Spacetime and Gravity
LEARNING GOALS
53.1 Einstein's 5econd Revolution 53.4 Testing General Relativity • What are the major ideas of • How do we test the predictions general relativity? of the general theory of relativity? • Is all motion relative? • What are gravitational waves? 53.2 Understanding 5pacetime 53.5 Hyperspace, Wormholes, • What is spacetime? and Warp Drive • What is curved spacetime? • Where does science end and science 53.3 A New View of Gravity fiction begin? • What is gravity? • What is a black hole? • How does gravity affect time?
~i?A The eternal mystery of the world is its comprehensibility. The fact that it is comprehensible is a miracle. Albert Einstein
hat is gravity! Newton considered gravity to be a mysterious force that somehow reaches W across vast distances of space to hold the Moon in orbit around Earth and the planets in orbit around the Sun. His law of gravity explained the actions and conse a Travelers goi ng in opposite b Two space probes launched quences of this mysterious force but said nothing about directions along paths that are in opposite di rections in Earth how the force is transmitted through space. as straight as possibl e will meet orbit will meet as they orbit Einstein removed the mystery of how gravity acts at a as they go around the Earth, a Earth. a fact that we usuailly distance. As he extended his theory of relativity, Einstein fact that we attribute to the attribute to the mysterious curvature of Earth's su rface. fo rce of gravity. found that he could explain gravity in terms of the struc ture of space and time. In his view, the orbits of the Moon Figure 53 .1 Travelers on Earth's surface and orbiting objects follow similar-shaped paths. but we usually explain these paths in and the planets are as natural as motion in a straight line. very different ways. As we investigate Einstein's revolutionary view of grav ity, we will see that the consequences of his discoveries abound in astronomy and explain phenomena ranging from posite directions and neither has ever fired its engines, the the peculiar orbit of Mercury to black holes. We will probes have somehow met. This might at first sound sur also see how space and time merge into a four-dimensional prising, but in fact this situation arises quite naturally with spacetime that determines the overall structure of the orbiting objects. If you launch two probes in opposite di universe. rections from a space station, they will meet as they orbit Earth (Figure S3. 1b). Since the time of Newton, we've generally explained 53.1 Einstein's Second the curved paths of the two probes as an effect caused by the Revolution force of gravity. However, by analogy with the explorers journeying in opposite directions on Earth, might we in Imagine that you and everyone around you believe the Earth stead conclude that the probes meet because space is some to be flat. As a wealthy patron of the sciences, you decide to how curved? The idea that space could be curved certainly sponsor an ex pedition to the far reaches of the world. You sounds strange at first. While it's easy to visualize a surface select two fearless explorers and give them careful instruc curving through space, our minds cannot visualize three tions. Each is to journey along a perfectl y straight path, but dimensional space itself as being curved. The idea that they are to travel in opposite directions. You provide each space can be curved lies at the heart of Einstein's second with a caravan for land-based travel and boats for water revolution-a revolutio nary view of gravity contained in crossings, and you tell each to turn back only after discov his general th eory of relativity, published in 1915. ering "something ex traordinary." Some time later, the two explorers return. You ask, "Did • What are the major ideas you discover something extraordinary?" To your surprise, they answer in unison, "Yes, but we both discovered the of general relativity? same thing: We ran into each other, despite having traveled Recall that the special theory of relativity applies only to in opposite directions along perfectly straight paths." situations in which we are not concerned with the effects Although this outcome would be extraordinarily sur of gravity. From this theory, we already know that space prising if you truly believed the Earth to be flat, we are not and time are inextricably linked. More specifically, special really surprised because we know that Earth is ro und (Fig relativity tells us that the three dimensions of space and the ure S3.1a) . In a sense, the explorers followed the straightest one dimension of time together form an inseparable,four possible paths, but these "straight" lines follow the curved dimensional combination called spacetime. When Einstein surface of the Earth. extended the theory of relativity to the general case that in Now let's consider a somewhat more modern scenario. cludes gravity, he discovered that matter shapes the "fabric" You are floating freely in a spaceship somewhere out in of spacetime in a manner analogous to the way heavy weights space. Hoping to learn more about space in your vicinity, distort a taut rubber sheet or trampoline (Figure S3.2). Of you launch two small probes along straight paths in oppo course, we cannot place weights "upon" spacetime because site directions. Each probe is equipped with a camera that all matter exists within spacetime, and we cannot visualize transmits pictures back to your spaceship. Imagine that, to distortions of spacetime. However, using a rubber sheet your astonishment, the probes one day transmit pictures as an analogy, we can begin to appreciate the principles of of each other! That is, although you launched them in op general relativity.
chapter S3 • Spacetime and Gravity 435 you accelerating away, with your speed growing ever faster, so she sends you a radio message saying, "Good-bye, have .~~.~":- ...... ,...I'~.Y'!."!r.'!r ."...... ,- • I ~I ,.. _. . . .' a nice trip!" . I . . ...Lr~, I~ .. i I ·1 THINK ABOUT_IT .:, .' ~kg .~...... f :. Suppose you start from rest in Jackie's re ference frame an d ~g j.' she se es you accelerate at Ig (= 9.8 m/s2). ApprOXimat ely ,." .. . .. ~ - __ ~~~I~~.~ how fast w ill Jackie see you going after I second? After 10 sec Figure 53.2 A rubber sh eet analogy to spacetime: Matter dis onds? A fter a minute? (Hint See Section 4. I t o review the torts the "fabric" of four- dimensiona l spacetime in a mann er analo meaning of acceleration.) gous to the w ay heavy weights distort a ta ut, two-dimensional r ub ber sheet The greater the mass. the greater the distortion of If all motion is relative, you should be free to claim that spacetime. you are still stationary and that it is Jackie who is receding into the distance at ever-faster speeds. You might therefore It is difficult to overstate the significance of general rela wish to reply: "Thanks, but I'm not going anywhere. You're tivity to our understanding of the universe. For example, the one accelerating into the distance." However, this situa the following ideas all come directly from Einstein's general tion has a new element that was not present when we dealt theory of relativity: with constant velocities: Because your rocket engines are firing, you feel a force inside your spaceship (Figure S3 .3). • Gravity arises from distortions of spacetime. It is not a In fact, because the engines are giving you an acceleration mysterious force that acts at a distance. The presence of of 1g, you'll feel a force holding you to the floor of your mass causes the distortions, and the resulting distortions spaceship that is the same as the force you fe el when you determine how other objects move through spacetime. stand on Earth's surface. In other words, you'll no longer be • Time runs slowly in gravitational fields. The stronger weightless but instead will be able to walk on the spaceship the gravity, the more slowly time runs. floor with your normal Earth weight Thus, Jackie may re • Black holes can exist in spacetime, and falling into a spond back: "Oh, yeah? If you're not going anywhere, why black hole means leaving the observable universe. are you stuck to the floor of your spaceship, and why do • The universe has no boundaries and no center, yet it you have your engines turned on? Furthermore, if I'm accel might still have a finite volume. erating, why am I weightless?" • Large masses that undergo rapid changes in motion or You must admit that Jackie is asking very good questions. structure emit gravitational waves that travel at the speed It certainly looks as if you really are the one who is acceler of light ating, in which case you cannot legitimately claim to be sta tionary. In other words, it seems that motion is no longer • Is all motion relative? relative when we introduce acceleration. This idea did not sit well with Einstein, because he believed that all motion Special relativity tells us that there is no single, absolute should be relative, regardless of whether the motion was answer to the question, "Who is moving?" when two people at constant velocity or included an acceleration. Einstein pass each other at a constant velocity in free-float frames. Each individual can claim to be at rest, and each claim is equally valid. However, the situation seems quite different if one of the reference frames is accelerating rather than F!fIny ,'(Jur ruckels Iilil ill:;u flJakes vou Iccl traveling at constant velocity. Understanding this apparent nDI ani, makes VOIl 3 Imw llOftlillY VOU 10 lile difference is a good way to begin our study of the general aceelerare 9.8 m l s2 Iloor 01 VOIII slJif.l theory of relativity. D . JJGkie colllllilles .Nllile ( ''I .. 10 final "ieliJhrleos V A Thought Experiment with Acceleration Imagine that you and your friend Jackie from Chapter S2 are both floating freely in space when you decide to fire your rocket engines. Be • " fore you fire the engines, you and Jackie are both in free .. . ~ float reference frames, which means you are both floating weightlessly in your spaceships. Thus, as we saw in Chap ter S2, you cannot determine who is really moving no Figure 53.3 Out in deep space with no engines fwing. you and matter what your relative velocity, because you will both Jackie would both fl oat freely in yo ur spaceships. But as soon as experience the laws of nature in exactly the same way. Now, you start firing your engi nes. you fee l a forc e that presses you agai nst the floor of your spaceship. Ja cki e sees you accelerate aw ay. suppose your rocket engines start to fire with enough thrust You might wish to claim that she is accelerating away from you, but 2 to give you an acceleration of 19, or 9.8 m/s -which is the how, then, do you explain the force that you feel and the fad that accelera tion of gravity on Earth [Section 4.1 J. Jackie sees Jackie remains w eightless?
436 part I V • A Deeper Look at Nature therefore needed some way to explain the force you feel due to firing your rocket engines without necessarily assuming that you are accelerating through space. (Einstein did not really do his thought experiments with spaceships, but the basic ideas of his thought experiments were the same.)
