Special theory of relativity through the Doppler effect

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2 INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys ) EUROPEAN JOURNAL OF PHYSICS doi: / /27/6/015 Speial theory of relativity through the Doppler effet M Morioni Departamento de Físia, Universidade Federal Fluminense, Av. Litorânea s/n, Boa Viagem CEP , Niterói, Rio de Janeiro, Brazil morioni@if.uff.br Reeived 27 June 2006, in final form 4 August 2006 Published 9 Otober 2006 Online at staks.iop.org/ejp/27/1409 Abstrat We present the speial theory of relativity taking the Doppler effet as the starting point, and derive several of its main effets, suh as time dilation, length ontration, addition of veloities and the mass energy relation, and assuming energy and momentum onservation, we disuss how to introdue the 4-momentum in a natural way. We also use the Doppler effet to explain the twin paradox, and its version on a ylinder. As a by-produt we disuss Bell s spaeship paradox, and the Lorentz transformation for arbitrary veloities in one dimension. 1. Introdution During 2005 we elebrated Einstein s annus mirabilis, a year in whih the ourse of physis hanged in a profound way. There were several events elebrating Einstein s ontributions, and this gave us the opportunity to go through some of the fundamentals of physis, following the path Einstein laid out for us, and rethinking ways to present some of his ideas, in partiular the speial theory of relativity STR) [1]. In this paper we present an elementary approah to the STR, taking the Doppler effet as the starting point. We will treat only one-dimensional motion for simpliity, but this is ertainly not a limiting assumption, sine one an easily generalize to arbitrary motions by ombining Lorentz boosts and rotations. Initially we derive the Doppler effet formula diretly from the two postulates of the STR, namely the following: the laws of physis are the same in all inertial frames of referene; the veloity of light is the same for all inertial observers. As a matter of fat one should onsider only one postulate, sine the priniple of the onstany of the speed of light is a onsequene of the first postulate, as Einstein points out in the E = M 2 paper see, for example, [1]): all one needs to do is to qualify what are the laws of physis that are supposed to be valid in all frames of referene. If one assumes Newton s /06/ $ IOP Publishing Ltd Printed in the UK 1409

3 1410 M Morioni laws, one obtains lassial kinematis. If one takes Maxwell s equations, then one is naturally led to a new view of spae and time. One we have derived the relativisti Doppler formula, we apply it to several different thought experiments, whih will allow us to derive all of relativisti kinematis: time dilation, length ontration, the addition of veloities, the Lorentz transformations and the mass energy relation. We also present a solution of the twin paradox through the Doppler effet, and its version on the ylinder. We disuss Bell s two spaeships paradox, and the Lorentz transformation for arbitrary veloities in one dimension. In deriving the mass energy relation we show, as a by-produt, that the energy of radiation suffers a Doppler shift too, without having to resort to the energy-frequeny relation of elementary quantum mehanis. This will allow us to introdue the four-momentum of a partile in a quite natural way. By now there is a great number of good books on the speial theory of relativity, suh as Frenh s or Taylor and Wheeler s [2], whih usually rely on spaetime diagrams. Our approah is based solely on the Doppler effet, and we believe it an be used as a tool to understand the essene of relativity. 2. The Doppler effet Consider a light soure moving along the x axis with veloity v and emitting light of frequeny ν 0, as measured in the inertial frame of the soure. Let us all Isabella an observer towards whih the light soure is moving to, and Marianna an observer from whih the light soure is moving away. Both, Isabella and Marianna, are at rest in the lab frame. The frequeny observed by any one of them should be a funtion of ν 0,v the veloity of the soure, as observed by them) and. Elementary dimensional analysis gives νv) = ν 0 fβ), where β = v/ and fβ) is an unknown funtion. From the point of view of Isabella, if a pulse is emitted at instant t 1 and a seond pulse at t 2, she will then observe a wavelength given by v) t, where t = t 2 t 1, and the observed frequeny is therefore νv). From the point of view of Marianna, we must replae v v, sine the light soure is moving away from her. Aording to the postulate of the onstany of the veloity of light, we have v) tνv) =, + v) tν v) =. 2.1) Note that we would have obtained the same equations if the observers were moving, instead of the soure, due to the first postulate all that matters is their relative motion). Dividing the first equation by the seond, we obtain fβ) f β) = 1+β. 2.2) We an now use the first postulate: if the soure and the observer move at the same veloity v, she will learly observe the same frequeny ν 0 emitted by the light soure, sine they are in the same inertial frame. But aording to our formula for νv), this is also equal to ν 0 fβ)f β): the light observed by someone at rest in the lab frame, between the soure and the travelling observer, has frequeny ν = ν 0 fβ), and this light will be seen by the moving observer to have the frequeny ν = νf β). Therefore fβ)f β) = 1. We an use this in 2.2) to obtain fβ) 2 = 1+β fβ)= This gives us the Doppler effet formula: 1+β νv) = ν 0 ) 1+β. 2.3) ). 2.4)

