PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

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1 Option A Relatiity A The beginnings of relatiity Learning objeties It is said that Albert Einstein, as a boy, asked himself what would happen if he held a mirror in front of himself and ran forward at the speed of light. With respet to the ground, the mirror would be moing at the speed of light. Rays of light leaing young Einstein s fae would also be moing at the speed of light relatie to the ground. This meant that the rays would not be moing relatie to the mirror, and hene there should be no refletion in it. This seemed odd to Einstein. He expeted that looking into the mirror would not reeal anything unusual. Some years later, Einstein would resole this puzzle with a reolutionary new theory of spae and time, the theory of speial relatiity. Use referene frames. Understand Galilean relatiity with Newton s postulates for spae and time. Understand the onsequenes of Maxwell s theory for the speed of light. Understand how magneti effets are a onsequene of relatiity. A. Referene frames In a physis experiment, an obserer reords the time and position at whih eents take plae. To do that, she uses a referene frame. y A referene frame is a set of oordinate axes and a set of loks at eery point in spae. If this set is not aelerating, the frame is alled an inertial referene frame (Figure A.). So if the eent is a lightning strike, an obserer will look at the reading of the lok at the point where lightning struk and reord that reading as the time of the eent. The oordinates of the strike point gie the position of the eent in spae. So, in Figure A., lightning strikes at time t s and position x 60 m and y 0. (We are ignoring the z oordinate.) The same eent an also be iewed by another obserer in a different frame of referene. Consider, therefore, the following situation inoling one obserer on the ground and another who is a passenger on a train. x Figure A. A two-dimensional referene frame. There are loks at eery point in spae (only a few are shown here). All loks show the same time. y lightning strikes at x 60 m and t s x0 x 0 x 0 x 0 x 40 x 50 x 60 x 70 x 80 m Figure A. In this frame of referene the obserer deides that lightning struk at time t s and position x 60 m. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

2 In Figure A., a train moes past the obserer on the ground (represented by the thin ertial line) at time t 0, and is struk by lightning s later. The obserer on the train is represented by the thik ertial line. The train is traelling at a eloity of 5 m s as far as the ground obserer is onerned. Let us see how the two obserers iew arious eents along the trip. Assume first that when the loks arried by the two obserers show zero, the origins of the rulers arried by the obserers oinide, as shown in Figure A.. The stationary obserer uses the symbol x to denote the distane of an eent from the origin of his ruler. The obserer on the train uses the symbol x. origins oinide when loks at both origins show zero x t x x t 9 x 0 6 the train has moed away; when the loks show s, lightning strikes at x 60 m Figure A. The origins of the two frames of referene oinide when loks in both frames show zero. The origins then separate. Lightning strikes a point on the train. The obserer on the train measures the point where the lightning strikes and finds the answer to be x. The ground obserer measures that the lightning struk at a distane x from his origin. x x t t t In time t s the train moes forward a distane t. These equations express the relationship between the oordinates of the same eent as iewed by two obserers who are in relatie motion. Thus, to the eent in Figure A. ( lightning strikes ) the ground obserer assigns the oordinates x 60 m and t s. The obserer on the train assigns to this same eent the oordinates t s and x m. We are assuming here what we know from eeryday experiene (a guide that, as we will see, may not always be reliable): that two obserers always agree on what the time oordinates are; in other words, time is ommon to both obserers. Or, as Newton wrote, Absolute, true and mathematial time, of itself, and from its own nature, flows equably without any relation to anything external. Of ourse, the obserer on the train may onsider herself at rest and the ground below her to be moing away with eloity. It is impossible for one of the obserers to laim that he or she is really at rest and that the other is really moing. There is no experiment that an be performed by, say, the obserer on the train that will onine her that she really moes (apart from looking out of the window). If we PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

3 onsider instead a spae station and a spaeraft in outer spae as our two frames, een looking out of the window will not help. Whateer results the obserer on the train gets out of her experiments, the ground obserer also gets out of the same experiments performed in his ground frame of referene. The equations x x t t t are alled Galilean transformation equations, in honour of Galileo (Figure A.4): the relationship between the oordinates of an eent when one frame moes past the other with uniform eloity in a straight line. Both obserers are equally justified in onsidering themseles to be at rest, and the desriptions they gie are equally alid. Galilean relatiity has an immediate onsequene for the law of addition of eloities. Consider a ball that rolls with eloity u as measured by the obserer on the train. Again assume that the two frames, train and ground, oinide when t t 0 and that the ball first starts rolling when t 0. Then, after time t the position of the ball is measured to be at x u t by the obserer on the train (Figure A.5). The ground obserer reords the position of the ball to be at x x + t (u + )t (reall t t ), so as far as the ground obserer is onerned the ball has a eloity (distane/time) gien by Figure A.4 Galileo Galilei (564 64). u u + x u t t x Figure A.5 An objet rolling on the floor of the moing frame appears to moe faster as far as the ground obserer is onerned. Worked example A. A ball rolls on the floor of a train at m s (with respet to the floor). The train moes with respet to the ground a to the right at m s, b to the left at m s. What is the eloity of the ball relatie to the ground? a The eloity is 4 m s. b The eloity is 0 m s. This apparently foolproof argument presents problems, howeer, if we replae the rolling ball in the train by a beam of light moing with eloity 08 m s as measured by the obserer on the train. Using the formula aboe implies that light would be traelling at a higher speed relatie to the ground obserer. At the end of the 9th entury, onsiderable efforts were made to detet ariations in the speed of light depending on the state of motion of the soure of light. The experimental result was that no suh ariations were deteted! PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

4 A. Maxwell and the speed of light In 864, James Clerk Maxwell orreted an apparent flaw in the laws of eletromagnetism by introduing his famous displaement urrent term into the eletromagneti equations. The result is that a hanging eletri flux produes a magneti field just as a hanging magneti flux produes an eletri field (as Faraday had disoered earlier). An immediate onlusion was that aelerated eletri harges produe a pair of self-sustaining eletri and magneti fields at right angles to eah other, whih eentually deouple from the harge and moe away from it at the speed of light. Maxwell disoered eletromagneti waes and thus demonstrated the eletromagneti nature of light. One predition of the Maxwell theory was that the speed of light is a uniersal onstant. Indeed, Maxwell was able to show that the speed of light is gien by the expression ε 0μ0 where the two onstants are the eletri permittiity and magneti permeability of free spae (auum): two onstants at the heart of eletriity and magnetism. speed Maxwell s theory predits that the speed of light in auum does not depend on the speed of its soure. Figure A.6 An obserer on the train measures the speed of light to be. An obserer on the ground would then measure a different speed, +, aording to Galileo. This is in diret onfl it with Galilean relatiity. Aording to Galilean relatiity, if the speed of light takes on the alue in the train frame of referene then it will hae the alue + in the ground frame of referene (Figure A.6). The situation faed by Einstein in 905 was that the Galilean transformation equations (whih were perfetly ompatible and onsistent with Newtonian mehanis) did not seem to be ompatible with Maxwell s theory. It turned out that, a bit before 905, the Duth physiist Hendrik Lorentz (Figure A.7) was also thinking about the same problem. He too realised that the Maxwell theory was not ompatible with the Galilean transformation equations. He set about trying to find the simplest set of transformation equations that would be ompatible with Maxwell s theory. Lorentz s answer was a strange-looking set of equations, the Lorentz transformation equations (Table A.): Galileo Lorentz x t x x t x t t Figure A.7 Hendrik Lorentz (85 98). t x t Table A. Galilean and Lorentz transformations. 4 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

5 As we will soon see, these equations predit two brand new phenomena, whih we will disuss in detail later: length ontration (a moing objet is measured to hae a smaller length than when it is at rest) and time dilation (moing loks run slow). But the new problem now was that these new equations were not ompatible with Newtonian mehanis! So there were two hoies aailable (Table A.). Choie A Aept Newtonian mehanis and the Galilean equations and then modify Maxwell s theory Choie B Aept Maxwell s theory and the Lorentz equations and then modify Newtonian mehanis Table A. The hoies Einstein was onfronted with. Einstein s hoie was B. He realised early on that all kinds of puzzles arise when one looks at eletri and magneti phenomena from two different inertial referene frames. These ould only be resoled if one made hoie B. A. Eletromagneti puzzles Imagine a wire at rest in a lab, in whih a urrent flows to the left. This means that the eletrons in the wire are moing with a drift speed to the right (Figure A.8). The wire also ontains positie harges that are at rest. The aerage distane between the eletrons and that between the positie harges are the same. The net harge of the wire is zero, so an obserer at rest in the lab measures zero eletri field around the wire. Now imagine a positie harge q that moes with the same eloity as the eletrons in the rod. The obserer in the lab has no doubt that there will be a magneti fore on this harge. The magneti field at the position of the harge is out of the page and so there will be a magneti fore repelling the harge q from the wire (use the right-hand rule for magneti fore). Now onsider things from the point of iew of another obserer who moes along with the harge q. For this obserer, the harge q and the eletrons in the rod are at rest, but the positie harges in the rod moe to the left with speed. These positie harges do produe a magneti field, but the magneti fore on q is zero sine F qb and the speed of the harge is zero q a Fmagneti q b Feletri Figure A.8 a An obserer at rest with respet to the wire measures a magneti fore on the moing harge. b An obserer moing along with the harge will measure an eletri fore. Relatiity and reality Relatiity says that the details of an experiment may appear different to different obserers in relatie motion. This does not mean that different obserers experiene a different reality. The physially signifiant aspets of the experiment are the same for both obserers. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 5

6 a Fmagneti b Figure A.9 a Two protons moe past an obserer with the same eloity. b The eletri and magneti fores on the harges look different in eah referene frame. Now, if the harge q feels a fore aording to one obserer, other obserers must reah similar onlusions. So the puzzle is: where does the fore on the harge q ome from? Notie that in Figure A.8b we hae made the distane between positie harges smaller and the distane between eletrons larger. This has to be the ase if Lorentz is right about length ontration. The separation between the positie harges is a distane that moes past the obserer and so has to get smaller. The separation between the eletrons used to be a moing distane so, now that it has stopped, it has to be bigger. The effet of this is that now, as far as the harge q is onerned, there is more positie harge than negatie harge in the wire near it. There will then be an eletri fore of repulsion. We hae learned that, as a result of length ontration, the fore that an obserer alls magneti in one referene frame may be an eletri fore in another frame. There are many suh puzzles whih an only be resoled if the phenomena of length ontration and time dilation are taken into aount. So onsider two positie harges that moe parallel to eah other with speed relatie to the blak frame (the lab); see Figure A.9a. For an obserer moing along with the harges (red frame) the harges are at rest and so there annot be a magneti fore between them. There is, howeer, an eletri fore of repulsion. For this obserer, the harges moe away from eah other along straight lines. For an obserer at rest in the lab there are repulsie eletri fores but there are also magneti fores; this is beause eah moing harge reates a magneti field and the other harge moes in that magneti field. The magneti field reated by the bottom harge at the position of the top harge is direted out of the page. The magneti fore on the top harge is therefore direted opposite to the eletri fore; the magneti fore is attratie. The net fore is still repulsie and the two harges moe away from eah other along ured paths. Examining the details of this situation shows that the two obserers will reah onsistent results only if time runs differently in the two different frames. This is more eidene that, to aoid these eletromagneti puzzles, ideas similar to Lorentz s must be true; it is not a surprise that Einstein s 905 paper is entitled On the eletrodynamis of moing bodies. Worked example A. A positie eletri harge q enters a region of magneti field B with speed (Figure A.0). Disuss the fores, if any, that the harge experienes aording to a a frame of referene at rest with respet to the magneti field (blak) and b a frame moing with the same eloity as the harge (red). a q B Figure A.0 a The harge will experiene a magneti fore gien by F qb. b The harge is at rest. Hene the magneti fore is zero. But there has to be a fore, and that an only be an eletri fore. 6 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

7 Nature of siene Paradigm shift A fundamental fat of relatiity theory is that the speed of light is onstant for all inertial referene frames. This simple postulate has farreahing onsequenes for our understanding of spae and time. The idea that time is an absolute, assumed to be orret for oer 000 years, was shown to be false. Aepting the new ideas was not easy, but they offered the best explanation of obserations and reolutionised physis.? Test yourself It was a ery hotly debated subjet enturies ago as to whether the Earth goes around the Sun or the other way around. Does relatiity make this whole argument irreleant sine all frames of referene are equialent? Disuss the approximations neessary in order to laim that the rotating Earth is an inertial frame of referene. Imagine that you are traelling in a train at onstant speed in a straight line and that you annot look at or ommuniate with the outside. Think of the first experiment that omes to your mind that you ould do to try to find out that you are indeed moing. Then analyse it arefully to see that it will not work. HL 4 Here is another experiment that ould be performed in the hope of determining whether you are moing or not. The oil shown below is plaed near a strong magnet, and a galanometer attahed to the oil registers a urrent. Disuss whether we an dedue that the oil moes with respet to the ground. N S 6 Outline how you would know that you find yourself in a rotating frame of referene. 7 An eletri urrent flows in a wire. A proton moing parallel to the wire will experiene a magneti fore due to the magneti field reated by the urrent. From the point of iew of an obserer traelling along with the proton, the proton is at rest and so should experiene no fore. Analyse the situation. I 8 An inertial frame of referene S moing to the right with speed 5 m s moes past another inertial frame S. At time zero the origins of the two frames oinide. a A phone rings at loation x 0 m and t 5.0 s. Determine the loation of this eent in the frame S. b A ball is measured to hae eloity 5.0 m s in frame S. Calulate the eloity of the ball in frame S. 9 An inertial frame of referene S moing to the left with speed 5 m s moes past another inertial frame S. At time zero the origins of the two frames oinide. a A phone rings at loation x 4 m and t 5.0 s. Determine the loation of this eent in the frame S. b A ball is measured to hae eloity 5 m s in frame S. Calulate the eloity of the ball in frame S. 5 Outline an experiment you might perform in a train that is aelerating along a straight line that would onine you that it is aelerating. Disuss whether you ould also determine the diretion of the aeleration. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 7

8 Learning objeties Understand the two postulates of speial relatiity. Desribe lok synhronisation. Understand and use the Lorentz transformations. Apply the eloity addition formula. Work with inariant quantities. Understand and apply time. dilation and length ontration. Use muon deay as eidene for relatiity. A The Lorentz transformations This setion introdues the postulates of the theory of relatiity and the transformation equations that relate the oordinates of an eent as seen from two different inertial frames. We disuss the two major onsequenes of these equations: time dilation and length ontration. We also disuss the muon deay experiment, whih proides eidene in support of the theory of relatiity. A. The postulates of speial relatiity As disussed in Setion A, the Lorentz transformation equations are the simplest set of transformations with whih the Maxwell theory is onsistent when looked at from different referene frames. Einstein (Figure A.) re-deried the same set of equations based on two muh simpler and more general assumptions. These two assumptions are known as the postulates of relatiity. They are: The laws of physis are the same in all inertial frames. The speed of light in auum is the same for all inertial obserers. Figure A. Albert Einstein ( ). So we see that it is not just eletromagnetism that must be onsistent with these transformations, but all the laws of physis. These two postulates of relatiity, although they sound simple, hae far-reahing onsequenes. The fat that the speed of light in a auum is the same for all obserers means that absolute time does not exist. Consider a beam of light. Two different obserers in relatie motion to eah other will measure different distanes oered by this beam. But if they are to agree that the speed of the beam is the same for both obserers, it follows that they must also measure different times of trael. Thus, obserers in motion relatie to eah other measure time differently. The onstany of the speed of light means that spae and time are now ineitably linked and are not independent of eah other, as they were in Newtonian mehanis. The strength of intuition Einstein was so onined of the onstany of the speed of light that he eleated it to one of the priniples of relatiity. But it was not until 964 that onlusie experimental erifiation of this took plae. In an experiment at CERN, neutral pions moing at deayed into a pair of photons moing in different diretions. The speed of the photons in both diretions was measured to be with extraordinary auray. The speed of light does not depend on the speed of its soure. 8 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

9 We hae seen that Galileo s transformation laws x x t, t t, u u need to be hanged. Einstein (and Lorentz) modified them to t x x t, t x, we may rewrite these as Using the gamma fator, γ x γ(x t), t γ t x These formulas are useful when we know x and t and we want to find x and t. If, on the other hand, we know x and t and want to find x and t, then x γ(x + t ), t γ t + x These equations may be used to relate the oordinates of a single eent in one inertial frame to those in another. We will denote by S some inertial frame. A frame that moes with speed to the right relatie to S we will all S. These equations assume that, when loks in both frames show zero (i.e. t t 0), the origins of the two frames oinide (i.e. x x 0). Note that γ >. A graph of the gamma fator γ ersus eloity is shown in Figure A.. We see that γ is approximately for eloities up to about half the speed of light, but approahes infinity as the speed approahes the speed of light. γ / Figure A. The gamma fator γ as a funtion of eloity. The alue of γ stays essentially lose to for alues of the eloity up to about half the speed of light, but approahes infinity as the eloity approahes the speed of light. A. Clok synhronisation The equations aboe also assume that the loks in eah frame are synhronised. This means that they show the same time at any gien instant of time, i.e. we hae lok synhronisation. How an this be ahieed? The idea is to use the fat that the speed of light is a uniersal onstant. Consider a lok that is a distane x from the origin. A light x signal leaing the origin at time zero will take a time to arrie at the lok. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 9

10 x and wait for a light signal from the origin to arrie. When the light signal does arrie, the lok is started (the lok is a stopwath). Doing this for all the loks in the referene frame ensures that they are all synhronised. So we set that lok to show a time equal to Worked examples A. Lightning strikes a point on the ground (frame S) at position x 500 m and time t 5.0 s. Determine where and when the lightning struk aording to a roket that fl ies to the right oer the ground at a speed of (Assume that when loks in both frames show zero, the origins of the two frames oinide.) This is a straightforward appliation of the Lorentz formulas (the gamma fator is γ x γ(x t) 5 ): ( ).0 09 m and t γ t x ( 08) 8. s The obserers disagree on the oordinates of the eent lightning strikes ; both are orret. A.4 A lok in a roket moing at 0.80 goes past a lab. When the origins of both frames oinide, all loks are set to show zero. Calulate the reading of the roket lok as it goes past the point with x 40 m. The eent lok goes past the gien point has oordinates x and t in the lab frame and x and t in the roket s. Hene, from the frame. We need to find t. We know that x 40 m. We alulate that t 0.80 Lorentz formula for time. t γ x ( 08) Exam tip It is important that you are able to use the Lorentz equations to go from frame S to frame S as well as from S to S μ s 0 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

11 Sometimes we will be interested in the differene in oordinates of a pair of eents. So, for eents and, we define Δx x x and Δt t t in S and Δx x x and Δt t t in S These differenes are related by Δx γ(δx Δt); Δt γ Δt Δx or, in reerse, Δx γ(δx + Δt ); Δt γ Δt + Δx Worked examples A.5 Consider a roket that moes relatie to a lab with speed 0.80 to the right. We all the inertial frame of the lab S and that of the roket S. The length of the roket as measured by an obserer on the roket is 60 m. A photon is emitted from the bak of the roket towards the front end. Calulate the time taken for the photon to reah the front end of the roket aording to obserers in the roket and in the lab roket front bak lab Figure A. Eent is the emission of the photon. Eent is the arrial of the photon at the front of the roket. Clearly, Δx x arrial x emission 60 m This distane is oered at the speed of light and so the time between emission and arrial is Δt s In frame S we use 5 Δx ( ) 890 m Δt s PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

12 A.6 a A roket approahes a spae station with speed (relatie to the spae station). Obserers in the roket reord the firing of a missile from the spae station and.0 s later (aording to roket loks) reord an explosion at another spae station behind the roket at a distane of m (also measured aording to the roket). Calulate the distane and the time interal between the eents firing of the missile and explosion aording to obserers in the spae station. Comment on your answers. b In frame S, eent A auses eent B and therefore ours before eent B. Show that in any frame, eent A always ours before eent B. roket missile fired spae station Figure A We will take the spae station as frame S and the roket as frame S. Then we know that Δx x expl x fire m; Δt t expl t fire.0 s. The explosion happens after the firing of the missile and so Δt > 0. We apply the (reerse) Lorentz transformations to find a The gamma fator for a speed of is γ Δx 5.0 ( ) m Δt ( ).5 s The negatie answer for the time interal means that the explosion happens before the firing of the missile. Therefore the missile ould not hae been responsible for the explosion! We ould hae predited that the missile had nothing to do with the explosion beause, aording to obserers in the roket, the missile would hae to oer a distane of m in.0 s, that is, with a speed of m s, whih exeeds the speed of light and so is impossible. b We are told that Δt t B ta > 0. Let x be the distane between the two eents. The fastest way information Δx <. In any other frame, from A an reah B is at the speed of light, and so Δt Δt t B t A Δx Δx γ Δt Δt > γ Δt γ Δt γ Δt >0 So eent A has to our before eent B in any frame. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

13 A. Time dilation Suppose that an obserer in frame S measures the time between suessie tiks of a lok. The lok is at rest in S and so the measurement of the tiks ours at the same plae, x 0. The result of his measurement is t. The obserer in S will measure a time interal equal to Δx Δt γ Δt γ(δt + 0) Δt γ Δt This shows that the interal between the tiks of a lok that is moing relatie to the obserer in S is greater than the interal measured in S where the lok is at rest. This is known as time dilation. In this ase, the time measured in S is speial beause it represents a time interal between two eents that happened at the same point in spae. The lok is at rest in S so its tiks happen at the same point in S. Suh a time interal is alled proper time interal. (This is just a name; it is not implied that this is a more orret measurement of time.) A proper time interal is the time in between two eents that take plae at the same point in spae. The time dilation formula is best remembered as time interal γ proper time interal proper time interal Exam tip Do not make the mistake of thinking that proper time interals are measured only in the frame S. A proper time interal is the time between two eents that take plae at the same point. Different obserers disagree in their measurements, but both are right Note that there is no question as to whih obserer is right and whih is wrong when it omes to measuring time interals. Both are right. Two inertial obserers moing relatie to eah other at onstant eloity both reah alid onlusions, aording to the priniple of relatiity. Worked examples A.7 The time interal between the tiks of a lok arried on a fast roket is half of what obserers on the Earth reord. Calulate the speed of the roket relatie to the Earth. From the time dilation formula it follows that PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

14 A.8 A roket moes past an obserer in a laboratory with speed An obserer in the laboratory measures that a radioatie sample of mass 50 mg (whih is at rest in the laboratory) has a half-life of.0 min. Calulate the half-life as measured by obserers in the roket. We hae two eents here. The first is that the laboratory obserer sees a ontainer with 50 mg of the radioatie sample. The seond eent is that the laboratory obserer sees a ontainer with 5 mg of the radioatie sample. These eents are separated by.0 min as far as the laboratory obserer is onerned. These two eents take plae at the same point in spae as far as the laboratory obserer is onerned, so the laboratory obserer has measured the proper time interal between these two eents. Hene the roket obserers will measure a longer halflife, equal to γ (proper time interal).0 min min The point of this example is that you must not make the mistake of thinking that proper time interals are measured by the moing obserer. There is no suh thing as the moing obserer: the roket obserer is free to onsider herself at rest and the laboratory obserer to be moing with eloity A.4 Length ontration Exam tip Note that it is only lengths in the diretion of motion that ontrat. Now onsider a rod that is at rest in frame S. An obserer in S measures the position of the ends of the rod, subtrats and finds a length Δx L 0. (Notie that, sine the rod is at rest, the measurements of the position of the ends an be done at different times: the rod is not going anywhere). But for an obserer in S the rod is moing, so to measure its length he must reord the position of the ends of the rod at the same time, that is, with t 0. Sine Δx γ(δx Δt), we obtain the result that L 0 γ(l 0) L L0 γ This shows that the rod, whih is moing relatie to the obserer in S, has a shorter length in S than the length measured in S, where the rod is at rest. This is known as length ontration. The length Δx L 0 is speial beause it is measured in a frame of referene where the rod is at rest. Suh a length is alled a proper length. The length of an objet measured by an inertial obserer with respet to whom the objet is at rest is alled a proper length. Obserers with respet to whom the objet moes at speed measure a shorter length: proper length length γ 4 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

15 We must aept experimentally erified obserations, no matter how strange they may appear Time dilation as desribed is a real effet. In the Hafele Keating experiment, aurate atomi loks taken for a ride aboard planes moing at ordinary speeds and then ompared with similar loks left behind show readings that are smaller by amounts onsistent with the formulas of relatiity. Time dilation is also a daily effet in the operation of partile aelerators. In suh mahines, partiles are aelerated to speeds that are ery lose to the speed of light and thus relatiisti effets must be taken into aount when designing these mahines. The time dilation formula has also been erified in muon- deay experiments. Worked example A.9 In the year 04, a group of astronauts embark on a journey towards the star Betelgeuse in a spaeraft moing at 0.75 with respet to the Earth. Three years after departure from the Earth (as measured by the astronauts loks) one of the astronauts announes that she has gien birth to a baby girl. The other astronauts immediately send a radio signal to the Earth announing the birth. Calulate when the good news is reeied on the Earth (aording to Earth loks). When the astronaut gies birth, three years hae gone by aording to the spaeraft s loks. This is a proper time interal sine the eents departure from Earth and astronaut gies birth happen at the same plae as far as the astronauts are onerned (inside the spaeraft). Thus, the time between these two eents aording to the Earth loks is time interal γ proper time interal.0 yr yr This is therefore also the time during whih the spaeraft has been traelling, as far as the Earth is onerned. The distane oered is (as far as the Earth is onerned) distane t yr.40 yr Exam tip ly year.40 ly This is the distane that the radio signal must then oer in bringing the message. This is done at the speed of light and so the time taken is 40 ly.40 yr.40 yr Hene, when the signal arries, the year on the Earth is (approximately). PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 5

16 A.5 Another look at time dilation We hae seen using Lorentz transformations that relatiity predits the phenomenon of time dilation. A simpler iew of this effet is the following. A diret onsequene of the priniple of relatiity is that obserers who are in motion relatie to eah other do not agree on the interal of time separating two eents. To see this, onsider the following situation: a train moes with eloity with respet to the ground as shown in Figure A.5. From point A on the train floor, a light signal is sent towards point B diretly aboe on the eiling. The time t it takes for light to trael from A to B and bak to A is reorded. Note that as far as the obserers inside the train are onerned, the light beam traels along the straightline segments AB and BA. From the point of iew of an obserer on the ground, howeer, things look somewhat different. In the time it takes for light to return to A, point B (whih moes along with the train) will hae moed forward. This means, therefore, that to this obserer the path of the light beam looks like that shown in Figure A.6. B L S A S Figure A.5 A signal is emitted at A, is refleted off B and returns to A again. The path shown is what the obserer on the train sees as the path of the signal. The emission and reeption of the signals happen at the same point in spae. B S x 0 A S x0 B B A x 0 A Δt x Δt Figure A.6 The ground obserer sees things differently. In the time it takes the signal to return, the train has moed forward. Thus, the emission and reeption of the signal do not happen at the same point in spae. (The diagram is exaggerated for larity.) Suppose that the stationary obserer measures the time it takes light to trael from A to B and bak to A to be t. What we will show now is that, if both obserers are to agree that the speed of light is the same, then the two time interals annot be the same. It is lear that Δt Δt 6 L L + ( Δt / ) PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

17 sine the train moes forward a distane t in the time interal t that the stationary obserer measures for the light beam to reah A. Thus, soling the first equation for L and substituting in the seond (after squaring both equations) (Δt) 4 (Δt ) (Δt) (Δt) (Δt ) + (Δt) ( )(Δt) (Δt ) (Δt) (Δt ) So, finally, Δt Δt This is the time dilation formula. It is ustomary to all the expression γ the gamma fator, in whih ase Δt γ Δt. Reall that γ >. The time interal for the trael of the light beam is longer on the ground obserer s lok. This is known as time dilation. If the train passengers measure a time interal of Δt 6.0 s and the train moes at a speed 0.80, then the time interal measured by the ground obserer is Δt 6.0 s s s s s whih is longer. It will be seen immediately that this large differene ame about only beause we hose the speed of the train to be extremely lose to the speed of light. Clearly, if the train speed is small ompared with the speed of light, then γ, and the two time interals agree, as we might expet them to from eeryday experiene. The reason that our eeryday experiene leads us astray is beause the speed of light is enormous ompared with eeryday speeds. Thus, the relatiisti time dilation effet we hae just disoered beomes releant only when speeds lose to the speed of light are enountered. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 7

18 A.6 Addition of eloities Consider a frame S (for example, a train) that moes at onstant speed in a straight line relatie to another frame S (for example, the ground). An objet slides on the train floor in the same diretion as the train (S ) and its eloity is measured, by obserers in S, to be u. What is the speed u of this objet as measured by obserers in S? (See Figure A.7.) S u S S S u Figure A.7 The speed of the moing objet is u in the frame S. What is its speed as measured in frame S? In pre-relatiity physis (i.e. Galilean Newtonian physis), the answer would be simply u +. This annot, howeer, be the orret relatiisti answer; if we replaed the sliding objet by a beam of light (u ), we would end up with an obserer (S) who measured a speed of light different from that measured by S. The orret answer for the speed u of the partile relatie to S is u + u + or, soling for u, u u u u It an easily be heked that, irrespetie of how lose u or is to the speed of light, u is always less than. For the ase in whih u, then u as well, as demanded by the priniple of the onstany of the speed of light (hek this). On the other hand, if the eloities inoled are small ompared with the speed of light, then we may neglet the u term in the denominator, in whih ase Einstein s formula redues to the familiar Galilean relatiity formula u u +. Worked examples A.0 An eletron has a speed of m s relatie to a roket, whih itself moes at a speed of m s with respet to the ground. What is the speed of the eletron with respet to the ground? Applying the formula aboe with u m s and m s, we find u m s. 8 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

19 A. Two rokets moe away from eah other with speeds of 0.8 and 0.9 with respet to the ground, as shown in Figure A.8. What is the speed of eah roket as measured from the other? What is the relatie speed of the two rokets as measured from the ground? A 0.9 u 0.8 roket A Applying the formula with the alues gien in the figure, we find u + u ( 0.90)( 0.80) + B Figure A.8 Let us first find the speed of A with respet to B. In the frame of referene in whih B is at rest, the ground moes to the left with speed 0.90 (i.e. a eloity of 0.90). The eloity of A with respet to the ground is This is illustrated in Figure A.9. u ground S ground S B s rest frame Figure A The minus sign means, of ourse, that roket A moes to the left relatie to B. Let us now find the speed of roket B as measured in A s rest frame. The appropriate diagram is shown in Figure A roket B S ground We thus find u u + u + u 0.9 S A s rest frame Figure A (0.90)(0.80) as expeted. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 9

20 A. A roket has a proper length of 50 m and traels at a speed relatie to the Earth. A missile is fired from the bak of the roket at a speed u relatie to the roket. a Calulate the time when the missile passes the front of the roket aording to i obserers in the roket and ii aording to obserers on the Earth. b Calulate the speed of the missile relatie to the Earth and show that your answer is onsistent with the answer in a. a i Calling, as usual, the Earth frame S and the roket frame S, we know that Δx 50 m and 50 Δt s, where we are onsidering the eents missile is fired and missile passes the front of the roket. Δx ii We need to find Δt. From the Lorentz formulas we know that Δt γ Δt +. The gamma fator is γ.0. Hene Δt s b From the relatiisti addition law for eloities, u u + u (0.900)(0.950) The distane traelled by the missile aording to Earth obserers is Δx γ(δx + Δt ).0 ( ) 645 m This distane was oered at a speed of and so the time taken, aording to Earth obserers, is s, as expeted from a PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

21 A.7 Simultaneity Another great hange introdued into physis as a result of relatiity is the onept of simultaneity. Imagine three rokets A, B and C traelling with the same onstant eloity (with respet to some inertial obserer) along the same straight line. Imagine that roket B is halfway between rokets A and C, as shown in Figure A.. signal signal A B C Figure A. B emits signals to A and C. These are reeied at the same time as far as B is onerned. But the reeption of the signals is not simultaneous as far as the ground obserer is onerned. Roket B emits light signals that are direted towards rokets A and C. Whih roket will reeie the signal first? The priniple of relatiity allows us to determine that, as far as an obserer in roket B is onerned, the signals are reeied by A and C simultaneously (i.e. at the same time). This is obious, sine we may imagine a big box enlosing all three rokets that moes with the same eloity as the rokets themseles. Then, for any obserer in the box, or in the rokets themseles, eerything appears to be at rest. Sine B is halfway between A and C, learly the two rokets reeie the signals at the same time (light traels with the same speed). Imagine, though, that we look at this situation from the point of iew of a different obserer, outside the rokets and the box, with respet to whom the rokets moe with eloity. This obserer sees that roket A is approahing the light signal, while roket C is moing away from it (again, remember that light traels with the same speed in eah diretion). Hene, it is obious to this obserer that roket A will reeie the signal before roket C does. In the next setion we will use the Lorentz equations to proe that: Eents that are simultaneous for one obserer and whih take plae at different points in spae are not simultaneous for another obserer in motion relatie to the fi rst. On the other hand, if two eents are simultaneous for one obserer and take plae at the same point in spae, they are simultaneous for all other obserers as well. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

22 Worked example A. Obserer T is in the middle of a train that is moing with onstant speed to the right with respet to the train station. Two light signals are emitted at the same time as far as the obserer T in the train is onerned (Figure A.). T A B G Figure A. a Determine whether the emissions are simultaneous for an obserer G on the ground. b The signals arrie at T at the same time as far as T is onerned. i Determine whether they arrie at T at the same time as far as G is onerned. ii Dedue, aording to G, whih signal is emitted first. a No, beause the eents (i.e. the emissions from A and from B) take plae at different points in spae, and so if they are simultaneous for obserer T they will not be simultaneous for obserer G. b i Yes, beause the reeption of the two signals by T takes plae at the same point in spae, so if they are simultaneous for T, they must also be simultaneous for G. ii From G s point of iew, T is moing away from the signal from A. So the signal from A has a larger distane to oer to get to T. If the signals are reeied at the same time, and moed at the same speed, it must be that the one from A was emitted before that from B. Simultaneity, like motion, is a relatie onept. Our notion of absolute simultaneity is based on the idea of absolute time: eents happen at speifi times that all obserers agree on. Einstein has taught us that the idea of absolute time, just like the idea of absolute motion, must be abandoned. The Lorentz equations allow for a more quantitatie approah to simultaneity. Suppose that as usual we hae frames S and S, with frame S moing to the right with speed relatie to S. Suppose that two eents take plae at the same time in frame S. The time interal between these two eents is thus zero: Δt 0. What is the time interal between the same two eents when measured in S? The Lorentz equations gie Δt γ Δt Δx Δx We hae the interesting obseration that, if Δx 0, i.e. if the simultaneous eents in S our at the same point in spae, then they are also simultaneous in all other frames. Howeer, if Δx 0, then the eents will not be simultaneous in other frames. γ PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

23 Worked examples A.4 Let us rework the preious example quantitatiely. Take the proper length of the train (frame S ) to be 00 m and let it moe with speed 0.98 to the right. Determine, aording to an obserer on the ground (frame S), whih light turns on first and by how muh. We are told that Δt 0 and Δx x B x A 00 m. We want to find Δt, the time interal between the eents representing the emission of light from A and from B. The gamma fator is γ 5.0. Then 0.98 Δt t B ta γ Δt + Δx s The positie sign indiates that t B > ta, that is, the light from A was emitted 4.9 μs before the light from B. Note: if we wanted to find the atual emission times of the lights and not just their differene, we would use ta γ t x ( 50) s and t B γ t x (+50) s giing the same answer for the differene. A.5 Let us look at the preious example again, but now we will assume that the emissions from A and B happened at the same time for obserer G on the ground, in frame S (Figure A.). Determine whih emission takes plae first in the train frame, frame S. We are now told that t 0. In frame S, light A is emitted from position xa a and light B from position xb +a Then We want to find t. The gamma fator is γ 0.98 Δt t B t A γ Δt Δx a. 0 8 a The negatie sign shows that aording to obserer T, light B was emitted first. But to get a numerial answer we need to know a. The light signals are emitted from the ends of the train, of proper length 00 m. In the frame S this length is Lorentz-ontrated to 60 m, and so a 0 m. (You an also obtain this result as follows. In the frame S, light B is emitted at t 0. The position of this eent in S is x 50 m m.) Using x γ(x t), (a 0) a 5.0 A x a B x +a Figure A. Hene, Δt s. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

24 μ A.8 Muon deay.0 km e detetor a.0 km μ detetor b Figure A.4 a If relatiity did not hold, muons reated at the top of the mountain would not hae enough time to reah the bottom as muons. b Beause of relatiisti effets, muons do reah the bottom of the mountain. Muons are partiles with properties similar to those of eletrons exept that they are more massie, are unstable and deay to eletrons; they hae an aerage lifetime of about. 0 6 s. (The reation is μ e + ν e + νμ.) This is the lifetime measured in a frame where the muon is at rest: the proper time interal between the reation of a muon and its subsequent deay. From the point of iew of an obserer in the laboratory, howeer, the muons are moing at high speed, and the lifetime is longer beause of time dilation. Consider a muon reated by a soure at the top of a mountain.0 km tall (Figure A.4). The muon traels at 0.99 towards the surfae of the Earth. Without relatiisti time dilation, the muon would hae traelled a distane (as measured by ground obserers) of only m 0.65 km before deaying to an eletron (Figure A.4a). Thus a detetor at the base of the mountain would reord the arrial of an eletron, not a muon. But experiments show the arrial of muons at the detetor. This is beause the lifetime of the muon as measured by ground obserers is time interal proper time. 0 6 s s In this time the muon traels a distane (as measured by ground obserers) of m 4.6 km This means that the muon reahes the surfae of the Earth before deaying (Figure A.4b). The fat that muons do make it to the surfae of the Earth is eidene in support of the time dilation effet. μ 0.4 km detetor Figure A.5 From the point of iew of an obserer on the muon, the mountain is muh shorter. The muon exists as a muon for only. 0 6 s in the muon s rest frame. So how does an obserer traelling along with the muon explain the arrial of muons (and not eletrons) at the surfae of the Earth (Figure A.5)? The answer is that the distane of.0 km measured by obserers on the Earth is a proper length for them but not for the obserer at rest with respet to the muon. This obserer laims that it is the Earth that is moing upwards, and so measures a length-ontrated distane of km 0.4 km In this sense, muon deay experiments are indiret onfi rmations of the length ontration effet. 4 to the surfae of the Earth. The Earth s surfae is oming up to this obserer with a speed of 0.99 and so the time when they will meet is s s that is, before the muon deays. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

25 Worked example A.6 At the Stanford Linear Aelerator, eletrons of speed moe a distane of.00 km. a Calulate how long this takes aording to obserers in the laboratory. b Calulate how long this takes aording to an obserer traelling along with the eletrons. Find the speed of the linear aelerator in the rest frame of the eletrons. a In the laboratory, the eletrons take a time of.00 0 s s b The arrial of the eletrons at the beginning of the aelerator trak and that the end happen at the same point in spae as far as the obserer traelling with the eletrons is onerned, so this is a proper time interal. Thus time interal γ proper time interal γ s.57 proper time interal That is, proper time interal s The speed of the aelerator is obiously in the opposite diretion. But this an be heked as follows. As far as the eletron is onerned, the length of the aelerator trak is moing past it and so is length-ontrated aording to length proper length γ.00 km km and so has a speed of speed ms m s 0.96 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 5

26 Nature of siene Pure siene applying general arguments to speial ases Some years before relatiity was introdued by Einstein, an experiment performed by Albert Mihelson and Edward Morley gae a ery puzzling result onerning the speed of light. The experimenters had expeted to find that light traelled faster when it was direted along the path of trael of the Earth than when it was at right angles to the Earth s motion. They found no differene. There were franti attempts to resole the diffiulties posed by this experiment. One attempt was to assume that moing lengths appear shorter. Lorentz showed that if one used his Lorentz transformation equations, ertain diffiulties with eletromagnetism went away. But it was Einstein who re-deried these transformation equations from far more general priniples, the postulates of relatiity. By demanding that all inertial obserers experiene the same laws of physis and measured the same eloity of light in auum, the Lorentz equations emerged as the simplest linear equations that ould ahiee this.? 6 Test yourself 0 An earthling sits on a benh in a park eating a sandwih. It takes him 5 min to finish it, aording to his wath. He is being monitored by inaders from planet Zenga who are orbiting at a speed of a Calulate how long the aliens rekon it takes an earthling to eat a sandwih. b The aliens in the spaeraft get hungry and start eating their sandwihes. It takes a Zengan 5 min to eat her sandwih aording to Zengan loks. They are atually being obsered by earthlings as they fly oer the Earth. Calulate how long it takes a Zengan to eat a sandwih aording to Earth loks. A ube has density ρ when the density is measured at rest. Suggest what happens to the density of the ube when it traels past you at a relatiisti speed. A pendulum in a fast train is found by obserers on the train to hae a period of.0 s. Calulate the period that obserers on a station platform would measure as the train moes past them at a speed of A spaeraft moes past you at a speed of 0.95 and you measure its length to be 00 m. Calulate the length you would measure if it were at rest with respet to you. 4 Two idential fast trains moe parallel to eah other. An obserer on train A tells an obserer on train B that by her measurements (i.e. by A s measurements) train A is 0 m long and train B is 8 m long. The obserer on train B takes measurements; alulate what he will he find for: a the speed of train A with respet to train B b the length of train A the length of train B. 5 An unstable partile has a lifetime of s as measured in its rest frame. The partile is moing in a laboratory with a speed of 0.95 with respet to the lab. a Calulate the lifetime of the partile aording to an obserer at rest in the laboratory. b Calulate the distane traelled by the partile before it deays, aording to the obserer in the laboratory. 6 The star Vega is about 50 ly away from the Earth. A spaeraft moing at is heading towards Vega. a Calulate how long it will take the spaeraft to get to Vega aording to loks on the Earth. b The rew of the spaeraft onsists of 8-year-old IB graduates. Calulate how old the graduates will be (aording to their loks) when they arrie at Vega. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

27 7 A roket traelling at 0.60 with respet to the Earth is launhed towards a star. After 4.0 yr of trael (as measured by loks on the roket) a radio message is sent to the Earth. Calulate when it will arrie on the Earth as measured by: a obserers on the Earth b obserers on the roket. 8 A spaeraft leaing the Earth with a speed of 0.80 sends a radio signal to the Earth as it passes a spae station 8.0 ly from the Earth (as measured from the Earth). a Calulate how long it takes the signal to arrie on the Earth aording to Earth obserers. b Calulate how long it takes the spaeraft to reah the spae station (aording to loks on the spaeraft). As soon as the signal is reeied on the Earth, a reply signal is sent to the spaeraft. Calulate how long the reply signal takes to arrie at the spaeraft, aording to Earth loks. d Aording to spaeraft loks, alulate how muh time goes by between the emission of the signal and the arrial of the reply. 9 A roket approahes a mirror on the ground at a speed of 0.90, as shown below. The distane D between the front of the roket and the mirror is.4 0 m, as measured by obserers on the ground, when a light signal is sent towards the mirror from the front of the roket. Calulate when the refleted signal is reeied by the roket as measured by: a obserers on the ground b obserers on the roket. 0.9 D 0 Two objets moe along the same straight line. Their speeds are as measured by an obserer on the ground. Find: a the eloity of B as measured by A b the eloity of A as measured by B. A B Repeat Question 0 for the arrangement below A B A partile A moes to the right with a speed of relatie to the ground. A seond partile, B, moes to the right with a speed of relatie to A. Calulate the speed of B relatie to the ground. Partile A moes to the left with a speed of relatie to the ground. A seond partile, B, moes to the right with a speed of relatie to A. Find the speed of B relatie to the ground. 4 A muon traelling at oers a distane of.00 km (as measured by an earthbound obserer) before deaying. a Calulate the muon s lifetime as measured by the earthbound obserer. b Calulate the lifetime as measured by an obserer traelling along with the muon. 5 The lifetime of the unstable pion partile is measured to be s (when at rest). This partile traels a distane of 0 m in the laboratory just before deaying. Calulate its speed. In the following questions, the frames S and S hae their usual meanings, that is, S moes past S with eloity and, when their origins oinide, both loks are set to zero. 6 In frame S an explosion ours at position x 600 m and time t.0 μ s. Frame S is moing at speed 0.75 in the positie diretion. Determine where and when the explosion takes plae aording to frame S. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 7

28 7 Frame S moes with speed An explosion ours at the origin of frame S when the loks in S read 6.0 μ s. Calulate where and when the explosion takes plae aording to frame S. 8 Frame S moes with speed Calulate the reading of the loks in frame S as the origin of S passes the point x 0 m. 9 Two eents in frame S are suh that Δx x x 00 m and Δt t t 6.00 μ s. a i The speed is Calulate Δx x x and Δt t t. ii Determine whether eent ould ause eent. b Determine whether there is a alue of suh that t 0. Comment on your answer. A Spaetime diagrams Learning objeties 0 Two eents in frame S are suh that Δx x x 00 m and Δt t t.0 μ s. a i The speed is Calulate Δx x x and Δt t t. ii Determine whether eent ould ause eent. b Determine whether there is a alue of suh that t < 0. Comment on your answer. Two simultaneous eents in frame S are separated by a distane x x x 00 m. Determine the time separating these two eents in frame S, stating whih one ours first. The speed is Understand, sketh and work with spaetime diagrams. Represent worldlines. Explain the twin paradox. This setion deals with a pitorial representation of relatiisti phenomena that makes the onepts of time dilation, length ontration and simultaneity partiularly transparent. A. The inariant hyperbola The Lorentz transformations hae the following important onsequene. Consider an eent that is measured to hae oordinates (x, t) in S and (x, t ) in S. As we know, these oordinates are different in the two frames; the obserers disagree about the spae and time oordinates. But they all agree on this: the quantity (x t ) is the same as (x t )! We an proe this easily as follows: (x t ) γ(x t) γ t x γ x xt + t t tx + γ x ( )t γ x )t ( γ Exam tip Remember that γ and so γ. (x t ) x t x The usefulness and signifiane of this result will beome apparent in the next setion. A. Spaetime diagrams (Minkowski diagrams) In Topi, we saw lots of motion graphs. In partiular, we saw graphs of position (ertial axis) ersus time (horizontal axis). In relatiity it is ustomary to show these graphs with the axes reersed, that is, with 8 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

29 time plotted on the ertial axis and position on the horizontal. These are alled spaetime diagrams. They are also alled Minkowski diagrams in honour of H. Minkowski, a mathematiian friend of Einstein who helped him with the mathematial formulation of the theory. For reasons we will see shortly, it is ery onenient to instead plot t on the ertial axis rather than time itself. The ertial axis then also has units of length. Sine the speed of light is a onstant eeryone agrees on, knowing the alue of t allows us to find the alue of t. Figure A.6 shows a spaetime diagram and an eent (marked by the dot). The spae and time oordinates of the eent are read from the graph in the usual way: we draw lines through the dot parallel to the axes and see where the lines interset the axes. This eent has x 5.0 m s. and t 6.0 m, giing t.0 08 Now onsider a partile that is at rest at position x a (Figure A.7). As time goes by we may think of a sequene of eents showing the position of the partile (whih does not hange) at different times. This sequene of eents traes the straight blue line on the spaetime diagram. This is alled the worldline of the partile. Another partile that starts from x 0 at t 0 and moes with onstant positie eloity would hae the worldline shown in red. The speed of this partile is gien by distane x x tan θ time t t so tan θ. The tangent of the angle of the wordline with the t-axis gies the speed, expressed in units of the speed of light. This is why it is onenient to plot t on the ertial axis: a photon moes at speed and so it makes an angle of θ tan 45 with both axes. Sine nothing an hae a speed that exeeds, the wordline of any partile will always make an angle less than 45 with the t-axis. In Figure A.8, the red worldline represents a partile moing to the right. The green worldline belongs to a partile moing to the left. The blue worldline is impossible sine it inoles a speed greater than. Now onsider a seond inertial referene frame S that moes to the right with speed. Let us assume that when loks in both frames show zero the origins of the frames oinide. The origin of frame S moes to the right with speed. Therefore, the worldline of the origin of S will make an angle θ tan with the t-axis of S. But this worldine is just the olletion of all eents with x 0, and is therefore the time axis of the frame S. Beause the speed of light is the same in all frames, the spae axis of the new frame must make the same angle with the old x-axis. Thus the new frame has axes as shown in red in Figure A.9. An eent will hae different oordinates in the two frames, as we know. To find the oordinates of an eent in the frame S, draw lines parallel to the slanted axes and see where they interset the axes. The oordinates in the two frames are onneted by Lorentz transformations. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 t / m x/m Figure A.6 The spae and time oordinates of an eent. t x θ t a x Figure A.7 The worldline of a partile at rest is the sequene of eents showing the position of the partile at different times. t photon photon impossible worldline x Figure A.8 Various worldlines. The one in blue is impossible beause it orresponds to a partile moing faster than light. t t worldline of origin of new frame θ x spae axis of new frame x Figure A.9 Two frames with a relatie eloity on the same spaetime diagram. The red axes represent a frame moing to the right. 9

30 Worked examples A.7 Use the spaetime diagram below to estimate the speed of frame S relatie to frame S. t / m 5.0 m 4 t / m 5.0 m x / m θ 0 4 x/m Figure A.0 The tangent of the angle θ is tan θ and so A.8 The spaetime diagram below shows three eents. List them from earliest to latest, as obsered in eah frame. t A t C x B x Figure A. In S (blue frame), eents A and C are simultaneous and our after eent B (A and C are on the same line parallel to the x-axis). In S (red frame), eents B and C are simultaneous and our before eent A (B and C are on the same line parallel to the x -axis). 0 PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

31 Let us now see the signifiane of the equation known as inariant hyperbola. Figure A. shows the inariant hyperbola passing through a point with oordinates (x, t 0). Its equation is t x. All eents on the blue ure hae the same alue of t x, namely. Furthermore these eents, as obsered in frame S, hae the same alue of t x, namely. If t x is plotted on the spaetime axes of frame S, the inariant hyperbola intersets the spae axis of frame S at the point (i.e. eent) P. The oordinates of P are (x, t 0). Sine t x t x, it follows that t x. Sine t 0, it follows that x m. This result shows that the sales on the two spae axes (x and x ) are not the same. This is the basis for length ontration, as we will see in the next setion. The sales are also different on the time axes. A simpler way to see that the sales on the axes are different is to note in Figure A. that the blue dashed line passes through the eent with oordinates x, t 0 (in S). It is parallel to the t -axis, and intersets the x -axis at Q. The oordinates of Q are learly t 0 and Exam tip The mathematial details of the inariant hyperbola are not required in examinations. Howeer, you should understand why the sales on the axes of frame S and frame S are different. The important thing to remember is that the sales are in fat different. t / m t / m.0 x γ(x t) γ( 0) x / m γ Q.0 This sets the sale on the x -axis. Consider now the inariant hyperbola that passes through the point with oordinates (t, x 0). Its equation is t x 0 ). This is plotted in Figure A.. The hyperbola intersets the time axis of frame S at eent P, whose oordinates are (t, 0). Sine (t ) (x ) t x, it follows that (t ) (x ), and therefore at eent P, (t ) 0 and so t m. We again see that the sales on the two time axes are different. This is the basis for time dilation. Again, a simpler way to see the differene in sale is to note in Figure A. that the blue dashed line passes through the eent with oordinates x 0, t m (in S). It intersets the t -axis at Q. Its oordinates are x 0 and t γ t.5 x P x γ x x/m Figure A. a The inariant hyperbola sets the sale on the x -axis (red frame). b This sale is also set by the dashed blue line, whih is parallel to the t -axis and passes through x m. t / m t / m.0 Q.5 P x t γ t.0 t γ( 0 ) 0.5 t γ t.0 x / m t γ This sets the sale on the t -axis x/m Figure A. The inariant hyperbola sets the sale on the t -axis (red frame). PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

32 t / m A. Length ontration and spaetime diagrams t / m x / m Length measured in S S in in m m 4 5 x/m Length measured in S is shorter Figure A.4 Rod at rest in S. Obserers in S measure a length of m for the rod. Obserers in S measure a shorter length. Exam tip For obserers in S, the rod is moing. So its length must be measured when the ends are reorded at the same time. Imagine a rod of length m at rest in the frame S. At t 0 the left end of the rod is at x 0 and the other end is at x m. The wordlines of the ends of the rod are shown in Figure A.4. The left end of the rod has a worldline that is the t -axis; the worldline of the other end is the blue line (whih is parallel to the t -axis). The wordlines of the two ends interset the x-axis at 0 and 0.8 m. These two intersetion points represent the positions of the moing rod s ends at the same time in frame S. Their differene therefore gies the length of the rod as measured in S. The length is 0.8 m, less than m, the length in S. The rod whih is moing aording to obserers in S has ontrated in length when measured in frame S. What if we had a rod of proper length m at rest in the frame S whih was iewed by obserers in the frame S? This is illustrated in Figure A.5. t / m t / m 5 4 x / m Length measured in S is shorter Exam tip Do not be led astray by your knowledge of Eulidean geometry! The length in S looks longer, but it isn t. The sales on the two axes are not the same. 0 Length measured in S is m 4 5 x/m Figure A.5 Rod at rest in S. Obserers in S measure a length of m for the rod. The blue lines are the worldlines of the ends of the rod. Obserers in S measure a shorter length. The two thik blue lines represent the worldlines of the ends of the rod. They interset the x -axis at 0 and 0.8 m. These two intersetion points represent the positions of the rod s ends at the same time in frame S. Their differene therefore gies the length of the rod as measured in S. The length is 0.8 m, less than m, the length in S. The rod whih is moing aording to obserers in S has ontrated in length when measured in frame S. A.4 Time dilation and spaetime diagrams t / m Q t / m x / m P 4 O 4 x/m Figure A.6 is a spaetime diagram showing the standard frames S and S. A lok at rest at the origin of S tiks at O and then at P. The time between tiks is t m. What is the time between tiks aording to frame S? We hae to draw a line parallel to the x-axis through P. This intersets the S time axis at point Q. This interal is greater than. Sine P has oordinates (t m, x 0) in S, this eent in the frame S has time oordinate t γ t + x γt Figure A.6 The green line is the worldline of a lok at rest in frame S. This lok shows t at P. t γt PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

33 Similarly, Figure A.7 an be used to analyse a lok at rest at the origin of frame S. It tiks at O and then at P. The time between tiks is suh that t. From P we draw a line parallel to the x -axis, whih intersets the t -axis at Q. The interal OQ is longer than m. A.5 The twin paradox We hae seen that the effet of time dilation is symmetri: if you moe relatie to me, I say that your loks are running slow ompared with mine but you say that my loks are running slow ompared with yours. This symmetry of time dilation is shown learly in the last two figures in the preious setion. An issue arises when two loks, initially at the same plae and showing the same time (say zero), are separated. Suppose one lok moes at a relatiisti speed away, suddenly reerses diretion and omes bak to its starting plae next to the lok that stayed behind. The readings of the two loks are ompared. What do they show? In the twin paradox ersion of the story, the loks are replaed by twins. Jane stays on the Earth and laims that sine her twin brother Joe is the one who moed away he must be younger. But Joe an laim that it is Jane and the Earth that moed away and then ame bak, so Jane should be younger. Who is the younger of the two when the twins are reunited? Deeply ingrained ideas are hard to get rid of Exam tip Do not be led astray by your knowledge of Eulidean geometry! In triangle OPQ the hypotenuse is the side OP and so it would seem to be the longest side. But the geometry of spaetime diagrams is not Eulidean. The Pythagorean theorem does not hold. t / m t / m x / m Q P 4 O 4 x/m Figure A.7 The purple line is the worldline of a lok at rest in frame S. This lok shows t at P. The physiist W. Rindler, disussing the twin paradox in his book Introdution to Speial Relatiity (Oxford Uniersity Press, 99), says: It is quite easily resoled, but seems to possess some hidden emotional ontent that makes it the subjet of interminable debate among the dilettantes of relatiity. So deeply ingrained is the idea of absolute time in us that we find this example hard to aept. The resolution of the paradox is that Jane has been in the same inertial frame throughout, whereas Joe hanged inertial frames in the interal of time it took for him to turn around. He was in an inertial frame moing outwards but he hanged to one moing inwards for the return trip. During the hangeoer from one frame to the other he must hae experiened aeleration and fores whih Jane neer did. The aeleration makes this situation asymmetri, and Joe is the one who is younger on his return to Earth. To understand this quantitatiely, suppose Joe leaes the Earth in a roket at a speed of 0.6, traels to a distant planet.0 ly away (as measured by Earth obserers) and returns. The gamma fator γ is.5..0 ly As Jane sees it, his outbound trip takes 5 yr, and another 5 yr to 0.6 return. She will hae aged 0 years when her brother returns. For Joe, time is running slower by the gamma fator, so he will age by yr on the outward trip and another 4 yr on the way bak. He.5 will be 8 years older. We an show all of this on a spae time diagram (Figure A.8). PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

34 t / yr t / yr t / yr 5 6 x / ly x / ly A.6 Causality 0 x / ly Figure A.8 Spaetime diagram used to resole the twin paradox. (Adapted from N. Daid Mermin, It s About Time, Prineton Uniersity Press, 005) t Eents that an be aused by E The power of diagrams again x Eents that an ause E Figure A.9 The past and future light ones of eent E. Figure A.9 shows an eent E on a spaetime diagram. The ones aboe and below E are defined respetiely by the timelines of photons leaing E and arriing at E. The one formed by the arriing photons (the past light one) inludes all eents that ould in priniple hae aused E to our. Sine nothing an moe faster than light, no eent in the past of E outside the shaded light one ould hae influened E. Similarly, the future light one of E onsists of those eents that ould in priniple be aused by E. E annot influene any eent outside this light one. Nature of siene E 4 The blak frame, S, is Jane s. The blue frame, S, is Joe s on the way out, and the red frame, S, is Joe s on the way bak. The irles show what the loks read in eah frame. We see that, at the moment before Joe turns around and hanges frame, the Earth lok shows 5 yr and Joe s lok shows 4 yr. When Joe s lok shows 4 yr, he determines (by drawing a line parallel to the x -axis) that Earth loks show. yr. (Time dilation is symmetri!) As soon as Joe hanges frame, he determines (by drawing a line parallel to the x -axis) that Earth loks show 6.8 yr. The sudden swith from one frame to another seems to reate a missing time of 0..6 yr. But this is not mysterious and nothing strange has happened to either Jane or Joe during this time; it has to do with the fat that we hae arious frames. The now of frame S is found by drawing lines parallel to the x -axis. The now of frame S is found by drawing lines parallel to the x -axis. Thus, what the two frames understand by now is different. Spaetime diagrams offer a simple and pitorial way of understanding relatiity. They an be used to unambiguously resole misunderstandings and paradoxes. Their power lies in their simpliity and their larity. In physis, using diagrams with appropriate notation to desribe situations has always helped understanding. Spaetime diagrams are espeially useful in showing what eent an or annot ause another. Feynman diagrams are another example of isualisation of a model, as is the use of etors to show magnitude and diretion. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

35 ? Test yourself In the questions that follow, the spaetime diagrams represent two inertial frames. The blak axes represent frame S. The red axes represent a frame S that moes past frame S with eloity. Use the spaetime diagram to alulate the eloity of frame S relatie to S. t / m t / m 4 a On a opy of the spaetime diagram, draw the worldline of a partile that is: i at rest in frame S at x.0 m ii moing with eloity 0.80 as measured in S at x.0 m at t 0. t / m.5 t / m x / m x / m x/m Use the spae time diagram to alulate the eloity of frame S relatie to S. t / m t / m b A photon is emitted from position x +.0 m at t 0. Use the spaetime diagram from part a to estimate when the photon arries at the eye of an obserer at rest at the origin of frame S, as obsered i in S and ii in S. 5 a S represents an alien attak ruise ship. Use the diagram below to determine the speed of the ruise ship relatie to S. b At t 0, a laser beam moing at the speed of light is launhed from x +.0 ly towards the ruise ship. By drawing the worldline of the laser beam, estimate the time when the beam hits the ruise ship, as obsered i in S and ii in S. t / ly x/m t / ly x/m x / m.0 x / ly PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 x / ly 5

36 6 The diagram shows two lamps at x ± 0.5 m that turn on at t 0 in frame S. t / m.0 t / m 0.5 x / m x/m a Determine if the spaeraft will be saed. b Using Lorentz transformations or otherwise, alulate, in the spaeraft frame: i the time of the explosion ii the position of the explosion. 8 a The dashed blue line in the spaetime diagram below is parallel to the t -axis and intersets the x -axis at P. Using Lorentz transformations, find the oordinates of P in the frame S, and hene label the eent with oordinates (x m, t 0). t / m.0 t / m a Determine whih lamp turns on first as obsered in frame S. b Draw the worldlines of the photons emitted from the two lamps towards an obserer at rest at the origin of S. Identify the lamp whose light reahes an obserer at the origin of frame S first. 7 In the spaetime diagram below, the red axes represent a spaeraft that is moing past a planet (blak axes). An explosion on the planet takes plae at x 500 m and t 000 m, emitting deadly photons. A shield around the spaeraft is turned on at the indiated point in spaetime. t / m x / m P x/m b Repeat part a, where now the speed is arbitrary. Express your answer in terms of the gamma fator, γ. t / m shield on explosion 000 x / m x/m PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

37 9 a The dashed blue line in the spaetime diagram 4 In the following spaetime diagram a lok is at is parallel to the x -axis and intersets the rest at the origin of frame S. t -axis at P. Find the oordinates of P in t / m the frame S, and hene label the eent with t / m 5 oordinates (x 0, t m). t / m.0 4 t / m.5 x / m P.0 x / m x/m Its first tik ours at t 0; mark this eent on the diagram. The seond tik of the lok ours when t mas measured in S ; mark this eent on the same spaetime diagram. By drawing appropriate lines, estimate the time in between tiks as measured in S..0 x/m b Repeat part a, where now the speed is arbitrary. Express your answer in terms of the gamma fator γ. 40 The purple line in the spaetime diagram below represents a rod of length.0 m at t 0 as measured in the frame S. t / m t / m 5 4 x / m x/m a On a opy of this diagram, draw appropriate lines to show that the length of the rod measured in frame S is less than m. b Draw a line to represent a rod of length.0 m as measured in the frame S. By drawing appropriate lines, show that the length of the rod measured in frame S is less than m. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 7

38 Learning objeties Understand and use the onepts of total and rest energy. Work with relatiisti momentum. Sole problems with partile aeleration. Appreiate the inariane of eletri harge. Appreiate photons as massless relatiisti partiles. Work with relatiisti units. A4 Relatiisti mehanis (HL) This setion introdues the hanges in Newtonian mehanis that are neessary as a result of the postulates of relatiity. We will hae to modify the onepts of total energy and momentum. A4. Relatiisti energy and rest energy One of the first onsequenes of Einstein s theories in mehanis is the equialene of mass and energy. The theory of relatiity predits that, to a partile of mass m that is at rest with respet to some inertial obserer, there orresponds an amount of energy E 0 that the obserer measures to be E 0 m. (We will not be able to gie a proof of this statement in this book.) This energy is alled the rest energy of the partile. Rest energy is the amount of energy needed to produe a partile at rest. Exam tip The phrase to reate a partile from the auum refers to partile ollisions in partile aelerators in whih energy, usually in the form of photons, materialises as partile antipartile pairs. Similarly, if the partile moes with speed relatie to some inertial obserer, the energy orresponding to the mass of the partile that this obserer will measure is gien by E γm m This is energy that the partile has beause it has mass and beause it moes. If the partile has other forms of energy, suh as graitational potential energy or eletrial potential energy, then the total energy of the partile will be γm plus the other forms of energy. We will mostly deal with situations where the partile has no other forms of energy assoiated with it. The total energy of the partile in that ase is then just γm. It is important to note immediately that, as the speed of the partile approahes the speed of light, the total energy approahes infinity (Figure A.40). This is a sign that a partile with mass annot reah the speed of light. Only partiles without mass, suh as photons, an moe at the speed of light. E/E / Figure A.40 A graph of the ariation with of the ratio of the total energy E to the rest energy E0 of a partile. As the speed of the partile approahes the speed of light, its energy inreases without limit. 8 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 8 FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

39 Worked examples A.9 Find the speed of a partile whose total energy is double its rest energy. We hae that E γm m γm γ A.0 Find a the rest energy of an eletron and b its total energy when it moes at a speed equal to a The rest energy is E m J J ev b The gamma fator at a speed of 0.80 is γ and so the total energy is E γm MeV 0.85 MeV 0.5 MeV A. A proton (rest energy 98 MeV) has a total energy of 70 MeV. Find its speed. Sine the total energy is gien by E γm, we hae that PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 9

40 If a partile is aelerated through a potential differene of V olts, its total energy will inrease by an amount qv, where q is the harge of the partile. If the partile is initially at rest, its initial total energy is the rest energy, E 0 m. After going through the potential differene, the total energy will be E m + qv. We an then find the speed of the partile, as the next example shows. Worked examples A. An eletron of rest energy 0.5 MeV is aelerated through a potential differene of 5.0 MV in a lab. a Find its total energy with respet to the lab. b Find its speed with respet to the lab. a The total energy of the eletron will inrease by qv e olts 5.0 MeV and so the total energy is E m 0 + qv 0.5 MeV MeV 5.5 MeV b We know that E γm 5.5 γ 0.5 γ Sine γ 0.785, it follows that ( 0.097) PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

41 A. a A proton is aelerated from rest through a potential differene V. Calulate the alue of V that will ause the proton to aelerate to a speed of (The rest energy of a proton is 98 MeV.) b Determine the aelerating potential required to aelerate a proton from a speed of 0.95 to a speed of a The gamma fator at a speed of 0.95 is γ The total energy of the proton after aeleration is thus E γm.0 98 MeV 00 MeV From E m + qv we find qv (00 98) MeV 064 MeV and so V. 09 V Notie how we hae aoided using SI units in order to make the numerial alulations easier. b The total energy of the proton at a speed of 0.95 is (from part a) E 00 MeV. The total energy at a speed of 0.99 is (working as in a) E 6649 MeV. The extra energy needed is then MeV, so the aelerating potential must be.6 09 V. Notie that a larger potential differene is needed to aelerate the proton from 0.95 to 0.99 than from rest to This is a sign that it is impossible to reah the speed of light. (See also the next example.) PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 4

42 A.4 A onstant fore is applied to a partile whih is initially at rest. Sketh a graph that shows the ariation of the speed of the partile with time for a Newtonian mehanis b relatiisti mehanis. In Newtonian mehanis, a onstant fore produes a onstant aeleration, and so the speed inreases uniformly without limit, exeeding the speed of light. In relatiisti mehanis, the speed inreases uniformly as long as the speed is substantially less than the speed of light, and is essentially idential with the Newtonian graph. Howeer, as the speed inreases, so does the energy. Beause it takes an infinite amount of energy for the partile to reah the speed of light, we onlude that the partile neer reahes the speed of light. The speed approahes the speed of light asymptotially. Note that the speed is always less than the orresponding Newtonian alue at the same time. Hene we hae the graph shown in Figure A.4. /.4 Newton Einstein t Figure A.4 A4. Momentum and energy In the last setion we saw the need to hange the definition of total energy. The new definition ensures that a partile annot be aelerated to the speed of light, beause that would take an infinite amount of energy. One other hange is required, to momentum, so that in relatiisti ollisions momentum is onsered as it is in ordinary mehanis. In lassial mehanis, momentum is gien by the produt of mass and eloity, but in relatiity this is modified to p m γm We still hae the usual law of momentum onseration, whih states that when no external fores at on a system the total momentum stays the same. The symbol m here stands for the rest mass of the partile and is a onstant for all obserers. Unlike Newtonian mehanis, a onstant fore on the partile will produe a dereasing aeleration in suh a way that the speed neer reahes the speed of light. 4 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 4 FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

43 Worked example A.5 A onstant fore F ats on an eletron that is initially at rest. Find the speed of the eletron as a funtion of time. Initially, for small t, the speed inreases uniformly, as in Newtonian mehanis. But as t beomes large, the speed tends to the speed of light, but does not reah or exeed it. This is beause as the speed inreases the aeleration beomes smaller and smaller, and the speed neer reahes the speed of light. This results in the graph shown in Figure A.4. dp Begin with Newton s seond law, F, where dt p γm is the momentum of the eletron. Sine dp d dt dt t Figure A.4 d + dt dp dt 0.8 m m /.0 m / m / d dt d dt it follows that d F dt m / or d / F dt m Integrating both sides, we find that (assuming the mass starts from rest) F t m Soling for the speed, we find (Ft) (m) + (Ft) The way the speed approahes the speed of light is shown in Figure A.4. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 4

44 A4. Kineti energy A mass moing with eloity has a total energy E gien by E m Its kineti energy E k is defined as the total energy minus the rest energy: Ek E m Newtonian mehanis is a good approximation to relatiity at low speeds This relatiisti definition of kineti energy does not look similar to the ordinary kineti energy, m. In fat, when is small ompared with, we an approximate the alue of the relatiisti using the binomial expansion for for small x: fator x + x+ x Applying this to E k with x, we find E k m + + m That is, Ek m In other words, for low speeds the relatiisti formula redues to the familiar Newtonian ersion. For higher speeds, the relatiisti formula must be used. This an be rewritten as Ek m m γm m (γ )m This definition ensures that the kineti energy is zero when 0, as an easily be heked. As in ordinary mehanis, the work done by a net fore equals the hange in kineti energy in relatiity as well. 44 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

45 Total energy, momentum and mass are related: from the definition of momentum, we find that p + m 4 m + m 4 m 4 E That is, 4 E m +p (This formula is the relatiisti ersion of the onentional formula p E from Newtonian mehanis.) This relation an be remembered m by using the Pythagorean theorem in the triangle in Figure A.4. The formula we deried aboe applies also to those partiles that hae zero mass, suh as the photon, in whih ase E p. Remembering that, hf h for a photon, E hf, we hae p λ. Exam tip In the last setion we saw, for partile aeleration through a potential differene V, that qv E. But E is also E k sine, for a partile aelerated from rest, E γm m (γ )m Ek A4.4 Relatiisti units We an use units suh as MeV or GeV for mass. This follows from the fat that the rest energy of a partile is gien by E 0 m and so allows us to express the mass of the partile in terms of its rest energy as m E 0/. Thus, the statement the mass of the pion is 5 MeV, means that the rest energy of this partile is 5 MeV 5 MeV. (To find the mass in kilograms, we would first hae to onert MeV to joules and then diide the result by the square of the speed of light.) Similarly, the momentum of a partile an be expressed in units of MeV or GeV. A partile of rest mass 5.0 MeV and total energy MeV has a momentum gien by p E m0 Figure A.4 The rest energy, momentum and total energy are related through the Pythagorean theorem for the triangle shown. E m 4 + p p (69 5) MeV 44 MeV p MeV p MeV PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 Exam tip It is absolutely essential that you are omfortable with these units. 45

46 The speed an be found from E m In onentional SI units, a momentum of MeV is kg m J 08 m s s m s kg m s Worked examples A.6 Find the momentum of a pion (rest mass 5 MeV ) whose speed is The total energy is E m MeV Using E m 4 + p We obtain p MeV p 80 MeV 46 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

47 A.7 Find the speed of a muon (rest mass 05 MeV ) whose momentum is 8 MeV. From p γm, we hae 8 MeV γ 05 MeV γ.7 Hene Exam tip You an also do this by first finding the total energy (5 MeV) and then the gamma fator (.9) and then the speed. and so 0.9. A.8 Find the kineti energy of an eletron whose momentum is.5 MeV. The total energy of the eletron is gien by E m 4 + p 0.5 MeV +.5 MeV.5 MeV E.58 MeV and so E k E m.58 MeV 0.5 MeV.07 MeV Nature of siene General priniples surie paradigm shifts usually Newtonian mehanis relies on basi quantities suh as mass, energy and momentum and the laws that relate these quantities. Einstein realised that laws suh as energy and momentum onseration are the result of deeper priniples and sought to presere them in relatiity. This meant modifying what we normally all energy and momentum in suh a way that these quantities are also onsered in relatiity. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 47

48 ? Test yourself 4 Calulate the energy needed to aelerate an eletron to a speed of a 0.50 b a Calulate the speed of a partile whose kineti energy is 0 times its rest energy. b Calulate the speed of a partile whose total energy is 0 times its rest energy. 44 Find the momentum of a proton whose total energy is 5 times its rest energy. 45 Find the total energy of an eletron with a momentum of 50 MeV. 46 Determine the momentum, in onentional SI units, of a proton with momentum 685 MeV. 47 Find the kineti energy of a proton whose momentum is 500 MeV. 48 The rest energy of a partile is 5 MeV and its total energy is 00 MeV. Find its speed. 49 Calulate the potential differene through whih a proton must be aelerated (from rest) so that its momentum is 00 GeV. 50 Find the momentum of a proton whose speed is Calulate the speed of a proton with momentum.5 GeV. 5 A proton initially at rest finds itself in a region of uniform eletri field of magnitude V m. The eletri field aelerates the proton oer a distane of.0 km. Calulate: a the kineti energy of the proton b the speed of the proton. 5 a Show that the speed of a partile with rest mass m and momentum p is gien by p 4 m +p b An eletron and a proton hae the same momentum. Calulate the ratio of their speeds when the momentum is i.00 MeV ii.00 GeV. What happens to the alue of the ratio as the momentum gets larger and larger? 54 A partile at rest breaks apart into two piees with masses 50 MeV and 5 MeV. The lighter fragment moes away at a speed of a Find the speed of the other fragment. b Determine the rest mass of the original partile that broke apart. 55 An eletron and a positron, eah with kineti energy.0 MeV, ollide head-on. The eletron and positron annihilate eah other, giing two photons. a Explain why the eletron positron pair annot reate just one photon. b Explain why the photons must be moing in opposite diretions. Calulate the energy of eah photon. 56 A neutral pion of mass 5 MeV traelling at 0.80 deays into two photons traelling in opposite diretions, π0 γ. Calulate the ratio of the frequeny of photon A to that of photon B. pion B A 57 Two idential bodies with rest mass.0 kg are moing diretly towards one another, eah with a speed of 0.80 relatie to the laboratory. They ollide and form one body. Determine the rest mass of this body. 58 State the formulas, in terms of the rest mass m of a partile, for a the relatiisti momentum p b the total energy E. Using these formulas, derie the formula p for the speed of the partile. E d The formula in applies to all partiles, een those that are massless. Dedue that, if the partile is a photon, then. 59 a Show that when a partile of mass m and harge q is aelerated from rest through a potential differene V, the speed it attains qv orresponds to a gamma fator γ +. m b A proton is aelerated from rest through a potential differene V. Calulate V suh that the proton reahes a speed of Show that a free eletron annot absorb or emit a photon. 48 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

49 A5 General relatiity (HL) Learning objeties The theory of general relatiity was formulated by Albert Einstein in 95. It is a theory of graitation that replaes the standard theory of graitation of Newton, and generalises Einstein s speial theory of relatiity. The theory of general relatiity stands as the rowning ahieement of Einstein s genius and is onsidered to be perhaps the most elegant and beautiful example of a physial theory eer onstruted. It is a radial theory in that it relates the distribution of matter and energy in the unierse to the struture of spae and time. The geometry of spaetime is a diret funtion of the matter and energy that spaetime ontains. A5. The priniple of equialene We saw in Topi that when a person stands on a sale inside a freely falling eleator ( Einstein s eleator ) the reading of the sale is zero. It is as if the person is weightless. This is what the sale would read if the eleator were moing at onstant eloity in deep spae, far from all masses. Similarly, onsider an astronaut in a spaeraft in orbit around the Earth. She too feels weightless and floats inside the spaeraft. But neither the person in the falling eleator nor the astronaut is really weightless. Graity does at on both. We an say that the aeleration of the freely falling eleator or the entripetal aeleration of the spaeraft has anelled out the fore of graity. The right aeleration an make graity disappear and make the frame of referene under onsideration look like one moing at onstant eloity. The right aeleration an also make graity appear. Consider an astronaut in a spaeraft that is moing with onstant eloity in deep spae, far from all masses. The astronaut really is weightless. The spaeraft engines are now ignited and the spaeraft aelerates forward. The astronaut feels pinned down to the floor. If he drops a oin, it will hit the floor, whereas preiously it would hae floated in the spaeraft. The oin falling to the floor and the sensation of being pinned down are what we normally assoiate with graity. These are two examples where the effets of aeleration mimi those of graity. Another expression for the effets of aeleration is inertial effets. Einstein eleated these obserations to a priniple of physis the equialene priniple (EP): Understand the priniple of equialene. Understand the bending of light. Desribe graitational redshift and the Pound Rebka experiment. Understand the nature of blak holes. Understand the meaning of the eent horizon. Understand time dilation near a blak hole. Understand the impliations of general relatiity for the unierse as a whole. Exam tip There is hardly an examination paper in whih you will not be asked to state this priniple. Graitational and inertial effets are indistinguishable. Applying this priniple to the two examples we disussed aboe, we may re-express it more preisely in two ersions: Version : A referene frame moing at onstant eloity far from all masses is equialent to a freely falling referene frame in a uniform graitational field. Exam tip You an use either of these ersions of the equialene priniple, but the original statement takes are of both. Version : An aelerating referene frame far from all masses is equialent to a referene frame at rest in a graitational field. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 49

50 Version B onstant eloity free fall in a graitational field A planet outer spae Figure A.44 A frame of referene moing at onstant eloity far from any masses (A) and a freely falling frame of referene in a graitational field (B) are equialent. Version onstant aeleration ag at rest in a graitational field B A planet outer spae Figure A.45 An aelerating frame of referene far from any masses (A) and a frame at rest in a graitational field (B) are equialent. free fall Figure A.44 shows a frame of referene A moing at onstant eloity in deep spae, and a frame of referene B falling freely in a graitational field. The EP says that the two frames are equialent. There is no experiment that the oupants of A an perform that will gie a different result from the same experiment performed in B, nor an the oupants of either frame deide whih of the two states of motion they are really in. Figure A.45 shows a frame of referene A aelerating in deep spae, and a frame of referene B at rest in a graitational field. Again, there is no experiment that the oupants of A an perform that will gie a different result from the same experiment performed in B, nor an the oupants deide whih is whih. This priniple has immediate onsequenes. A5. Consequenes of the EP: the bending of light One onsequene of the EP is that light bends towards a massie body. Imagine a box falling freely in the graitational field of a planet (Figure A.46). A ray of light is emitted from inside the box initially parallel to the floor. Sine, by the EP, this frame is equialent to a frame moing at onstant eloity in outer spae, there an be no doubt that the ray of light will hit the opposite side of the box at point X. But an obserer, (who must also see the ray hit at X), says that the ray is bending towards the planet. But this means that, from the point of iew of an obserer at rest on the surfae of the massie body, light has bent towards it. A ray of light bends towards a massie body. X Massie objets an thus at as a kind of graitational lens. A5. Consequenes of the EP: graitational redshift planet Figure A.46 A ray of light bends towards a massie objet. ag outer spae graitational field planet Figure A.47 In the aelerated frame the light ray will hae a lower frequeny at the top beause of the Doppler effet. Exam tip In an exam you should be able to use the EP to dedue the bending of light, graitational red-shift and time dilation. A seond onsequene of the EP is that a ray of light emitted upwards in a graitational field will hae its frequeny redued as it limbs higher. In Figure A.47 a referene frame is at rest in a graitational field of a planet. A ray of light is emitted upwards. This frame is equialent to a frame in outer spae with an aeleration equal to the graitational field strength on the planet. In this aelerated frame, an obserer at the top of the frame is moing away from the light emitted at the base. So, aording to the Doppler effet, he will obsere a lower frequeny than that emitted at the base. By the EP, the same thing happens in a frame at rest in a graitational field. The frequeny of a ray of light is redued as it moes higher in a graitational field. Notie that if the frequeny dereases as the ray moes upwards, the period must inrease (the speed of light is onstant). But the period may be used as a lok. The period is the time in between tiks of a lok. Graitational red-shift is therefore equialent to saying that we hae a graitational time dilation effet. Consider two idential loks. 50 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

51 One is plaed near a massie body and the other far from it. When the lok near the massie body shows that s has gone by, the faraway lok will show that more than s has gone by. Time slows down near a massie body. A5.4 The tests of general relatiity There are many experimental tests that the general theory of relatiity has passed. These inlude: The bending of light and radio signals near massie objets: this was measured by Eddington in 99, in agreement with Einstein s predition. Graitational frequeny shift: this was obsered by Pound and Rebka in 960 (see Setion A5.5). The Shapiro time delay experiment: radio signals sent to a planet or spaeraft and refleted bak to the Earth take slightly longer to return when they pass near the Sun than when the Sun is out of the way. This delay is predited by the theory of relatiity. The preession of the perihelion of Merury: Merury has an abnormality in its orbit that Newtonian graitation annot aount for. Einstein s theory orretly predits this anomaly. Graitational lensing: massie galaxies may bend light from distant stars and een galaxies, reating multiple images. This has been obsered; see Figure A.48. Figure A.48 Multiple images of a distant quasar, aused by an interening galaxy whose graitational field has defleted the light from the quasar. A5.5 The Pound Rebka experiment The phenomenon of graitational red-shift was experimentally erified by the Pound Rebka experiment in 960. In this experiment, performed at Harard Uniersity, a beam of gamma rays of energy 4.4 kev from a nulear transition in iron-57 was emitted from the top of a tower.6 m high and deteted at ground leel; see Figure A.49. Just as the frequeny of light dereases as the light moes upwards in a graitational field, it inreases if the light is moing downwards: we hae a blue-shift (whih is what Pound and Rebka measured). Theory predits that the frequeny shift f between the emitted and reeied frequenies is gien by Δf f gh Here, f stands for the emitted or the obsered frequeny and H is the height from whih the photons are emitted. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 H Figure A.49 A photon is emitted from the top of a tower and obsered at the base, where its frequeny is measured and found to be higher than that emitted. 5

52 Worked example A.9 A photon of energy 4.4 kev is emitted from the top of a 0 m-tall tower towards the ground. Calulate the shift in frequeny expeted at the base of the tower. On emission, the photon has a frequeny gien by E hf f E h Hz The shift (it is a blue-shift) is thus Δf f gh Δf f gh Hz Hz Note how sensitie this experiment atually is: the shift in frequeny is only about 04 Hz, ompared with the emitted frequeny of about 08 Hz, a frational shift of only 04 Hz Hz A5.6 The struture of the theory Credit where redit is due Einstein s general theory of relatiity would hae been impossible to deelop were it not for 9th-entury mathematiians who were bold enough to question Eulid s fi fth axiom of geometry. Modifying that axiom meant that new kinds of geometry beame aailable. Deeloping the mathematial mahinery to desribe these geometries was one of the greatest ahieements of 9th-entury mathematis. The theory of general relatiity is a physial theory different from all others, in that: The mass and energy ontent of spae determines the geometry of that spae and time. The geometry of spaetime determines the motion of mass and energy in the spaetime. That is, geometry mass energy Spaetime is a four-dimensional world with three spae oordinates and one time oordinate. In the absene of any fores, a body moes in this four-dimensional world along paths of shortest length, alled geodesis. If a single mass M is the only mass present in the unierse, the solutions of the Einstein equations imply that, far from this mass, the geometry of spae is the usual Eulidean flat geometry, with all its familiar rules (for example, the angles of a triangle add up to 80 ). As we approah the neighbourhood of M, the spae beomes ured, as illustrated in 5 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 5 FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

53 Figure A.50. The rules of geometry then hae to hange. Large masses with small radii produe extreme bending of the spaetime around them. Thus, the motion of a planet around the Sun is, aording to Einstein, not the result of a graitational fore ating on the planet (as Newton would hae it) but rather due to the ured geometry in the spae and time around the Sun reated by the large mass of the Sun. Similarly, light retains its familiar property of traelling from one plae to another in the shortest possible time. Sine the speed of light is onstant, this means that light traels along paths of shortest length: geodesis. In the flat Eulidean geometry we are used to, geodesis are straight lines. In the ured non-eulidean geometry of general relatiity, something else replaes the onept of a straight line. A ray of light traelling near the Sun looks bent to us beause we are used to flat spae. But the geometry near the Sun is ured and the bent ray is atually a geodesi: it is the straight line appropriate to that geometry. apparent Earth Sun atual Figure A.50 The mass of the Sun auses a urature of the spae around it. Light from a star bends on its way to the Earth and this makes the star appear to be in a different position from its atual position. A5.7 Blak holes The theory of general relatiity also predits the existene of objets that ontrat under the influene of their own graitation, beoming eer smaller. No mehanism is known for stopping this ollapse, and the objet is expeted to beome a hole in spaetime, a point of infinite density. This reates an immense bending of spaetime around this point. This point is alled a blak hole, a name oined by John Arhibald Wheeler (Figure A.5), sine nothing an esape from it. Massie stars an, under appropriate onditions, ollapse under their own graitation and end up as blak holes (see Option D, Astrophysis). Powerful theorems by Stephen Hawking and Roger Penrose show that the formation of blak holes is ineitable and not dependent too muh on the details of how the ollapse itself proeeds. Figure A.5 John A. Wheeler, a legendary figure in blak-hole physis and the man who oined the term. A5.8 The Shwarzshild radius and the eent horizon A distane known as the Shwarzshild radius of a blak hole is of importane in understanding the behaiour of blak holes. Karl Shwarzshild (Figure A.5) was a German astronomer who proided the first solution of the Einstein equations. The Shwarzshild radius is gien by GM RS where M is the mass and the speed of light. This radius is not the atual radius of the blak hole (the blak hole is a point), but the distane from the hole s entre that separates spae into a region from whih an objet an esape and a region from whih no objet an esape (Figure A.5). Figure A.5 Karl Shwarzshild was the man who proided the first solution of Einstein s equations. esape possible RS blak hole esape impossible Figure A.5 The Shwarzshild radius splits spae into two regions. From within this radius nothing not een light an esape. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 5

54 Quantum blak holes are different from lassial blak holes The statement that nothing esapes from a blak hole is true only if we ignore quantum effets. Stephen Hawking showed in 974 that, when quantum effets are taken into aount, blak holes radiate just like a blak body does, with the reiproal of the mass playing the role of temperature. For massie blak holes this is not signifiant, sine in that ase the temperature is ery small. Howeer, it is interesting that, one the onept of temperature was introdued, knowledge of thermodynamis ould be transferred to blak-hole physis in a formal way, een though blak holes do not hae a temperature in the normal sense. Any objet loser to the entre of the blak hole than R S will fall into the hole; no amount of energy supplied to this body will allow it to esape from the blak hole. The esape eloity at a distane R S from the entre of the blak hole is the speed of light, so nothing an esape from within this radius. The Shwarzshild radius is also alled the eent horizon radius. The latter name is apt, sine anything taking plae within the eent horizon annot be seen by or ommuniated to the outside. The area of the eent horizon is taken as the surfae area of the blak hole (but, remember, the blak hole is a point). The Shwarzshild radius an be deried in an elementary way without reourse to general relatiity if we assume that a photon has a mass m on whih the blak hole s graity ats. Then, as in the alulation of esape eloity in Topi 0 on graitation, GM m m RS Thus m anels and we an sole for R S. It an be readily alulated from this formula that, for a star of one solar mass (M 00 kg), the Shwarzshild radius is about km: RS GM ( 08) 0 m Figure A.54 The eent horizon on a ollapsing star is rising; that is, only rays within a steadily shrinking ertial one an reah an obserer on the star. This means, for example, that if the Sun were to beome a blak hole its entire mass would be onfined within a sphere with a radius of km or less. For a blak hole of one Earth mass, this radius would be just 9 mm. Figure A.54 shows that an obserer on the surfae of a star that is about to beome a blak hole would only be able to reeie light through a one that gets smaller and smaller as the star approahes its Shwarzshild radius. This is beause rays of light oming sideways will bend towards the star and will not reah the obserer. 54 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

55 A5.9 Time dilation in general relatiity We hae seen that the EP predits that time runs slow near massie bodies. A speial ase of this phenomenon takes plae near a blak hole. Consider a lok that is at a distane r from the entre of a blak hole of Shwarzshild radius R S (the lok is outside the eent horizon, i.e. r > R S); see Figure A.55. An obserer who is stationary with respet to the lok measures the time interal between two tiks of the lok to be t 0. An idential lok ery far away from the blak hole will measure a time interal t eent horizon near lok far lok Figure A.55 A lok near a blak hole runs slowly ompared with a lok far away. t 0 RS r This formula is the general relatiisti analogue of time dilation in speial relatiity. It applies near a blak hole. Thus, onsider a (theoretial) obserer approahing a blak hole. This obserer sends signals to a faraway obserer in a spaeraft, informing the spaeraft of his position. When his distane from the entre of the blak hole is r.50r S, the obserer stops and sends two signals s apart (as measured by his loks). The spaeraft obserers will reeie signals separated in time by t t 0 RS r s The extreme ase of time dilation is when the obserer is just an infinitesimally small distane outside the eent horizon. If he stops there and sends two signals s apart, the faraway obserer will reeie the signals separated by an enormous interal of time. In partiular, if the obserer is at the eent horizon when the seond signal is emitted, that signal will neer be reeied. A5.0 General relatiity and the unierse Soon after he published his general theory of relatiity, Einstein applied his equations for general relatiity to the unierse as a whole. The result looks like this: R μ ( R Λ)g μ 8 πg T 4 μ and will not be examined! On a large sale, the unierse looks like a loud of dust of density ρ. Under arious simplifying assumptions, the equations redue to an equation for a quantity that we will loosely all the radius of the unierse, R. Soling these equations gies the dependene of R on time. Einstein himself belieed in a stati unierse, whih would hae R onstant. His alulations, howeer, did not gie a onstant R. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 55

56 R t no big ba big ng ba ng Figure A.56 A model of the unierse with a onstant radius. Einstein introdued the osmologial onstant in order to make the unierse stati. g tin era l e ing a rat ele e d Ωʌ, 0 expanding ollapsing 0 fla t Ω 0 Ω < Ω So he modified them, adding the famous osmologial onstant term Λ. This term made R onstant! This is shown in Figure A.56. In this model there is no Big Bang and the unierse always has the same size. This was before Hubble disoered that the unierse is expanding. Einstein missed the hane to theoretially predit an expanding unierse; he later alled the addition of the osmologial onstant the greatest blunder of my life. This onstant may be thought of as being related to a auum energy, energy that is present in all spae. This energy is now alled dark energy. The osmologial onstant went into obsurity for many deades but it did not go away: it was to make a omebak with a engeane muh later! The first serious attempt to find how R depends on time was made by the Russian mathematiian Alexander Friedmann (888 95), in work later taken up by Lemaître, Robertson and Walker. Friedmann applied the Einstein equations and realised that there were a number of possibilities: the solutions depend on how muh matter and energy the unierse ontains (Figure A.7). We define the density parameter for matter, Ωm, and for dark energy, ΩΛ, as Ωm > Ωm, 0 where ρm is the atual density of matter in the unierse and ρrit is a referene density alled the ritial density, about 0 6 kg m ; ρλ is the density of dark energy. The Friedmann equations gie arious solutions, depending on the alues of Ωm and ΩΛ. Deiding whih solution to pik depends ruially on these alues, whih is why osmologists hae expended enormous amounts of energy and time trying to aurately measure Ωm and ΩΛ. In Figure A.57 the subsript 0 indiates the alues of these parameters at the present time. There are four main regions in this diagram. Models aboe the red dashed line are ones in whih the geometry of the unierse resembles the surfae of a sphere (Figure A.58). Points below the line hae a geometry like that of the surfae of a saddle (Figure A.58a). Points on the line imply a flat unierse where the rules of Eulidean geometry apply (Figure A.58b). (a ele rat io relatie size of unierse (or aerage intergalati distane) n) Figure A.57 There are arious possibilities for the eolution of the unierse, depending on how muh energy and mass it ontains. (From M. Jones and R. Lambourne, An Introdution to Galaxies and Cosmology, Cambridge Uniersity Press, in assoiation with The Open Uniersity, 004. The Open Uniersity, used with permission) ρm ρ and ΩΛ Λ ρrit ρrit rk da gy er en ρ ure at ur ρ ie t < a ρ g ne flat ρ ρ> ρ positie ur atu re 0 0 Big Bang (lookbak time depends on model) now 0 Billions of years 0 Big runh 0 Figure A.59 Solutions of Einstein s equations for the eolution of the sale fator. The present time is indiated by now. Notie that the age of the unierse aries depending on whih solution is hosen in other words, different solutions imply different ages. Three models assume zero dark energy; the red line does not. negatie urature zero urature positie urature Figure A.58 Three models of the unierse with different urature. a In negatieurature models, the angles of a triangle add up to less than 80 and initially parallel lines eentually dierge. b Ordinary flat geometry. Positie urature, in whih the angles of a triangle add up to more than 80 and initially parallel lines eentually interset. (From M. Jones and R. Lambourne, An Introdution to Galaxies and Cosmology, Cambridge Uniersity Press, in assoiation with The Open Uniersity, 004. The Open Uniersity, used with permission) Figure A.59 shows how the radius R of the unierse aries with time for arious alues of the parameters. In all ases the radius starts from zero, implying a Big Bang. In one possibility, R(t) starts from zero, inreases to a maximum alue and then dereases to zero again the unierse ollapses after an initial period of expansion. This is alled the losed model, and orresponds to Ωm >, 56 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

57 that is, ρm > ρrit. A seond possibility applies when Ωm <, that is, ρm < ρrit. The present data from the Plank satellite obseratory indiate that Ωm 0. and ΩΛ 0.68, so Ωm + ΩΛ, on the red dashed line in Figure A.57. This implies a flat unierse, meaning that at present our unierse has a flat geometry, % of its mass energy ontent is matter, 68% is dark energy and it is expanding foreer at an aelerating rate. This is shown by the red line in Figure A.59. Nature of siene Creatie and ritial thinking The theory of general relatiity is an almost magial theory in whih the energy and mass ontent of spaetime determines the geometry of spaetime. In turn, the kind of geometry spaetime has determines how the mass and energy of the spaetime moe about. To onnet these ideas in the theory of general relatiity, Einstein used intuition, reatie thinking and imagination. Initial solutions of Einstein s equations suggested that no light ould esape from a blak hole. Then a further imaginatie leap the deelopment of quantum theory showed an unexpeted onnetion between blak holes and thermodynamis.? Test yourself 6 Disuss the statement a ray of light does not atually bend near a massie objet but follows a straight-line path in the geometry of the spae around the massie objet. 6 A spaeraft fi lled with air at ordinary density and pressure is far from any large masses. A helium-fi lled balloon floats inside the spaeraft, whih now aelerates to the right. Determine whih way (if any) the balloon moes. 6 The spaeraft of question 6 now has a lighted andle in it. The raft aelerates to the right. Disuss what happens to the flame of the andle. 64 a Desribe what is meant by the equialene priniple. b Explain how this priniple leads to the preditions that: i light bends in a graitational field ii time runs slower near a massie objet. 65 Desribe what the general theory of relatiity predits about a massie objet whose radius is getting smaller. 66 In a referene frame falling freely in a graitational field, an obserer attahes a mass m to the end of a spring, extends the spring and lets the mass go. He measures the period of osillation of the mass. Disuss whether he will find the same answer as an obserer doing the same thing: a on the surfae of a ery massie star b in a true inertial frame far from any masses. 67 Calulate the shift in frequeny of light of waelength nm emitted from sea leel and deteted at a height of 50.0 m. 68 A ollapsed star has a radius that is 5.0 times larger than its Shwarzshild radius. An obserer on the surfae of the star arries a lok and a laser. Eery seond, the obserer sends a short pulse of laser light of duration.00 ms and waelength m (as measured by her instruments) towards another obserer in a spaeraft far from the star. Disuss qualitatiely what the obserer in the spaeraft an expet to measure for: a the waelength of the pulses b the frequeny of reeption of onseutie pulses the duration of the pulses. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 57

58 69 Two idential loks are plaed on a rotating dis. One is plaed on the irumferene of the dis and the other halfway towards the entre. Explain why the lok on the irumferene will run slowly relatie to the other lok. 70 Calulate the density of the Earth if its entire mass were onfined within a radius equal to its Shwarzshild radius. 7 Calulate the Shwarzshild radius of a blak hole with a mass equal to 0 solar masses. 7 Explain what is meant by geodesi. 7 A mass m moes past a massie body along the path shown below. Explain the shape of the path aording to: a Newtonian graity b Einstein s theory of general relatiity. 74 A ball is thrown with a eloity that is initially parallel to the floor of a spaeraft, as shown below. Draw and explain the shape of the ball s path when: a the spaeraft is moing with onstant eloity in deep spae, far from any large masses b the spaeraft is moing with onstant aeleration in deep spae, far from any large masses. 75 A ray of light parallel to the floor of a spaeraft enters the spaeraft through a small window, as shown below. Draw and explain the shape of the ray s path when: a the spaeraft is moing with onstant eloity in deep spae, far from any large masses b the spaeraft is moing with onstant aeleration in deep spae, far from any large masses. 76 An obserer is standing on the surfae of a massie objet that is ollapsing and is about to form a blak hole. Desribe what the obserer sees in the sky: a before the objet shrinks past its eent horizon b after the objet goes past its eent horizon. 77 A plane flying from southern Europe to New York City will fly oer Ireland, aross the North Atlanti, oer the east oast of Canada and then south to New York. Suggest why suh a path is followed. 58 PHYSICS PHYSICS FOR THE 04 IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS

59 78 Einstein s birthday present. A olleague of Einstein at Prineton presented him with the following birthday present. A long tube was onneted to a bowl at its top end. A spring was attahed to the base of the bowl and onneted to a heay brass ball that hung out of the bowl. The spring was ery weak and ould not pull the ball into the bowl. The exerise was to find a sure-fire method for putting the ball into the bowl without touhing it. Einstein immediately found a way of doing it. Can you? 79 An obserer approahing a blak hole stops and sends signals to a faraway spaeraft eery.0 s as measured by his loks. The signals are reeied.0 s apart by obserers in the spaeraft. Determine how lose he is to the eent horizon. 80 A spaeraft aelerates in the auum of outer spae far from any masses. An obserer in the spaeraft sends a radio message to a stationary spaeraft far away. The duration of the radio transmission is 5.0 s, aording to the obserer s lok in the moing spaeraft. Explain whether the transmission will take less than, exatly or more than 5.0 s when it is reeied by the faraway stationary spaeraft. 8 A blak hole has a mass of kg. a State what is meant by a blak hole. b Calulate the Shwarzshild radius R S of the blak hole. Explain why the Shwarzshild radius is not in fat the atual radius of the blak hole. d Blue light of frequeny Hz is emitted by a soure that is stationary at a distane of 0.0 R S aboe the eent horizon of a blak hole. Calulate the period of this blue light aording to an obserer next to the soure. e Determine the frequeny measured by a distant obserer who reeies light emitted by this soure. 8 a State the formula for the graitational (Shwarzshild) radius R S of a blak hole of mass M. b The sphere of radius R S around a blak hole is alled the eent horizon. State the area of the eent horizon of a blak hole of mass M. Suggest why, oer time, the area of the eent horizon of a blak hole always inreases. d In physis, there is one other physial quantity that always inreases with time. Can you state what that quantity is? (You will learn a lot of interesting things if you pursue the analogy implied by your answers to and d.) 8 a A ray of light is emitted from within the eent horizon (dashed irle) of a blak hole, as shown below. Copy the diagram and draw a possible path for this ray of light. b Light from a distant star arries at a theoretial obserer within the eent horizon of a blak hole, as shown in part b of the figure. Explain how it is possible for the ray shown to enter the obserer s eye. light from distant star a PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 ray of light obserer blak hole blak hole eent horizon b eent horizon 59

60 Exam-style questions a State what is meant by a referene frame. b Disuss the speed of light in the ontext of Maxwell s theory. [] [] A positie harge q moes parallel to a wire that arries eletri urrent. The eloity of the harged partile is the same as the drift eloity of the eletrons in the wire q Disuss the fore experiened by the harged partile from the point of iew of an obserer: i at rest with respet to the wire. ii at rest with respet to the harged partile. [] [] A roket moes past a spae station at a speed of The proper length of the spae station is 480 m. The origins of the spae station and roket frames oinide when the roket is moing past the middle of the spae station. At that instant, loks in both frames are set to zero. a State what is meant by proper length. [] b Determine the length of the spae station aording to an obserer in the roket. [] Two lamps at eah end of the spae station turn on simultaneously aording to spae station obserers, at time zero. Determine i whih lamp turns on first, aording to an obserer in the roket ii the time interal between the lamps turning on, aording to an obserer in the roket. [4] [] d The spaetime diagram below applies to the frames of the spae station and the roket. The thin axes represent the spae station referene frame. t / m t / m x / m x/m Use the diagram to erify your answer to i. e By drawing appropriate lines on a opy of the spaetime diagram, show the eents: i light from the lamp at x 40 m reahes the roket (the roket is assumed to be a point) ii light from the lamp at x 40 m reahes the roket (the roket is assumed to be a point). 60 [] [] [] PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

61 A roket moes to the right with speed 0.78 relatie to a spae station. The roket emits two missiles, eah at a speed of 0.50 relatie to the roket. One missile, R, is emitted to the right and the other, L, to the left. a Calulate the eloity of missile i R relatie to the spae station ii L relatie to the spae station iii R relatie to missile L. [] [] [] b Show by expliit alulation that the speed of light is independent of the speed of its soure. [] 4 The spaetime diagram below may be used to disuss the twin paradox. The thin blak axes represent the referene frame of the twin on the Earth. The dark grey axes are for the frame of the traelling twin on her way away from the Earth. The light grey axes represent her homeward frame. The time axes in eah frame may be thought of as the worldlines of loks at rest in eah frame. The traelling twin moes at 0.80 on both legs of the trip. She turns around at eent T when she reahes a planet ly away, as measured by the Earth obserers. She returns to Earth at eent R. R C B T A a Desribe what is meant by the twin paradox. [] b Outline how the paradox is resoled. [] Calulate the following lok readings: i the Earth lok at B ii the outgoing lok at T iii the Earth lok at A i the Earth lok at C the Earth lok at R i the inoming lok at R. [] [] [] [] [] [] d State by how muh eah twin has aged when they are reunited. PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 [] 6

62 5 A roket of proper length 690 m moes with speed 0.85 past a lab. A partile is emitted from the bak of the roket and is reeied at the front. The partile has speed 0.75 as measured in the roket frame. a Calulate the time it takes for the partile to reah the front of the roket aording to i obserers in the roket ii obserers in the lab. [] [] b Suggest whether either of the times alulated in a is a proper time interal. [] Calulate the speed of the partile aording to the lab. [] d i Without using Lorentz transformations, determine the distane traelled by the partile aording to the lab. ii Verify your answer to i by using an appropriate Lorentz transformation. [] [] 6 A spaeraft leaes the Earth at a speed of 0.60 towards a spae station 6.0 ly away (as measured by obserers on the Earth). a Calulate the time the spaeraft will take to reah the spae station, aording to i Earth obserers ii spaeraft obserers. [] [] b As the spaeraft goes past the spae station it sends a radio signal bak towards the Earth. Calulate the time it will take the signal to arrie on the Earth aording to i Earth obserers ii spaeraft obserers. 7 The aerage lifetime of a muon as measured in the muon s rest frame is about. 0 6 s. Muons deay into eletrons. A muon is reated at a height of.0 km aboe the surfae of the Earth (as measured by obserers on the Earth) and moes towards the surfae at a speed of The gamma fator γ for this speed is 5.0. By appropriate alulations, explain how muon deay experiments proide eidene for i time dilation ii length ontration. [] [4] [] [] HL 8 a A proton is aelerated from rest through a potential differene of.5 GV. Determine, for the aelerated proton, i the total energy ii the momentum iii the speed. [] [] [] b In a hypothetial experiment, two partiles, eah of rest mass 5 MeV and moing in opposite diretions with speed 0.98 relatie to the laboratory, ollide and form a single partile. Calulate the rest mass of this partile. 6 [] PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05

63 HL 9 a A neutral pion with rest energy 5 MeV and moing at 0.98 relatie to a lab deays into two photons. One photon is emitted in the diretion of motion of the pion and the other in the opposite diretion. i State what is meant by the rest energy of a partile. ii Calulate the momentum and total energy of the pion as measured in the lab. [] [] b By applying the laws of onseration of energy and momentum to this deay, determine: i the energy of the forward-moing photon ii the momentum of the bakward-moing photon. [] [] i Show that the speed of a partile with momentum p and total energy E is gien by p. E ii Dedue that a partile of zero rest mass must moe at the speed of light. [] [] HL 0 a Desribe what is meant by the equialene priniple. b [] i Use the equialene priniple to dedue that a ray of light bends towards a massie body. ii Einstein would laim that the ray does not in fat bend. By referene to the urature of spae, suggest what Einstein might mean by this. [] [] The image below shows a multiple image of a distant quasar. The image has been formed beause light from the quasar went past a massie galaxy on its way to the Earth. HL a Explain how the multiple image is formed. [] State what is meant by graitational red-shift. [] b Explain graitational red-shift using the equialene priniple. [4] In the Pound Rebka experiment, a gamma ray was emitted from the top of a tower m high. i Calulate the frational hange in frequeny obsered at the bottom of the tower. ii Explain why, for this experiment to be suessful, frequeny must be measured ery preisely. iii Outline why this experiment is eidene for graitational time dilation. [] [] [] d The experiment in is repeated in an eleator of height m that is falling freely aboe the Earth. The gamma ray is emitted from the top of the eleator. Predit the frequeny of the gamma ray as measured at the base of the eleator. [] PHYSICS FOR THE IB DIPLOMA CAMBRIDGE UNIVERSITY PRESS 05 6