The Equivalence Principle In 1907, Einstein hit upon what he later called "the happiest thought of my life." His revela tion consisted of the idea that, whenever you feel weight (as opposed to weightlessness), you can equally well attrib ute it to effects of either acceleration or gravity. This idea is called the equivalence principle. Stated more precisely, it says:
The effects of gravity are exactly equivalent to the effects of acceleration.* Figure 53.5 According to the eqUiva lence pnnciple, you can claim to be stationary in a gravitational field, using your engines to To clarify the meaning of the equivalence principle, prevent you from falling. You feel weight due to gravity, w hile Jackie imagine that you are sitting inside with doors closed and is w eight less because she is in free-fall. window shades pulled down when your room is magically removed from Earth and sent hurtling through space with an acceleration of 19 (Figure 53.4). According to the equiv ing you stationary), and you feel weight just as you would alence principle, you would have no way of knowing that if you were hovering in a helicopter on Earth. And how do you've left Earth. Any experiment you performed, such as you explain Jackie's weightlessness? Easy: Because she is not dropping balls of different weights, would yield the same using her engines, she and her spaceship are falling through results you'd get on Earth. the gravitational field that fills space around you-and Now let's go back to Jackie's questions. She claims that anyone in free-fall feels weightless [Section 4.1]. In essence, you must be accelerating because you feel weight as you you can claim that the situation is much as it would be if stand in your spaceship. However, the equivalence princi you were hovering over a cliff while Jackie had fallen over ple tells us that, with equal validity, you can claim to feel the edge (Figure S3.5). Thus, you can respond: "Sorry, Jackie, weight because of gravity. From this perspective, space is but 1 still say that you have it backward. I'm using my en filled with a gravitational field pointing "downward" toward gines to prevent my spaceship from falling, and 1 feel weight the floor of your spaceship. You are stationary only because because of gravity. You're weightless because you're in free your rocket engi ne prevents you from falling (thereby keep falL 1 hope you won't be hurt by hitting whatever lies at the bottom of this gravitational field l " Because the equivalence principle says that acceleration ' Technica ll y, this eguivalence holds only within sm all regions o f space. affects the laws of physics in exactly the same way as grav Over la rger regions, the gravity o f a massive objec t varies in ways that would no t occur due to acceleration; fo r exa mple, such va riati on gives rise ity, it allows us to treat all motion as relative. Thus, it is the to tidal force s that do no t arise from acceleration. starting point for general relativity. Just as the consequences of special relativity follow directly from the ideas that the laws of nature are the same for everyone and that everyone The Equivalence Principle always measures the same speed of light, the astounding You callnot tell the difference between predictions of general relativity follow from the equivalence bemg IT!.•~ closed roam 011 Eartl! , and bemg ITI iJ closed room principle. Moreover, just as special relativity led us to rec ac~eleretmg througll spac . al ,g. ognize some underlying truths about nature-such as that space and time are different for observers in different refer NIJl - ence frames-general relativity also leads us to a new and , '. • E deeper understanding of the universe.
,( \iIi·.] S3.2 Understanding Spacetime It's easy to say that you can equally well attribute your .~< . . ~ weight to effects of gravity or of acceleration, but the two effects tend to look very different. A person standing on the Figure 53.4 The equivalence principle sta les that he effects of surface of the Earth appears to be motionless, while an as gravity are exactly equiva len 0 t he ef ects of acceleration, Thu s, according t o the equivalence principle , yo cannot tell the differ tronaut accelerating through space continually gains speed. en ce between being in a closed room on Earth and being in a How can gravity and acceleration produce such similar closed I'oom cce erdtmg through space at Ig effects when they look so different? According to the general
chapter S3 • Spacetime and Gravity 437 SPECIAL TOPIC Einstein's Leap
Given that the similarities in the effects of gravity and of accelera or experimental evidence for it at the time. This leap of faith sent tion were well known to scientists as far back as the time of Newton, him on a path far ahead of his scientific colleagues. you may be wondering why the equivalence principle is so surpris From a historical viewpoint, special relativity was a "theory ing. The answer is that the similarities were generally attributed waiting to happen" because it was needed to explain two significant to coincidence-although a very puzzling coincidence. It was as if problems left over from the nineteenth century: the perplexing other scientists imagined that nature was showing them two boxes, constancy of the speed of light, demonstrated in the Michelson one labeled "effects of gravity" and the other labeled "effects of ac Morley experiment ISe ction S2. 3], and some seeming peculiarities celeration." They shook, weighed, and kicked the boxes but could of the laws of electromagnetism. Indeed, several other scientists never find any obvious differences between them. They concluded: were very close to discovering the ideas of special relativity when "Vlhat a strange coincidence! The boxes seem the same from the Einstein published the theory in 1905, and someone was bound outside even though they contain different things." Einstein's reve to come up with special relativity around that time. lation was, in essence, to look at the boxes and say that it is not a General relativity, in contrast, was a tour de force by Einstein. He coincidence at all. The boxes appear the same from the outside be recognized that unsolved problems remained after completing the cause they co ntain the same thing. theory of special relativity, and he alone took the leap of faith re In many ways, Einstein's assertion of the equivalence principle quired to accept the equivalence principle. Without Einstein, gen represented a leap of faith, although it was a faitb he would will eral relativity probably would have remained undiscovered for at ingly test through scientific experiment. He proposed the equiva least a couple of decades beyond 1915, the year he completed and lence principle because he thought the universe would make more published the theory. sense if it were true, not because of any compelling observational
theory of relativity, the answer is that they look different space, with the three independent directions of length, only because we're not seeing the whole picture. Instead of width, and depth. looking just at the three dimensions of space, we must learn We live in three-dimensional space and thus cannot vi to "look" at the four dimensions of spacetime. sualize any direction that is distinct from length, width, and depth (and combinations thereof). However, just because • What is spacetime? we cannot see "other" directions doesn't mean they don't exist. Thus, we can imagine sweeping space back and forth The first step in understanding spacetime is understanding in some "other" direction to generate a four-dimensional what we mean when we say, for example, that something space. is two dimensional or three dimensional. The concept of Although we have no hope of visualizing a four dimension describes the number of independent directions dimensional space, we can describe it mathematically. In in which movement is possible (Figure 53.6). A point has algebra, we do one-dimensional problems with the single zero dimensions. If you were a geometric prisoner confined variable x, two-dimensional problems with the variables x to a point, you'd have no place to go. Sweeping a point back and y, and three-dimensional problems with the variables and forth along one direction generates a line. The line x, y, and z. A four-dimensional problem simply requires is one dimensional because only one direction of motion adding a fourth variable, as in x, y, z, and w. We could con is possible (going backward is considered the same as going tinue to five dimensions, six dimensions, and many more. forward by a negative distance). Sweeping a line back and Any space with more than three dimensions is called a forth generates a two-dimensional plane. The two direc hyperspace, which means "beyond space." tions of possible motion are, say, lengthwise and widthwise. Any other direction is just a combination of these two. If Spacetime Spacetime is a four-dimensional space in which we sweep a plane up and down, it fills three-dimensional the four directions of possible motion are length, width,
Figure 53.6 An object's number of Apain/ /las a Sweepmg iJ pomt beck Sweeping iJ line back Sweeping aplane up dime nsions is the number of independent dllllensions and lOrlh i/enerates a and forth generates. 11 and down generates a directions in whi ch movement is possible l - dimenS!~~al line. 2·dimensiqnal plane. 3.dimeTona,space within the object. It is zero for a point. one for a li ne, two for a plane, and three for space. " .. 0;. '" /. ~y y • z x
x
438 par t I V • A Deeper Look at Nature depth, and time. Note that time is not "the" fourth dimen sion. It is simply one of the four. (However, time differs in an important way from the other three dimensions. 5ee Mathematical Insight 53.1.) We cannot picture all four dimensions of spacetime at once, but we can imagine what things would look like if we could. In addition to the three spatial dimensions of space T 2 in time that we ordinarily see, every object would be stretched out through time. Objects that we see as three-dimensional 1 in our ordinary lives would appear as four-dimensional objects in spacetime. If we could see in four dimensions, we could look through time just as easily as we look to our left or right. If we looked at a person, we could see every a A book has an unambiguous three-dimensional shape. event in that person's life. If we wondered what really hap pened during some historical event, we'd simply look to . . find the answer. ..._- " , -:-..,., .:" ,
- --. 04
11':. _." H • •• THINK ABOUT IT _
.- 6 __ _ Try to imagine how you would look in four dimensions. How ~~ ~ would your body, stretched through t ime, appear? Imagine that you bumped into someone on the bus yesterday. W hat would b Two-dimensional pictures of the book can look very different. this event look like in spacetime) Figure 53 .7 Two-dimensional views of a three-dimensional. object (like a book) ca n appear di fferent. even though the object This spacetime view of objects provides a new way of has only a Single real shape. In a similar way, observers In di ffer understanding why different observers can disagree about ent reference frames may measure space and time differently measurements of time and distance. Because we can't visu (because they perceive only thnee dimen sions at once) even alize four dimensions, we'll use a three-dimensional anal though they are all observing the same four-dimenSional space time real ity ogy. 5uppose you give the same book to many different people and ask each person to measure the book's dimen sions. Everyone will get the same results, agreeing on the Spacetime Diagrams 5uppose you drive your car along a three-dimensional structure of the book (Figure 53.7a). Now, straight road from home to work as shown in Figure 53.8a. suppose instead that you show each person only a two At 8:00 A.M., you leave your house and accelerate to 60 krn/hr. dimensional picture of the book rather than the book itself. You maintain this speed until you come to a red light, where The pictures may look very different, even though they show you decelerate to a stop. After the light turns green, you the same book in all cases (Figure 53 .7b). If the people be accelerate again to 60 krn/hr, which you maintain until you lieved that the two-ctimensional pictures reflected reality, they slow to a stop when you reach work at 8:10. What does might argue endlessly about what the book really looks like. your trip look like in spacetime? In our ordinary lives, we perceive only three dimensions, If we could see all four dimensions of spacetime, we'd and we assume that this perception reflects reality. But space see all three dimensions of your car and your trip stretched time is actually four dimensional. Just as different people out through the 10 minutes of time taken for your trip. We can see different two-dimensional pictures of the same can't visualize aU four dimensions at once, but in this case three-dimensional book, different observers can see dif we have a special situation: Your trip progressed along only ferent three-dimensional "pictures" of the same spacetime one dimension of space because you took a straight road. reality. These different "pictures" are the differing percep Therefore, we can represent your trip in spacetime by draw tions of time and space of observers in different reference ing a graph showing your path through one dimension of frames. Thus, different observers will get different results space on the horizontal axis and your path through time on when they measure time, length, or mass, even though the vertical axis (Figure 53.8b). This type of graph is called they are all actually looking at the same spacetime reality. a spacetime diagram. In the words of a famous textbook on relativity: The car's path through four-dimensional spacetime is called its worldline. Any particular point along a worldline Space is different for different observers. represents a particular event. That is, an event is a specific Time is different for different observers. place and time. For example, the lowest point on the world Spacetime is the same for everyone. ,. line in Figure 53.8b represents the event of your leaving your house: The place is 0 kilometers from home, and the ' From E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2d ed., Freeman, time is 8:00 A.M . You can see three very important proper 1992. ties of any worldline in Figure 53.8b:
chapter S3 • Spacetime and Gravity 439 8:10 Car comes to a stop at work. 8:10 When the car IS moving 8:09:30 Car begins to decelerate. at constant velocity, its 809 worldline IS straight but slanted ...... 808 Car maintains 60 km/hr When the car is stopped, 8:07 its worldline is vertical.
8:06 Car reaches 60 km/hr QJ 8:06 .~ 8:05 Car begins to accelerate from res t. 8:05 ...... When the car accelerates, Car at rest. its worldline curves to the 8:04 Car comes to stop at stop sign. 8 04 nght 8:03:30 Car begins to decelerate. ro•• •• 8 03 ...... When the car decelerates, Car maintains 60 km/hr. its worldllne curves 8:02 upward.
8:01 Car reaches 60 km/hr. 8:01
8:00 Car accelerates away from home. 8:00 0 2 3 4 5 6 7 8 space (km) a This diagram sh ows the events that occur during a I O- minute car t rip b W e make a spacetime diagram for the trip by putting space from home to work on a straight road. (i n t his case, the car's distance from home) on t he horizontal axis and time on the vertical axis.
Figure 53.8 A sp acetime diagram all ows us to represent one dimenSion of space and t he dimen sion of time on a Single graph.
1. The worldline of an object at rest is vertical (that is, seconds for distance. In that case, light follows 45° lines on a parallel to the time axis). The object is going nowhere spacetime diagram because it travels 1 light-second of dis in space, but it still moves through time. tance with each second of time. For example, suppose you 2. The worldline of an object moving at constant ve are sitting still in your chair, so that your worldline is ver locity is straight but slanted. The more slanted the tical (Figure S3.9a). If at some particular time you flash worldline, the faster the object is moving. a laser beam pointed to your right, the worldline of the 3. The worldline of an accelerating object is curved. If light goes diagonally to the right. If you flash the laser to the object's speed is increasing, its worldline curves the left a few seconds later, its worldline goes diagonally toward the horizontal. If its speed is decreasing, its to the left. Worldlines for several other objects are shown worldline gradually becomes more vertical. in Figure S3 .9b. In Figure S3.Sb, we use units of minutes for time and THINK ABOUT IT_ kilometers for distance. In relativity, it is usually easier to work with spacetime diagrams in which we use units related Explain why, in Figure 53.9, the worldl ines of all t he objects we to the speed of light, such as seconds for time and light- see in our everyday lives would be nearly vert ica l.
time (seconds)
o~_ tt, 1 0 ~ ~ Event 2: You turn on a ~ laser light pointed to the ~"o , 8 ,' :;;; 0/'0; 6 g left; the worldline tracks 0'0 0: 45 0 to the left. "'0. ' ~ 0<$l I, 1' 1" I I ' .. , space space 10 8 6 4 2 2 (light-se conds) 10 (light-seconds) Straight worldllnes·::····· The slope can never be more Event 7: You turn on a 4 Indicate motion at '.. " laser light pointed to 6 than 45 0 from the vertical, , constant velocity because that would mean a the right; ~he worldline 8 .') tracks 45 to the nght...... t ...... speed faster than light 10 a Light foll ows 45' lin es on a spacetime diagram t hat uses b T hiS spaceti me diagram shows several sample worldll nes. Object s at rest units of seconds for t ime and light-seconds for space , have vertical w o rldlines, objects moving at constant velocity have straight but slanted worldllnes, and accelerating objects have curved worldllnes. Figure 53.9 Spacetime diagrams marked with units of seconds for time and light-seconds for space . 440 part IV A Deeper Look at Nature time (seconds) time (seconds) 10 10 8 ~0 6 (5 ~'l>(j c 4 space space 10 8, 6 4 10 (light-seconds) 10 8 6 4 4 6 8 10 (light-seconds) 4 4 6 6 8 8 10 10 Figure 53.10 Spacetime diagrams for a The spacetime diagram from b The spacetime diagram from the situation in which Jackie is moving by your point of view. Jackie's point of view. you at 0.9c. We can use spacetime diagrams to clarify the relativity that you would see Jackie's time running slowly, while she of time and space. Suppose you see Jackie moving past you sees your time running slowly (along with effects on length in a spaceship at 0.9c. Figure S3.10a shows the spacetime and mass). We know there is no contradiction here, but diagram from your point of view: You are at rest and there simply a problem with our old common sense about space fore have a vertical worldline, while Jackie is moving and and time. has a slanted worldline. Of course, Jackie claims that you From a four-dimensional perspective, the problem is are moving by her and therefore would draw the spacetime that the large angle between your worldline and Jackie's diagram shown in Figure S3.10b, in which her worldline means that neither of you is looking at the other "straight is vertical and yours is slanted. Special' relativity tells us on" in spacetime. Thus, you and Jackie are both looking MATHEMATICAL INSIGHT S3 . I Spacetime Geometry Spacetime geometry is easy in principle, because it simply requires tion is x = 5 and the vertical separation is y = O. However, the adding a fourth dimension (time) to our usual three dimensions of distance ~ between the points is still the same. space. However, spacetime has some surprising properties, because This fact should not be surprising: Distance is a real, physical time enters the equations of spacetime geometry differently than quantity, while the x and y separations are artifacts of a chosen do the three dimensions of space. coordinate system. The same idea holds if we add a z-axis, perpen To gain insight into the nature of spacetime geometry, consider dicular to both x and y (you can represent the z-axis with a pencil two points in a plane separated by amounts x = 3 along the hori that sticks straight up out of the page), to make a three-dimensional zontal axis and y = 4 along the vertical axis (see the left side of the coordinate system. Different stationary observers using different figure below). You may recall from geometry that the distance be coordinate systems can disagree about the x, y, and z separations, tween the two points is given by ~. Now consider the but they will always agree on the distance Vx2 + / + z2. same two points viewed from a coordinate system that happens to We can think of spacetime as having a fourth axis, which we be rotated so that both points lie along the x-axis, as in the right will call the' t-axis, for time. We might expect that, just as different side of the figure. In this coordinate system, the horizontal separa- observers always agree on the three-dimensional distance between two points, they will also agree on some kind of four-dimensional "distance" that has the formula Vx 2 + / + z2 + t 2. However, The x- and y-coordinates of two points can it turns out that different observers will instead agree on the value be different In different coordinate systems . of the quantity V x 2 + l + z2 - t 2. This value is called the y interval. (Technically, the interval formula should use ct rather than t t so that all the terms .have units of length.) That is, different ob I; servers can disagree about the values of x, y, z, and t separating two ,~ y=4 events, but all will agree on the interval between the two events. The minus sign that goes with the time dimension in the interval formula is what makes the geometry of spacetime surprisingly complex. For example, the three-dimensional distance between two fJx = 3 £. points can be zero only if the two points are in the same place, but x the interval between two events can be zero even if they are in dif ferent places in spacetime, as long as x 2 + l + z2 = t 2. For ex . . . but the distance between them (red line) ample, the interval is zero between any two events connected by is the same either way a light path on a spacetime diagram. If you study general relativity The distance between two points in a plane is the same regardless further, you will see many more examples of how this strange geom of how we set up a coordinate system. etry comes into play. chapter S3 Spacetime and Gravity 441 at the same four-dimensional reality but from different distance between two points in a flat plane is always a three-dimensional perspectives. It's not surprising that, straight line. Clearly, this rule cannot be true on Earth's like two people looking at each other cross-eyed, if you surface, because there is no such thing as a straight line! see Jackie's time running slowly, she sees the same thing What rule replaces this shortest-distance rule on Earth's when she looks at you. surface? If you experiment by measuring pieces of string stretched in different ways between two points on a globe, • What is curved spacetime? you'll find that the shortest and straightest possible path between two points on Earth's surface is a piece of a great So far, we've been viewing spacetime diagrams drawn on circle-a circle whose center is at the center of Earth (Fig the flat pages of this book. However, as we discussed in the ure S3.11a). For example, the equator is a great circle, and beginning of this chapter, spacetime can be curved. What any "line" of longitude is part of a great circle. Note that do we mean by curved spacetime? circles of latitude (besides the equator) are not great circles It's easy to visualize the curvature of a two-dimensional because their centers are not at the center of Earth. Thus, if surface, such as the surface of a bent sheet of paper or the sur you are seeking the shortest and straightest route between face of the Earth. These surfaces are still two-dimensional two cities, you must follow a great-circle route. For example, despite their curvature because they still allow only two Philadelphia and Beijing are both at about 40 0 N latitude, independent directions of travel. For example, we usually but the shortest route between them does not follow the identify the two independent directions of travel on Earth's circle of 40 0 N latitude. Instead, it follows a great-circle route surface as north-south (changes in latitude) and east-west that extends far to the north (Figure S3.11 b). (changes in longitude). Unfortunately, we have no hope of visualizing the curvature of three-dimensional space, let _ THINK ABOUT IT _ alone of four-dimensional spacetime. After all, we can vi sualize the curvature of two-dimensional surfaces because Find a globe and locate New Orleans and Katmandu (N epal). they curve through the third dimension of space. By anal Explain why the shortest route between these two Ci ties goes ogy, we'd therefore need extra dimensions to visualize the almost directly over the NO I-th Pole. Why do you think air curvature of space or spacetime, but we cannot see any pl anes try. to follow great -circle routes as closely as poss ible? dimensions beyond those of three-dimensional space. Nevertheless, we can determine whether space or spacetime Other familiar rules of geometry in a flat plane also are is curved by identifying the rules of geometry that apply. different on Earth's curved surface. Figure S3.12a shows the straight-line rule and several other geometrical rules on a Three Basic Types afGeametry Because we cannot visual flat surface, and Figure S3.12b shows how these rules differ ize curved space or spacetime, we'll instead use two for a spherical surface like that of Earth; in the latter case, dimensional surfaces as an analogy. Consider Earth's we must draw "lines" as portions of great circles, because curved, two-dimensional surface. Because Earth's surface those are the shortest and straightest possible paths. Notice, is curved everY'.vhere, there really is no such thing as for example, that lines that are anywhere parallel on a flat a "straight" line on the Earth. For example, if you take a plane stay parallel forever, while lines that start out parallel piece of string and lay it across a globe, it will inevitably on a sphere eventually converge (just as lines of longitude curve around the surface. Thus, we can immediately see start out parallel at Earth's equator but all converge at the a fundamental difference between the rules of geometry North and South Poles). Similarly, the sum of the angles in on a curved surface like that of Earth and the more familiar a triangle is always 1800 in a flat plane but is greater than rules of geometry on a flat plane. Recall that the shortest 1800 on the surface of a sphere, and the circumference of The three CIIcles shown in red are Apatir followlIlg a great cl/ele IS shOr1er great circles because their centers and straighter than any other North Pole are at the center of Earth pach lJeCWeen CWo points . ". on Ealtl(s surface. great-Circle route center of Earth Bei jing Philadelphia route along Except far the equ8COr, Imes 01 400 N latitude lat/rude (shown In black! are nol greal circles a A great circle is any circle on the surface of Earth that b The shortest and straightest possible path beNleen two points has its center at the ce nter of Earth. on Earth is always a piece 0' a great Circle. Figure 53 .1 1 The straightest possible path bet-."Ieen any two pO ints on a sphere must be a segment of a great circl e. 442 part IV • A Deeper Look at Nature the surface of a sphere. In a similar way, but in two more Triangle: sum of ...... ~ ...... Parallel Lines: angles is IBO'. '. ~ remain parallel. dimensions, the four-dimensional spacetime of our uni Y!J\ verse obeys different geometrical rules in different regions ~~. but presumably has some overall shape. This overall shape Straightest Possible ...... '>/ '...... Circle: C = 2rv. Pat~ is a straight line.··· / must be one of the three general types of geometry shown -----' in Figure 53.12: flat, spherical, or saddle shaped. This fact a Rules of flat ge ometry. explains how it is possible for the universe to have no cen ter and no edges, an idea we first discussed in Chapter l. In geometry, a plane is infinite in extent, which means Triangle : sum of ...... Parallel Lmes. it has no center or edges. An idealized, saddle-shaped (hy angles is greater...... A eventually converge. than lBO' ..•~ perbolic) surface is also infinite in extent. Thus, if the uni ------verse has either a flat or a saddle-shaped geometry overall, then spacetime is infinite and the universe has no center and no edges. Straightest Possible Path: IS ap,ece of a.. ·...... Circle: C< 2rv. In contrast, if the overall geometry of the universe is great circle. spherical, then spacetime is finite, much like the surface of Earth. However, it would still have no center or edges. Just b Rules of spherical ge ometry. as you can sailor fly around Earth's surface endlessly, you could fly through the universe forever and never encounter an edge. And just as the surface of Earth has no center Triangle : sum of ...... ~ .. ··Parallel Lines: angles is less New York is no more "central" than Beijing or any other place than lBO ' "~ J. '~"'''"V d;~,g,. on Earth's surface-there would be no center to the uni verse. (Of course, the three-dimensional Earth does have a Straightest Possible !\ .... center, but this center is not part of the two-dimensional Path: IS apIece of a···· · surface of Earth and therefore plays no role in our analogy.) hyperbola. .. .. Circle. C > 2rrr c Rules of sadd le-shaped geometry. "Straight" Lines in Curved Spacetime The rules of geometry give us a way to determine the geometry of any localized Figure 53.12 These diagrams contrast three bas ic types of region of spacetime. In particular, we can learn the geome geometry. try of spacetime by observing the paths of objects that are following the straightest possible paths between two points a circle is 27fr in a flat plane but is less than 27fr ona in spacetime. For example, if the straightest possible path spherical surface. Generalizing these ideas to more than is truly straight, we know that spacetime is flat in that re two dimensions, we say that space, or spacetime, has a flat gion. If the straightest possible path is curved, then the geometry if the rules of geometry for a flat plane hold. shape of the curve must be telling us the shape of space (Flat geometry is also known as Euclidean geometry, after time. However, given that we can visualize neither the time the Greek mathematician Euclid [c. 325-270 B.C.].) For part of spacetime nor the curvature of spacetime, how can example, if the circumference of a circle in space really is we know whether an object is traveling on the straightest 27fr, then space has a flat geometry. However, if the circum possible path through spacetime? ference of a circle in space turns out to be less than 27fr, we Einstein used the equivalence principle to provide the say that space has a spherical geometry because the rules answer. According to the equivalence principle, we can are those that hold on the surface of a sphere. attribute a feeling of weight either to experiencing a force Flat and spherical geometries are two of three general generated by acceleration or to being in a gravitational types of geometry. The third general type of geometry is field. Similarly, any time we feel weightless, we may attribute called saddle-shaped geometry (also called hyperbolic geom it either to being in free-fall or to traveling at constant ve etry) because its rules are most easily visualized on a two locity far from any gravitational fields. Because traveling dimensional surface shaped like a saddle (Figure S3.12c). at constant velocity means traveling in a straight line, In this case, lines that start out parallel eventually diverge, Einstein reasoned that objects experiencing weightlessness the sum of the angles in a triangle is less than 180°, and the for any reason must be traveling in a "straight" line-that circumference of a circle is greater than 27fr. is, along a line that is the straightest possible path between two points in spacetime. In other words: The Geometry ofthe Universe The actual geometry of space Ifyou are floating freely, then your worldline is time turns out to be a mixture of all three general types. following the straightest possible path through space Earth's surface again provides a good analogy. When we time. Ifyou feel weight, then you are not on the view only small portions of Earth's surface, some regions straightest possible path. appear flat while others are curved with hills and valleys. However, when we expand our view to the entire Earth, This fact provides us with a remarkable way to examine it's clear that the overall geometry of Earth's surface is like the geometry of spacetime. Recall that any orbit is a free-fall chapter S3 Spacetime and Gravity 443 trajectory [Section 4.1]. The Space Station is always free falling toward Earth, but its forward velocity always moves it ahead just enough to "miss" hitting the ground. Earth is constantly free-falling toward the Sun, but our planet's orbital speed keeps us going around and around instead of ever hitting the Sun. According to the equivalence prin ciple, all orbits must therefore represent paths of objects that are following the straightest possible path through spacetime. Thus, the shapes and speeds of orbits reveal the geometry of spacetime, which leads us to an entirely new view of gravity. _ THINK ABOUT IT _ Suppose you are standing on a scale in your bathroom. Is your wor ldline following the straightest possible path through space time? Explain.