SPECIAL RELATIVITY. MATH2410 KOMISSAROV S.S

SPECIAL RELATIVITY. MATH2410 KOMISSAROV S.S 2012

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Contents Contents 2 1 Spae and Time in Newtonian Physis 9 1.1 Spae............................................ 9 1.1.1 Einstein summation rule.............................. 11 1.2 Time............................................. 12 1.3 Galilean relativity..................................... 12 1.4 Newtonian Mehanis................................... 13 1.5 Galilean transformation.................................. 13 1.6 The lak of speed limit................................... 15 1.7 Light............................................. 15 1.8 Advaned material: Maxwell equations, eletromagneti waves, and Galilean invariane 17 1.8.1 Maxwell equations................................. 17 1.8.2 Some relevant results from vetor alulus.................... 17 1.8.3 Wave equation in eletromagnetism....................... 18 1.8.4 Plane waves..................................... 18 1.8.5 Wave equation is not Galilean invariant..................... 19 2 Basi Speial Relativity 21 2.1 Einstein s postulates.................................... 21 2.2 Einstein s thought experiments.............................. 21 2.2.1 Experiment 1. Relativity of simultaneity..................... 21 2.2.2 Experiment 2. Time dilation........................... 22 2.2.3 Experiment 3. Length ontration........................ 24 2.2.4 Synronization of loks.............................. 26 2.3 Lorentz transformation................................... 28 2.3.1 Derivation...................................... 28 2.3.2 Newtonian limit.................................. 29 2.4 Relativisti veloity addition.............................. 30 2.4.1 One-dimensional veloity addition....................... 30 2.4.2 Three-dimensional veloity addition...................... 31 2.5 Aberration of light..................................... 31 2.6 Doppler effet........................................ 33 2.6.1 Transverse Doppler effet............................. 33 2.6.2 Radial Doppler effet................................ 34 2.6.3 General ase.................................... 35 3 Spae-time 37 3.1 Minkowski diagrams.................................... 37 3.2 Spae-time......................................... 38 3.3 Light one.......................................... 40 3.4 Causal struture of spae-time.............................. 41 3.5 Types of spae-time intervals............................... 43 3

4 CONTENTS 3.6 Vetors........................................... 44 3.6.1 Definition...................................... 44 3.6.2 Operations of addition and multipliation.................... 46 3.6.3 Coordinate transformation............................ 46 3.6.4 Infinitesimal displaement vetors........................ 47 3.7 Tensors........................................... 48 3.7.1 Definition...................................... 48 3.7.2 Components of tensors............................... 48 3.7.3 Coordinate transformation............................ 48 3.8 Metri tensor........................................ 49 3.8.1 Definition...................................... 49 3.8.2 Classifiation of spae-time vetors and other results speifi to spae-time.. 50 4 Relativisti partile mehanis 53 4.1 Tensor equations and the Priniple of Relativity.................... 53 4.2 4-veloity and 4-momentum................................ 55 4.3 Energy-momentum onservation............................. 57 4.4 Photons........................................... 59 4.5 Partile ollisions...................................... 60 4.5.1 Nulear reoil.................................... 60 4.5.2 Absorption of neutrons.............................. 62 4.6 4-aeleration and 4-fore................................. 62

CONTENTS 5 Figure 1: Charlie Chaplin: They heer me beause they all understand me, and they heer you beause no one an understand you.

6 CONTENTS Figure 2: Arthur Stanley Eddington, the great English astrophysiist. From the onversation that took plae in the lobby of The Royal Soiety: Silverstein -... only three sientists in the world understand theory of relativity. I was told that you are one of them. Eddington - Emm.... Silverstein - Don t be so modest, Eddington! Eddington - On the ontrary. I am just wondering who this third person might be.

CONTENTS 7 Figure 3: Einstein: Sine the mathematiians took over the theory of relativity I do no longer understand it.

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Chapter 1 Spae and Time in Newtonian Physis 1.1 Spae The abstrat notion of physial spae reflets the properties of physial objets to have sizes and physial events to be loated at different plaes relative to eah other. In Newtonian physis, the physial spae was onsidered as a fundamental omponent of the world around us, whih exists by itself independently of other physial bodies and normal matter of any kind. It was assumed that 1) one ould interat with this spae and unambiguously determine the motion of objets in this spae, in addition to the easily observed motion of physial bodies relative to eah other, 2) that one may introdue points of this spae, and determine at whih point any partiular event took plae. The atual ways of doing this remained mysterious though. It was often thought that the spae is filled with a primordial substane, alled ether or plenum, whih an be deteted one way or another, and that atoms of ether orrespond to points of physial spae and that motion relative to these atoms is the motion in physial spae. This idea of physial spae was often alled the absolute spae and the motion in this spae the absolute motion. There also was a onsensus that the best mathematial model for the absolute spae was the 3-dimensional Eulidean spae. By definition, in suh spae one an onstrut uboids, retangular parallelepipeds, suh that the lengths of their edges, a, b, and, and the diagonal l satisfy the following equation l 2 = a 2 + b 2 + 2, (1.1) no matter how big the uboid is. This was strongly supported by the results of pratial geometry. b l a Figure 1.1: A uboid Given this property on an onstrut a set of Cartesian oordinates, {x 1, x 2, x 3 } (the same meaning as {x, y, z}). These oordinates are distanes between the origin and the point along the 9

10 CHAPTER 1. SPACE AND TIME IN NEWTONIAN PHYSICS oordinate axes. x x x x x x Figure 1.2: Cartesian oordinates In Cartesian oordinates, the distane between point A and point B with oordinates {x 1 a, x 2 a, x 3 a} and {x 1 b, x2 b, x3 b } respetively is where x i = x i a x i b. For infinitesimally lose points this beomes l 2 ab = ( x 1 ) 2 + ( x 2 ) 2 + ( x 3 ) 2, (1.2) dl 2 = (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2, (1.3) where dx i are infinitesimally small differenes between Cartesian oordinates of these points. This equation allows us to find distanes along urved lines by means of integration. N x 3 x 1 O r Figure 1.3: Spherial oordinates Other types of oordinates an also used in Eulidean spae. oordinates, {r, θ φ}, defined via One example is the spherial x 1 = r sin θ os φ; (1.4) x 2 = r sin θ sin φ; (1.5) x 3 = r os θ. (1.6) Here r is the distane from the origin, θ [0, π] is the polar angle, and φ [0, 2π) is the asymuthal angle. The oordinate lines of these oordinates are not straight lines but urves. Suh oordinate systems are alled urvilinear. The oordinate lines of spherial oordinates are perpendiular to

1.1. SPACE 11 eah other at every point. Suh oordinate systems are alled orthogonal (There are non-orthogonal urvilinear oordinates). In spherial oordinates, the distane between infinitesimally lose points is dl 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2. (1.7) Sine, the oeffiients of dθ 2 and dφ 2 vary in spae l 2 ab ( r) 2 + r 2 θ 2 + r 2 sin 2 θ φ 2 (1.8) for the distane between points with finite separations r, θ, φ. In order to find this distane one has to integrate l ab = b a dl (1.9) along the line onneting these points. For example the irumferene of a irle of radius r 0 is l = dl = 2 π 0 r 0 dθ = 2πr 0. (1.10) (Notie that we seleted suh oordinates that the irle is entered on the origin, r = r 0, and it is in a meridional plane, φ =onst. As the result, along the irle dl = r 0 dθ.) In the generi ase of urvilinear oordinates, the distane between infinitesimally lose points is given by the positive-definite quadrati form dl 2 = 3 i=1 j=1 3 g ij dx i dx j, (1.11) where g ij = g ji are oeffiients that are funtions of oordinates. Suh quadrati forms are alled metri forms. g ij are in fat the omponents of so-alled metri tensor in the oordinate basis of utilised oordinates. It is easy to see that in a Cartesian basis { 1 if i = j g ij = δ ij = (1.12) 0 if i j Not all positive definite metri forms orrespond to Eulidean spae. If there does not exist a oordinate transformation whih redues a given metri form to that of Eq.1.3 then the spae with suh metri form is not Eulidean. This mathematial result is utilized in General Relativity. 1.1.1 Einstein summation rule In the modern mathematial formulation of the Theory of Relativity it is important to distinguish between upper and lower indexes as the index position determines the mathematial nature of the indexed quantity. E.g. a single upper index indiates a vetor (or a ontravariant vetor), as in b i, whereas b i stands for a mathematial objet of a different type, though uniquely related to the vetor b, the so-alled one-form (or ovariant vetor). This applies to oordinates, as the Cartesian oordinates an be interpreted as omponents of the radius-vetor. Our syllabus is rather limited and we will not be able to explore the differene between ovariant and ontravariant vetors in full, as well as the differene between various types of tensors. However, we will keep using this modern notation. The Einstein summation rule is a onvention on the notation for summation over indexes. Namely, any index appearing one as a lower index and one as an upper index of the same indexed objet or in the produt of a number of indexed objets stands for summation over all allowed values of this index. Suh index is alled a dummy, or summation index. Indexes whih are not dummy are alled free indexes. Their role is to give a orret equation for any allowed index value, e.g. 1,2, and 3 in three-dimensional spae.

12 CHAPTER 1. SPACE AND TIME IN NEWTONIAN PHYSICS Aording to this rule we an rewrite Eq.1.11 in a more onise form: dl 2 = g ij dx i dx j. (1.13) This rule allows to simplify expressions involving multiple summations. Here are some more examples: 1. a i b i stands for n i=1 a ib i ; here i is a dummy index; 2. In a i b i i is a free index. 3. a i b kij stands for n i=1 a ib kij ; here k and j are free indexes and i is a dummy index; 4. a i f x stands for n i lower index. 1.2 Time i=1 ai f x i. Notie that index i in the partial derivative x i is treated as a The notion of time reflets our everyday-life observation that all events an be plaed in a partiular order refleting their ausal onnetion. In this order, event A appears before event B if A aused B or ould ause B. This ausal order seems to be ompletely independent on the individual analysing these events. This notion also reflets the obvious fat that one event an last longer then another one. In Newtonian physis, time was onsidered as a kind of fundamental ever going proess, presumably periodi, so that one an ompare the rate of this proess to rates of all other proesses. Although the nature of this proess remained mysterious it was assumed that all other periodi proesses, like the Earth rotation, refleted it. Given the fundamental nature of time it would be natural to assume that this proess ours in ether. This understanding of time has lead naturally to the absolute meaning of simultaneity. That is one ould unambiguously deide whether two events were simultaneous or not. Similarly, physial events ould be plaed in only one partiular order, so that if event A preedes event B aording to the observation of some observer, this has to be the same for all other observers, unless a mistake is made. Similarly, any event ould be desribed by only one duration, when the same unit of time is used to quantify it. These are the reasons for the time of Newtonian physis to be alled the absolute time. There is only one time for everyone. Both in theoretial and pratial terms, a unique temporal order of events ould only be established if there existed signals propagating with infinite speed. In this ase, when an event ours in a remote plae everyone an beome aware of it instantaneously by means of suh super-signals. Then all events immediately divide into three groups with respet to this event: (i) The events simultaneous with it they our at the same instant as the arrival of the super-signal generated by the event; (ii) The events preeding it they our before the arrival of this super-signal and ould not be aused by it. But they ould have aused the original event; (iii) The events following it they our after the arrival of this super-signal and an be aused by it. But they annot ause the original event. If, however, there are no suh super-signals, things beome highly ompliated as one needs to know not only the distanes to the events but also the motion of the observer and how exatly the signals propagated through the spae separating the observer and these events. Newtonian physis assumes that suh infinite speed signals do exist and they play fundamental role in interation between physial bodies. This is how in the Newtonian theory of gravity, the gravity fore depends only on the urrent loation of the interating masses. 1.3 Galilean relativity Galileo, who is regarded to be the first true natural sientist, made a simple observation whih turned out to have far reahing onsequenes for modern physis. He notied that it was diffiult to

1.4. NEWTONIAN MECHANICS 13 tell whether a ship was anhored or oasting at sea by means of mehanial experiments arried out on board of this ship. It is easy to determine where a body is moving through air in the ase of motion, it experienes the air resistane, the drag and lift fores. But here we are dealing only with a motion relative to air. What about the motion relative to the absolute spae and the interation with ether? If suh an interation ourred then one ould measure the absolute motion. Galileo s observation tells us that this must be at least a rather week interation. No other mehanial experiment, made after Galileo, has been able to detet suh an interation. Newtonian mehanis states this fat in its First Law as: The motion of a physial body whih does not interat with other bodies remains unhanged. It moves with onstant speed along straight line. This means that one annot determine the motion through absolute spae by mehanial means. Only the relative motion between physial bodies an be determined this way. 1.4 Newtonian Mehanis Newton (1643-1727) founded the lassial mehanis - a basi set of mathematial laws of motion based on the ideas of spae and time desribed above. One of the key notions he introdued is the notion of an inertial referene frame. A referene frame is a solid oordinate grid (usually Cartesian but not always), used to quantitatively desribe the physial motion. In general, suh a frame an be in arbitrary motion in spae and one an introdue many different referene frames. However, not all referene frames are equally onvenient. Some of them are muh more onvenient than others as the motion of bodies that are not interating with other bodies is partiularly simple in suh frames, namely they move with zero aeleration, a = dw dt = 0, (1.14) where w = dr/dt is the body veloity. Suh frames are alled inertial frames. We stress that existene of suh frames is a basi assumption (postulate) of the theory. It is alled the first law of Newtonian mehanis. If the motion of a body as measured in inertial frame is in fat aelerated then it is subjet to interation with other bodies. Mathematially suh interation is desribed by a fore vetor f. The aeleration is then given by the seond law of Newtonian mehanis ma = f (1.15) where m is a salar quantity alled the inertial mass of the body (this is the property desribing body s ability to resist ation of external fores). Eah kind of interation should be desribed by additional laws determining the fore vetor as a funtion of other parameters (e.g the law of gravity). The third law of Newtonian mehanis deals with binary interations, or interations involving only two bodies, say A and B. It states that f a = f b, (1.16) where f a is the fore ating on body A and f b is the fore ating on body B. 1.5 Galilean transformation Consider two Cartesian frames, S and S, with oordinates {x i } and {xĩ} respetively. Assume that (i) their orresponding axes are parallel, (ii) their origins oinide at time t = 0, (iii) frame S is moving relative to S along the x 1 axis with onstant speed v, as shown in Figure 1.4. This will be alled the standard onfiguration.

14 CHAPTER 1. SPACE AND TIME IN NEWTONIAN PHYSICS x x ~ x A vt x O O ~ ~ x v x = x +vt ~ ~ ~ x Figure 1.4: Measuring the x 1 oordinate of event A in two referene frames in the standard onfiguration. For simpliity, we show the ase where this event ours in the plane x 3 = 0. The Galilean transformation relates the oordinates of events as measured in both frames. Given the absolute nature of time Newtonian physis, it is the same for both frames. So this may look over-elaborate if we write t = t. (1.17) However, this make sense if we wish to stress that both fiduial observers, who ride these frames and make the measurements, use loks made to the same standard and synhronised with eah other. Next let us onsider the spatial oordinates of some event A. The x 2 oordinate of this event is its distane in the absolute spae from the plane given by the equation x 2 = 0. Similarly, the x 2 oordinate of this event is its distane in the absolute spae from the plane given by the equation x 2 = 0. Sine these two planes oinide both these distanes are the same and hene x 2 = x 2. Similarly, we onlude that x 3 = x 3. As to the remaining oordinate, the distane between planes x 1 = 0 and x 1 = 0 at the time of the event is vt, and hene x 1 = x 1 + vt (see Figure 1.4). Summarising, x 1 = x 1 + vt, x 2 = x 2, x 3 = x 3. (1.18) This is the Galilean transformation for the standard onfiguration. If we allow the frame S to move in arbitrary diretion with veloity v i then a more general result follows, From this we derive the following two important onlusions: The veloity transformation law: x i = xĩ + v i t. (1.19) w i = wĩ + v i, (1.20) where w i = dx i /dt and wĩ = dxĩ/dt are the veloities of a body as measured in frames S and S respetively.

1.6. THE LACK OF SPEED LIMIT 15 The aeleration transformation law: a i = aĩ, (1.21) where a i = d 2 x i /dt 2 and aĩ = d 2 xĩ/dt 2 are the aelerations of a body as measured in frames S and S respetively. Thus, in both frames the aeleration is exatly the same. These results are as fundamental as it gets, beause they follow diretly from the notions of absolute spae and absolute time. Equation 1.21 tells us that the body aeleration is the same in all inertial frames and so must be the fore produing this aeleration. Thus, the laws of Newtonian mehanis are the same (or invariant) in all inertial frames. This is exatly what was disovered by Galileo and is now known as the Galilean priniple of relativity. In partiular, if the aeleration measured in the frame S vanishes then it also vanishes in the frame S. This tells us that there exist infinitely many inertial frames and they all move relative to eah other with onstant speed. Only one of these frames is at rest in the absolute spae (here we do not differentiate between frames with different orientation of their axes or/and loations of their origins) but we annot tell whih one. This make the absolute spae a very elusive if not ghostly objet. 1.6 The lak of speed limit Is there any speed limit a physial body an have in Newtonian mehanis? The answer to this question is No. To see this onsider a partile of mass m under the ation of onstant fore f. Aording to the seond law of Newton its speed then grows linearly, w = w 0 + f m t, without a limit. This onlusion also agrees with the Galilean priniple of relativity. Indeed, suppose the is a maximum allowed speed, say w max. Aording to this priniple it must be the same for all inertial frames. Now onsider a body moving with suh a speed to the right of the frame S. This frame an also move with speed w max relative to the frame S. Then aording to the Galilean veloity addition this body moves relative to frame S with speed 2w max. This ontradits to our assumption that there exist a speed limit, and hene this assumption has to be disarded. 1.7 Light The nature of light was a big mystery in Newtonian physis and a subjet of heated debates between sientists. One point of view was that light is made by waves propagating in ether, by analogy with sound whih is made by waves in air. The speed of light waves was a subjet of great interest to sientists. The most natural expetation for waves in ether is to have infinite speed. Indeed, waves with infinite speed fit niely the onept of absolute time, and if suh waves exist then there is no more natural medium for suh wave as the ether of absolute spae. However, the light turned out to have finite speed. Duth astronomer Roemer notied that the motion of Jupiter s moons had systemati variation, whih ould be easily explained only if one assumed that light had finite, though very large, speed. Sine then, many other measurements have been made whih all agree on the value for the speed of light 3 10 10 m/s. The development of mathematial theory of eletromagnetism resulted in the notions of eletri and magneti fields, whih exist around eletrially harged bodies. These fields do not manifest themselves in any other ways but via fores ating on other eletrially harged bodies. Attempts do desribe the properties of these fields mathematially resulted in Maxwell s equations, whih agreed with experiments most perfetly.

16 CHAPTER 1. SPACE AND TIME IN NEWTONIAN PHYSICS What is the nature of eletri and magneti field? They ould just reflet some internal properties of matter, like air, surrounding the eletrially harged bodies. Indeed, it was found that the eletri and magneti fields depended on the hemial and physial state of surrounding matter. However, the experiments learly indiated that the eletromagneti fields ould also happily exist in vauum (empty spae). This fat prompted suggestions that in eletromagnetism we are dealing with ether. Analysis of Maxwell equations shows that eletri and magneti fields hange via waves propagating with finite speed. In vauum the speed of these waves is the same in all diretion and equal to the known speed of light! When this had been disovered, Maxwell immediately interpreted light as eletromagneti waves or ether waves. Sine aording to the Galilean transformation the result of any speed measurement depends on the seletion of inertial frame, the fat that Maxwell equations yielded a single speed ould only mean that they are valid only in one partiular frame, the rest frame of ether and absolute spae. On the other hand, the fat that the astronomial observations and laboratory experiments did not find any variation of the speed of light as well seemed to indiate that Earth was almost at rest in the absolute spae. Earth Sun V orb Earth Sun V orb 2V orb V orb Figure 1.5: Left panel: Earth s veloity relative to the Sun at two opposite points of its orbit. Right panel: Earth s absolute veloity at two opposite points of its orbit and the Sun absolute veloity, assuming that at the left point the Earth veloity vanishes. However, Newtonian mehanis learly shows that Earth annot be exatly at rest in absolute spae all the time. Indeed, it orbits the Sun and even if at one point of this orbit the speed of Earth s absolute motion is exatly zero it must be nonzero at all other points, reahing the maximum value equal to twie the orbital speed at the opposite point of the orbit. This simple argument shows that during one alendar year the speed of light should show variation of the order of the Earth orbital speed and that the speed of light should be different in different diretions by at least the orbital speed. Provided the speed measurements are suffiiently aurate we must be able to see these effets. Amerian physiists Mikelson and Morley were first to design experiments of suh auray (by the year 1887) and to everyone s amazement and disbelief their results were negative. Within their experimental errors, the speed of light was the same in all diretions all the time! Sine then, the auray of experiments has improved dramatially but the result is still the same, learly indiating shortomings of Newtonian physis with its absolute spae and time. Moreover, no objet has shown speed exeeding the speed of light. In his ground-braking work On the eletrodynamis of moving bodies, published in 1905, Albert Einstein paved way to new physis with ompletely new ideas on the nature of physial spae and time, the Theory of Relativity, whih aommodates these remarkable experimental findings.

1.8. ADVANCED MATERIAL: MAXWELL EQUATIONS, ELECTROMAGNETIC WAVES, AND GALILEAN INV 1.8 Advaned material: Maxwell equations, eletromagneti waves, and Galilean invariane 1.8.1 Maxwell equations Maxwell (1831-1879) ompleted the mathematial theory of eletrodynamis. After his work, the evolution of eletromagneti field in vauum is desribed by B = 0, (1.22) 1 B + E = 0, t (1.23) E = 0, (1.24) 1 E + B = 0, (1.25) t where is a onstant with dimension of speed. Laboratory experiments with eletromagneti materials allowed to measure this onstant it turned out to be equal to the speed of light! Comment: Later, the works by Plank(1858-1947) and Einstein(1879-1955) lead to the onlusion that eletromagneti energy is emitted, absorbed, and propagate in disrete quantities, or photons. Thus, Newton s ideas have been partially onfirmed as well. Suh partile-wave duality is a ommon property of miro-partiles that is aounted for in quantum theory. 1.8.2 Some relevant results from vetor alulus Notation: {x k } - Cartesian oordinates (k = 1, 2, 3); {ê k } are the unit vetors along the x k axes; r = x k ê k is the position vetor (radius vetor) of the point with oordinates {x k }; A(r) = A k (r)ê k is a vetor field in Eulidean spae (vetor funtion); A k are the omponents of vetor A in the basis {ê k }; f(r) is a salar field in Eulidean spae (salar funtion). The divergene of vetor field A is defined as A = Ak x k. (1.26) (Notie use of Einstein summation onvention in this equation!). This is a salar field. One an think of as a vetor with omponents / x k and onsider A as a salar produt of and A. The url of vetor field A is defined via the determinant rule for vetor produt ê 1 ê 2 ê 3 A = x 1 x 2 x 3 A 1 A 3 A 3 = (1.27) = ê 1 ( A3 x 2 A2 x 3 ) + ê 2 ( A1 x 3 A3 x 1 ) + ê 3 ( A2 x 1 A1 ). (1.28) x2 This is a vetor field. The gradient of a salar field is defined as This is a vetor field. The Laplaian of salar field f is defined as From this definition and eqs.(1.26,1.29) one finds that f = f x 1 ê1 + f x 2 ê2 + f x 3 ê3. (1.29) 2 f = 2 f = f. (1.30) 2 f (x 1 ) 2 + 2 f (x 2 ) 2 + 2 f (x 3 ) 2. (1.31)

18 CHAPTER 1. SPACE AND TIME IN NEWTONIAN PHYSICS This is a salar field. The Laplaian of vetor field A is defined as 2 A = ê k 2 A k. (1.32) (Notie use of Einstein summation onvention in this equation!). This is a vetor field. The following vetor identity is very handy ( A) = ( A) 2 A. (1.33) 1.8.3 Wave equation in eletromagnetism Apply to eq.1.23 to obtain 1 B + ( E) = 0. t Using eq.1.33 and ommutation of partial derivatives this redues to 1 t B + ( E) 2 E = 0. The seond term vanishes due to eq.1.24 and eq.1.25 allows us to replae B with E t in the first term. This gives us the final result 1 2 E 2 t 2 2 E = 0. (1.34) In a similar fashion one an show that These are examples of the anonial wave equation where ψ(r, t) is some funtion of spae and time. 1 2 2 B t 2 2 B = 0. (1.35) 1 2 ψ 2 t 2 2 ψ = 0, (1.36) 1.8.4 Plane waves Look for solutions of eq.1.36 that depend only on t and x 1. Then and eq.1.36 redues to ψ x 2 = ψ x 3 = 0 1 2 ψ 2 t 2 2 ψ (x 1 = 0. (1.37) ) 2 This is a one-dimensional wave equation. It is easy to verify by substitution that it has solutions of the form ψ ± (t, x 1 ) = f(x 1 ± t), (1.38) where f(x) is an arbitrary twie differentiable funtion. ψ + (t, x 1 ) = f(x 1 + t) desribes waves propagating with speed in the negative diretion of the x 1 axis and ψ (t, x 1 ) = f(x 1 t) desribes waves propagating with speed in the positive diretion of the x 1 axis. Thus, equations (1.34,1.35), tell us straight away that Maxwell equations imply eletromagneti waves propagating with speed, the speed of light.

1.8. ADVANCED MATERIAL: MAXWELL EQUATIONS, ELECTROMAGNETIC WAVES, AND GALILEAN INV 1.8.5 Wave equation is not Galilean invariant Can the eletromagneti phenomena be used to determine the absolute motion, that is motion relative to the absolute spae. If like the equations of Newtonian mehanis the equations of eletrodynamis are the same in all inertial frames then they annot. Thus, it is important to see how the Maxwell equations transform under the Galilean transformation. However, it is suffiient to onsider only the wave equation, eq.1.36, whih is a derivative of Maxwell s equations. For simpliity sake, one an deal only with its one-dimensional version, eq.1.37. Denoting x 1 as simply x we have 1 2 2 ψ t 2 + 2 ψ = 0. (1.39) x2 The Galilean transformation reads x = x vt. It is easy to see that in new variables, {t, x}, equation 1.39 beomes 1 2 ψ 2 t 2 + 2v 2 ) ψ (1 2 t x + v2 2 ψ 2 = 0. (1.40) x 2 Sine eq.1.40 has a different form ompared to eq.1.39 we onlude that the wave equation, and hene the Maxwell equations, are not invariant under Galilean transformation! One should be able to detet motion relative to the absolute spae! In order to eluidate this result onsider the wave solutions of eq.1.40. Diret substitution shows that this equation is satisfied by Ψ ± (t, x) = f( x + a ± t), (1.41) where f(x) is again an arbitrary twie differentiable funtion and a ± = v ±. (1.42) These are waves propagating with speeds a ±. In fat, Ψ + (t, x) = f( x + (v + )t) desribes wave propagating with speed = v and Ψ (t, x) = f( x + (v )t) desribes wave propagating with speed = v + Comparing these results with eq.1.20 shows us that what we have got here is simply the Galilean veloity transformation for eletromagneti waves. Thus, we arrive to the following onlusions 1. Eletromagneti waves an propagate with speed in all diretions only in one very speial inertial frame, namely the frame that is in rest in absolute spae. In any other frame it will be different in different diretions, as ditated by the Galilean transformation, and equal to only in the diretions normal to the frame veloity relative to the absolute spae. 2. The Maxwell equations are not general. They hold only in the frame at rest in the absolute spae. In spite of looking very onvining these onlusions however do not omply with physial experiments whih show beyond any doubt that in all frames the eletromagneti waves propagate with the same speed in all diretions! These experimental results show that the Galilean transformation is not that general as thought before Einstein, and hene the notions of spae and time as desribed of Newtonian physis are not orret.

20 CHAPTER 1. SPACE AND TIME IN NEWTONIAN PHYSICS

Chapter 2 Basi Speial Relativity 2.1 Einstein s postulates Paper On the eletrodynamis of moving bodies by Einstein (1905). Postulate 1 (Priniple of Relativity): All physial laws are the same (invariant) in all inertial frames. Postulate 2: The speed of light (in vauum) is the same in all inertial frames. It is the same in magnitude and does not depend on diretion of propagation. Postulate 1 implies that no physial experiment an be used to measure the absolute motion. In other words the notions of absolute motion and absolute spae beome redundant. As far as physis is onerned the absolute spae does not exist! Postulate 2 is fully onsistent with Postulate 1 (and hene may be onsidered as a derivative of Postulate 1). Indeed, if Maxwell s equations (1.22-1.25) are the same in all inertial frames then the eletromagneti waves propagate with the same speed whih is given by the onstant in these equations. Sine Postulate 2 is in onflit with the Galilean veloity addition it shows that the very basi properties of physial spae and time have to be reonsidered. In the next setion we arry out a number of very simple thought experiments whih show how dramati the required modifiations are. 2.2 Einstein s thought experiments In these experiments we assume that when the speed of the same light signal is measured in any inertial frame the result is always the same, namely. 2.2.1 Experiment 1. Relativity of simultaneity A arriage is moving with speed v past a platform. A ondutor, who stands in the middle of the arriage, sends simultaneously two light pulses in the opposite diretions along the trak. Both passengers and the rowd waiting on the platform observe how the pulses hit the ends of the arriage. The arriage passengers agree that both pulses reah the ends simultaneously as they propagate with the same speed and have to over the same distane (see left panel of figure 2.1). If L is the arriage length as measured by its passengers then the required time is t left = t right = L/2. 21

22 CHAPTER 2. BASIC SPECIAL RELATIVITY ~ t=0 ~ L L t=0 L t=l/2(+v) ~~ t=l/2 t=l/2(-v) L Figure 2.1: Thought experiment number 1. Left panel: Events as seen in the arriage frame. Right panel: Events as seen in the platform frame. The rowd on the platform, however, see things differently. As both ends of the arriage move to the right the pulse sent to the left has to over shorter distane and reahes its end earlier than the pulse sent to the right (right panel of figure 2.1). In fat, the required times are t left = L 2 ( 1 ) + v and t right = L 2 ( ) 1, v where L is the arriage length as measured by the rowd (obtain these results.) As we shell see later L L). Thus, two events, whih are simultaneous in the arriage frame are not simultaneous in the platform frame. This implies that temporal order of events depends of the frame of referene and hene that the absolute time does no longer exists. Instead, eah inertial frame must have its own time. The next experiment supports this onlusion, showing that the same events may have different durations in different frames. 2.2.2 Experiment 2. Time dilation This time a single pulse is fired from one side of the arriage perpendiular to the trak, reflets of the other side, and returns bak. In the arriage frame (left panel of figure 2.2) the pulse overs the distane 2W, where W is the arriage width, and this takes time t = 2W. In order to find t, the elapsed time as measured in the platform frame, we notie that we an write the distane overed by the pulse as t and as 2 W 2 + (v t/2) 2 (right panel of figure 2.2). Thus, ( t/2) 2 = W 2 + (v t/2) 2.

2.2. EINSTEIN S THOUGHT EXPERIMENTS 23 ~ t=0 t=0 ~ t=w/ t= t/2 ~ t=2w/ W t= t W v t Figure 2.2: Thought experiment number 2. Left panel: Events as seen in the arriage frame. Right panel: Events as seen in the platform frame. From this we find t = 2W 1 1 v2 / = t 1 2 1 v2 /. 2 Introduing the so-alled Lorentz fator γ = 1/ 1 v 2 / 2. (2.1) we an write this result as t = γ t. (2.2) This shows us that not only simultaneity is relative but also the duration of events. (Notie that we assumed here that the arriage width is the same in both inertial frames. We will ome bak to this assumption later.) Proper time and time of inertial frames. The best way of measuring time is via some periodi proess. Eah standard lok is based on suh a proess. Its time is alled the proper time of the lok. When instead of a lok we have some physial body it is also useful to introdue the proper time of this body this an be defined as the time that would be measured by a standard lok moving with this body. Often, the proper time is denoted using the Greek letter τ. As to the time of inertial frame, it an be based on the time of a single standard lok o-moving with this frame. It is easy to see how to use suh a lok for measuring time of events whih our at the same loation as this lok. For remote events one has to have a system of ommuniation between various loations of the frame. The best type of signal for suh a system is light, as its speed is given before hand. The lok reords the reeption time of the light signal sent by the event and the time of the event is this time minus the distane to the event divided by the speed of light. However, an idential result will be obtained if a whole grid of losely spaed standard loks is

24 CHAPTER 2. BASIC SPECIAL RELATIVITY build for the frame, eah at rest in this frame, and synhronised with other loks using light signals. When an event ours one simply has to use the reading of the lok at the same loation as the event. When it omes to our thought experiment, then t ould measured by means of ondutor s standard lok only the pulse is fired from its loation and then omes bak to its loation. Thus, t is the proper time of the ondutors lok. In ontrast, t is the time of the inertial frame of the platform and not a proper time of any of its loks. Indeed, two loks, loated at points A and B, are required to determine this time interval. Given these definitions, we onlude that the result (2.2) implies that t = γ τ, (2.3) where τ is the proper time of some standard lok or other physial body and t is the time of the inertial frame where this lok moves with the Lorentz fator γ. Sine γ > 1 for any v 0 this shows us that t > τ. What does this imply? One an say that aording to the time system of a given inertial frame, any moving lok slows down. Sine this result does not depend on the physial nature of the lok mehanism it inevitably implies that all physial proesses within a moving body slow down ompared to the proesses of a similar body at rest, when heked against the time system of the inertial frame where these time measurements are made. This effet is alled the time dilation. 2.2.3 Experiment 3. Length ontration This time the light pulse is fired from one end of the arriage along the trak, gets refleted of the other end and omes bak. In the arriage frame the time of pulse journey in both diretions is equal to L/ where L is the arriage length as measured in the arriage frame (left panel of figure 2.3). Thus, the total time of the pulse journey is t = 2 L/. ~ t=0 ~ L t=0 ~ ~ ~ t = t/2 = L/ t= t v t 1 1 L ~ ~ ~ t = t = 2L/ t= t t 1 v t 2 2 L Figure 2.3: Thought experiment number 3. Left panel: Events as seen in the arriage frame. Right panel: Events as seen in the platform frame.

2.2. EINSTEIN S THOUGHT EXPERIMENTS 25 In the platform frame the first leg of the journey (before the refletion) takes some time t 1 and the seond leg takes t 2 whih is less then t 1 beause of the arriage motion. To find t 1 we notie that the distane overed by the pulse during the first leg an be expressed as t 1 and also as L + v t 1, where L is the length of the arriage as measured in the platform frame (see the right panel of figure 2.3). Thus, t 1 = L/( v). To find t 2 we notie that the distane overed by the pulse during the seond leg an be expressed as t 2 and also as L v t 2, (see the right panel of figure 2.3). Thus, The total time is t = t 1 + t 2 = t 2 = L/( + v). L v + L + v = 2L γ2. Sine t is atually a proper time interval we an apply eq.2.3 and write Combining the last two results we obtain whih gives us t = γ t = γ 2 L. 2L γ2 = γ 2 L L = L/γ (2.4) Thus, the length of the arriage in the platform frame is different from that in the arriage frame. This shows that if we aept that the speed of light is the same in all inertial frames then we have to get rid of the absolute spae as well! Similarly to the definition of the proper time interval, the proper length of an objet, whih we will denote as L 0, is defined as the length measured in the frame where this objet is at rest. In this experiment the proper length is L. Thus we an write L = L 0 /γ (2.5) In this equation the lengths are measured along the diretion of relative motion of two inertial frames. Sine, γ > 1 we onlude that L is always shorter than L 0. Hene the name of this effet length ontration. Consider two idential bars. When they are rested one alongside the other they have exatly the same length. Set them in relative motion in suh a way that they are aligned with the diretion of motion (see figure 2.4). In the frame where one bar is at rest the other bar is shorter, and the other way around. At first this may seem ontraditory. However, this is in full agreement with the Priniple of Relativity. In both frames we observe the same phenomenon the moving bar beomes shorter. Moreover, the relativity of simultaneity explains how this an be atually possible. In order to measure the length of the moving bar the observer should mark the positions of its ends simultaneously and then to measure the distane between the marks. This way the observer in the rest frame of the upper bar in Figure 2.4 finds that the length of the lower bar is L = L 0 /γ. However, the observer in the rest frame of the lower bar finds that the positions are not marked simultaneously, but the position of the left end is marked before the position of the right end (Reall the thought experiment 1 in order to verify this onlusion.). As the result, the distane between the marks, L is even smaller than L 0 /γ, in fat the atual alulations give L = L 0 /γ 2 = L/γ, in agreement with results obtained in the frame of the upper bar. Lengths measured perpendiular to the diretion of relative motion of two inertial frames must be the same in both frames. To show this, onsider two idential bars perpendiular to the diretion of motion (see Fig. 2.5). In this ase there is in no need to know the simultaneous positions of

26 CHAPTER 2. BASIC SPECIAL RELATIVITY L = L 0 L = L / 0 V V L = L / 0 L = L 0 Figure 2.4: Two idential bars are aligned with the diretion of their relative motion. In the rest frames of both bars the same phenomenon is observed a moving bar is shorter. V V Figure 2.5: Two idential bars are aligned perpendiular to the diretion of their relative motion. If the bars did not retain equal lengths then the equivalene of inertial frames would be broken. In one frame a moving bar gets shorter, whereas in the other it gets longer. This would ontradit to the Priniple of Relativity. the ends as they do not move in the diretion along whih the length is measured and, thus, the relativity of simultaneity is no longer important. For example, one ould use two strings strethed parallel to the x axis so that the ends of one of the bars slide along these strings (Fig. 2.5). By observing whether the other bar fits between these strings or not one an deide if it is longer or shorter in the absolute sense it does not matter whih inertial observer makes this observation, the result will be the same. Let us say that the right bar in Figure 2.5 is shorter. Then in the frame of the left bar moving bars ontrat, whereas in the frame of the right bar moving bars lengthen. This breaks the equivalene of inertial frames postulated in the Relativity Priniple. Similarly, we show that the Relativity Priniple does not allow the right bar to be longer then the left one. There is no onflit with this priniple only if the bars have the same length. 2.2.4 Synronization of loks Consider a set of standard loks plaed on the arriage dek along the trak. To make sure that all these loks an be used for onsistent time measurements the arriage passengers should synhronise them. This an be done by seleting the lok in the middle to be a referene lok and then by making sure that all other loks show the same time simultaneously with the referene one. One

2.2. EINSTEIN S THOUGHT EXPERIMENTS 27 way of doing this is by sending light signals at time t 0 from the referene lok to all the others. When a arriage lok reeives this signal it should show time t = t 0 + l/, where l is the distane to the referene lok. Now onsider set of standard loks plaed on the platform along the trak. This set an also be synhronised using the above proedure (now the referene lok will be in the middle of the platform). Realling the result of the thought experiment 1 we are fored to onlude that to the platform rowd the arriage loks will appear desynhronised (see fig.2.6) and the other way around to the arriage passengers the platform loks will appear desynhronised (see fig.2.7). That is if the passengers inside the arriage standing next to eah of the arriage loks are asked arriage V platform Figure 2.6: Cloks synhronised in the arriage frame appear desynhronised in the platform frame. to report the time shown by the platform lok whih is loated right opposite to his lok when his lok shows time t 1 their reports will all have different readings, and the other way around. So in general, a set of loks synhronised in one inertial frame will be appear as desynhronised in another inertial frame moving relative to the first one. arriage V platform Figure 2.7: Cloks synhronised in the platform frame appear desynhronised in the arriage frame.

28 CHAPTER 2. BASIC SPECIAL RELATIVITY 2.3 Lorentz transformation In Speial Relativity the transition from one inertial frame to another (in standard onfiguration) is no longer desribed by the Galilean transformation but by the Lorentz transformation. This transformation ensures that light propagates with the same speed in all inertial frames. y ~ y z O O ~ ~ z v x ~ x Figure 2.8: Two inertial frames in standard onfiguration 2.3.1 Derivation The Galilean transformation t = t x = x + v t y = ỹ z = z t = t x = x vt ỹ = y z = z is inonsistent with the seond postulate of Speial Relativity. The new transformation should have the form t = f( t, x, v) t = f(t, x, v) x = g( t, x, v) x = g(t, x, v) (2.7) y = ỹ ỹ = y z = z z = z just beause of the symmetry between the two frames. Indeed, the only differene between the frames S and S is the diretion of relative motion: If S moves with speed v relative to S then S moves with speed v relative to S. Hene omes the hange in the sign of v in the equations of diret and inverse transformations (2.7). Y and z oordinates are invariant beause lengths normal to the diretion of motion are unhanged (see Se.2.2.3). Now we need to find funtions f and g. (2.6) is a linear transformation. Assume that (2.7) is linear as well (if our derivation fall through we will ome bak and try something less restritive). Then Clearly, we should have x = v t for any t if x = 0. Thus, (2.6) x = γ(v) x + δ(v) t + η(v). (2.8) ( vγ + δ) t + η(v) = 0 for any t. This requires η = 0 and δ = vγ.

2.3. LORENTZ TRANSFORMATION 29 Thus, x = γ( x + v t), (2.9) x = γ(x vt). (2.10) In priniple, the symmetry of diret and inverse transformation is preserved both if γ( v) = γ(v) and γ( v) = γ(v). However, it is lear that for x + we should have x + as well. This ondition selets γ( v) = γ(v) > 0. Now to the main ondition that allows us to fully determine the transformation. Suppose that a light signal is fired at time t = t = 0 in the positive diretion of the x axis. (Note that we an always ensure that the standard time-keeping loks loated at the origins of S and S show the same time when the origins oinide.) Eventually, the signal loation will be x = t and x = t. Substitute these into eq.(2.10,2.9) to find Now we substitute t from eq.2.12 into eq.2.11 and derive t = γ t(1 + v/), (2.11) t = γt(1 v/). (2.12) t = γ 2 t(1 v/)(1 + v/) γ = 1/ (1 v 2 / 2 ). (2.13) We immediately reognise the Lorentz fator. In order to find funtion g of Eq.2.7 we simply substitute x from Eq.2.9 into Eq.2.10 and then express t as a funtion of t and x: x = γ[γ x + vγ t vt], γvt = γ 2 v t x(1 γ 2 ). It is easy to show that Thus, and finally (1 γ 2 ) = v 2 γ 2 / 2. γvt = γ 2 v t + v2 γ 2 x 2 t = γ ( t + v ) 2 x. (2.14) Summarising, the oordinate transformations that keep the speed of light unhanged are t = γ( t + (v/ 2 ) x) x = γ( x + v t) y = ỹ z = z t = γ(t (v/ 2 )x) x = γ(x vt) ỹ = y z = z (2.15) They are due to Lorentz(1853-1928) and Larmor(1857-1942). 2.3.2 Newtonian limit Consider the Lorentz transformations in the ase of v. This is the realm of our everyday life. In fat even the fastest rokets fly with speeds muh less than the speed of light. In this limit γ = (1 v 2 / 2 ) 1/2 1

30 CHAPTER 2. BASIC SPECIAL RELATIVITY and the transformation law for the x oordinate redues from to the old good Galilean form x = γ( x + v t) x = x + v t x = γ(x vt) x = x vt. This is why the Galilean transformation appears to work so well. Similarly, we find that the time equation of the Lorentz transformation redues to t = t + (v/ 2 ) x If T and L are the typial time and length sales of our everyday life, then t T and (v/ 2 ) x (v/)(l/). Let us ompare these two terms. If L = 1 km then (v/)(l/) (v/)10 5 s 10 5 s. This is a very short time indeed and muh smaller then the timesale T we normally deal with. Hene, with very high auray we may put t = t. This is what has led to the onept absolute time of Newtonian physis. Formally, the Newtonian limit an be reahed by letting +, or just replaing with +. 2.4 Relativisti veloity addition For v > the Lorentz fator γ = (1 v 2 / 2 ) 1/2 beomes imaginary and the equations of Lorentz transformation, as well as the time dilation and Lorentz ontration equations, beome meaningless. This suggests that is the maximum possible speed in nature. How an this possibly be the ase? If in the frame S we have a body moving with speed w > /2 to the right and this frame moves relative to the frame S with speed v > /2, then in the frame S this body should move with speed w > /2 + /2 =. However, in this alulation we have used the veloity addition law of Newtonian mehanis, w = w + v, whih is based on the Galilean transformation, not the Lorentz transformation! So what does the Lorentz transformation tells us in this regard? 2.4.1 One-dimensional veloity addition Consider a partile moving in the frame S with speed w along the x axis. Then we an write d x d t = w, dỹ d t = d z d t = 0, and dx dt = w, dy dt = dz dt = 0. From the first two equations of the Lorentz transformation (2.15) one has { dt = γ(d t + (v/ 2 )d x). (2.16) dx = γ(d x + vd t) Thus, w = dx dt = d x + vd t d t + (v/ 2 )d x =

2.5. ABERRATION OF LIGHT 31 = d x/d t + v 1 + (v/ 2 )d x/d t = w + v 1 + (v w/ 2 ). The result is the relativisti veloity addition law for one-dimensional motion w = w + v 1 + (v w/ 2 ). (2.17) Obviously, this differs from the Newtonian result, but redues to it when v, w 1. Now let us onsider the numbers v = w = /2. In this ase w but aording to Eq.2.17 w = /2 + /2 1 + 1/4 = 4 5 <! Simple analysis of Eq.2.17 shows that for any < v, w < we obtain < w <. Thus indeed, the speed of light annot be exeeded! Now we an use eqs.(2.17,2.18 in order to verify, that the speed of light is indeed the same in both frames. Substitute w = in eq.2.17 for any v. Next substitute w = w = w = + v 1 + v/ 2 = + v + v = + v 1 v/ 2 = + v v = for any v. In fat, it is easy to show that for any < w < and < v < eq.2.17 gives < w <. This result is onsistent with our earlier onlusion that must be the highest possible speed. The inverse to Eq.2.17 law is the same as eq.2.17 up to the sign of v: w = w v 1 (vw/ 2 ). (2.18) The hange of sign of v in this equation follows diretly from the hange of sign of v in the inverse Lorentz transformation (2.15). (Another way of deriving eq.2.18 is via finding w from eq.2.17) 2.4.2 Three-dimensional veloity addition In Se.2.4.1 we onsidered the motion only along the x axis. If instead we allow the partile to move in arbitrary diretions then using the same method we an derive the following result w x = w x + v 1 + vw x / 2, wy = for the diret transformation and w x = wx v 1 vw x / 2, wỹ = wỹ γ(1 + vw x / 2 ), wz = w y γ(1 vw x / 2 ), w z = w z γ(1 + vw x / 2 ). (2.19) w z γ(1 vw x / 2 ). (2.20) for the inverse one. (Notie again that the only differene in the diret and the inverse equations is the sign of v.) 2.5 Aberration of light A light signal propagates at the angle θ to the x axis of frame S. What is the orresponding angle measured in frame S.

32 CHAPTER 2. BASIC SPECIAL RELATIVITY y ~ y w~ z O O ~ ~ z v ~ ~ x Figure 2.9: We an always rotates both frames about the x-axis so that the signal is onfined to the XOY plane as in figure 2.9. Then the veloity vetor of the signal has the following omponents in frame S and w x = os θ, wỹ = sin θ, w z = 0, w x = os θ, w y = sin θ, w z = 0, in frame S. Substitute these into the first two equations in (2.19) and find that where β = v/ From these one also finds os θ = os θ + β 1 + β os θ Let us analyse the results. Differentiate eq.2.21 with respet to β (2.21) sin sin θ = θ, (2.22) γ(1 + β os θ) sin tan θ = θ. (2.23) γ(β + os θ) sin θ dθ dβ = and the substitute sin θ from eq.2.22 to obtain sin 2 θ (1 + β os θ) 2 dθ dβ = γ sin θ 1 + β os θ. This derivative is negative or zero ( θ = 0, π). Thus, θ dereases with β and the diretion of light signal is loser to the x diretion in frame S. To visualise this effet imagine an isotropi soure of light moving with speed lose to the speed of light. Its light emission will be beamed in the diretion of motion (see fig.2.10) In fat, eq.2.21 shows that in the limit v (β 1) we have os θ 1 and hene θ 0 (with exeption of θ = π whih always gives θ = π). In the Newtonian limit ( ) Equations (2.21-2.23) redue to θ = θ, showing that there is no abberation of light effet.

2.6. DOPPLER EFFECT 33 ~ Frame of the soure, S Observer's frame, S v Figure 2.10: The effet of aberration of light on the radiation of moving soures. If in the rest frame of some light soure its radiation is isotropi (left panel) then in the frame where the soure is moving with relativisti speed its radiation is beamed in the diretion of motion (right panel). 2.6 Doppler effet The Doppler effet desribes the variation of periodi light signal due to the motion of emitter relative to reeiver. 2.6.1 Transverse Doppler effet Problem setup: A soure (emitter) of periodi light signal (e.g. monohromati wave) moves perpendiular to the light of sight of some observer with speed v. Find the frequeny ν of the signal reeived by the observer if in the soure frame the frequeny of emitted light is ν 0. v reeiver emitter Figure 2.11: Solution: T 0 = 1/ν 0 is the period of the emitter in its rest frame. Suppose, that in the frame of the emitter this signal is emitted during the time t 0 = NT 0, where N is the number of produed wave rests. By definition, both these times are the proper times of the emitter. In the frame of reeiver the emission proess takes a longer time, t e, given by the time dilation formula t e = γ t 0. Assuming that the distane between the emitter and the reeiver is so large that its hange during the emission is negligible, the signal is reeived during the same time interval of the same length, t r = t e. Sine the number of rests remains the same, the period of the reeived light is

34 CHAPTER 2. BASIC SPECIAL RELATIVITY Thus, the frequeny of the reeived signal is T r = t r /N = γt 0. (2.24) ν = 1 T r = 1 γt 0 = ν 0 γ. (2.25) Sine γ > 1 we have ν < ν 0. One an see that the transverse Doppler effet is entirely due to the time dilation. 2.6.2 Radial Doppler effet Problem setup: Now we onsider the ase where the soure is moving along the line of sight and so the distane between the emitter and the reeiver is inreasing during the proess of emission. This has an additional effet on the frequeny of the reeived signal. v reeiver emitter Figure 2.12: Solution: The emission time as measured in the frame of the observer is still given by the time dilation formula t e = γ t 0. During the time t e the emitter moves away from the observer by the distane l = t e v. This leads to the inrease in the period of reeption time by l/. Thus, the total reeption time in this ase is t r = t e + t ev 1 + β = t 0 γ(1 + β) = t 0 1 β. This yields the observed period of the reeived signal T r = t r N = t 0 1 + β N 1 β = T 1 + β 0 1 β, and the observed frequeny ν = ν 0 1 β 1 + β. (2.26) Notie that in this ase ν < ν 0 if v > 0 (the emitter is moving away from the reeiver) and ν > ν 0 if v < 0 (the emitter is approahing the reeiver).

2.6. DOPPLER EFFECT 35 v reeiver emitter Figure 2.13: 2.6.3 General ase Problem setup: In this ase, the soure is moving at angle θ with the line of sight. This inludes the transverse (θ = π/2) and the radial ases (θ = 0, π). Solution: The only differene ompared to the radial ase is in the rate of inrease of the distane between the emitter and the reeiver. Now l = t e v os θ, where we assume that v > 0 and that the emitter is far away from the reiever. In order to see this, onsider the triangle with sides l 0, l, and v t e, where l 0 is the distane between the emitter and the reiever at the beginning of emission, l is this distane at the end of emission and v t e is the distane overed by the emitter during the time of emission. From this triangle, l 2 = l 2 0 + (v t e ) 2 2l 0 v t e os θ, or ( ( ) ( ) ) 2 l 2 = l0 2 v te v te 1 2 os θ +. l 0 l 0 For a distant emitter v t e /l 0 1 and we have ( ( )) l 2 l0 2 v te 1 2 os θ. l 0 Moreover, ( )) 1/2 ( ( )) v te v te l l 0 (1 2 os θ l 0 1 os θ = l 0 os θv t e. l 0 l 0 and Repeating the alulations of the radial ase we then find that T r = T 0 γ (1 β os θ), ν = ν 0 γ (1 β os θ). (2.27) Now it is less lear whether ν > ν 0 or otherwise, and further analysis is required. In the Newtonian limit ( ) one has ν = ν 0, so no effet is seen. In the ase of β 1, the Taylor expansion of Eq.2.27 gives ν = ν 0 (1 + β os θ + O(β 2 )).

36 CHAPTER 2. BASIC SPECIAL RELATIVITY

Chapter 3 Spae-time 3.1 Minkowski diagrams The differene between the Newtonian and relativisti visions of spae and time is niely illustrated with the help of Minkowski diagrams. The idea is to represent instantaneous loalised events as points of a two dimensional graph. Surely, some information is lost beause of we really need four dimensions to fully represent spae time events. So what is atually shown is time and one of the spatial oordinates (usually x) as measured in some inertial frame, and it is assumed that other oordinates of displayed events vanish, y = z = 0. The left panel of Fig.3.1 is a Newtonian version of Minkowski diagram. Along the horizontal axis we show the x oordinate of some inertial frame (frame S) and along the vertial axis its time multiplied by the speed of light (t has the dimension of length, just like x). Any two events, like A and B in this figure, that belong to a line parallel to the time axis, are simultaneous in this frame. The whole suh line represents all real and potential events ourring at the same time t = t 0 and shows the spatial separation between them. One may say that it represents the spae at time t = t 0. Any partile is represented by a line on this diagram, alled the world line of the partile. For example, the time axis is the world line of the frame s origin. The world line of a partile moving with onstant speed along the x axis of the frame is also straight but now it is inlined to the time axis. The inlination angle varies between 0 and π/2, depending on the partile speed. Consider another inertial frame (frame S) whih has the same orientation of axes and moves along the x-axis with onstant speed v. (As always, we also assume that the origins of both frames oinide at time t = t = 0, where t is the time of the frame S. The line t is the world line of the origin of the frame S it an be onsidered as the time axis of this frame. The angle between the t and t axes follows from the equations of motion of the origin x = βt, where β = v/. From this we have tan θ = x = β. (3.1) t Aording to the Galilean transformation, the events A and B, shown in the left panel of Fig.3.1, are also simultaneous in the frame S and our at exatly the same time, t = t 0. Thus, the spae of this frame at time t = t 0 is the same as the spae of the frame S at time t = t 0. Both spaes are represented by the same line parallel to the x axis. For the same reason the x axis oinides with the x axis in the diagram - they show the spae at t = t = 0. The right panel of Fig.3.1 is a Minkowski diagram of speial relativity for the same ombination of two inertial frames. One obvious differene is that x axis does no longer oinide with the x axis. Indeed, the axis shows all events that our at time t = 0. They are simultaneous in the moving frame ( S). However, they are not simultaneous in the stationary frame (S) and thus annot be represented by the x axis of the diagram. To find the equation of the x axis we should use the equation of Lorentz transformation 37

38 CHAPTER 3. SPACE-TIME A B t ~ t spae at time t=t 0 A B t ~ t ~ ~ spae at time t=t 0 spae at time t=t 0 ~ ~ O spae at time t=t 0 x Newtonian version ~ x O Relativisti version ~ x x Figure 3.1: Newtonian (left panel) and Minkowskian (right panel) visions of spae and time. Putting t = 0 we find t = γ(t βx). t = βx, whih is the equation of a straight line passing through the origin at the angle θ = artan β (see Eq.3.1) with the x axis. This line shows the spae of frame S at the time t = 0, the set of all atual and potential events at time t = 0 and their spaial separation. Similarly, one dedues that the spae of frame S at t 0 is represented by a line parallel to the x axis. It is obvious from this diagram that these two frames have different spaes and different times! Sine β < 1 we have θ < π/4. Thus, the time and the x axis of frame S never merge, with the time axis staying above and the x axis below the line t = x whih is the world line of a light signal sent from the origin at time t = 0. 3.2 Spae-time In Newtonian physis the physial spae Eulidean. By definition, a spae is Eulidean if one an introdue suh oordinates x, y, z that its metri form beomes dl 2 = dx 2 + dy 2 + dz 2 (3.2) at every point of the spae. Suh oordinates are alled Cartesian oordinates. There are infinitely many different Cartesian oordinates. For example, one an obtain one system of Cartesian oordinates from another via rotation about the z axis by angle θ: x = x os θ + y sin θ, ỹ = x sin θ + y os θ, z = z. This transformation keeps the metri form invariant; dl 2 = d x 2 + dỹ 2 + d z 2. (3.3) In Newtonian Physis it is assumed that the lengths of objets and distanes between them do not depend on the seletion of inertial frame where the length measurements are arried out. The geometry of Physial Spae is reassuringly the same for every one. In Speial Relativity the lengths of objets and distanes between them do depend on the inertial frame used for the length measurements. The metri form (3.2) is not invariant under the Lorentz transformation. The

3.2. SPACE-TIME 39 geometry of the Physial Spae is no longer reassuringly fixed. In mathematial terms the Physial Spae an no longer be modelled as a metri spae. This almost amounts to saying that the Physial Spae is not quite a proper physial reality. Einstein s teaher at Zurih Polytehni, professor Minkowski (1864-1909), disovered that the following generalised metri form ds 2 = 2 dt 2 + dx 2 + dy 2 + dz 2 (3.4) is in fat invariant under the Lorentz transformations. Obviously dy 2 + dz 2 = dỹ 2 + d z 2, so we only need to show that 2 dt 2 + dx 2 = 2 d t 2 + d x 2. The alulations are most onsize when we use the Lorentz transformation written in terms of hyperboli funtions t = t osh θ + x sinh θ, (3.5) x = x osh θ + t sinh θ. To see that the Lorentz transformation an indeed be written this way, we first write it as t = γ( t + β x) x = γ( x + β t), (3.6) and then introdue θ via sinh θ = γβ, from whih it follows that γ = osh θ. From the first two equations in (3.5) we have Then dt = d t osh θ + d x sinh θ, dx = d x osh θ + d t sinh θ. 2 dt 2 + dx 2 = (d t osh θ + d x sinh θ) 2 + (d x osh θ + d t sinh θ) 2 = 2 d t 2 osh 2 θ 2d xd t osh θ sinh θ d x 2 sinh 2 θ+ + 2 d t 2 sinh 2 θ + 2d xd t osh θ sinh θ + d x 2 osh 2 θ = 2 d t 2 (osh 2 θ sinh 2 θ) + d x 2 (osh 2 θ sinh 2 θ) = = 2 d t 2 + d x 2. To make the last step we use the well known result osh 2 θ sinh 2 θ = 1. Thus, the quadrati form (3.4) is indeed Lorentz invariant and this fat suggests that spae and time should be united into a single 4-dimensional metri spae with metri form (3.4)!!! Like in the ase of Minkowski diagrams points of this spae are alled events. This spae is alled the spae-time or the Minkowski spae-time. ds 2 is the generalised distane between points of this spae it is alled the spae-time interval. Denoting x 0 = t, x 1 = x, x 2 = y, x 3 = z (3.7) we an rewrite eq.3.4 as ds 2 = (dx 0 ) 2 + (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2. (3.8) Given the similarity of this metri form with the metri form of Eulidean spae in Cartesian oordinates the spae-time of Speial relativity is often alled pseudo-eulidean (where pseudo refers to the sign - in front of the first term) and the oordinates (3.7) are often alled pseudo-cartesian. Thus, there is a one-to-one orrespondene between inertial frames with their time and Cartesian grid and systems of pseudo-cartesian oordinates in spae-time! The Lorentz transformation an now be onsidered as a transformation from one system of pseudo-cartesian oordinates to another system of pseudo-cartesian oordinates in spae-time. Minkowski diagrams are simply maps of spae-time onto a 2D Eulidean plane of a paper sheet. Notie that the usual Eulidean distanes between the points on these diagrams do not reflet the generalised distanes between the orresponding events in spae-time.

40 CHAPTER 3. SPACE-TIME In arbitrary urvilinear oordinates the spae-time metri beomes ds 2 = g µν dx µ dx ν, (3.9) where the oeffiients g µν are quite arbitrary as well. They an be written as omponents of a 4 4 matrix g 00 g 01 g 02 g 03 g 10 g 11 g 12 g 13 g 20 g 21 g 22 g 23 g 30 g 31 g 32 g 33 Here and in the rest of the notes we will assume that Greek indexes run from from 0 to 3 whereas Latin indexes run from 1 to 3. One an always impose the symmetry ondition on the metri form g µν = g νµ, (3.10) and this is what we will always assume here. In pseudo-cartesian oordinates g 00 = 1, g 11 = g 22 = g 33 = 1, and g µν = 0 if µ ν, (3.11) or in the matrix form g µν = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (3.12) 3.3 Light one x 0 t /4 A /4 A x 1 x 2 x Figure 3.2: Left panel: The light one of event A. Right panel: Appearane of the light one in Minkowski diagram. Selet some event, say event A, in spae-time. Let x ν A to be the pseudo-cartesian oordinates of this event. Consider the surfae determined by s 2 = 0

3.4. CAUSAL STRUCTURE OF SPACE-TIME 41 or ( x 1 ) 2 + ( x 2 ) 2 + ( x 3 ) 2 = ( x 0 ) 2, (3.13) where x ν = x ν x ν A. This surfae is alled the light one of event A. It is made out of all events that are onneted to A by world lines of light signals. It is easy to see that this is indeed a fourdimensional one whose half-opening angle is π/4 = 45 o, whose symmetry axis is parallel to the x 0 axis, and whose tip is loated at A (see the left panel of fig.3.2). t < t A t = t A t > t A A Figure 3.3: In 3-dimensional physial spae of any inertial frame the lower half of the light one of event A orresponds to a spherial light front onverging of the loation of event A (left panel, t < t A ), and its upper half orresponds to a spherial light front diverging from the loation of this event (right panel, t > t A ). In Minkowski diagrams the light one of event A gives two straight lines passing through A at the angle of π/4 to the x 0 axis (time axis). They are the world lines of two light signals passing trough A (see the right panel of fig.3.2). In the physial spae of any inertial frame the bottom half of the light one orresponds to a spherial light front onverging onto the spaial loation of A (t t A ) = (x x A ) 2 + (y y A ) 2 + (z z A ) 2 (3.14) and its upper half orresponds to a spherial light front diverging from the spaial loation of A (t t A ) = + (x x A ) 2 + (y y A ) 2 + (z z A ) 2 (3.15) (see fig.3.3). Due to the length ontration effet shapes and sizes of physial objets as well as distanes between them differ in different inertial frames. However, the shape and size of these light fronts remain the same, being desribed by eqs.(3.14,3.15) in all inertial frames. Consider a partile travelling with onstant veloity in some inertial frame. Then its world line is a straight line. Provided that it passes through A its equation is (x x A ) 2 + (y y A ) 2 + (z z A ) 2 = v 2 (t t A ) 2 < 2 (t t A ) 2. Thus, suh a world line is loated inside the light one of A (see fig.3.4). 3.4 Causal struture of spae-time Physial objets interat with eah other by means of waves or partiles they produe. Let us all these interation agents as signals. If an event A is aused or triggered by some other event B then this is arranged via a signal that B sends to A and that reahes the loation of A before A ours. Sine the signal speed must be less or equal than the speed of light its world line is inlined at an angle to the time axis that is less than or equal to π/4. Thus the event B must lie inside the lower half of the light one of event A (see the left panel of fig.3.5). On the other hand, the world lines

42 CHAPTER 3. SPACE-TIME x 0 /4 A x 1 x 2 Figure 3.4: The world line of any physial objet that passes through A lies inside the light one of A. of all possible signals sent by the event A fill the upper half of the light one of A. Thus, the events that may be aused by A are loated inside the upper half of the light one of A (see the left panel of fig.3.5). As to the events that lie outside of the light one of A they an neither ause not be aused by A they are ausally unrelated to A. x 0 events ausally unrelated to A events that may be aused by A A x 0 events that may our before or after A depending on the inertial frame events in the absolute future of A A x 0 events that in some inertial frame our at the same loation as A events that in any inertial frame our A at a different loation than A x 1 events that may have aused A x 2 x 1 events in the absolute past of A x 2 x 1 events that in some inertial frame our at the same loation as A x 2 Figure 3.5: The light one of event A, and the ausal, spatial, and temporal onnetion of this event to all other events. Consider an event B inside the light one of A. Construt a straight line onneting A and B. Sine this line is insight the light one it is inlined at an angle less than π/4 with the time axis and an be a world line of some physial objet. For example this an be a grid point of some inertial frame. In this frame A and B have the same spaial loation. For any event C loated outside the light one of A the line onneting A and C annot be a world line of any physial objet as this would imply motion with speed exeeding the speed of light. Thus, in any inertial frame A and C will have different spaial loations (see the right panel of fig.3.5). Comparing this with the asual struture of spae time we onlude that the events that an be asually onneted are also the events that an have the same loation in some inertial frame. On the ontrary, the events that annot be ausally onneted are spatially separated in all inertial frames.

3.5. TYPES OF SPACE-TIME INTERVALS 43 The asual struture is also related with the temporal order of events. The events that an be aused by A are in the future of A (t > t A ) in all inertial frames and the events that ould ause A are in the past of A (t < t A ) in all inertial frames (This is beause θ in the right panel of Figure 3.1 annot exeed π/2.). Thus, the events that have unambiguous temporal loation with respet to A are all inside the light one of A. On the other, hand the events that annot be ausally onneted to A (they are outside of the light one of A) an preeed A in some frames and follow it in other frames (See Figure 3.5). 3.5 Types of spae-time intervals For any event B that lies outside of the light one of event A the spae-time interval separating these two events is positive, 3 s 2 AB = ( x 0 AB) 2 + ( x i AB) 2 > 0, (3.16) where x ν AB = xν B xν A. Suh intervals are alled spae-like. It is easy to show that there always exists an inertial frame where A and B are simultaneous and s 2 AB = i=1 3 ( x i AB) 2. i=1 In some inertial frames moving relative to this one A ours before B, in others B ours before A. Thus, the temporal order of A and B depends on the hoie of inertial frame. For any event B on the surfae of the light one of event A the spae-time interval separating these two events is zero, s 2 AB = 0. (3.17) Suh intervals are alled null. For any event B that lies inside the light one of event A the spae-time interval separating these two events is negative, 3 s 2 AB = ( x 0 AB) 2 + ( x i AB) 2 < 0. (3.18) Suh intervals are alled time-like. It is easy to show that there always exists an inertial frame where A and B are separated only in time but not in spae and hene i=1 s 2 AB = ( x 0 AB) 2. (3.19) If B lies inside the upper half of the light one of A then in any inertial frame A preedes B and otherwise if B lies inside the lower half of the light one of A then in any inertial frame B preedes A. To see we use the Minkowski diagram (figure 3.6) that shows A, B and the axes of two inertial frames entred on the event A (this an always be done simply by resetting the zero time and hanging the loation of origin.). As we have already seen the angle θ is always less than π/4. Sine the lines of simultaneity of frame S also make angle θ with the x axis the event A preedes the event B in this frame as well as in frame S. One an say that B belongs to the absolute future of A. Similarly, we see that any event C that lies inside the lower half of the light one of A always preedes A and thus belongs to the absolute past of A. These temporal properties are summarised in the middle panel of fig.3.5.

44 CHAPTER 3. SPACE-TIME x 0 x 0~ B ~ x 1 A x 1 C Figure 3.6: 3.6 Vetors The visualisation/interpretation of vetors as straight arrows whose length desribes the vetor magnitude works fine for Eulidean spae but is not fully suitable for urved manifolds (e.g. surfaes in Eulidean spae) and spae-time. To see the shortomings of this interpretation in spae-time onsider the light one of event A and a straight line on the surfae of this one (see fig.3.7). Take two points, B and C, on this line. They seem to define two different vetors AB and AC. However, the generalised distanes (squared) between these points are equal to zero s 2 AB = s 2 AC = 0 and so are the generalised lengths of the arrows. Thus, arrows AB and AC have the same lengths and the same diretions! Moreover, within this interpretation it is impossible to address the differene between vetors that onnet points with spae-like separation and vetors that onnet points with time-like separation. Below we give the modern generalised definition of geometri vetors due to Cartan, whih is very robust and suits very well the Theory of Relativity, both Speial and General. 3.6.1 Definition Consider a point (we denote it as A) in spae-time (in fat this theory applies to any other kind of spae or manifold) and a urve (a one-dimensional ontinuous string of points) passing through this point. Let λ to be a parameter of this urve (oordinate of its points). This parameter defines a diretional derivative d/dλ at point A. Indeed, if F is a funtion defined on this spae (spae-time, or manifold) then on this urve it is a funtion of λ and an be differentiated with respet to it: F = F (λ) and F = df/dλ. Next introdue some oordinates in the spae-time {x ν }. Now the urve an be desribed by the funtions x µ = x µ (λ). Moreover, F = F (x ν ) and an be differentiated with respet to x ν as well. Aording to the hain rule df dλ = dxν dλ F x ν, where we applied the Einstein summation onvention. Thus, d dλ = dxν dλ x ν. (3.20)

3.6. VECTORS 45 x 0 C B x 1 /4 A x 2 Figure 3.7: Figure 3.8: It is easy to verify that all diretional derivatives defined in this way at point A form a 4-dimensional abstrat vetor spae, whih we will denote as T A. The set of operators { } x ν, ν = 0,..., 3 is alled the oordinate basis in T A, indued by the oordinates {x ν }, and dx ν, ν = 0,..., 3 dλ are the omponents of the diretional derivative d/dλ in this basis. These diretional derivatives are identified with vetors in modern geometry. They are often alled Cartan s vetors. d/dλ is also alled the tangent vetor to the urve with parameter λ at point A. As it has been stressed this definition is very robust. It an be applied to the Eulidean spae of Newtonian physis equally well. To see this onsider the trajetory of a partile in Eulidean spae and parametrise it using the Newtonian absolute time, t. The diretional derivative d/dt is in fat the veloity vetor of this partile, v = d dt. Indeed, d dt = dxi dt x i

46 CHAPTER 3. SPACE-TIME and thus the omponents of v in the oordinate basis are v i = dxi dt, whih is the familiar result of Partile Kinematis. 3.6.2 Operations of addition and multipliation The operations of addition and multipliation by real number for vetors in T A are defined in terms of their omponents. Let Then we say that and a = a ν x ν, b = b ν x ν, = ν x ν, α R1. = a + b iff ν = a ν + b ν (3.21) = α a iff ν = αa ν. (3.22) These definitions ensure all the familiar properties of vetor addition and multipliation (ommutative law, assoiative law et.) Notie that we use arrow to indiate 4-dimensional vetors of spae-time, e.g. a, b, whereas underline is reserved for 3-dimensional usual vetors of spae, e.g. a, b. 3.6.3 Coordinate transformation Introdue new oordinates, {x ν }, in spae-time. Obviously, x ν = x ν (x µ ) as well as x µ = x µ (x ν ) The transformation from {x ν } to {x ν } delivers the transformation matrix A ν µ = x ν x µ. (3.23) We agree to onsider the upper index in this expression as the row index and the lower one as the olumn index. The inverse transformation, from {x ν } to {x ν }, delivers another transformation matrix xµ A µ ν =. (3.24) x ν These two matries are inverse to eah other. Indeed their multipliation results in the unit matrix Here δ ν β is Kroneker s delta symbol: A ν µa µ β x ν x µ x ν = x µ = = x β x β δ ν β. δ ν β = { 1 if ν = β 0 if ν β. The new oordinates will introdue new oordinate basis at every point of spae-time and in this new basis vetors will have different omponents. First we find the transformation of the oordinate basis of Cartan s vetors. Aording to the hain rule f x ν = f x µ x µ x ν.

3.6. VECTORS 47 Thus, x ν = A µ ν. (3.25) x µ Similarly one finds = Aν µ x µ x ν. (3.26) Notie that in these expressions we automatially selet the orret transformation matrix when we apply the Einstein summation rule. Now we an find the transformation law for the omponents of vetors. On one hand u = u ν x ν = uνa µ ν µ. On the other hand u = u µ µ. Comparing these two results we immediately obtain Similarly we find u µ = A µ ν u ν. (3.27) u ν = A ν µu µ. (3.28) In these equations, we automatially selet the orret transformation matrix when we apply the Einstein summation rule. In the old fashion textbooks on the Theory of Relativity, it is the transformation laws 3.27 and 3.28 that define vetors (or to be more preise the so-alled ontravariant vetors). 3.6.4 Infinitesimal displaement vetors Consider a urve with parameter λ and its tangent vetor d/dλ at the point with oordinates x ν. The infinitesimal inrement of parameter due to infinitesimal displaement along the urve dλ is a salar by whih we an multiply the tangent vetor and obtain the infinitesimal displaement vetor The omponents of this vetor in the oordinate basis an be found via ds = dλ d dλ. (3.29) ds ν = dλ dxν dλ = dxν. (3.30) Thus, the omponents of ds are the infinitesimal inrements of oordinates dx ν orresponding to Figure 3.9: the infinitesimal inrement of the urve parameter λ, ds = dx ν x ν. (3.31)

... 48 CHAPTER 3. SPACE-TIME 3.7 Tensors 3.7.1 Definition In brief, tensors defined at point A are linear salar operators ating on vetors from T A. In other words, a tensor is a linear salar funtion of, in general, several vetor variables. The number of vetors the tensor needs to be fed with in order to produe a salar is alled its rank. vetor 1 vetor 2 tensor of rank m salar vetor m Figure 3.10: Let, for example, T to be a seond rank tensor defined at point A, u, v, w T A, and α, β R 1. Then aording to the definition T ( u, v ) R 1 ; (3.32) 3.7.2 Components of tensors T (α u + β w, v ) = αt ( u, v ) + βt ( w, v ); (3.33) T ( v, α u + β w ) = αt ( v, u ) + βt ( v, w ). (3.34) Components of tensors are defined via their ation on the basis vetors. For example ( ) T νµ = T x ν, x µ. The ation of tensors on vetors an be fully desribed in terms of their omponents. For example, Thus, T ( u, v ) = T (u ν x ν, vµ x µ ) = uν v µ T ( x ν, x µ ) = uν v µ T νµ. T ( u, v ) = u ν v µ T νµ. (3.35) This is often desribed as a ontration of T νµ with u ν (over index ν) followed by a ontration with v µ (over index µ). 3.7.3 Coordinate transformation Introdue new oordinates {x µ }. The new omponents T ν µ of tensor T are ( ) ( ) T ν µ = T x ν, x µ = T A α ν x α, Aβ µ x β = ( ) A α ν A β µ T x α, x β = A α ν A β µ T αβ

3.8. METRIC TENSOR 49 Thus, T ν µ = A α ν A β µ T αβ (3.36) Note that the transformation matrix A appears one per eah index of T. One of the indexes of A is always the summation index (also known as dummy or ontration index) this is α in the first transformation matrix in eq.3.36 and β in the seond one. Whether this index is upper or lower one is determined by the Einstein summation onvention in general it must be the opposite ase to that of the transformed tensor ontrated with A. The other index (known as free index beause it is not involved in the ontration) is the same both in ase and letter as the orresponding index of T on the left-hand side this is ν in the first transformation matrix in eq.3.36 and µ in the seond one. 3.8 Metri tensor 3.8.1 Definition Consider the infinitesimal displaement vetor onneting points x ν and x ν + dx ν. A seond rank tensor g whih is symmetri, Figure 3.11: g( u, v ) = g( v, u ), (3.37) and when ating on ds produes the generalised distane between the points x ν and x ν + dx ν g( ds, ds) = ds 2 (3.38) is alled the metri tensor. Sine the omponents of ds are dx ν we have ds 2 = g νµ dx ν dx µ, (3.39) where g νµ are the omponents of the metri tensor. We immediately reognise this as the metri form. The metri tensor is also used to define the salar produt of vetors and their magnitudes, ( u v ) = g( u, v ) (3.40) and In omponents these results read and u 2 = g( u, u ). (3.41) ( u v ) = g νµ u ν v µ (3.42) u 2 = g νµ u ν u µ. (3.43)

50 CHAPTER 3. SPACE-TIME 3.8.2 Classifiation of spae-time vetors and other results speifi to spae-time The theory desribed above in Setions 3.2-3.5 is very general and does not use properties speifi to spae-time. Here we begin to apply this theory to spae-time. Sine the spae-time of Speial Relativity is four dimensional salars, vetors and tensors of spae-time are often alled 4-salars, 4-vetors and 4-tensors. In ontrast the usual 3-dimensional salars, vetors and tensors are alled 3-salars, 3-vetors and 3-tensors. A spae-time vetor a is alled time-like if a 2 < 0, spae-like if a 2 > 0 null if a 2 = 0 To ompute a 2 we have to use eq.3.43. The alulations are easiest in pseudo-cartesian oordinates {x ν } as in these oordinates 1 if ν = µ = 0; g νµ = +1 if ν = µ = 1, 2, 3; 0 if ν µ. and hene a 2 = g νµ a ν a µ = 3 g νν a ν a ν = (a 0 ) 2 + (a 1 ) 2 + (a 2 ) 2 + (a 3 ) 2. (3.44) ν=0 For example, onsider spae-time vetor a = (1, 1, 0, 0) whose omponents are those in the oordinate basis of some pseudo-cartesian oordinates. Then a 2 = 1 2 + 1 2 + 0 2 + 0 2 = 0 and thus this vetor is null. In fat all sorts of alulations also easiest in pseudo-cartesian oordinates. For example, the salar produt of two vetors is simply ( a b ) = g νµ a ν b µ = a 0 b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3. (3.45) What are the magnitudes of pseudo-cartesian basis vetors and the angles between them? This is easy to answer as 2 ( ) x ν = g x ν, x ν = g νν. Thus / x 0 is a unit time-like vetor as x 0 and / x i (i = 1, 2, 3) are unit spae-like vetor as Similarly we find that 2 x i = g 00 = 1, (3.46) 2 = g ii = 1. (3.47) ( ) x ν x µ = g νµ = 0 if ν µ, and thus these unit vetors are orthogonal to eah other.

3.8. METRIC TENSOR 51 Finally, we onstrut the transformation matrix of Lorentz transformations. In the ase of pseudo-cartesian oordinates the Lorentz transformation reads x 0 = γ(x 0 βx 1 ) x 1 = γ(x 1 βx 0 ) x 2 = x 2. (3.48) x 3 = x 3 From this we find that and thus A µ ν = dx 0 = γ(dx 0 βdx 1 ) dx 1 = γ(dx 1 βdx 0 ) dx 2 = dx 2 dx 3 = dx 3 γ βγ 0 0 βγ γ 0 0 0 0 1 0 0 0 0 1 (3.49). (3.50) Thus, pseudo-cartesian omponents of any 4-vetor transform aording to the same law as eq.3.48: a 0 = γ(a 0 βa 1 ) a 1 = γ(a 1 βa 0 ) a 2 = a 2 (3.51) a 3 = a 3 In fat eq.3.49 is a partiular example of this rule as dx ν are the omponents of infinitesimal displaement vetor ds.

52 CHAPTER 3. SPACE-TIME

Chapter 4 Relativisti partile mehanis 4.1 Tensor equations and the Priniple of Relativity The Einstein s Priniple of Relativity requires all laws of Physis to be the same in all inertial frames and hene to be invariant under the Lorentz transformation. Sine the laws of Newtonian physis are Galilean invariant they annot be Lorentz invariant and have to be modified. How an this be done? Before, we do this we should try to answer the related question: How to write Lorentz invariant equations? The great value of the onept of spae-time and of the tensor alulus in the Theory of Relativity is that they suggest a straightforward way of onstruting Lorentz invariant equations. Indeed, the Lorentz transformation, whih in physial terms desribes the transformation between two inertial frames in standard onfiguration, orresponds to a partiular type of oordinate transformations in spaetime, namely a sub-lass of transformations between two systems of pseudo-cartesian oordinates. However, vetor equations involving spae-time vetors like a = m b, (4.1) where m is a spae-time salar, are fully meaningful without any use of oordinates, and hene fully independent on the hoie of oordinates in spae-time. One we introdue some oordinates we an write the representation of this equations in terms of omponents of vetors a and b a ν = mb ν (ν = 0, 1, 2, 3). (4.2) Clearly, the form of equation (4.2) is the same for any hoie of oordinates in spae-time and hene this equation is Lorentz invariant too (We also note that the left and the right hand sides of eq.4.2 transform in exatly the same way; see eq.3.27.). The same applies to all proper tensor equations in general, like G νµ = at νµ. However, the detailed study of tensor operations is out of the sope of this ourse and we will deal only with vetor equations. Thus, we now know how to onstrut Lorentz invariant equations. But we also need to figure out how to relate suh equations with the orresponding equations of Newtonian physis. The Newtonian equations do not omply with the Einstein s Relativity Priniple but they are very aurate when the involved speeds are muh lower ompared to the speed of light. Thus, the relativisti equations must redue to the Newtonian ones in the limit of low speeds. In order, to ompare the relativisti and Newtonian equation we need to find out how the 4-vetor equations an be redued to the 3-vetor and 3-salar equations harateristi for Newtonian Physis. Equations like eq.4.2 differ from the vetor equations of lassial physis in that they have not 3 but 4 omponents. The following simple analysis shows that any 4-vetor equation orresponds to a pair of traditional Newtonian-like equations, one them being a 3-salar equation and the other one a 3-vetor equation. Indeed, let us selet an arbitrary inertial frame. This frame introdues 53

54 CHAPTER 4. RELATIVISTIC PARTICLE MECHANICS hyper-surfaes ( hyper beause they are 3-dimensional) of simultaneity, t = t 0, in spae-time. Eah suh hypersurfae is the spae of this inertial frame at a partiular time. (Another inertial frame introdues different hypersurfaes of this sort and has different spae and time.) Now onsider a transformation of spaial oordinates (x i, i = 1, 2, 3) in the spae of this frame. The transformation matrix of this transformation is Aĩj = xĩ x j, (i, j = 1, 2, 3). Quantities that remain invariants under suh transformations are the usual spaial salars (or 3- salars). The omponents of usual spaial vetors (or 3-vetors) vary aording to the vetor transformation law aĩ = Aĩja j (i = 1, 2, 3). This transformation an also be viewed as a transformation of spae-time oordinates but of a rather speifi sort, namely x 0 = x 0, xĩ = xĩ(x j ), (i, j = 1, 2, 3). Thus, and so we have A 0 0 = 1, A 0 i = 0, Aĩ0 = 0, (4.3) A µ ν = 1 0 0 0 0 A 1 1 A 1 2 A 1 3 0 A 2 1 A 2 2 A 2 3 0 A 3 1 A 3 2 A 3 3. (4.4) Next we apply this transformation law to a spae-time vetor. Whereas in general a µ = A µ ν a ν in our ase we have and a 0 = a 0 (4.5) aĩ = Aĩja j (i = 1, 2, 3). (4.6) Thus, the a 0 is invariant and hene behaves as a 3-salar, whereas a i behave as omponents of a 3-vetor. This result is summarised in the following notation a = (a 0, a). (4.7) Thus, any 4-vetor equations splits into a 3-salar equation and a 3-vetor equation. For example, the 4-vetor equation a = m b splits into the 3-salar equation and the 3-vetor equation a 0 = mb 0 a = mb. This result shows that Newtonian physis missed important onnetions between various physial quantities and between various equations. These are unovered in Speial Relativity.

4.2. 4-VELOCITY AND 4-MOMENTUM 55 4.2 4-veloity and 4-momentum Before we attempt to write relativisti versions of the most important laws of partile dynamis, we need to introdue relevant 4-salars and 4-vetors. The 3-veloity vetor of Newtonian physis is the tangent vetor along the partile trajetory with the absolute time as a parameter v = d dt (v i = dx i /dt) (see the end of Se.3.2.1). In the same manner we an introdue the 4-veloity vetor of a partile. Let us see how this is done. First instead of a urve in spae, whih is the partile trajetory, we need a urve in spae-time. The only obvious hoise is the world-line of the partile. Seond, we need a time-like parameter along this urve. However, we no longer have the absolute time and eah inertial frame has its own time. The only natural hoie whih gives us a unique parameter is the proper time of the partile, τ, whih is the time as measured by a fiduial standard lok moving with this partile (the omoving lok). From this definition, it follows that τ is a 4-salar. Indeed, it is defined without any referene to a oordinate system in spae-time. Thus, the 4-veloity of a partile an be defined as the diretional derivative d u = (4.8) dτ along the world line of this partile. One a oordinate system is introdued for the spae-time, we an find the omponents of u in the orresponding oordinate basis, u ν = dxν dτ. (4.9) Sine dx ν are also the omponents of the infinitesimal displaement 4-vetor ds along the world line, in the same basis, we an also write u = ds dτ. (4.10) This an be understood as the 4-vetor ds divided by the 4-salar dτ, the orresponding inrement of the proper time. Consider an arbitrary inertial frame with arbitrary spatial oordinates x i and the normalised time oordinate x 0 = t, where t is the time of this inertial frame. Then all the basis 4-vetors / x i are orthogonal to / x 0 but in general they are not unit 4-vetors. Whereas the basis 4-vetor is unit. In this basis, x 0 = 1 t u 0 = dx0 dτ = dt dτ. Sine dt, dτ, and the partile Lorentz fator Γ are related via the time dilation formula we an write dt = Γdτ, u 0 = Γ. As to the other omponents, u i = dxi dτ = dt dx i dτ dt = Γvi, where v i = dx i /dt are the omponents of the 3-veloity vetor of this partile in this inertial frame in the oordinate basis of its spae. These results are summarised using the following notation u = (Γ, Γv). (4.11)

56 CHAPTER 4. RELATIVISTIC PARTICLE MECHANICS Notie that Γ is a 3-salar. (The most frequently used notation for the Lorentz fator is γ. However, in these notes we reserve γ speifially for the Lorentz fator of the relative motion of the two inertial frames involved in the Lorentz transformation and use Γ for the Lorentz fator assoiated with any other kind of motion.) Given the above results, we an find the magnitude of the 4-veloity: u 2 = Γ 2 2 + Γ 2 v 2 = 2 Γ 2 (1 v 2 / 2 ) = 2. Thus, 4-veloity is a time-like vetor. Reall, that u 2 is a 4-salar and in any inertial frame and for any seletion of spae-time oordinates, the alulations of this quantity will yield exatly the same result, 2. For example, in the partile rest frame (the inertial frame where this partile is instantaneously at rest) Γ = 1, v = 0 and u = (, 0, 0, 0) (4.12) From this we immediately find that, u 2 = 2. (4.13) This example demonstrates how easy the alulations an beome when we make a good hoise of oordinate system/inertial frame. Another important partile parameter in Newtonian mehanis is its inertial mass, whih is assumed to be the same in all inertial frames. However, we annot be sure that this assumption, whih makes the inertial mass a 4-salar, will be onsistent with the relativisti mehanis. (In fat, as we will see very soon, it is not.) Yet, we need a 4-salar to be used in 4-tensor equations. To make sure, we an use the same trik as with the proper time. We introdue the partile rest mass (or proper mass), m 0, whih is defined as the inertial mass measured in the partile rest frame. This defines m 0 uniquely, yielding the same quantity no matter whih inertial frame is atually used to desribe the partile motion, or whih oordinates we introdue in spae-time for this purpose. In Newtonian mehanis, the produt of partile s inertial mass and its 3-veloity is another important 3-vetor the 3-vetor of momentum, p = mv. This suggests to introdue the 4-vetor of momentum (or the 4-momentum vetor) via P = m0 u. (4.14) Using eq.4.13 we immediately obtain P 2 = m 2 0 2 (4.15) and thus the 4-momentum vetor is also time-like. Using eq.4.11 we find that P = (m0 Γ, m 0 Γv). (4.16) The relativisti generalisation of the seond law of Newton is also quite straightforward d P dτ = F, (4.17) where F is a 4-vetor of fore or just 4-fore. We will ome bak to this law later after we onsider the relativisti laws of onservation of energy and momentum.

4.3. ENERGY-MOMENTUM CONSERVATION 57 4.3 Energy-momentum onservation Consider an isolated system of N partiles. The term isolated means that these partiles may interat with eah other but not with their surrounding. Moreover, our analysis here is restrited to only short-range interation between partiles (ollisions) whereas the long-range interations and the orresponding potential energy are ignored. The Newtonian mehanis states that the total energy and the total momentum of suh a system are onserved, that is they do not hange in time: and N E (k) = onst (4.18) k=1 N p (k) = onst, (4.19) k=1 where E (k) = m (k) v 2 (k) /2 and p (k) = m (k)v (k) are the energy and momentum of k-th partile respetively, m (k) is its inertial mass and v (k) is its 3-veloity. Note that we use round brakets here to indiate that k is the partile number (or name). This helps to avoid potential onfusion with tensor indexes. We should start writing the relativisti laws of mehanis as 4-tensor equations and the equation N P (k) = onst (4.20) k=1 seems to be a good andidate for the 4-vetor ounterpart of momentum onservation. If this is a good hoie then we should reover the Newtonian law of momentum onservation in the limit of slow speeds (v ). Let us split this equation into a 3-salar and a 3-vetor equations following the disussion in Se.4.1. Using eq.4.16 we obtain and N N P(k) 0 = onst or m 0,(k) Γ (k) = onst (4.21) k=1 k=1 N N P(k) i = onst or m 0,(k) Γ (k) v (k) = onst. (4.22) k=1 k=1 Quik inspetion shows that the last equation appears to be the relativisti version of 3-momentum onservation and suggests to define the relativisti 3-momentum of a partile as p = m 0 Γv. (4.23) In fat, we an make the expression for p to be idential to the Newtonian one if we define the inertial mass of moving partile as p = mv (4.24) m = m 0 Γ. (4.25) This definition make the inertial mass to be different from the rest mass and depending of the partile speed. Only at small speeds (v and hene Γ 1) we have m (k) = m 0,(k). In order to understand the meaning of eq.4.21 we onsider its Newtonian limit (v ). If we simply replae eah Γ in eq.4.21 with unity then we obtain N m 0,(k) = onst. k=1

58 CHAPTER 4. RELATIVISTIC PARTICLE MECHANICS Clearly this is the mass onservation law the total mass of the system is onserved. However, we get deeper insight into the nature of eq.4.21 if we onsider the Taylor expansion of Γ Γ = ( ( v ) ) 2 1/2 1 = 1 + 1 ( v ) 2 3 ( v ) 4 + +... 2 4 and keep not only the first but also the seond term in this expansion. Then and eq.4.21 reads ( m 0 Γ m 0 1 + 1 v 2 ) 2 2 = 1 (m 0 2 + 12 ) m 0v 2 (4.26) N (m 0,(k) 2 + 12 ) m 0,(k)v 2 = onst. (4.27) k=1 Sine in Newtonian mehanis the mass onservation is a separate law then eq.4.27 redues to the Newtonian energy onservation N 1 2 m 0,(k)v 2 = onst. (4.28) This implies that k=1 the 3-salar equation (4.21) is the relativisti generalisation of the energy onservation law; the 4-vetor equation (4.20) unites the energy and momentum onservation laws into a single energy-momentum onservation law; the 4-momentum vetor desribes not only the partile 3-momentum but also its energy (a 3-salar). For this reason it is often alled the energy-momentum vetor. What should we onsider as the partile energy in the relativisti ase? Whatever is the definition it should deliver a onserved quantity! Given the results (4.26,4.28) we seem to have only two options. The first one is K = E m 0 2. (4.29) As we have seen, in the Newtonian limit K redues to the usual expression for the kineti energy, whih seems to make this a sensible hoie. If, however, we use this option then the total energy of the system will be onserved only provided that the total rest mass of the system is onserved. In fat, the laboratory experiments with atoms and sub-atomi partiles show that the total rest mass of partiles is not always onserved in partile ollisions N m 0,(k) 2 onst. k=1 For this reason this option is unsatisfatory. The seond option is E = P 0 = m 0 Γ 2 = m 2 (4.30) and eq.4.21 shows that in this ase the total energy of an isolated system is always onserved N E (k) = onst. k=1 Thus, E = m 2, is the only satisfatory definition of partile energy. In the ase, v = 0, it redues to E 0 = m 0 2, (4.31)

4.4. PHOTONS 59 whih is alled the rest mass-energy of the partile. In general, E E 0 and the differene between them K = E E 0 (4.32) is alled the partile kineti energy. The lak of rest mass-energy onservation in partile ollisions means that the total kineti energy of an isolated system may inrease or derease provided its total rest mass-energy dereases or inreases respetively. In fat it is the redution of the total rest mass-energy of nulear fuel in nulear reators during the proess of radioative deay that is the soure of heat in nulear power stations. The definitions (4.24,4.30) allow us to write P = (E/, p). (4.33) That is given some inertial frame the energy-momentum vetor splits into the partile energy E (3-salar) and momentum p as measured in this frame. Thus, Speial Relativity unovers the deep onnetion between energy and momentum. In Se.4.2, eq.4.15, we derived an expression for the magnitude of 4-momentum vetor of a partile with rest mass m 0. Using eq.4.33 we an find another expression for it. Utilising pseudo-cartesian oordinates we find that P 2 = E2 2 + (p1 ) 2 + (p 2 ) 2 + (p 3 ) 2 = E2 2 + p2. Combining this result with the one of eq.4.15 we find the following identity where E 0 = m 0 2 is the rest mass-energy. E 2 = p 2 2 + m 2 0 4 or E 2 = E 2 0 + p 2 2, (4.34) 4.4 Photons In 1900 German physiist Max Plank onluded that eletromagneti energy is emitted not in arbitrary amounts but in quanta of energy E = hν, (4.35) where ν is the frequeny of radiation and h is a onstant, known as the Plank onstant. Later, in 1905, Einstein onluded that the eletromagneti energy is not only emitted but also absorbed in quanta and proposed that it must atually propagate in quanta. Thus, in some respets light an be onsidered as a olletion of partiles, alled photons. It is natural to assume that the speed of a photon equals to the speed of light. This makes world lines of photons null and we annot introdue 4-veloity of photons in the same fashion as we have done for massive partiles. Indeed, the 4-veloity is defined as ds/dτ where τ is the proper time of the partile. This time is measured by the standard lok moving with the same speed as the partile. However, no suh lok an be fored to move with the speed of light. Indeed, if the rest mass of the lok is m 0 and it is moving with the Lorentz fator Γ, then aording to eq.4.30 the total energy of this lok is E = m 0 Γ 2. Thus, as v we have Γ and hene E and given the finite energy resoures available to man the speed of light annot be reahed by the lok. This reasoning invites the following question: How an photons move with the speed of light and still have the finite energy given by eq.4.35? The only possibility is to assume that for a photon m 0 = 0. (4.36) This makes sense as photons annot be found at rest in any inertial frame they always move with the speed of light.

60 CHAPTER 4. RELATIVISTIC PARTICLE MECHANICS Partiles with zero rest mass are usually alled massless (in ontrast to massive partiles whih have m 0 0). However, this does not mean that the inertial mass of suh massless partiles is also zero. In fat, ombining Einstein s E = m 2 with Plank s E = hν we find Using this result we an find the 3-momentum vetor of a photon where n is the unit vetor in the diretion of motion. Indeed, m = hν/ 2. (4.37) p = E n, (4.38) p = mv = E 2 n = E n. This allows us to ompile the 4-momentum vetor of photon as P = ( E, E n) = E hν (1, n) = (1, n) (4.39) even if we annot introdue 4-veloity for partiles of this sort. From this equation we find that 4.5 Partile ollisions P 2 = 0 (4.40) The simplest and yet very important lass of problems in mehanis is the partile ollisions. In suh problems a number of partiles interat with eah other by means of some short-range fores whih an be rather omplex and the atual interation an be diffiult to desribe. However, the onservation of energy and momentum allows to answer a number of important questions on the final outome of ollisions. Here we onsider few simple examples. 4.5.1 Nulear reoil Problem: If not exited via ollisions with other partiles atomi nulei reside in the ground states. However, via ollisions they an be moved to states with higher rest-mass energy, the exited states, and than spontaneously return to the ground state. The energy exess is then passed to another partile emitted by the nuleus during the transition. This partile an be a photon. Denote the rest mass-energy of the ground state as E g and the rest mass-energy of exited state E g + δe. Find the energy of emitted photon in the frame where the exited nuleus is initially at rest. Solution: In the rest frame of the exited nuleus its 4-momentum is ( ) Eg + δe Q 0 =, 0. (4.41) In the same frame its 4-momentum after the transition to the ground state is ( ) E Q 1 =, p, (4.42) where p is its 3-momentum and E > E g is its energy whih inludes the kineti energy as well. In the same frame the 4-momentum of the emitted photon is ( Ep P =, E ) p n, (4.43)

4.5. PARTICLE COLLISIONS 61 where n is the diretion of motion of the photon. The energy-momentum onservation law requires Q 0 = Q 1 + P. (4.44) From this point there are two ways to proeed, one is more elegant than the other. The less elegant way: Here we simply use the fat that the orresponding omponents of 4-vetors on the left and right hand sides of Eq.4.44 must be the same: Form this we find and { Eg + δe = E + E p, 0 = p + Ep n (4.45) E = E g + δe E p, p 2 = E2 p 2. The next step is to utilize Eq.4.34, whih in our ase reads E 2 = E 2 g + p 2 2. Substituting the above expressions for E and p into this equation we obtain (E g + δe g E p ) 2 = E 2 g + E2 p 2 2. From this we an find E p as a funtion of E g and δe only. