RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY Objectives Describe how an object s momentum changes as it approaches the speed of light. (16.1) Describe how mass and energy are related. (16.2) Describe how the correspondence principle applies to special relativity. (16.3) State the principle of equivalence. (16.4) Describe the relationship between the presence of mass and the curvature of spacetime. (16.5) Describe Einstein s predictions based on his theory of general relativity. (16.6) discover! MATERIALS plastic garbage bag, trash can, pool ball, marble ANALYZE AND CONCLUDE 1. 2. 3. 302 Slow moving marbles rapidly spiral into the central ball. Faster moving marbles should orbit about the central ball. Depending on the direction of launch, orbits may be circular or elliptical. A heavier ball will make a steeper well, simulating a stronger gravitational field. Done with care, the marble may nicely simulate a planet orbiting the sun or a probe orbiting Earth. 16 302 No material object, particle or future spaceship, can be accelerated to the speed of light. Why this is so has to do with momentum and energy, which, in relativity theory, have new definitions. One of the most celebrated outcomes of special relativity is the discovery that mass and energy are one and the same thing as described by E mc 2. Einstein s general theory of relativity, developed a decade after his special theory of relativity, offers another celebrated outcome, an alternative to Newton s theory of gravity. Both theories of relativity have changed the way we see the universe. discover! RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY... THE BIG IDEA How Can Space-Time be Modeled? 1. Stretch a plastic garbage bag tightly across the top of a trash can. Tape the edges of the bag to the side of the can. 2. Place a pool ball or other heavy sphere in the center of the garbage bag. This should cause the bag to sag in the center. 3. Launch a marble by giving it a velocity tangent to the circumference of the can. 4. Try launching the marble with a variety of initial velocities. According to special relativity, mass and energy are equivalent. According to general relativity, gravity causes space to become curved and time to undergo changes. Analyze and Conclude 1. Observing Describe the motion of the marble. What effect does changing the initial speed and direction of the marble have on the shape of the orbit? 2. Predicting How might changing the mass of the heavy central sphere affect the motion of the marble? 3. Making Generalizations How closely does this model represent the motion of Earth satellites?
16.1 Momentum and Inertia in Relativity If we push an object that is free to move, it will accelerate. If we maintain a steady push, it will accelerate to higher and higher speeds. If we push with a greater and greater force, we expect the acceleration in turn to increase. It might seem that the speed should increase without limit, but there is a speed limit in the universe the speed of light. In fact, we cannot accelerate any material object enough to reach the speed of light, let alone surpass it. Newtonian and Relativistic Momentum We can understand this from Newton s second law, which Newton originally expressed in terms of momentum: F mv/ t (which reduces to the familiar F ma, or a F/m). The momentum form, interestingly, remains valid in relativity theory. Recall from Chapter 8 that the change of momentum of an object is equal to the impulse applied to it. Apply more impulse and the object acquires more momentum. Double the impulse and the momentum doubles. Apply ten times as much impulse and the object gains ten times as much momentum. Does this mean that momentum can increase without any limit, even though speed cannot? Yes, it does. We learned that momentum equals mass times velocity. In equation form, p mv (we use p for momentum). To Newton, infinite momentum would mean infinite speed. Not so in relativity. Einstein showed that a new definition of momentum is required. It is mv p 1 v 2 c 2 where v is the speed of an object and c is the speed of light. This is relativistic momentum, which is noticeable at speeds approaching the speed of light. Notice that the square root in the denominator looks just like the one in the formula for time dilation in Chapter 15. It tells us that the relativistic momentum of an object of mass m and speed v is larger than mv by a factor of 1/ 1 (v 2 /c 2 ). As an object approaches the speed of light, its momentum increases dramatically. As v approaches c, the denominator of the equation approaches zero. This means that the momentum approaches infinity! An object pushed to the speed of light would have infinite momentum and would require an infinite impulse, which is clearly impossible. So nothing that has mass can be pushed to the speed of light, much less beyond it. Hence, we see that c is the speed limit in the universe. At least one thing reaches the speed of light light itself! But a particle of light has no rest mass. A material particle can never be brought to the speed of light. Light can never be brought to rest. For: Links on speed of light Visit: www.scilinks.org Web Code: csn 1601 16.1 Momentum and Inertia in Relativity Key Terms relativistic momentum, rest mass Common Misconception The momentum of an object is always simply its mass 3 velocity. FACT The relativistic momentum of an object of mass m and speed v is actually larger than mv. Teaching Tip State that if you push an object that is free to move, it accelerates in accord with Newton s second law, a 5 F/m. The momentum version of Newton s second law, F 5 Dp/Dt, says that if you push an object that is free to move, its momentum increases. Both the acceleration and the change-of-momentum versions of the second law give the same result. However, for very high speeds, the momentum version is more accurate. F 5 Dp/Dt holds for all speeds, even those near the speed of light as long as the relativistic expression for momentum is used. CHAPTER 16 RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY 303 303
Teaching Tip Write the expression for relativistic momentum on the board. Point out that it differs from the classical expression for momentum by its denominator. A common interpretation is that of a relativistic mass, m 5 m o / 1 2 v 2 /c 2, times velocity v. Because the increase in m with speed is directional (as is length contraction), and P rather than m is a vector, the concept of momentum increase rather than mass increase is preferred in advanced physics courses. Either treatment of relativistic mass or relativistic momentum, however, leads to the same description of rapidly moving objects in accord with observations. Teaching Tip Show that for small speeds the relativistic momentum equation reduces to the familiar mv (just as for small speeds t 5 t o in time dilation). Then show that when v approaches c, the denominator of the equation approaches zero. This means that the momentum approaches infinity! An object pushed to the speed of light would have infinite momentum and would require an infinite impulse (force 3 time), which is clearly impossible. Nothing material can be pushed to the speed of light. The speed of light c is the upper speed limit in the universe. CONCEPT As an object CHECK approaches the speed of light, its momentum increases dramatically. 304 Teaching Resources Reading and Study Workbook PresentationEXPRESS Interactive Textbook Conceptual Physics Alive! DVDs Special Relativity II FIGURE 16.1 If the momentum of the electrons were equal to the Newtonian value mv, the beam would follow the dashed line. But because the relativistic momentum, or inertia in motion, is greater, the beam follows the stiffer trajectory shown by the solid line. At ordinary speeds, an object s momentum is simply its classical value, mv. For example, at 30 m/s (0.0000001c), the relativistic momentum differs from the classical value by less than one trillionth of a percent. Newton s definition of momentum is valid at low speeds. 304 What if v is much less than c? Then the denominator of the equation is nearly equal to 1 and p is nearly equal to mv. Newton s definition of momentum is valid at low speed. Trajectory of High-Speed Particles We often say that a particle pushed close to the speed of light acts as if its mass were increasing, because its momentum its inertia in motion increases more than its speed increases. The rest mass of an object, represented by m in the equation on the previous page, is a true constant, a property of the object no matter what speed it has. Subatomic particles are routinely pushed to nearly the speed of light. The momenta of such particles may be thousands of times more than the Newton expression mv predicts. One way to look at the momentum of a high-speed particle is in terms of the stiffness of its trajectory. The more momentum it has, the harder it is to deflect it the stiffer is its trajectory. If it has a lot of momentum, it more greatly resists changing course. This can be seen when a beam of electrons is directed into a magnetic field, as shown in Figure 16.1. Charged particles moving in a magnetic field experience a force that deflects them from their normal paths. For a particle with a small momentum, the path curves sharply. For a particle with a large momentum, the path curves only a little its trajectory is stiffer. Even though one particle may be moving only a little faster than another one say 99.9% of the speed of light instead of 99% of the speed of light its momentum will be considerably greater and it will follow a straighter path in the magnetic field. Through such experiments, physicists working with subatomic particles at atomic accelerators verify every day the correctness of the relativistic definition of momentum and the speed limit imposed by nature. CONCEPT CHECK How does an object s momentum change as it approaches the speed of light?
16.2 Equivalence of Mass and Energy A remarkable insight of Einstein s special theory of relativity is his conclusion that mass is simply a form of energy. A piece of matter, even if at rest and even if not interacting with anything else, has an energy of being called its rest energy. Einstein concluded that it takes energy to make mass and that energy is released when mass disappears. Rest mass is, in effect, a kind of potential energy. Mass stores energy, just as a boulder rolled to the top of a hill stores energy. When the mass of something decreases, as it can do in nuclear reactions, energy is released, just as the boulder rolling to the bottom of the hill releases energy. Conversion of Mass to Energy The amount of rest energy E is related to the mass m by the most celebrated equation of the twentieth century, E mc 2 where c is again the speed of light. This equation gives the total energy content of a piece of stationary matter of mass m. Mass and energy are equivalent anything with mass also has energy. In ordinary units of measurement, the speed of light c is a large quantity and its square is even larger. This means that a small amount of mass stores a large amount of energy. The quantity c 2 is a conversion factor. It converts the measurement of mass to the measurement of equivalent energy. It is the ratio of rest energy to mass: E/m c 2. Its appearance in either form of this equation has nothing to do with light and nothing to do with motion. The magnitude of c 2 is 90 quadrillion (9 10 16 ) joules per kilogram. One kilogram of matter has an energy of being equal to 90 quadrillion joules. Even a speck of matter with a mass of only 1 milligram has a rest energy of 90 billion joules. (This is equivalent to the kinetic energy of a 3-ton truck moving at over 20 times the speed of sound!) Examples of Mass-Energy Conversions Rest energy, like any form of energy, can be converted to other forms. When we strike a match, for example, a chemical reaction occurs and heat is released. Phosphorus atoms in the match head rearrange themselves and combine with oxygen in the air to form new molecules. The resulting molecules have very slightly less mass than the separate phosphorus and oxygen molecules. From a mass standpoint, the whole is slightly less than the sum of its parts, but not by very much by only about one part in a billion. For all chemical reactions that give off energy, there is a corresponding decrease in mass. E = mc 2 says that mass is congealed energy. Mass and energy are two sides of the same coin. think! Can we look at the equation E mc 2 in another way and say that matter transforms into pure energy when it is traveling at the speed of light squared? Answer: 16.2 16.2 Equivalence of Mass and Energy Key Term rest energy The mass energy equivalence is important and usually generates high interest. Common Misconception E 5 mc 2 means that energy is mass traveling at the speed of light squared. FACT The equation gives the total energy content of a piece of stationary matter of mass m. Teaching Tip Write E 5 mc 2 on the board. State that this is the most celebrated equation of the twentieth century. It relates energy and mass. Every material object is composed of energy energy of being. This energy of being is appropriately called rest energy. Teaching Tip Stress that c 2 is a constant conversion factor and is NOT the speed of the mass. Also, stress that the equation E 5 mc 2 is NOT restricted to chemical and nuclear reactions. Teaching Tip E 5 mc 2 says that mass is congealed energy. Mass and energy are two sides of the same coin. CHAPTER 16 RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY 305 305
Teaching Tip State that the mass of something is actually the internal energy within it and that this energy can be converted to other forms of energy, such as light. Teaching Tip Mention that the 4.5 million tons of matter that is converted to radiant energy by the sun each second is carried (as radiant energy) through space, so when we speak of matter being converted to energy, we are merely converting from one form to another from a form with one set of units, perhaps, to another. Because of the mass energy equivalence, in any reaction that takes into account the whole system, the total amount of mass 1 energy does not change. FIGURE 16.2 In one second, 4.5 million tons of rest mass are converted to radiant energy in the sun. In nuclear reactions, the decrease in rest mass is considerably more than in chemical reactions about one part in a thousand. This decrease of mass in the sun by the process of thermonuclear fusion bathes the solar system with radiant energy and nourishes life. The sun is so massive that in a million years only one tenmillionth of the sun s rest mass will have been converted to radiant energy. The present stage of thermonuclear fusion in the sun has been going on for the past 5 billion years, and there is sufficient hydrogen fuel for fusion to last another 5 billion years. It is nice to have such a big sun! Nuclear power plants, such as the one shown in Figure 16.3, make use of the equivalence of mass and energy to produce enormous amounts of energy. FIGURE 16.3 Saying that a power plant delivers 90 million megajoules of energy to its consumers is equivalent to saying that it delivers 1 gram of energy to its consumers, because mass and energy are equivalent. CONCEPT Mass and energy are CHECK equivalent anything with mass also has energy. 306 Teaching Resources Reading and Study Workbook Problem-Solving Exercises in Physics 9-2 PresentationEXPRESS Interactive Textbook 306 The equation E mc 2 is not restricted to chemical and nuclear reactions. A change in energy of any object at rest is accompanied by a change in its mass. The filament of a lightbulb has more mass when it is energized with electricity than when it is turned off. A hot cup of tea has more mass than the same cup of tea when cold. A wound-up spring clock has more mass than the same clock when unwound. But these examples involve incredibly small changes in mass too small to be measured by conventional methods. No wonder the fundamental relationship between mass and energy was not discovered until the 1900s. The equation E mc 2 is more than a formula for the conversion of rest mass into other kinds of energy, or vice versa. It states that energy and mass are the same thing. Mass is simply congealed energy. If you want to know how much energy is in a system, measure its mass. For an object at rest, its energy is its mass. Shake a massive object back and forth; it is energy itself that is hard to shake. CONCEPT CHECK What is the relationship between mass and energy?
16.3 The Correspondence Principle If a new theory is to be valid, it must account for the verified results of the old theory. The correspondence principle states that new theory and old must overlap and agree in the region where the results of the old theory have been fully verified. It was advanced as a principle by the Danish physicist Niels Bohr earlier in this century when Newtonian mechanics was being challenged by both quantum theory and relativity. According to the correspondence principle, if the equations of special relativity (or any other new theory) are to be valid, they must correspond to those of Newtonian mechanics classical mechanics when speeds much less than the speed of light are considered. The relativity equations for time dilation, length contraction, and momentum are t 0 t 1 L L 0 v 2 c 2 1 v2 mv p 1 v 2 c 2 We can see that each of these equations reduces to a Newtonian value for speeds that are very small compared with c. Then, the ratio (v/c) 2 is very small, and for everyday speeds may be taken to be zero. The relativity equations become t 0 t t 0 1 0 c 2 L L 0 1 0 L 0 mv p mv 1 0 So for everyday speeds, the time scales and length scales of moving objects are essentially unchanged. Also, the Newtonian equation for momentum holds true (and so does the Newtonian equation for kinetic energy). But when the speed of light is approached, things change dramatically. Near the speed of light Newtonian mechanics change completely. The equations of special relativity hold for all speeds, although they are significant only for speeds near the speed of light. Equations remind us that you can never change only one thing. Change a term on one side of an equation and you change something on the other side. Much of nature is built on patterns, and looking for those patterns is the primary preoccupation of both artists and scientists. We connect things that were always there but never put together in our thinking. 16.3 The Correspondence Principle Key Terms correspondence principle, general theory of relativity Teaching Tip The equations in this section serve only to illustrate the correspondence principle. It is not necessary that your students memorize them. Teaching Tip Show your students that when small speeds are involved, the relativity formulas reduce to the everyday observation that time, length, and the momenta of things do not appear any different when moving. This is because the differences are too tiny to detect. The correspondence principle is one of the neater principles of physics and is a guide to clear and rational thinking not only about the ideas of physics, but for all good theory even in areas as far removed from science as government, religion, and ethics. Simply put, if a new idea is valid, then it ought to be in harmony with ideas that are valid in the region it overlaps. CHAPTER 16 RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY 307 307
CONCEPT According to the CHECK correspondence principle, if the equations of special relativity (or any other new theory) are to be valid, they must correspond to those of Newtonian mechanics classical mechanics when speeds much less than the speed of light are considered. Teaching Resources Reading and Study Workbook Concept-Development Practice Book 16-1 PresentationEXPRESS Interactive Textbook 16.4 General Relativity Key Term principle of equivalence Einstein actually imagined himself in elevators, certainly more common at the time than spaceships. So we see that advances in science take place not by discarding the current ideas and techniques, but by extending them to reveal new implications. Einstein never claimed that accepted laws of physics were wrong, but instead showed that the laws of physics implied something that hadn t before been appreciated. The special theory of relativity is about motion observed in uniformly moving frames of reference, which is why it is called special. Einstein s conviction that the laws of nature should be expressed in the same form in every frame of reference, accelerated as well as non-accelerated, was the primary motivation that led him to develop the general theory of relativity a new theory of gravitation, in which gravity causes space to become curved and time to slow down. CONCEPT CHECK How does the correspondence principle apply to special relativity? 16.4 General Relativity Einstein was led to a new theory of gravity by thinking about observers in accelerated motion. He imagined himself in a spaceship far away from gravitational influences, as shown in Figure 16.4. In such a spaceship at rest or in uniform motion relative to the distant stars, Einstein and everything within the ship would float freely; there would be no up and no down. But if rocket motors were activated to accelerate the ship, things would be different; phenomena similar to gravity would be observed. The wall adjacent to the rocket motors (the floor ) would push up against any occupants and give them the sensation of weight. If the acceleration of the spaceship were equal to g, the occupants could well be convinced the ship was not accelerating, but was at rest on the surface of Earth. FIGURE 16.4 Imagine being on a spaceship far away from gravitational influences. a. Everything inside is weightless when the spaceship isn t accelerating. b. When the spaceship accelerates, an occupant inside feels gravity. 308 308
The Principle of Equivalence Einstein concluded, in what is now called the principle of equivalence, that gravity and accelerated motion through space-time are related. The principle of equivalence states that local observations made in an accelerated frame of reference cannot be distinguished from observations made in a Newtonian gravitational field. There is no way you can tell whether you are being pulled by gravity or being accelerated. The effects of gravity and the effects of acceleration are equivalent. To examine this new gravity in the accelerating spaceship, Einstein considered the consequence of dropping two balls, say one of wood and the other of lead. Figure 16.5 shows that when released, the balls would continue to move upward side by side with the velocity that the ship had at the moment of release. If the ship were moving at constant velocity (zero acceleration), the balls would appear to remain suspended in the same place since both the ship and the balls move the same amount. But if the ship were accelerating, the floor would move upward faster than the balls, which would soon be intercepted by the floor. Both balls, regardless of their masses, would meet the floor at the same time. Occupants of the spaceship might attribute their observations to the force of gravity. FIGURE 16.5 To an observer inside the accelerating ship, a lead ball and a wood ball accelerate downward together when released, just as they would if pulled by gravity. Teaching Tidbit Space elevator: Satellites in synchronous orbit can drop vertical cables to the surface of Earth where they can be attached. Rather than rocketing material to the satellite, material can be lifted in elevator fashion! Both interpretations of the falling balls are equally valid. Einstein incorporated this equivalence, or impossibility of distinguishing between gravitation and acceleration, in the foundation of his general theory of relativity. The principle of equivalence would be interesting but not revolutionary if it applied only to mechanical phenomena. But Einstein went further and stated that the principle holds for all natural phenomena, including optical, electromagnetic, and mechanical phenomena. CHAPTER 16 RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY 309 309
Teaching Tip Tell your students that the three most important theories of physics in the twentieth century are the special theory of relativity (1905), the general theory of relativity (1915), and the theory of quantum mechanics (1926). The first and third theories have been focal points of interest and research since their inceptions, yet the second, general relativity, has been largely ignored by physicists until recently. New interest stems from the interest of pulsars, quasars, compact X-ray sources, and black holes. All these indicate the existence of very strong gravitational fields described only by general relativity. The move is now on to a quantum theory of gravitation that will agree with general relativity for macroscopic objects. FIGURE 16.6 A ball is thrown sideways in an accelerating spaceship in the absence of gravity. a. An outside observer sees the ball travel in a straight line. b. To an inside observer, the ball follows a parabolic path as if in a gravitational field. Bending of Light by Gravity Just as a tossed ball curves in a gravitational field, so does a light beam. Consider a ball thrown sideways in a stationary spaceship in the absence of gravity. The ball will follow a straight-line path relative to both an observer inside the ship and to a stationary observer outside the spaceship. But if the ship is accelerating, the floor overtakes the ball and it hits the wall below the level at which it was thrown. An observer outside the ship still sees a straight-line path, as illustrated in Figure 16.6a, but to an observer in the accelerating ship, the path is curved; it is a parabola, as shown in Figure 16.6b. Figure 16.7 illustrates that the same holds true for a beam of light. The only difference is in the amount of path curvature. As shown in Figure 16.8, if a ball were thrown at nearly the speed of light, the curvature of its path would be nearly the same as that of the light beam. 310 310 FIGURE 16.7 A light ray enters the spaceship horizontally through a side window. a. Like the ball in Figure 16.6, light appears, to an outside observer, to be travelling horizontally in a straight line. b. To an inside observer, the light appears to bend.
Using his principle of equivalence, Einstein took another giant step that led him to the general theory of relativity. He reasoned that since acceleration (a space-time effect) can mimic gravity (a force), perhaps gravity is not a separate force after all; perhaps it is nothing but a manifestation of space-time. From this bold idea he derived the mathematics of gravity as being a result of curved space-time. According to Newton, tossed balls curve because of a force of gravity. According to Einstein, tossed balls and light don t curve because of any force, but because the space-time in which they travel is curved. CONCEPT CHECK What does the principle of equivalence state? 16.5 Gravity, Space, and a New Geometry Space-time has four dimensions three space dimensions (such as length, width, and height) and one time dimension (past to future). Einstein perceived a gravitational field as a geometrical warping of four-dimensional space-time. Four-dimensional geometry is altogether different from the three-dimensional geometry introduced by Euclid centuries earlier. Euclidean geometry (the ordinary geometry taught in school) is no longer valid when applied to objects in the presence of strong gravitational fields. Four-Dimensional Geometry The familiar rules of Euclidean geometry pertain to various figures that can be drawn on a flat surface. In Euclidean geometry, the ratio of the circumference of a circle to its diameter is equal to ; all the angles in a triangle add up to 180 ; and the shortest distance between two points is a straight line. The rules of Euclidean geometry are valid in flat space, but if you draw circles or triangles on a curved surface like a sphere or a saddle-shaped object, as shown in Figure 16.9, the Euclidean rules no longer hold. If you measure the sum of the angles for a triangle drawn on the outside of a ball (positive curvature), the sum of the angles is greater than 180. For a triangle drawn on a saddle (negative curvature), the sum is less than 180. FIGURE 16.8 The trajectory of a baseball tossed at nearly the speed of light closely follows the trajectory of a light beam. FIGURE 16.9 The sum of the angles of a triangle is not always 180. a. On a flat surface, the sum is 180. b. On a spherical surface, the sum is greater than 180. c. On a saddleshaped surface, the sum is less than 180. CONCEPT The principle of CHECK equivalence states that local observations made in an accelerated frame of reference cannot be distinguished from observations made in a Newtonian gravitational field. Teaching Resources Reading and Study Workbook PresentationEXPRESS Interactive Textbook 16.5 Gravity, Space, and a New Geometry Key Terms geodesic, gravitational wave CHAPTER 16 RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY 311 311
Teaching Tip One important point to make is that relativity doesn t mean that everything is relative, but rather that no matter how you view a situation, the physical outcome is the same. There is a general misconception about this. Point out that in special and general relativity the fundamental truths of nature look the same from every point of view not different from different points of view! FIGURE 16.10 The geometry of Earth s two-dimensional curved surface differs from the Euclidean geometry of a flat plane. Look at an airplane s flight path drawn on a flat map and you ll see that the line is curved. The same line drawn on the surface of a globe would be a geodesic a straight (shortest-distancebetween-two-points) line on Earth s curved surface. FIGURE 16.11 The light rays joining the three planets form a triangle. Since the sun s gravity bends the light rays, the sum of the angles of the resulting triangle is greater than 180. Similarly, the geometry of Earth s two-dimensional curved surface differs from the Euclidean geometry of a flat plane. As shown in Figure 16.10a, the sum of the angles for an equilateral triangle (the one here has the sides equal 1 Earth s circumference) is greater than 4 180. Earth s circumference is only twice its diameter, as illustrated in Figure 16.10b, instead of 3.14 times its diameter. Of course, the lines forming the triangles in Figures 16.9 and 16.10 are not straight from the three-dimensional view, but are the straightest or shortest distances between two points if we are confined to the curved surface. These lines of shortest distance are called geodesics. The path of a light beam follows a geodesic. Suppose three experimenters on planets Earth, Venus, and Mars measure the angles of a triangle formed by light beams traveling between them. The light beams bend when passing the sun, resulting in the sum of the three angles being larger than 180, as illustrated in Figure 16.11. So the three-dimensional space around the sun is positively curved. The planets that orbit the sun travel along four-dimensional geodesics in this positively curved space-time. Freely falling objects, satellites, and light rays all travel along geodesics in four-dimensional space-time. 312 312 The Shape of the Universe Although space-time is curved locally (within a solar system or within a galaxy), recent evidence shows that the universe as a whole is flat. This is a striking knifeedge condition. There are an infinite number of possible positive curvatures to space-time, and an infinite number of possible negative curvatures, but only one condition of zero curvature. A universe of zero or negative curvature is open-ended and extends without limit.
If the universe had positive curvature, it would close in on itself, just as the surface of Earth closes in on itself. If you march straight ahead on Earth, never turning, you will eventually return to your starting point. And if you shine a flashlight into a space of positive curvature, the light will eventually illuminate the back of your head (if you wait long enough!). No one knows why the universe is actually flat or nearly flat. The leading theory is that this is the result of an incredibly large and near-instantaneous inflation that took place as part of the Big Bang some 13.7 billion years ago. General relativity, then, calls for a new geometry: a geometry not only of curved space but of curved time as well a geometry of curved four-dimensional space-time. 16.5.1 Even if the universe at large has no average curvature, there s very much curvature near massive bodies. The presence of mass produces a curvature or warping of space-time; conversely, a curvature of space-time reveals the presence of mass. Instead of visualizing gravitational forces between masses, we abandon altogether the idea of force and think of masses responding in their motion to the curvature or warping of the spacetime they inhabit. General relativity tells us that the bumps, depressions, and warpings of geometrical space-time are gravity. 16.5.2 We cannot visualize the four-dimensional bumps and depressions in space-time because we are three-dimensional beings. We can get a glimpse of this warping by considering a simplified analogy in two dimensions: a heavy ball resting on the middle of a waterbed, which is illustrated in Figure 16.12. The more massive the ball, the more it dents or warps the two-dimensional surface. A marble rolled across such a surface may trace an oval curve and orbit the ball. The planets that orbit the sun similarly travel along four-dimensional geodesics in the warped space-time about the sun. Gravitational Waves Every object has mass, and therefore makes a bump or depression in the surrounding space-time. When an object moves, the surrounding warp of space and time moves to readjust to the new position. These readjustments produce ripples in the overall geometry of space-time, similar to moving a ball that rests on the surface of a waterbed. A disturbance ripples across the waterbed surface in waves; if we move a more massive ball, then we get a greater disturbance and the production of even stronger waves. The ripples that travel outward from the gravitational sources at the speed of light are gravitational waves. think! Whoa! We learned previously that the pull of gravity is an interaction between masses. And we learned that light has no mass. Now we say that light can be bent by gravity. Isn t this a contradiction? Answer: 16.5 FIGURE 16.12 Space-time near a star is curved in a way similar to the surface of a waterbed when a heavy ball rests on it. Teaching Tidbit Gravity bends around lumps of matter like light bends in lenses. Light emitted from a source travels along multiple geodesic paths to an observer who sees multiple distorted images of the source projected onto the sky. So like a light lens, a gravitational lens can produce multiple images the way a fun-house mirror does with light. The amount of gravitational bending depends on the mass. Teaching Tidbit Planets in our solar system don t crash into the sun only because their tangential velocities are sufficient for orbit. Likewise for stars in galaxies: Stars with sufficient tangential velocities orbit about the galactic center. But slower stars are pulled into and gobbled up by the galactic nucleus, which, if massive enough, is usually a black hole. CHAPTER 16 RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY 313 313
CONCEPT The presence of mass CHECK produces a curvature or warping of space-time; conversely, a curvature of space-time reveals the presence of mass. Teaching Resources Reading and Study Workbook PresentationEXPRESS Interactive Textbook 16.6 Tests of General Relativity Key Term gravitational red shift Any accelerating object produces a gravitational wave. In general, the more massive the object and the greater its acceleration, the stronger the resulting gravitational wave. But even the strongest waves produced by ordinary astronomical events are the weakest known in nature. For example, the gravitational waves emitted by a vibrating electric charge are a trillion-trillion-trillion times weaker than the electromagnetic waves emitted by the same charge. Detecting gravitational waves is enormously difficult, but physicists think they may be able to do it, and searches are under way at present. CONCEPT CHECK What is the relationship between the presence of mass and the curvature of space-time? 16.6 Tests of General Relativity Upon developing the general theory of relativity, Einstein predicted that the elliptical orbits of the planets precess about the sun, starlight passing close to the sun is deflected, and gravitation causes time to slow down. Later, his predictions were successfully tested and confirmed. FIGURE 16.13 Einstein s theory predicted that elliptical orbits of the planets should precess. Precession of the Planetary Orbits Using four-dimensional field equations, Einstein recalculated the orbits of the planets about the sun. Planets and comets travel along curved paths because of the curvature of space-time. With only one minor exception, his theory gave almost exactly the same results as Newton s law of gravity. The exception was that Einstein s theory predicted that the elliptical orbits of the planets should precess independent of the Newtonian influence of other planets, as shown in Figure 16.13. This precession would be very slight for distant planets and more pronounced close to the sun. Mercury is the only planet close enough to the sun for the curvature of space to produce an effect big enough to measure. Precession in the orbits of planets caused by perturbations of other planets was well known. Since the early 1800s astronomers measured a precession of Mercury s orbit about 574 seconds of arc per century. Perturbations by the other planets were found to account for the precession except for 43 seconds of arc per century. Even after all known corrections due to possible perturbations by other planets had been applied, the calculations of scientists failed to account for the extra 43 seconds of arc. Either Venus was extra massive or a never-discovered other planet (called Vulcan) was pulling on Mercury. And then came the explanation of Einstein, whose general relativity equations applied to Mercury s orbit predict the extra 43 seconds of arc per century! 314 314
Deflection of Starlight As a second test of his theory, Einstein predicted that starlight passing close to the sun would be deflected by an angle of 1.75 seconds of arc large enough to be measured. This deflection of starlight can be observed during an eclipse of the sun. (Measuring this deflection has become a standard practice at every total eclipse since the first measurements were made during the total eclipse of 1919.) A photograph taken of the darkened sky around the eclipsed sun reveals the presence of the nearby bright stars. The positions of the stars are compared with those in other photographs of the same part of the sky taken at night with the same telescope. In every instance, the deflection of starlight, which is illustrated in Figure 16.14, has supported Einstein s prediction. More support is provided by gravitational lensing, a phenomenon in which light from a distant galaxy is bent as it passes by a nearer galaxy in such a way that multiple images of the distant galaxy appear. Gravitational Red Shift Einstein made a third prediction that gravity causes clocks to run slow. He predicted that clocks on the first floor of a building should tick slightly more slowly than clocks on the top floor, which are farther from Earth and at a higher gravitation potential energy. As shown in Figure 16.15, if you move from a distant point down to the surface of Earth, you move in the direction that the gravitational force acts toward lower potential energy, where clocks run more slowly. From the top to the bottom of the tallest skyscraper, the difference is very small only a few millionths of a second per decade because the difference in Earth s gravitation at the bottom and top of the skyscraper is very small. For larger differences, like those at the surface of the sun compared with the surface of Earth, the clock-slowing effect is more pronounced. A clock in the deeper potential well at the surface of the sun should run measurably slower than a clock at the surface of Earth. Einstein suggested a way to measure this. FIGURE 16.14 Starlight bends as it grazes the sun. Point A shows the apparent position; point B shows the true position. (The deflection is exaggerated.) FIGURE 16.15 Gravity causes clocks to run slow. A clock at the surface of Earth runs slower than a clock farther away. Teaching Tip Explain to your students that gravitational lensing is a consequence of general relativity. The gravity of massive objects distorts the fabric of space-time and thereby the paths of light rays passing the objects. How much bending depends on the mass of the object. By measuring the bending and having a measure of how much visible matter the object possesses, investigators can infer how much dark matter must also be present in the object. Teaching Tidbit Gravitational lensing was first noticed with the famed solar eclipse of May 29, 1919, off the west coast of Africa. In 1979 the second example of gravitational lensing was found around a massive foreground cluster galaxy. Special and general relativity are my favorite parts of physics I don t understand. your honest students! CHAPTER 16 RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY 315 315
Teaching Tidbit A doublepulsar system of two radio pulsars that orbit one another quickly and with high acceleration has recently been identified for tests of general relativity. Research reports precision timing observations for a 3-year period. With mass measurements possible, four independent tests confirm the validity of general relativity at the 0.05% level in the strong-field regime. Link to SPACE SCIENCE Newton s and Einstein s Gravity Compared From Newton s law, one can calculate the orbits of comets and asteroids and even predict the existence of undiscovered planets. Even today, when computing the trajectories of space probes throughout the solar system and beyond, only ordinary Newtonian theory is used. This is because the gravitational fields of these bodies are very weak, and from the viewpoint of general relativity, the surrounding space-time is essentially flat. But for regions of more intense gravitation, where space-time is more appreciably curved, Newtonian theory cannot adequately account for various phenomena like the precession of Mercury s orbit close to the sun and, in the case of stronger fields, the gravitational red shift and other apparent distortions of space and time. These distortions reach their limit in the case of a star that collapses to a black hole, where space-time completely folds over on itself. Only Einsteinian gravitation reaches into this domain. CONCEPT Upon developing the CHECK general theory of relativity, Einstein predicted that the elliptical orbits of the planets precess about the sun, starlight passing close to the sun is deflected, and gravitation causes time to slow down. 316 Teaching Resources Reading and Study Workbook PresentationEXPRESS Interactive Textbook Next-Time Questions 16-1, 16-2 Why do we not notice the bending of light by gravity in our everyday environment? Answer: 16.6 316 think! The medieval philosopher William of Occam said that when deciding between two competing theories, choose the simpler explanation don t make more assumptions than are necessary when describing phenomena. Light traveling against gravity is observed to have a slightly lower frequency due to an effect called the gravitational red shift. Because red light is at the low-frequency end of the visible spectrum, a lowering of frequency shifts the color of the emitted light toward the red. Although this effect is weak in the gravitational field of the sun, it is stronger in more compact stars with greater surface gravity. An experiment confirming Einstein s prediction was performed in 1960 with high-frequency gamma rays sent between the top and bottom floors of a laboratory building at Harvard University. 16.6 Incredibly precise measurements confirmed the gravitational slowing of time. So measurements of time depend not only on relative motion, as we learned in special relativity, but also on gravity. In special relativity, time dilation depends on the speed of one frame of reference relative to another one. In general relativity, the gravitational red shift depends on the location of one point in a gravitational field relative to another one. It is important to note the relativistic nature of time in both special relativity and general relativity. In both theories, however, there is no way that you can extend the duration of your own experience. Others moving at different speeds or in different gravitational fields may see you aging slowly, but your aging is seen from their frame of reference never your own. As mentioned earlier, changes in time and other relativistic effects are always attributed to the other guy. CONCEPT CHECK What three predictions did Einstein make based on his general theory of relativity?
16 REVIEW For: Self-Assessment Visit: PHSchool.com Web Code: csa 1600 REVIEW Teaching Resources TeacherEXPRESS Conceptual Physics Alive! DVDs Special Relativity II Concept Summary As an object approaches the speed of light, its momentum increases dramatically. Mass and energy are equivalent anything with mass also has energy. According to the correspondence principle, if the equations of special relativity (or any other new theory) are to be valid, they must correspond to those of Newtonian mechanics when speeds much less than the speed of light are considered. The principle of equivalence states that local observations made in an accelerated frame of reference cannot be distinguished from observations made in a Newtonian gravitational field. Key Terms relativistic momentum (p. 303) rest mass (p. 304) rest energy (p. 305) correspondence principle (p. 307) general theory of relativity (p. 308) think! Answers principle of equivalence (p. 309) geodesic (p. 312) gravitational wave (p. 313) gravitational red shift (p. 316) 16.2 No, no, no! Matter cannot be made to move at the speed of light, let alone the speed of light squared (which is not a speed!). The equation E mc 2 simply means that energy and mass are two sides of the same coin. 16.5 There is no contradiction when the massenergy equivalence is understood. It s true that light is massless, but it is not energyless. The fact that gravity deflects light is evidence that gravity pulls on the energy of light. Energy indeed is equivalent to mass! The presence of mass produces a curvature or warping of space-time; conversely, a curvature of space-time reveals the presence of mass. Upon developing the general theory of relativity, Einstein predicted that the elliptical orbits of the planets precess about the sun, starlight passing close to the sun is deflected, and gravitation causes time to slow down. 16.6 Earth s gravity is too weak to produce a measurable bending. Even the sun produces only a tiny deflection. It takes a whole galaxy to bend light appreciably. CHAPTER 16 RELATIVITY MOMENTUM, MASS, ENERGY, AND GRAVITY 317 317
ASSESS Check Concepts 1. Infinite 2. The mass of an object or particle at rest 3. It doesn t bend as much. It has a stiffer trajectory. 4. Mass and energy are two sides of the same coin. 5. c 2, or 9 3 10 16 J/kg 6. No; it is universal. 7. Solar, chemical, and nuclear power (Check students work for other examples.) 8. It emits energy. 9. For nuclear reactions, about one part per thousand; for chemical reactions, about one part per billion 10. Old and new laws agree in the region of overlap. 11. The same results as with the simpler classical formulas 12. Overlap smoothly 13. Local observations made in an accelerated frame of reference cannot be distinguished from observations made in a Newtonian gravitational field. 14. Both are attracted by gravity. Baseballs are noticeably deflected only because they travel with less speed than photons. 15. A line of shortest distance between two points 16. Planets travel along 4-dimensional geodesics in warped space-time about the sun. 17. Displacement of stars whose light grazes the sun during a solar eclipse 18. The higher clock; the one at the top of the skyscraper runs faster. 16 ASSESS Check Concepts Section 16.1 1. What would be the momentum of an object if it were pushed to the speed of light? 2. What is meant by rest mass? 3. What relativistic effect is evident when a beam of high-speed charged particles bends in a magnetic field? Section 16.2 4. What is meant by the equivalence of mass and energy? That is, what does the equation E mc 2 mean? 5. What is the numerical quantity of the ratio rest energy/rest mass? 6. Does the equation E mc 2 apply only to reactions that involve the atomic nucleus? Explain. 7. What evidence is there for the equivalence of mass and energy? 8. When the mass of something decreases, does it emit or absorb energy? 9. Compare the relative amounts of mass lost in nuclear reactions and in chemical reactions. Section 16.3 10. What is the correspondence principle? 11. What results when low everyday speeds are used in the relativistic equations for time and length? 12. Do the equations of Newton and Einstein overlap, or is there a sharp break between them? Section 16.4 13. State the principle of equivalence. 14. Compare the bending of the paths of baseballs and of photons by a gravitational field. Section 16.5 15. What is a geodesic? 16. According to general relativity, in what paths do planets travel as they orbit the sun? Section 16.6 17. What is the evidence for light bending near the sun? 18. Which runs faster, a clock at the top of the Sears Tower in Chicago or a clock on the shore of Lake Michigan? 318 318