The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space (2024)

42–1Curved spaces with two dimensions

According to Newtoneverything attracts everything else with a force inversely proportionalto the square of the distance from it, and objects respond to forceswith accelerations proportional to the forces. They are Newton’slaws of universal gravitation and ofmotion. As you know, they account for the motions of balls, planets,satellites, galaxies, and so forth.

Einstein had a differentinterpretation of the law of gravitation. Accordingto him, space and time—which must be put together as space-time—arecurved near heavy masses. And it is the attempt of things to goalong “straight lines” in this curved space-time which makes them movethe way they do. Now that is a complex idea—very complex. It is theidea we want to explain in this chapter.

Our subject has three parts. One involves the effects of gravitation.Another involves the ideas of space-time which we already studied. Thethird involves the idea of curved space-time. We will simplify oursubject in the beginning by not worrying about gravity and by leavingout the time—discussing just curved space. We will talk later aboutthe other parts, but we will concentrate now on the idea of curvedspace—what is meant by curved space, and, more specifically, what ismeant by curved space in this application ofEinstein. Now even that muchturns out to be somewhat difficult in three dimensions. So we will firstreduce the problem still further and talk about what is meant by thewords “curved space” in two dimensions.

In order to understand this idea of curved space in two dimensions youreally have to appreciate the limited point of view of the character wholives in such a space. Suppose we imagine a bug with no eyes who liveson a plane, as shown in Fig.42–1. He can move only on theplane, and he has no way of knowing that there is anyway to discover any“outside world.” (He hasn’t got your imagination.) We are, of course,going to argue by analogy. We live in a three-dimensional world,and we don’t have any imagination about going off our three-dimensionalworld in a new direction; so we have to think the thing out by analogy.It is as though we were bugs living on a plane, and there was a space inanother direction. That’s why we will first work with the bug,remembering that he must live on his surface and can’t get out.

As another example of a bug living in two dimensions, let’s imagine onewho lives on a sphere. We imagine that he can walk around on the surfaceof the sphere, as in Fig.42–2 but that he can’t look“up,” or “down,” or “out.”

Now we want to consider still a third kind of creature. He isalso a bug like the others, and also lives on a plane, as our first bugdid, but this time the plane is peculiar. The temperature is differentat different places. Also, the bug and any rulers he uses are all madeof the same material which expands when it is heated. Whenever he puts aruler somewhere to measure something the ruler expands immediately tothe proper length for the temperature at that place. Wherever he putsany object—himself, a ruler, a triangle, or anything—the thingstretches itself because of the thermal expansion. Everything is longerin the hot places than it is in the cold places, and everything has thesame coefficient of expansion. We will call the home of our third bug a“hot plate,” although we will particularly want to think of a specialkind of hot plate that is cold in the center and gets hotter as we goout toward the edges (Fig.42–3).

Now we are going to imagine that our bugs begin to study geometry.Although we imagine that they are blind so that they can’t see any“outside” world, they can do a lot with their legs and feelers. Theycan draw lines, and they can make rulers, and measure off lengths.First, let’s suppose that they start with the simplest idea in geometry.They learn how to make a straight line—defined as the shortest linebetween two points. Our first bug—see Fig.42–4—learnsto make very good lines. But what happens to the bug on the sphere? Hedraws his straight line as the shortest distance—forhim—between two points, as in Fig.42–5. It may looklike a curve to us, but he has no way of getting off the sphere andfinding out that there is “really” a shorter line. He just knows thatif he tries any other path in his world it is always longer thanhis straight line. So we will let him have his straight line as theshortest arc between two points. (It is, of course an arc of a greatcircle.)

Finally, our third bug—the one in Fig.42–3—will alsodraw “straight lines” that look like curves to us. For instance, theshortest distance between $A$ and$B$ in Fig.42–6 would beon a curve like the one shown. Why? Because when his line curves outtoward the warmer parts of his hot plate, the rulers get longer (fromour omniscient point of view) and it takes fewer “yardsticks” laidend-to-end to get from $A$ to$B$. So for him the line isstraight—he has no way of knowing that there could be someone out in astrange three-dimensional world who would call a different line“straight.”

We think you get the idea now that all the rest of the analysis willalways be from the point of view of the creatures on the particularsurfaces and not from our point of view. With that in mind let’ssee what the rest of their geometries looks like. Let’s assume that thebugs have all learned how to make two lines intersect at right angles.(You can figure out how they could do it.) Then our first bug (the oneon the normal plane) finds an interesting fact. If he starts at thepoint$A$ and makes a line $100$inches long, then makes a right angleand marks off another $100$inches, then makes another right angle andgoes another $100$inches, then makes a third right angle and a fourthline $100$inches long, he ends up right at the starting point as shownin Fig.42–7(a). It is a property of his world—one of thefacts of his “geometry.”

Then he discovers another interesting thing. If he makes a triangle—afigure with three straight lines—the sum of the angles is equalto$180^\circ$, that is, to the sum of two right angles. SeeFig.42–7(b).

Then he invents the circle. What’s a circle? A circle is made this way:You rush off on straight lines in many many directions from a singlepoint, and lay out a lot of dots that are all the same distance fromthat point. See Fig.42–7(c). (We have to be careful how wedefine these things because we’ve got to be able to make the analogs forthe other fellows.) Of course, its equivalent to the curve you can makeby swinging a ruler around a point. Anyway, our bug learns how to makecircles. Then one day he thinks of measuring the distance around acircle. He measures several circles and finds a neat relationship: Thedistance around is always the same number times the radius$r$ (whichis, of course, the distance from the center out to the curve). Thecircumference and the radius always have the sameratio—approximately$6.283$—independent of the size of the circle.

Now let’s see what our other bugs have been finding out abouttheir geometries. First, what happens to the bug on the spherewhen he tries to make a “square”? If he follows the prescription wegave above, he would probably think that the result was hardly worth thetrouble. He gets a figure like the one shown in Fig.42–8.His endpoint$B$ isn’t on top of the starting point$A$. It doesn’t workout to a closed figure at all. Get a sphere and try it. A similar thingwould happen to our friend on the hot plate. If he lays out fourstraight lines of equal length—as measured with his expandingrulers—joined by right angles he gets a picture like the one inFig.42–9.

Now suppose that our bugs had each had their own Euclid who had told them what geometry “should” be like, andthat they had checked him out roughly by making crude measurements on asmall scale. Then as they tried to make accurate squares on alarger scale they would discover that something was wrong. The point is,that just by geometrical measurements they would discover thatsomething was the matter with their space. We define a curvedspace to be a space in which the geometry is not what we expect for aplane. The geometry of the bugs on the sphere or on the hot plate is thegeometry of a curved space. The rules of Euclidean geometry fail. And itisn’t necessary to be able to lift yourself out of the plane in order tofind out that the world that you live in is curved. It isn’t necessaryto circumnavigate the globe in order to find out that it is a ball. Youcan find out that you live on a ball by laying out a square. If thesquare is very small you will need a lot of accuracy, but if the squareis large the measurement can be done more crudely.

Let’s take the case of a triangle on a plane. The sum of the angles is$180$degrees. Our friend on the sphere can find triangles that are verypeculiar. He can, for example, find triangles which have threeright angles. Yes indeed! One is shown in Fig.42–10.Suppose our bug starts at the north pole and makes a straight line allthe way down to the equator. Then he makes a right angle and anotherperfect straight line the same length. Then he does it again. For thevery special length he has chosen he gets right back to his startingpoint, and also meets the first line with a right angle. So there is nodoubt that for him this triangle has three right angles, or$270$degrees in the sum. It turns out that for him the sum of theangles of the triangle is always greater than $180$degrees. Infact, the excess (for the special case shown, the extra $90$degrees) isproportional to how much area the triangle has. If a triangle on asphere is very small, its angles add up to very nearly $180$degrees,only a little bit over. As the triangle gets bigger the discrepancy goesup. The bugs on the hot plate would discover similar difficulties withtheir triangles.

Let’s look next at what our other bugs find out about circles. They makecircles and measure their circumferences. For example, the bug on thesphere might make a circle like the one shown inFig.42–11. And he would discover that the circumference isless than $2\pi$times the radius. (You can see that because fromthe wisdom of our three-dimensional view it is obvious that what hecalls the “radius” is a curve which is longer than the trueradius of the circle.) Suppose that the bug on the sphere had readEuclid, and decided to predict a radiusby dividing the circumference$C$ by$2\pi$, taking\begin{equation}\label{Eq:II:42:1}r_{\text{pred}}=\frac{C}{2\pi}.\end{equation}Then he would find that the measured radius was larger than thepredicted radius. Pursuing the subject, he might define the differenceto be the “excess radius,” and write\begin{equation}\label{Eq:II:42:2}r_{\text{meas}}-r_{\text{pred}}=r_{\text{excess}},\end{equation}and study how the excess radius effect depended on the size of the circle.

Our bug on the hot plate would discover a similar phenomenon. Suppose hewas to draw a circle centered at the cold spot on the plate as inFig.42–12. If we were to watch him as he makes the circlewe would notice that his rulers are short near the center and get longeras they are moved outward—although the bug doesn’t know it, of course.When he measures the circumference the ruler is long all the time, sohe, too, finds out that the measured radius is longer than the predictedradius, $C/2\pi$. The hot-plate bug also finds an “excessradius effect.” Andagain the size of the effect depends on the radius of the circle.

We will define a “curved space” as one in which these types ofgeometrical errors occur: The sum of the angles of a triangle isdifferent from $180$degrees; the circumference of a circle dividedby$2\pi$ is not equal to the radius; the rule for making a squaredoesn’t give a closed figure. You can think of others.

We have given two different examples of curved space: the sphere and thehot plate. But it is interesting that if we choose the right temperaturevariation as a function of distance on the hot plate, the twogeometries will be exactly the same. It is rather amusing. We canmake the bug on the hot plate get exactly the same answers as the bug onthe ball. For those who like geometry and geometrical problems we’lltell you how it can be done. If you assume that the length of the rulers(as determined by the temperature) goes in proportion to one plus someconstant times the square of the distance away from the origin, then youwill find that the geometry of that hot plate is exactly the same in alldetails1 as the geometryof the sphere.

There are, of course, other kinds of geometry. We could ask about thegeometry of a bug who lived on a pear, namely something which has asharper curvature in one place and a weaker curvature in the otherplace, so that the excess in angles in triangles is more severe when hemakes little triangles in one part of his world than when he makes themin another part. In other words, the curvature of a space can vary fromplace to place. That’s just a generalization of the idea. It can also beimitated by a suitable distribution of temperature on a hot plate.

We may also point out that the results could come out with the oppositekind of discrepancies. You could find out, for example, that alltriangles when they are made too large have the sum of their anglesless than $180$degrees. That may sound impossible, but it isn’tat all. First of all, we could have a hot plate with the temperaturedecreasing with the distance from the center. Then all the effects wouldbe reversed. But we can also do it purely geometrically by looking atthe two-dimensional geometry of the surface of a saddle. Imagine asaddle-shaped surface like the one sketched in Fig.42–13.Now draw a “circle” on the surface, defined as the locus of all pointsthe same distance from a center. This circle is a curve that oscillatesup and down with a scallop effect. So its circumference is larger thanyou would expect from calculating$2\pi r_{\text{meas}}$. So $C/2\pi$ isnow greater than$r_{\text{meas}}$. The “excessradius” would benegative.

Spheres and pears and such are all surfaces of positivecurvatures; and the others are calledsurfaces of negative curvature.In general, a two-dimensional world will have a curvature which variesfrom place to place and may be positive in some places and negative inother places. In general, we mean by a curved space simply one in whichthe rules of Euclidean geometry break down with one sign of discrepancyor the other. The amount of curvature—defined, say, by the excessradius—may vary fromplace to place.

We might point out that, from our definition of curvature, a cylinderis, surprisingly enough, not curved. If a bug lived on a cylinder, asshown in Fig.42–14, he would find out that triangles,squares, and circles would all have the same behavior they have on aplane. This is easy to see, by just thinking about how all the figureswill look if the cylinder is unrolled onto a plane. Then all thegeometrical figures can be made to correspond exactly to the way theyare in a plane. So there is no way for a bug living on a cylinder(assuming that he doesn’t go all the way around, but just makes localmeasurements) to discover that his space is curved. In our technicalsense, then, we consider that his space is not curved. What wewant to talk about is more precisely called intrinsiccurvature; that is, a curvature whichcan be found by measurements only in a local region. (A cylinder has nointrinsic curvature.) This was the sense intended byEinstein when he said that ourspace is curved. But we as yet only have defined a curved space in twodimensions; we must go onward to see what the idea might mean in threedimensions.

42–2Curvature in three-dimensional space

We live in three-dimensional space and we are going to consider the ideathat three-dimensional space is curved. You say, “But how can youimagine it being bent in any direction?” Well, we can’t imagine spacebeing bent in any direction because our imagination isn’t good enough.(Perhaps it’s just as well that we can’t imagine too much, so that wedon’t get too free of the real world.) But we can still define acurvature without getting out of our three-dimensional world. All wehave been talking about in two dimensions was simply an exercise to showhow we could get a definition of curvature which didn’t require that webe able to “look in” from the outside.

We can determine whether our world is curved or not in a way quiteanalogous to the one used by the gentlemen who live on the sphere and onthe hot plate. We may not be able to distinguish between two such casesbut we certainly can distinguish those cases from the flat space, theordinary plane. How? Easy enough: We lay out a triangle and measure theangles. Or we make a great big circle and measure the circumference andthe radius. Or we try to lay out some accurate squares, or try to make acube. In each case we test whether the laws of geometry work. If theydon’t work, we say that our space is curved. If we lay out a bigtriangle and the sum of its angles exceeds $180$degrees, we can say ourspace is curved. Or if the measured radius of a circle is not equal toits circumference over$2\pi$, we can say our space is curved.

You will notice that in three dimensions the situation can be much morecomplicated than in two. At any one place in two dimensions there is acertain amount of curvature. But in three dimensions there can beseveral components to the curvature. If we lay out a triangle insome plane, we may get a different answer than if we orient the plane ofthe triangle in a different way. Or take the example of a circle.Suppose we draw a circle and measure the radius and it doesn’t checkwith$C/2\pi$ so that there is some excess radius. Now we draw another circle at rightangles—as in Fig.42–15. There’s no need for the excessto be exactly the same for both circles. In fact, there might be apositive excess for a circle in one plane, and a defect (negativeexcess) for a circle in the other plane.

Perhaps you are thinking of a better idea: Can’t we get around all ofthese components by using a sphere in three dimensions? We canspecify a sphere by taking all the points that are the same distancefrom a given point in space. Then we can measure the surface area bylaying out a fine scale rectangular grid on the surface of the sphereand adding up all the bits of area. According to Euclid the totalarea$A$ is supposed to be $4\pi$times the square of the radius; so wecan define a “predicted radius” as$\sqrt{A/4\pi}$. But we can alsomeasure the radius directly by digging a hole to the center andmeasuring the distance. Again, we can take the measured radius minus thepredicted radius and call the difference the radius excess,\begin{equation*}r_{\text{excess}}=r_{\text{meas}}-\biggl(\frac{\text{measured area}}{4\pi}\biggr)^{1/2},\end{equation*}which would be a perfectly satisfactory measure of the curvature. It hasthe great advantage that it doesn’t depend upon how we orient a triangleor a circle.

But the excess radiusof a sphere also has a disadvantage; it doesn’t completely characterizethe space. It gives what is called the meancurvature of the three-dimensional world,since there is an averaging effect over the various curvatures. Since itis an average, however, it does not solve completely the problem ofdefining the geometry. If you know only this number you can’t predictall properties of the geometry of the space, because you can’t tell whatwould happen with circles of different orientation. The completedefinition requires the specification of six “curvature numbers” ateach point. Of course the mathematicians know how to write all thosenumbers. You can read someday in a mathematics book how to write themall in a high-class and elegant form, but it is first a good idea toknow in a rough way what it is that you are trying to write about. Formost of our purposes the average curvature will be enough.2

42–3Our space is curved

Now comes the main question. Is it true? That is, is the actual physicalthree-dimensional space we live in curved? Once we have enoughimagination to realize the possibility that space might be curved, thehuman mind naturally gets curious about whether the real world is curvedor not. People have made direct geometrical measurements to try to findout, and haven’t found any deviations. On the other hand, by argumentsabout gravitation, Einsteindiscovered that space is curved, and we’d like to tell you whatEinstein’s law is for the amount of curvature, and also tell you alittle bit about how he found out about it.

Einstein said that space iscurved and that matter is the source of the curvature. (Matter is alsothe source of gravitation, so gravity is related to the curvature—butthat will come later in the chapter.) Let us suppose, to make things alittle easier, that the matter is distributed continuously with somedensity, which may vary, however, as much as you want from place toplace.3 The rule thatEinstein gave for the curvatureis the following: If there is a region of space with matter in it and wetake a sphere small enough that the density$\rho$ of matter inside itis effectively constant, then the radius excess for the sphere is proportional tothe mass inside the sphere. Using the definition of excessradius, we have\begin{equation}\label{Eq:II:42:3}\text{Radius excess}=r_{\text{meas}}-\sqrt{\frac{A}{4\pi}}=\frac{G}{3c^2}\cdot M.\end{equation}Here, $G$ is the gravitational constant (of Newton’s theory), $c$ is thevelocity of light, and $M=4\pi\rho r^3/3$ is the mass of the matterinside the sphere. This is Einstein’s law for the mean curvature ofspace.

Suppose we take the earth as an example and forget that the densityvaries from point to point—so we won’t have to do any integrals.Suppose we were to measure the surface of the earth very carefully, andthen dig a hole to the center and measure the radius. From the surfacearea we could calculate the predicted radius we would get from settingthe area equal to$4\pi r^2$. When we compared the predicted radius withthe actual radius, we would find that the actual radius exceeded thepredicted radius by the amount given in Eq.(42.3). Theconstant$G/3c^2$ is about $2.5\times10^{-29}$cm per gram, so for eachgram of material the measured radius is off by $2.5\times10^{-29}$cm.Putting in the mass of the earth, which is about $6\times10^{27}$grams,it turns out that the earth has $1.5$millimeters more radius than itshould have for its surface area.4 Doing the samecalculation for the sun, you find that the sun’s radius is one-half akilometer too long.

You should note that the law says that the average curvatureabove the surface area of the earth is zero. But that doesnot mean that all the components of the curvature are zero. Theremay still be—and, in fact, there is—some curvature above the earth.For a circle in a plane there will be an excess radius of one sign for some orientationsand of the opposite sign for other orientations. It just turns out thatthe average over a sphere is zero when there is no mass insideit. Incidentally, it turns out that there is a relation between thevarious components of the curvature and the variation of theaverage curvature from place to place. So if you know the averagecurvature everywhere, you can figure out the details of the curvaturecomponents at each place. The average curvature inside the earth varieswith altitude, and this means that some curvature components are nonzeroboth inside the earth and outside. It is that curvature that we see as agravitational force.

Suppose we have a bug on a plane, and suppose that the “plane” haslittle pimples in the surface. Wherever there is a pimple the bug wouldconclude that his space had little local regions of curvature. We havethe same thing in three dimensions. Wherever there is a lump of matter,our three-dimensional space has a local curvature—a kind ofthree-dimensional pimple.

If we make a lot of bumps on a plane there might be an overall curvaturebesides all the pimples—the surface might become like a ball. It wouldbe interesting to know whether our space has a net average curvature aswell as the local pimples due to the lumps of matter like the earth andthe sun. The astrophysicists have been trying to answer that question bymaking measurements of galaxies at very large distances. For example, ifthe number of galaxies we see in a spherical shell at a large distanceis different from what we would expect from our knowledge of the radiusof the shell, we would have a measure of the excess radius of a tremendously large sphere. Fromsuch measurements it is hoped to find out whether our whole universe isflat on the average, or round—whether it is “closed,” like a sphere,or “open” like a plane. You may have heard about the debates that aregoing on about this subject. There are debates because the astronomicalmeasurements are still completely inconclusive; the experimental dataare not precise enough to give a definite answer. Unfortunately, wedon’t have the slightest idea about the overall curvature of ouruniverse on a large scale.

42–4Geometry in space-time

Now we have to talk about time. As you know from the special theory ofrelativity, measurements of space and measurements of time areinterrelated. And it would be kind of crazy to have something happeningto the space, without the time being involved in the same thing. Youwill remember that the measurement of time depends on the speed at whichyou move. For instance, if we watch a guy going by in a spaceship we seethat things happen more slowly for him than for us. Let’s say he takesoff on a trip and returns in $100$seconds flat by our watches;his watch might say that he had been gone for only $95$seconds. Incomparison with ours, his watch—and all other processes, like hisheart beat—have been running slow.

Now let’s consider an interesting problem. Suppose you are the one inthe spaceship. We ask you to start off at a given signal and return toyour starting place just in time to catch a later signal—at, say,exactly $100$seconds later according to our clock. And you arealso asked to make the trip in such a way that your watch willshow the longest possible elapsed time. How should you move? Youshould stand still. If you move at all your watch will read less than$100$sec when you get back.

Suppose, however, we change the problem a little. Suppose we ask you tostart at point$A$ on a given signal and go to point$B$ (both fixedrelative to us), and to do it in such a way that you arrive back just atthe time of a second signal (say $100$seconds later according to ourfixed clock). Again you are asked to make the trip in the way that letsyou arrive with the latest possible reading on your watch. How would youdo it? For which path and schedule will your watch show thegreatest elapsed time when you arrive? The answer is that you will spendthe longest time from your point of view if you make the trip bygoing at a uniform speed along a straight line. Reason: Any extramotions and any extra-high speeds will make your clock go slower. (Sincethe time deviations depend on the square of the velocity, whatyou lose by going extra fast at one place you can never make up by goingextra slowly in another place.)

The point of all this is that we can use the idea to define “a straightline” in space-time. The analog of a straight line in space is forspace-time a motion at uniform velocity in a constant direction.

The curve of shortest distance in space corresponds in space-time not tothe path of shortest time, but to the one of longest time,because of the funny things that happen to signs of the $t$-terms inrelativity. “Straight-line” motion—the analog of “uniform velocityalong a straight line”—is then that motion which takes a watch fromone place at one time to another place at another time in the way thatgives the longest time reading for the watch. This will be ourdefinition for the analog of a straight line in space-time.

42–5Gravity and the principle of equivalence

Now we are ready to discuss the laws of gravitation.Einstein was trying to generatea theory of gravitation that would fit with the relativity theory thathe had developed earlier. He was struggling along until he latched ontoone important principle which guided him into getting the correct laws.That principle is based on the idea that when a thing is falling freelyeverything inside it seems weightless. For example, a satellite in orbitis falling freely in the earth’s gravity, and an astronaut in it feelsweightless. This idea, when stated with greater precision, is calledEinstein’s principle ofequivalence. It depends on the fact that all objects fall with exactlythe same acceleration no matter what their mass, or what they are madeof. If we have a spaceship that is “coasting”—so it’s in a freefall—and there is a man inside, then the laws governing the fall ofthe man and the ship are the same. So if he puts himself in the middleof the ship he will stay there. He doesn’t fall with respect tothe ship. That’s what we mean when we say he is “weightless.”

Now suppose you are in a rocket ship which is accelerating. Acceleratingwith respect to what? Let’s just say that its engines are on andgenerating a thrust so that it is not coasting in a free fall. Alsoimagine that you are way out in empty space so that there arepractically no gravitational forces on the ship. If the ship isaccelerating with “$1$g” you will be able to stand on the “floor”and will feel your normal weight. Also if you let go of a ball, it will“fall” toward the floor. Why? Because the ship is accelerating“upward,” but the ball has no forces on it, so it will not accelerate;it will get left behind. Inside the ship the ball will appear to have adownward acceleration of “$1$g.”

Now let’s compare that with the situation in a spaceship sitting at reston the surface of the earth. Everything is the same! You would bepressed toward the floor, a ball would fall with an acceleration of$1$g, and so on. In fact, how could you tell inside a space shipwhether you are sitting on the earth or are accelerating in free space?According to Einstein’sequivalence principle there is no way to tell if you only makemeasurements of what happens to things inside!

To be strictly correct, that is true only for one point inside the ship.The gravitational field of the earth is not precisely uniform, so afreely falling ball has a slightly different acceleration at differentplaces—the direction changes and the magnitude changes. But if weimagine a strictly uniform gravitational field, it is completelyimitated in every respect by a system with a constant acceleration. Thatis the basis of the principle of equivalence.

42–6The speed of clocks in a gravitational field

Now we want to use the principle of equivalence for figuring out astrange thing that happens in a gravitational field. We’ll show yousomething that happens in a rocket ship which you probably wouldn’t haveexpected to happen in a gravitational field. Suppose we put a clock atthe “head” of the rocket ship—that is, at the “front” end—and weput another identical clock at the “tail,” as inFig.42–16. Let’s call the two clocks $A$ and$B$. If wecompare these two clocks when the ship is accelerating, the clock at thehead seems to run fast relative to the one at the tail. To see that,imagine that the front clock emits a flash of light each second, andthat you are sitting at the tail comparing the arrival of the lightflashes with the ticks of clock$B$. Let’s say that the rocket is in theposition$a$ of Fig.42–17 when clock$A$ emits a flash,and at the position$b$ when the flash arrives at clock$B$. Later onthe ship will be at position$c$ when the clock$A$ emits its nextflash, and at position$d$ when you see it arrive at clock$B$.

The first flash travels the distance$L_1$ and the second flash travelsthe shorter distance$L_2$. It is a shorter distance because the ship isaccelerating and has a higher speed at the time of the second flash. Youcan see, then, that if the two flashes were emitted from clock$A$ onesecond apart, they would arrive at clock$B$ with a separation somewhatless than one second, since the second flash doesn’t spend as much timeon the way. The same thing will also happen for all the later flashes.So if you were sitting in the tail you would conclude that clock$A$ wasrunning faster than clock$B$. If you were to do the same thing inreverse—letting clock$B$ emit light and observing it atclock$A$—you would conclude that $B$ was running slowerthan$A$. Everything fits together and there is nothing mysterious aboutit all.

But now let’s think of the rocket ship at rest in the earth’s gravity.The same thing happens. If you sit on the floor with one clockand watch another one which is sitting on a high shelf, it will appearto run faster than the one on the floor! You say, “But that is wrong.The times should be the same. With no acceleration there’s no reason forthe clocks to appear to be out of step.” But they must if the principleof equivalence is right. And Einstein insisted that the principle was right, and wentcourageously and correctly ahead. He proposed that clocks at differentplaces in a gravitational field must appear to run at different speeds.But if one always appears to be running at a different speed withrespect to the other, then so far as the first is concerned the otheris running at a different rate.

But now you see we have the analog for clocks of the hot ruler we weretalking about earlier, when we had the bug on a hot plate. We imaginedthat rulers and bugs and everything changed lengths in the same way atvarious temperatures so they could never tell that their measuringsticks were changing as they moved around on the hot plate. It’s thesame with clocks in a gravitational field. Every clock we put at ahigher level is seen to go faster. Heartbeats go faster, all processesrun faster.

If they didn’t you would be able to tell the difference between agravitational field and an accelerating reference system. The idea thattime can vary from place to place is a difficult one, but it is the ideaEinstein used, and it iscorrect—believe it or not.

Using the principle of equivalence we can figure out how much the speedof a clock changes with height in a gravitational field. We just workout the apparent discrepancy between the two clocks in the acceleratingrocket ship. The easiest way to do this is to use the result we found inChapter34 of Vol.I for the Dopplereffect. There, we found—seeEq.(34.14)—that if $v$ is the relative velocityof a source and a receiver, the received frequency$\omega$ isrelated to the emitted frequency$\omega_0$ by\begin{equation}\label{Eq:II:42:4}\omega=\omega_0\,\frac{1+v/c}{\sqrt{1-v^2/c^2}}.\end{equation}Now if we think of the accelerating rocket ship inFig.42–17 the emitter and receiver are moving with equalvelocities at any one instant. But in the time that it takes the lightsignals to go from clock$A$ to clock$B$ the ship has accelerated. Ithas, in fact, picked up the additional velocity$gt$, where $g$ is theacceleration and $t$ is time it takes light to travel the distance$H$from $A$ to$B$. This time is very nearly$H/c$. So when the signalsarrive at$B$, the ship has increased its velocity by$gH/c$. Thereceiver always has this velocity with respect to the emitter atthe instant the signal left it. So this is the velocity we should use inthe Doppler shift formula, Eq.(42.4). Assuming that theacceleration and the length of the ship are small enough that thisvelocity is much smaller than$c$, we can neglect the term in$v^2/c^2$.We have that\begin{equation}\label{Eq:II:42:5}\omega=\omega_0\biggl(1+\frac{gH}{c^2}\biggr).\end{equation}So for the two clocks in the spaceship we have the relation\begin{equation}\label{Eq:II:42:6}(\text{Rate at the receiver})=(\text{Rate of emission})\biggl(1+\frac{gH}{c^2}\biggr),\end{equation}\begin{equation}\label{Eq:II:42:6}\begin{pmatrix}\text{Rate}\\[-.75ex]\text{at the}\\[-.75ex]\text{receiver}\end{pmatrix}=\begin{pmatrix}\text{Rate of}\\[-.75ex]\text{emission}\end{pmatrix}\!\biggl(\!1+\frac{gH}{c^2}\biggr),\end{equation}where $H$ is the height of the emitter above the receiver.

From the equivalence principle the same result must hold for twoclocks separated by the height$H$ in a gravitational field with thefree fall acceleration$g$.

This is such an important idea we would like to demonstrate that it alsofollows from another law of physics—from the conservation of energy.We know that the gravitational force on an object is proportional to itsmass$M$, which is related to its total internal energy$E$by$M=E/c^2$. For instance, the masses of nuclei determined from theenergies of nuclear reactions which transmute one nucleus intoanother agree with the masses obtained from atomic weights.

Now think of an atom which has a lowest energy state of totalenergy$E_0$ and a higher energy state$E_1$, and which can go from thestate$E_1$ to the state$E_0$ by emitting light. The frequency$\omega$of the light will be given by\begin{equation}\label{Eq:II:42:7}\hbar\omega=E_1-E_0.\end{equation}

Now suppose we have such an atom in the state$E_1$ sitting on thefloor, and we carry it from the floor to the height$H$. To do that wemust do some work in carrying the mass$m_1=E_1/c^2$ up against thegravitational force. The amount of work done is\begin{equation}\label{Eq:II:42:8}\frac{E_1}{c^2}\,gH.\end{equation}Then we let the atom emit a photon and go into the lower energystate$E_0$. Afterward we carry the atom back to the floor. On thereturn trip the mass is$E_0/c^2$; we get back the energy\begin{equation}\label{Eq:II:42:9}\frac{E_0}{c^2}\,gH,\end{equation}so we have done a net amount of work equal to\begin{equation}\label{Eq:II:42:10}\Delta U=\frac{E_1-E_0}{c^2}\,gH.\end{equation}

When the atom emitted the photon it gave up the energy$E_1-E_0$. Nowsuppose that the photon happened to go down to the floor and beabsorbed. How much energy would it deliver there? You might at firstthink that it would deliver just the energy$E_1-E_0$. But that can’t beright if energy is conserved, as you can see from the followingargument. We started with the energy$E_1$ at the floor. When we finish,the energy at the floor level is the energy$E_0$ of the atom in itslower state plus the energy$E_{\text{ph}}$ received from the photon. Inthe meantime we have had to supply the additional energy$\Delta U$ ofEq.(42.10). If energy is conserved, the energy we end upwith at the floor must be greater than we started with by just the workwe have done. Namely, we must have that\begin{equation*}E_{\text{ph}}+E_0=E_1+\Delta U,\end{equation*}or\begin{equation}\label{Eq:II:42:11}E_{\text{ph}}=(E_1-E_0)+\Delta U.\end{equation}It must be that the photon does not arrive at the floor with justthe energy$E_1-E_0$ it started with, but with a little moreenergy. Otherwise some energy would have been lost. If we substitute inEq.(42.11) the$\Delta U$ we got in Eq.(42.10)we get that the photon arrives at the floor with the energy\begin{equation}\label{Eq:II:42:12}E_{\text{ph}}=(E_1-E_0)\biggl(1+\frac{gH}{c^2}\biggr).\end{equation}But a photon of energy$E_{\text{ph}}$ has thefrequency$\omega=E_{\text{ph}}/\hbar$. Calling the frequency of theemitted photon$\omega_0$—which is by Eq.(42.7)equal to$(E_1-E_0)/\hbar$—our result in Eq.(42.12) givesagain the relation of(42.5) between the frequency of thephoton when it is absorbed on the floor and the frequency with which itwas emitted.

The same result can be obtained in still another way. A photon offrequency$\omega_0$ has the energy$E_0=\hbar\omega_0$. Since theenergy$E_0$ has the relativistic mass$E_0/c^2$ the photon has a mass(not rest mass)$\hbar\omega_0/c^2$, and is “attracted” by theearth. In falling the distance$H$ it will gain an additionalenergy$(\hbar\omega_0/c^2)gH$, so it arrives with the energy\begin{equation*}E=\hbar\omega_0\biggl(1+\frac{gH}{c^2}\biggr).\end{equation*}But its frequency after the fall is$E/\hbar$, giving again the resultin Eq.(42.5). Our ideas about relativity, quantum physics,and energy conservation all fit together only ifEinstein’s predictions aboutclocks in a gravitational field are right. The frequency changes we aretalking about are normally very small. For instance, for an altitudedifference of $20$meters at the earth’s surface the frequencydifference is only about two parts in$10^{15}$. However, just such achange has recently been found experimentally using the Mössbauereffect.5 Einstein was perfectly correct.

42–7The curvature of space-time

Now we want to relate what we have just been talking about to the ideaof curved space-time. We have already pointed out that if the time goesat different rates in different places, it is analogous to the curvedspace of the hot plate. But it is more than an analogy; it means thatspace-time is curved. Let’s try to do some geometry inspace-time. That may at first sound peculiar, but we have often madediagrams of space-time with distance plotted along one axis and timealong the other. Suppose we try to make a rectangle in space-time. Webegin by plotting a graph of height$H$ versus$t$ as inFig.42–18(a). To make the base of our rectangle we take anobject which is at rest at the height$H_1$ and follow its worldline for $100$seconds. We get the line$BD$ in part(b) of the figurewhich is parallel to the $t$-axis. Now let’s take another object whichis $100$feet above the first one at$t=0$. It starts at the point$A$in Fig.42–18(c). Now we follow its world line for$100$seconds as measured by a clock at$A$. The object goes from $A$to$C$, as shown in part(d) of the figure. But notice that since timegoes at a different rate at the two heights—we are assuming that thereis a gravitational field—the two points $C$ and$D$ are notsimultaneous. If we try to complete the square by drawing a line to thepoint$C'$ which is $100$feet above$D$ at the same time, as inFig.42–18(e), the pieces don’t fit. And that’s what wemean when we say that space-time is curved.

42–8Motion in curved space-time

Let’s consider an interesting little puzzle. We have two identicalclocks, $A$ and$B$, sitting together on the surface of the earth.Now we lift clock$A$ to some height$H$, holdit there awhile, and return it to the ground so that it arrives at justthe instant when clock$B$ has advanced by $100$seconds. Then clock$A$will read something like $107$seconds, because it was running fasterwhen it was up in the air. Now here is the puzzle. How should we moveclock$A$ so that it reads the latest possible time—always assumingthat it returns when $B$ reads $100$seconds? You say, “That’s easy.Just take $A$ as high as you can. Then it will run as fast as possible,and be the latest when you return.” Wrong. You forgot something—we’veonly got $100$seconds to go up and back. If we go very high, we have togo very fast to get there and back in $100$seconds. And you mustn’tforget the effect of special relativity which causes moving clocks toslow down by the factor$\sqrt{1-v^2/c^2}$. This relativityeffect works in the direction of making clock$A$ read less timethan clock$B$. You see that we have a kind of game. If we stand stillwith clock$A$ we get $100$seconds. If we go up slowly to a smallheight and come down slowly we can get a little more than $100$seconds.If we go a little higher, maybe we can gain a little more. But if we gotoo high we have to move fast to get there, and we may slow down theclock enough that we end up with less than $100$seconds. What programof height versus time—how high to go and with what speed to get there,carefully adjusted to bring us back to clock$B$ when it has increasedby $100$seconds—will give us the largest possible time reading onclock$A$?

Answer: Find out how fast you have to throw a ball up into the air sothat it will fall back to earth in exactly $100$seconds. The ball’smotion—rising fast, slowing down, stopping, and coming back down—isexactly the right motion to make the time the maximum on a wrist watchstrapped to the ball.

Now consider a slightly different game. We have two points $A$ and$B$both on the earth’s surface at some distance from one another, as inFig.42–19. We playthe same game that we did earlier to find what we call the straightline. We ask how we should go from $A$ to$B$ so that the time on ourmoving watch will be the longest—assuming we start at$A$ on a givensignal and arrive at$B$ on another signal at$B$ which we will say is$100$seconds later by a fixed clock. Now you say, “Well we found outbefore that the thing to do is to coast along a straight line at auniform speed chosen so that we arrive at$B$ exactly $100$secondslater. If we don’t go along a straight line it takes more speed, and ourwatch is slowed down.” But wait! That was before we took gravity intoaccount. Isn’t it better to curve upward a little bit and then comedown? Then during part of the time we are higher up and our watch willrun a little faster? It is, indeed. If you solve the mathematicalproblem of adjusting the curve of the motion so that the elapsed time ofthe moving watch is the most it can possibly be, you will find that themotion is a parabola—the same curve followed by something that moveson a free ballistic path in the gravitational field, as inFig.42–19. Therefore the law of motion in a gravitationalfield can also be stated: An object always moves from one place toanother so that a clock carried on it gives a longer time than it wouldon any other possible trajectory—with, of course, the same startingand finishing conditions. The time measured by a moving clock is oftencalled its “proper time.” In free fall, the trajectory makes theproper time of an object a maximum.

Let’s see how this all works out. We begin with Eq.(42.5)which says that the excess rate of the moving watch is\begin{equation}\label{Eq:II:42:13}\frac{\omega_0gH}{c^2}.\end{equation}Besides this, we have to remember that there is a correction of theopposite sign for the speed. For this effect we know that\begin{equation*}\omega=\omega_0\sqrt{1-v^2/c^2}.\end{equation*}Although the principle is valid for any speed, we take an example inwhich the speeds are always much less than$c$. Then we can write thisequation as\begin{equation*}\omega=\omega_0(1-v^2/2c^2),\end{equation*}and the defect in the rate of our clock is\begin{equation}\label{Eq:II:42:14}-\omega_0\,\frac{v^2}{2c^2}.\end{equation}Combining the two terms in (42.13) and(42.14)we have that\begin{equation}\label{Eq:II:42:15}\Delta\omega=\frac{\omega_0}{c^2}\biggl(gH-\frac{v^2}{2}\biggr).\end{equation}Such a frequency shift of our moving clock means that if we measure atime$dt$ on a fixed clock, the moving clock will register the time\begin{equation}\label{Eq:II:42:16}dt\biggl[1+\biggl(\frac{gH}{c^2}-\frac{v^2}{2c^2}\biggr)\biggr],\end{equation}The total time excess over the trajectory is the integral of the extraterm with respect to time, namely\begin{equation}\label{Eq:II:42:17}\frac{1}{c^2}\int\biggl(gH-\frac{v^2}{2}\biggr)\,dt,\end{equation}which is supposed to be a maximum.

The term$gH$ is just the gravitational potential$\phi$. Suppose wemultiply the whole thing by a constant factor$-mc^2$, where $m$ is themass of the object. The constant won’t change the condition for themaximum, but the minus sign will just change the maximum to a minimum.Equation(42.17) then says that the object will move so that\begin{equation}\label{Eq:II:42:18}\int\biggl(\frac{mv^2}{2}-m\phi\biggr)\,dt=\text{a minimum}.\end{equation}But now the integrand is just the difference of the kinetic andpotential energies. And if you look in Chapter19 ofVolumeII you will see that when we discussed the principle of leastaction we showed that Newton’s lawsfor an object in any potential could be written exactly in the form ofEq.(42.18).

42–9Einstein’s theory of gravitation

Einstein’s form of theequations of motion—that the proper time should be a maximum in curvedspace-time—gives the same results as Newton’s laws for low velocities.As he was circling around the earth, Gordon Cooper’s watch was readinglater than it would have in any other path you could have imagined forhis satellite.6

So the law of gravitation can be stated in terms of the ideas of thegeometry of space-time in this remarkable way. The particles always takethe longest proper time—in space-time a quantity analogous to the“shortest distance.” That’s the law of motion in a gravitationalfield. The great advantage of putting it this way is that the lawdoesn’t depend on any coordinates, or any other way of defining thesituation.

In the laws (1) and(2) you have a precise statement ofEinstein’s theory ofgravitation—although you will usually find it stated in a morecomplicated mathematical form. We should, however, make one furtheraddition. Just as time scales change from place to place in agravitational field, so do also the length scales. Rulers change lengthsas you move around. It is impossible with space and time so intimatelymixed to have something happen with time that isn’t in some wayreflected in space. Take even the simplest example: You are riding pastthe earth. What is “time” from your point of view ispartly space from our point of view. So there must also bechanges in space. It is the entire space-time which is distortedby the presence of matter, and this is more complicated than a changeonly in time scale. However, the rule that we gave inEq.(42.3) is enough to determine completely all the laws ofgravitation, provided that it is understood that this rule about thecurvature of space applies not only from one man’s point of view but istrue for everybody. Somebody riding by a mass of material sees adifferent mass content because of the kinetic energy he calculates forits motion past him, and he must include the mass corresponding to thatenergy. The theory must be arranged so that everybody—no matter how hemoves—will, when he draws a sphere, find that the excessradius is$G/3c^2$times the total mass (or, better, $G/3c^4$times the totalenergy content) inside the sphere. That this law—law(1)—should betrue in any moving system is one of the great laws of gravitation,called Einstein’s fieldequation. The other great law is(2)—that things must move sothat the proper time is a maximum—and is calledEinstein’s equation ofmotion.

To write these laws in a complete algebraic form, to compare them withNewton’s laws, or to relate them to electrodynamics is difficultmathematically. But it is the way our most complete laws of the physicsof gravity look today.

Although they gave a result in agreement with Newton’s mechanics for thesimple example we considered, they do not always do so. The threediscrepancies first derived by Einstein have been experimentally confirmed: The orbit of Mercuryis not a fixed ellipse; starlight passing near the sun is deflectedtwice as much as you would think; and the rates of clocks depend ontheir location in a gravitational field. Whenever the predictions ofEinstein have been found todiffer from the ideas of Newtonian mechanics Nature has chosenEinstein’s.

Let’s summarize everything that we have said in the following way.First, time and distance rates depend on the place in space you measurethem and on the time. This is equivalent to the statement thatspace-time is curved. From the measured area of a sphere we can define apredicted radius, $\sqrt{A/4\pi}$, but the actual measured radius willhave an excess over this which is proportional (the constant is$G/3c^2$)to the total mass contained inside the sphere. This fixes the exactdegree of the curvature of space-time. And the curvature must be thesame no matter who is looking at the matter or how it is moving. Second,particles move on “straight lines” (trajectories of maximum propertime) in this curved space-time. This is the content ofEinstein’s formulation of thelaws of gravitation.