4 Speial theory of relativity through the Doppler effet 1411 There is another elementary derivation of this formula, diretly from the postulates of the STR, whih we leave for appendix A in order not to interrupt the flow of the paper. We now move on to apply this formula to different set-ups and obtain the main onsequenes of relativisti kinematis. 3. The addition of veloities The addition of veloities may easily be derived with the aid of the Doppler effet formula. Let us suppose Isabella moves with veloity v from left to right, and Marianna, who owns a light soure of frequeny ν 0, moves with veloity u from right to left. For simpliity assume they are approahing eah other. Both veloities are measured relative to their father s laboratory frame, who is in between the two. He will measure a frequeny ν f = ν 0 1+u/)/1 u/)) for Marianna s light. This means that Isabella should observe ν = ν f 1+v/)/1 v/)). But sine we an write, in Isabella s frame, that ν = ν 0 1+w/)/1 w/)), where w is the relative veloity, we have 1 u/ 1+u/ ) 1 v/ 1+v/ ) = ) 1 w/ 3.1) 1+w/ from whih we readily derive w = u + v 1+uv/. 3.2) 2 We will use this equation again when deriving the mass energy relation. 4. Time dilation We an use the Doppler effet to derive the time dilation effet. While deriving the Doppler effet, we found that ) 1+β v) tν 0 = τν 0, 4.1) where the right-hand side is the veloity of light in the light-soure frame, that is, the produt of the wavelength and frequeny as measured by an observer that moves along with the light soure. τ is the time interval between two pulses in that frame. One obtains immediately τ t =, 4.2) 2 ) whih is the time dilation effet. Note that in most elementary presentations of relativity one derives the time dilation first and then uses it to derive the Doppler effet expression. 5. Length ontration Suppose now that Marianna takes off from rest, till she reahes veloity v <, of ourse) from the origin of Isabella s inertial frame S 0, goes to a point x 0 at distane L 0, as measured by Isabella, and omes bak to the origin with the veloity v, and stops at the origin. Let us assume that there is a far-away light soure in the same line that Marianna is moving, emitting light pulses with frequeny ν 0, and that Marianna s aeleration and deeleration times are short ompared to the total travelling time, even though this is not ruial in the derivation, as we will see. Initially we should note that eah light pulse that passes through Marianna will neessarily reah Isabella before she omes bak to the origin, sine v<.

5 1412 M Morioni So, if even after taking note of a given pulse Marianna immediately deides to go bak to the origin, that pulse will reah Isabella before Marianna reahes her. Moreover, eah pulse that reahes Isabella neessarily passed through Marianna. Therefore the number of pulses ounted by Marianna is exatly the same number of pulses ounted by Isabella. Let us ount how many pulses Isabella observes: the total travel time measured by her is 2L 0 /v, and so the number of pulses she ounts is given by N I = ν 0 2L 0 /v. In order to ompute the number of pulses Marianna observes, we must use the Doppler formula. On her way to x 0 she observes N + = ν 0 1+β)/1 β)) L /v. Here we are onsidering that she observes the point x 0 at an unknown distane L, and that this point moves toward her with veloity v. On her way bak she will observe N = ν 0 )/1+β)) L /v. Note that on her way bak she observes the same distane L to the origin. The fat that N M = N + + N = N I gives 2L /v 2 ) = 2L 0/v L = L 0 2 ), 5.1) whih is the length ontration effet 1 FitzGerald Lorentz ontration). 6. The twin paradox Inidentally this ounting proedure also solves the famous twin paradox, or in this ase, the sister s paradox : two sisters, Isabella and Marianna, are separated. Isabella stays in an inertial frame and Marianna goes on a round trip just like the one we desribed. Eah one of them sees the other s lok run slower than her own. When they meet again, who is right? Aording to Isabella, there where 2Tν 0 pulses, and aording to Marianna there were T 1 +β)/)) ν 0 + T )/1+β)) ν 0. Sine they observe the same number of pulses, we must have T = T 2 ), whih means that Marianna s lok indeed ran slower than Isabella s. It is also possible to use this method to derive the time differene for an arbitrary motion. The ounting argument requires only that the veloity of the traveller is always smaller than the veloity of the light. Therefore, let us onsider an arbitrary veloity funtion for the distant light soure as a funtion of time vτ), where τ is Marianna s proper time. Between instants τ and τ +dτ, Marianna observes dnτ) = dτ 1+β)/)) ν 0 pulses. During the whole trip she will have observed T ) 1+β N = dτ ν 0 6.1) 0 pulses, whih should be equal to Isabella s ounting, Tν 0. This gives the relation T ) 1+β T 1+β T = dτ = dτ. 6.2) ) In the ase where the observer returns to the origin, the seond term on the right-hand side vanishes, T 0 β dτ = 0, 6.3) 2 ) 1 If one feels uneasy with the round trip in this argument, we may provide the following variation: when Marianna reahes x 0, she has observed ν 0 1 +β)/)) L /v pulses, whereas Isabella has observed ν 0 L 0 /v pulses. The differene between these two numbers should be exatly the number of pulses between x 0 and Isabella: these are the pulses that have rossed Marianna, but not Isabella. This number is given by ν 0 L 0 /. Therefore we have ν 0 1+β)/)) L /v = ν 0 L 0 /v + ν 0 L 0 /, whih gives the result 5.1) again.

6 Speial theory of relativity through the Doppler effet 1413 sine this is the total distane overed by the travelling observer. We will show this more formally in setion 10, where we disuss the Lorentz transformations for arbitrary veloities. Therefore we obtain the more familiar looking result T 1 T = dτ. 6.4) 0 2 ) Sine 2 ) < 1, we see that T>T, always. 7. The twin paradox on a ylinder There is an interesting variation of the twin paradox, where one onsiders the spae to be a ylinder, that is, the spatial dimension is ompat. The two paradoxes seem very similar, but there is one ruial differene. In the previous paradox, ultimately, the explanation was due to the fat that whereas one of the observers is always reeiving pulses at the same frequeny the other observes blue-shifted and then red-shifted signals, due to the fat that she hanged her veloity in order to go bak to the origin. This is equivalent to the more usual explanation where it is pointed out that the situation is not really symmetri, sine one of the observers had to aelerate, whereas the other did not, breaking the symmetry between the two. In the ase of a ylinder one annot use this breaking of the symmetry explanation. This is so beause there is more than one geodesi path that takes the travelling observer bak to the origin, whih is always an inertial frame of referene. Therefore there is no apparent asymmetry in this ase. We an use our ounting method, similarly to what we have done in the lassi twin paradox ase. Suppose Isabella and Marianna are loated at the origin, and that Marianna goes on a round trip, winding around the ylinder, whih has radius R. Let us onsider a light soure at distane L 1 from the origin, measured in the lokwise diretion, whih sends light pulses with frequeny ν 0. This soure has been sending pulses for a long time, and we an assume that at the beginning of Marianna s trip both observers start ounting pulses. We will also assume that the soure is sending pulses towards the origin along the path of length L 1 until Marianna reahes it, and then the light soure will send signals along the path of length 2πR L 1. Let us ount the number of pulses in the two frames of referene. Aording to Marianna, in the first leg of her trip she observes pulses at a higher rate, given by ) 1+β N1 m = τ 1 ν 0 7.1) where τ 1 is the time she takes to reah the light soure. On her way bak to the origin, light is observed at a lower rate, and the number of pulses is simply given by ) N2 m = τ 2 ν 0 7.2) 1+β where τ 2 is now the time Marianna takes to go bak to the origin. The total number of pulses that Marianna has ounted is N m = τ 1 1+β ) ) ) + τ 2 ν ) 1+β Aording to Isabella, we have the following. Let T 1 T 2 ) be the time Marianna takes to reah the light soure reah the origin from the light soure). During the first leg of Marianna s trip, Isabella observes T 1 ν 0, but in order that their ounting mathes, that is, they

7 1414 M Morioni ount the same physial pulses emitted from the soure, we have to add all the pulses that are between Isabella and the soure at the time Marianna reahed it, whih is equal to L 1 ν 0 /. This gives N1 i = T 1 + L ) 1 ν ) During the seond leg of Marianna s trip, eah pulse that she ounts rosses her and reahes Isabella before she reahes the origin, and eah pulse ounted by Isabella neessarily rosses Marianna. We are assuming that Marianna takes a time T 2 to arrive at the origin, but sine the light soure only starts sending pulses after Marianna rosses it, Isabella will have to wait a time equal to L 2 / to start reeiving pulses. This gives N2 I = T 2 L 2 ) ν 0 7.5) and the total number of pulses ounted by Isabella is given by N i = T 1 + L 1 + T 2 L ) 2 ν ) Piking L 1 = L 2 gives, by symmetry, τ 1 = τ 2 = τ/2 and T 1 = T 2 = T/2, where τ and T are the times of how long the round trip takes aording to eah observer. Substituting these in 7.6) and 7.3), we obtain τ = T 2 ) 7.7) whih is the same onlusion as in the lassi twin paradox 2. In this analysis we see how useful the ounting approah is. Whereas in the usual solutions of the twin paradox in a ylinder one needs to do quite involved arguments, in this ase one hardly sees any differene between the analysis of the lassial twin paradox and this one. The reader may still feel a little uneasy in the searh for the hidden asymmetry between the two systems of referene. In this ase one an onvine oneself by notiing the following: one of the observers is always reeiving pulses at frequeny ν 0, whereas the other one sees a higher frequeny during one leg and a lower frequeny during the other. We an see, then, that ultimately the reason for the asymmetry between the two observers is that there is a preferred global inertial frame [5 7]. 8. Bell s two spaeship paradox J Bell has presented a quite interesting paradox in the speial theory of relativity, whih has aused and apparently it still auses!) heated debate among physiists. The paradox is explained in his book [8], and we summarize it here. Suppose two spaeships, I and M, are at rest in a given inertial frame, separated by a distane L 0, as measured in the lab frame, and suppose that an observer, loated exatly at the midpoint joining the two spaeships, emits a light signal towards them whih serves as a signal for them to start their trip. Eah spaeship is equipped with a program that tells the rew how to aelerate it, and the programs in I and M are exatly the same. The first question is: will the distane between the two spaeships be Lorentz-ontrated aording to the lab frame? The answer to this question is, learly, no. The seond question is, and here is the apparent paradox: if there was a thin thread joining the 2 If we take L 1 = L 2, equation 7.3) hanges to N m = τ1+βl 1 L 2 )/L)/ 2 ), sine, by symmetry, τ 1 /τ 2 = L 1 /L 2, and equation 7.6) beomes N i = T1+βL 1 L 2 )/L), where we used that T = L 1 + L 2 )/v,and we get the same result as 7.7).

8 Speial theory of relativity through the Doppler effet 1415 two spaeships, then, sine the measurement in the lab frame gives L 0 for their separation, then how ome the thread should be Lorentz-ontrated aording to, say, the observers in I? In other words, if the measurement in the lab frame gives L 0, then the length of the thread as measured by the spaeships should be bigger than L 0 in order to ompensate the Lorentz ontration. There is something learly wrong in this onlusion. To answer the seond question onsider the following: if we observe, in the lab frame, that their distane is L 0, then in I s frame their distane should be bigger than L 0, but, sine I and M have exatly the same veloity programs, one ould be led to think that their distane should be onstant aording to them. This is where the solution of the paradox stems, and an be easily understood by realling one of the fundamental aspets of relativity: simultaneity is not an absolute onept. The lab measurement of the positions of the two spaeships ours at a given time t in the lab frame, but this is not the ase in the spaeship s frames. Let us denote the measurements of the positions of the I and M spaeships, in the lab frame, by events E I and E M. The orresponding events as seen by the I frame are denoted by EI 1 and E1 M.Atthe moment that a measurement is performed at I, what are the loations in spaetime of events EI 1 and E1 M? We postpone this to setion 11, where we find the Lorentz transformation for an arbitrary motion. 9. The Lorentz transformation We an also derive the Lorentz transformation of oordinates using a variation of these ounting arguments. Let us suppose that in the inertial frame S 0 there is an observer, Isabella, at the origin of S 0. A seond inertial frame, S, where Marianna is, moves along the x axis with veloity v. A given physial event E will have oordinates x, t) aording to Isabella, and oordinates x,t ) aording to Marianna. The relation between these oordinates is given by the Lorentz transformation. Let us suppose that there is a light soure that has been emitting light for a long time, and that it is loated very far away, in suh a way that there are light pulses passing through the origin of S 0 sine before t = 0 whih we assume is when both origins oinide). We will take as the physial event E, the rossing of a light pulse at x at time t. When this happens, Isabella has already observed tν 0 pulses, and there are ν 0 x/ pulses between her and x, totalling N 1 = t + x/)ν 0 pulses. Aording to Marianna, though, she has ounted t 1+β)/1 β)) ν 0 pulses, and there are 1+β)/1 β)) ν 0 x / pulses between her and the physial loation of E, totalling N 2 = t 1+β)/1 β)) ν β/)) ν 0 x / pulses 3. The two numbers N 1 and N 2 have to be equal, whih gives the following equation: t 1+β ) + x 1+β ) = t + x. 9.1) Note that his equation is the statement of the fat that the phase of a plane wave is a relativisti invariant. Before we proeed, we an show that 9.1) implies the end of absolute simultaneity. In order to show that, onsider two physial events E 1 and E 2, separated by a distane L 0, and our at the same time t aording to Isabella. Assuming that Marianna also observes simultaneous physial events that is, t = 0 for her too), we would have that the distane between E 1 and E 2 is given by x = L 0 2 ), and so 9.1) implies L 0 1+β) = L 0 9.2) 3 This an be derived using the priniple of relativity: sine the total number of pulses Isabella measured is given by N I = t + x/)ν 0, then the total number of pulses measure by Marianna must be N M = t + x /)ν, whih is the expression we found.

9 1416 M Morioni whih is true only if β = 0: Isabella and Marianna must be at rest in relation to eah other. This is the end of absolute simultaneity. In order to find the Lorentz transformation for x and t we still need one more equation. Suppose, then, that another event E takes plae at x,t) aording to Isabella, and that Marianna moves with veloity v. What are the oordinates of E aording to Marianna? They are simply given by x,t ), sine all we have done was to perform a parity transformation 4, and so we an write ) ) t x = t x 1+β 1+β. 9.3) Adding and subtrating 9.1) and 9.3), we obtain the Lorentz transformation x = x + vt t = t + βx /. 9.4) 2 ) 2 ) Note that multiplying 9.1) with 9.3) we obtain the invariane of the interval between two events x ) 2 x t 2 = t 2 ) ) 10. The Lorentz transformation for arbitrary veloities Usually one thinks of the STR as a theory that deals with inertial frames only, where aeleration has no plae. This is not orret, and there are, indeed, transformation laws for the oordinates from a frame in arbitrary motion to a given inertial frame the lab frame ). These were found by Nelson in [9], and we derive here their one-dimensional version. Consider two systems of oordinate in a similar way to that done in setion 9, where we derived the Lorentz transformations. The differene now is that Marianna s veloity an depend on time in an arbitrary way. At t = t = 0 their origins oinide, and they are reeiving light pulses from a distant light soure. Consider a physial event E that happens at some point in spaetime. Let us ount how many light pulses are there between the first pulse, whih defined the t = t = 0 instant, and the pulse that rossed the event E. Aording to Isabella, there are N I = t + x ) ν ) To perform Marianna s ounting we have to move along with her. Between τ and τ +dτ she observes ) 1+β dn = dτ ν ) where now β = βτ). She reords the spaetime oordinates of the event E as being x,t ), and therefore, between the first pulse and the pulse that defines the event, there are x /λ = x νt )/. Integrating 10.2) from 0 to t and adding x νt )/, we get the total number of pulses N M = x 1+βt ) ) t ) 1+β ν t 0 + dτ ν ) ) 0 4 If we want to do stritly one-dimensional physis, than we have to assume STR is parity invariant. In the real, three-dimensional ase, this is a onsequene of rotational invariane, sine a transformation x x an be obtained through a rotation ofπ around the z-axis. This is not a parity transformation in three dimensions, sine y y,but it is enough for our onsiderations.

10 Speial theory of relativity through the Doppler effet 1417 Equating 10.1) and 10.3), and after some little algebra, we obtain t + x = γt ) x + γt )βt ) x + dτ γτ)+ dτ γ τ)βτ) 10.4) 0 0 where we introdued γτ) = 2 τ)). As in the ase of the Lorentz transformation, we still need one more equation. This is easily done by observing that in Isabella s frame, between t and t +dt, Marianna has travelled vt) dt, whih will orrespond to γτ)βτ)dτ in Marianna s oordinates, and the final portion, whih aording to Marianna, is x, will be γt )x, whih gives x = γt )x + t Substituting 10.5)in10.4), we obtain, finally, 0 t t dτ γ τ)βτ). 10.5) t t = γt )βt ) x + dτ γ τ). 10.6) 0 It is not easy to invert equations 10.5) and 10.6), exept in a few ases, suh as in the hyperboli motion. It is straightforward to see that these transformations redue to Lorentz transformation in the ase where β is onstant. It is instrutive to apply these transformations to the ase of hyperboli motion, and use it to solve Bell s paradox. Before doing that, it is onvenient to parametrize the veloity as v = tanhφ), where φ is an arbitrary funtion of the proper time. The Lorentz transformation beomes t t x = oshφ)x + dτ sinhφ) t = sinhφ)x + dτ oshφ). 10.7) 0 0 The uniform aeleration problem is instrutive. In this ase φ = aτ/, where a is the aeleration measured in a frame of referene where the spaeship is instantaneously at rest. In general, one approahes this problem by finding the oordinate x,ini s frame, of the M spaeship in order to see that it is indeed pulling apart. We will do the opposite here: given that M is at a ertain spaetime point, whih haraterizes a physial event E, with oordinates x, t) aording to the lab frame, what are its oordinates in I s frame? As we will see, the omputations are muh simpler in this ase. In the lab we have, for the spaeship M, { xm = αoshτ/α) 1) + L ) t M = α sinhτ/α). The oordinates of a physial event are, in I s oordinates, x,τ ), whih are onneted to the lab oordinates by { x = x + α) oshτ /α) α t = x + α) sinhτ 10.9) /α). It is easy to see that from 10.8) and 10.9) we obtain x + α) 2 = x M + α) 2 t M ) 2 = α L αL 0 oshτ/α) sinhτ /α) = α sinhτ/α) α2 2 + L αL 0 oshτ/α) ), 10.10) from whih we an extrat the asymptoti behaviour expτ /α) α/l 0 ) expτ/2α) and x αl 0 ) expτ/2α) L 0 expτ /2α). We see, then, that the spaeship M distanes itself from I, whih is essentially due to the failure of simultaneity.

11 1418 M Morioni Figure 1. The same proess as seen from two different referene frames. 11. The mass energy relation Finally, let us derive the mass energy relation. Consider a body of mass M that emits two bundles of radiation in opposite diretions, eah bundle arries an energy equal to δe/2. By symmetry the body will stay put. Before we proeed, let us make an observation onerning the linear momentum of the body and of the radiation. We will assume that momentum and energy are onserved, and that the momentum of the body is given by P = ξv/)mv, based on dimensional grounds. Moreover, notiing that as v vwe should have P P, we dedue that ξv/) = ξ v/). Sine as v 0, P = Mv, wealsohaveξ0) = 1. These onditions imply that ξv/) = 1+αv/) 2 +, so that, whatever the relativisti momentum is, it equals Mv up to fators of third order in v/. Sine in the following we will be looking at fators up to seond order only, we may use P = Mv. We will also be using the fat that the linear momentum arried by a bundle of radiation of energy δe is δe/. This relation preedes the mass energy relation and is usually derived [3] using onservation of total momentum mehanial and eletromagneti) but does not assume an expliit form for the linear momentum 5. Let us analyse this is a frame that moves with veloity v along the same line as the emitted radiation, as shown in figure 1. In this frame, the energy of eah bundle will hange, and, sine the body was at rest in the original frame, it will be moving with veloity v in the seond frame. If its mass is M now, then the linear momentum of the body is M v. In order to aount the momentum arried by the radiation we have to find the expression for the energy of the radiation bundles in the moving frame. In priniple, we ould use the quantum relation that states that to a photon of energy E orresponds the frequeny ν = E/h, and by applying the Doppler formula find E = E1 +β)/)). This is orret, of ourse, but has one aspet that is less than satisfatory: the speial theory of relativity is a ompletely lassial theory in the sense that its starting point is Maxwell s equations, and therefore we should not resort to any quantum relations in order to derive its results. Moreover, it is not obvious at all that the two theories ould be put together in a simple and as diret manner as it is done in general; after all, why should these two theories be ompatible? Therefore, instead of using quantum mehanis, we will take a somewhat longer route, but whih is more elementary and has the advantage of not appealing to quantum relations. Using dimensional analysis, we an write an expression for the transformation of the energy of a bundle of radiation from one inertial frame to another, whih moves with veloity u in relation to it: it is simply E = Ef β), where fβ)is an unknown universal funtion, and β = u/. This funtion must satisfy a omposition law. Consider three inertial frames, S 1,S 2 and S 3, suh that S 2 moves with veloity v with relation to S 1, and S 3 moves with veloity u 5 Stritly speaking, we should remind the reader that in these derivations one uses the Lorentz fore expression d p/dt = q E + v B), whih is relativisti, not manifestly, though.

12 Speial theory of relativity through the Doppler effet 1419 with relation to S 2. Suppose there is a bundle of radiation in S 1 with energy E 1, then we must have E 3 = E 2 fu/) = E 1 f v/)f u/) and E 3 = E 1 f u/ + v/)/1+uv/ 2 )), where E i is the energy of the radiation measured in S i, and we have used the formula for the addition of veloities. This means that ) u/ + v/ fu/)fv/)= f. 11.1) 1+uv/ 2 It is not diffiult to solve this equation, and we find see appendix 2) that ) 1+u/ a/2 fu/)= 11.2) 1 u/ where a is a onstant. Later we will show that a = 1. Proeeding with our argument, we an ompute the momentum arried by light in the moving frame. Denoting by δe ± the pulses that move to the right and to the left, we obtain δe ± = δe ) 1 ± β a/ ) 2 1 β Therefore, momentum onservation gives Mv = M v + δe ) 1+β a/2 δe ) a/ ) β This equation implies, up to seond order in v/, that M M)v = δea v 2 M M) = a E ) This is not quite the final result sine we still need to fix a, whih will done by using energy onservation. One again, we an write the total energy of the body as UM,v) = ηv/)m 2 based on dimensional grounds. The funtion ηv/) should be suh that, to seond order in v/, the differene in energy of a body of mass M at rest and the same body with veloity v should be the kineti energy, that is ηv/) η0) = v/) 2 /2. And sine the energy should not depend on the diretion of the veloity, ηv/) = η v/), and therefore ηv/) = η 0 + v/) 2 /2+ov/) 4 ). Let us look at the energy onservation in both frames of referene. In the lab frame we have UM,0) = UM, 0) + δe 11.6) whereas in the moving frame we have, up to seond order in v/) 2, UM,v) = UM,v)+ δe + + δe UM,0) + Mv2 2 = UM, 0) + M v 2 + δe1+a 2 β 2 ). 11.7) 2 Equations 11.6) and 11.7) imply that M M )v 2 /2 = a 2 δeβ 2, whih together with 11.5), fixes a = 1, and establishes the well-known mass energy relation δe = M M ) ) Before losing this setion, let us find the exat forms of the funtions ξv/) and ηv/). Let us find ηv/) first. Suppose the body emits two bundles of radiation with the same energy, to the left and right, eah with energy δe. After this emission the mass of the body is M = M δe/ 2.In

13 1420 M Morioni the lab frame we have from energy onservation, UM,0) = UM, 0) + δe/2+δe/2, while in the moving frame we have UM,v) = UM,v)+ δe + + δe, and therefore ηv/)m 2 = ηv/)m 2 + δe ) 1+β ) ) δe + = = 2 1+β 2 γδm2 ) 11.9) whih implies that ηv/) = γ, and the energy is given by UM,v) = γm 2. Finally, for momentum we have, following a set-up similar to the one just desribed, before the emission of radiation, P = ξv/)mv, and after ξv/)m v + δe + / δe /, whih gives ξv/)mv = ξv/)m v + δe ) 1+β ) ) 2 1+β = ξv/)m v + δev/2 2 ) = ξv/)m v + γδmv 11.10) whih fixes the funtion ξv/) = γ. Note that we an write these expressions in terms of the derivatives of t and x with respet to the proper time of the partile τ, sine dt/dτ = γ. We are then led naturally to the definition of the four-momentum in this ase the twomomentum ) P µ = P 0,P 1 ), with P 0 = U/ = Mdt)/dτ and P 1 = P = M dx/dτ, whih are expressed in terms of the four-veloity by P µ = M dx µ /dτ. These two relations an be used as the starting point to the study of relativisti dynamis. This ompletes our disussion of speial relativity. 12. Conlusions We have derived all relativisti kinematis, taking the Doppler effet as the starting point. The novelty in this presentation is that we derive the Doppler effet diretly from the relativity postulates, and from there obtain everything else: time dilation, length ontration, addition of veloities, Lorentz transformations for onstant veloity and for arbitrary veloities, the mass energy relation, and the introdution of relativisti energy and momentum for a massive body. We also used it to explain the twin paradox. Our approah is partiularly useful in the disussion of the twin paradox on a ylinder. In deriving the mass energy relation we used only fairly general arguments, staying always in the realm of elementary lassial physis: we did not use the energy/frequeny relation for photons from quantum mehanis, nor used the lassial expression of a wave paket in terms of eletromagneti fields, to derive the fat that the energy will also be Doppler shifted. We should stress the fat that, in deriving the relativisti energy and momentum, we assumed that there are onserved quantities suh as energy and momentum, and that light interats with matter in suh a way that all energy an be absorbed or emitted. Another feature of this approah is that all arguments presented are truly one-dimensional: we did not use light rays moving perpendiularly to its motion, for example, as is done in some elementary presentations of the speial theory of relativity. After the ompletion of this paper, it has been brought to my attention that the approah to the kinematis of the STR presented here is related to Bondi s K-alulus [10]. The main differene between the two approahes is that in K-alulus one uses geometrial arguments in spaetime diagrams to establish the main results of the STR, whereas in our approah the main tool is the relativisti invariane of the number of pulses between two events, whih is very easy to understand and aept. Moreover, it is quite simple to obtain other results, suh

14 Speial theory of relativity through the Doppler effet 1421 as the Lorentz transformations for arbitrary veloities, whih is not so straightforward using the K-alulus. Aknowledgments I would like to thank L Morioni for several useful disussions, a ritial reading of this note and for insisting in making the presentation of the mass energy relation elementary, helping along the way with several ruial insights. A very nie exhange with N D Mermin is gratefully aknowledged, as well as his sending me hapter 7 of his forthoming book [9], whih has some overlap with the arguments presented in this paper, espeially the derivation of the Doppler shift expression and the addition of veloities. Thanks are also due to M Parikh for telling me about the twin paradox on a ylinder, and N Lemos for a ritial reading of this paper. This work has been partially supported by Faperj. Appendix A In this appendix we present another elementary derivation of the Doppler effet. Suppose that Isabella is in an inertial frame of referene, and that she is sending light pulses at a frequeny ν 0, and that Marianna is moving towards her with a mirror. As explained in the paper, she will observe pulses at frequeny ν 1 = fβ)ν 0, and sine her mirror reflets them, this is equivalent to her arrying a light soure sending pulses at frequeny ν 1. Finally, Isabella will observe pulses at a frequeny ν 2 = fβ)ν 1 = fβ) 2 ν 0. Let us analyse two sequential pulses refleting from Mariannas mirror, all them pulse-1 and pulse-2, as seen in Isabellas frame. Let the distane between these two pulses be λ 0.It is easy to see that pulse-2 hits the mirror a time t after pulse-1, given by t + v t = λ 0. A.1) During this time interval, pulse-1 has travelled t and the mirror has travelled v t; therefore the distane between the two pulses, whih is the wave length measured by Isabella, is given by λ = v) t. A.2) Now we use the onstany of the speed of light: we know that λ 0 ν 0 = and that λν 2 =, whih imply v) tν = + v) tν 0 ν = + v v ν 0 A.3) and therefore fβ)= ) 1+β. A.4) Appendix B We want to solve the funtional equation ) u + v fu/)fv/)= f. B.1) 1+uv/ 2

