment cannot extend beyond the boundary of the cone, either in the past or the future.

One of the visual instruments used to depict the space-time continuum is the conic section representing allowable activities within the continuum. In Figure 4.7, any event which occurs at the "present" point of existence will have no effect on matter or space outside of the "future" part of the cone. Likewise, no event in the present can be affected by any activity outside of the "past" part of the cone. Since c (the speed of light) is the governing speed limit of matter and energy in the universe, the surface of the conic section is defined by that velocity, where

Note that this is a "three-space" representation, as opposed to a four-space representation. In this example, one of the three coordinates of x, y and z is missing. In other words, this figure is depicted in two-dimensional space, and the third dimension is t, for time. A four-space (space-time) version of this diagram would be very difficult to draw or represent on paper.

Einstein eventually ended up arguing in favor of the space-time continuum. In one of his arguments, he suggested somewhat of the following: Suppose there was a box (constructed of whatever we wish), initially devoid of matter, and within that box, another box, also devoid of matter. Since the first (large) box, could be used to define a region of "empty" space, then we no longer would necessarily need the box to have thick walls or even be made of anything; the walls of the box could be infinitely thin.

The second (smaller) box within would have similar properties and would define a region of space within the larger box and could move around within it. Also within the large box could now be placed various bits and pieces of matter which could be used to define the contents of the box, and likewise within the smaller box. Finally, he suggested that, if the larger box could be used to define a particular volume of space, it could be enlarged to envelope an even bigger volume of space, and furthermore, as such, it could be finally enlarged to include all of space. From this, he attempted to show that the non-existence of space was an untenable notion.

In his next argument, he suggested that since light would bend in a gravitational field, then the gravitational field had to exist, but not so much in the way that we seem to understand it.

One of the results of the General Theory of Relativity was depicted as the Gaussian curvature of space in a gravitational field. In Gaussian curvature, spatial (volumetric) elements (x, y and z coordinates) are said to have g_m,n (gravitational) factors representing the "bending" of space due to the presence of a gravitational body. A two dimensional analog of this is to place a heavy metal ball on a square sheet of strong rubber, pinned at the corners and bearing criss-crossed lines drawn on it-forming squares which define a Cartesian coordinate system (x and y components). The heavy ball represents a gravitating body of mass in space and the amount of lengthening in the lines are represented by the g_m,n factors. This is shown in figure 4.8.

Figure 4.8 Two dimensional picture of space-time. The heavy metal ball at the center represents the effect of gravity in local space.

Einstein lauded Descartes in his expression of the "continuous" universe. From Einstein's point of view, the concept of "empty space" belonged to "pre-scientific thought."

One of the most important byproducts of Einstein's Theories was the equating of matter to energy. This relationship is reflected most distinctively in Equation 1.1 found on the first page of the introduction to this book. It is undoubtedly the most famous equation ever written.

Unfortunately, few people even know what the equation actually means. What it means, essentially, is that, properly operated upon, ordinary matter can be directly converted into pure energy. Of course, again, most people do not realize that, not only is matter convertible to energy, but only a small amount of matter can be converted into a fantastic amount of energy-which led us into the construction of the very first atomic bomb.

The most common example of this type of energy release is the collision of an electron with a positron, in which both particles are annihilated, thereby releasing a total of 1.022 MeV of energy (each one containing .511 MeV separately). These types of collisions take place very often (more often than most of us realize), which means that there are actually lots of positrons around.

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Home Begin Preface Acknowledgements Contents Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Appendix A Appendix B1 Appendix B2 Appendix C1 Appendix C2 Appendix D Appendix E Appendix F Appendix G General References Future Books About the Front Cover About the Author Index