15 1422 M Morioni Let us take u/ arbitrary and v/ infinitesimal, equal to ɛ. Expanding both sides of B.1) in the Taylor series, we obtain f u/)f 0) + ɛa + ) = f u/ + ɛ)1 ɛu/ 2 )) = fu/)+ ɛf u/)1 u 2 / 2 ) + B.2) where we introdued a = f 0), and an set f0) = 1 for physial reasons: if the observer is at rest with the light soure, then the energy measured by the observer should be the same as the one emitted. This expansion gives us the elementary differential equation f af u/) u/) = 1 u 2 /. B.3) 2 This an be easily solved, and we obtain ) 1+u/ a/2 fu/)=. B.4) 1 u/ If the reader feels this is a little too involved, or would rather avoid the use of alulus, there is another derivation we an provide 6. Replae v/ by tanhx), and u/ by tanhy), and define gx) = ln ftanhx)). Equation B.1) beomes gx) + gy) = gx + y) B.5) whose solution is trivially seen to be gx) = ax, or, equivalently, ftanhx)) = ax, whih is easy to show to be the same as B.4). Appendix C This is an alternative derivation of the Lorentz transformation using time dilation and length ontration. Its method does lie outside the main line of reasoning of the paper, but we found it ould be useful to have an alternative way to derive the Lorentz transformation, using time dilation and length ontration as the starting point. Initially, let us all the laboratory frame S, and an inertial frame moving with veloity v along the x diretion, S 0. We want to find the transformation law of the oordinates x 0,t 0 ) of a given physial event E 0 in S 0 to the oordinates x, t) of this same event, as observed in S and whih we refer to as E). We assume that x 0,t 0 ) = 0, 0) orresponds to x, t) = 0, 0). In order to find this transformation law, let us assume that at the same time and position of E 0 a light flash was emitted. This light flash will arrive at the origin of S 0 at time t 0 + x 0 /. Due to the time dilation formula, this orresponds to time γt 0 +x 0 /) as measured in S, where γ 1 = 2 ) 1 2. In order to find the time the event E took plae when measured in S, we have to roll the film bakwards. Sine the light arrived at the origin of S 0 at time measured in S) γt 0 + x 0 /), it means that it left the event E at time γt 0 + x 0 /) γ 1 x 0 / + v): we have to subtrat the time light takes to over the distane γ 1 x 0 to the origin, as the origin moves with veloity v. This gives t = γ t 0 + x 0 ) γ 1 x 0 + v t = t 0 + βx 0 /. C.1) 2 ) We an perform a similar argument to obtain the transformation law for the spatial oordinate. At the time light from the event E 0 arrives at the origin of S 0, the origin itself has travelled a distane vγ 1 t 0 + x 0 /), aording to S. We have to subtrat from this value the distane 6 I thank N Mermin for this suggestion.

16 Speial theory of relativity through the Doppler effet 1423 travelled by the origin of S 0 during the time light is going from E to the origin of S 0, whih is given by vγ 1 x 0 / + v), and add the length ontrated distane γ 1 x 0. This gives x = vγ t + x ) 0 vγ 1 x 0 + v + γ 1 x 0 x = x 0 + vt 0. C.2) 2 ) Equations C.2) and C.1) are the Lorentz transformation of oordinates for one-dimensional relativisti motion. Referenes [1] See, for example, Davis F and Einstein A ed) 1952 The Priniple of Relativity New York: Dover) reprint) [2] Frenh A P 1968 Speial Relativity New York: Norton) Taylor E and Wheeler J A 1992 Spaetime Physis San Franiso: Freeman) [3] Jakson J D 1998 Classial Eletrodynamis 3rd edn New York: Wiley) [4] Mermin N D 2005 It s About Time: Understanding Einstein s Relativity Prineton, NJ: Prineton University Press) [5] Dray T 1990 The twin paradox revisited Am.J.Phys [6] Weeks J 2001 The twin paradox in a losed universe Am. Math. Mon. Aug/Sep [7] Bansal D, Laing J and Sriharan A 2005 On the twin paradox in a universe with a ompat dimension Preprint gr-q/ [8] Bell J S 2004 Speakable and Unspeakable in Quantum Mehanis Cambridge: Cambridge University Press) [9] Nelson R A 1987 Generalized Lorentz transformation for an aelerated, rotating frame of referene J. Math. Phys Nelson R A 1994 J. Math. Phys erratum) [10] Bondi H 1980 Relativity and Common Sense New York: Dover)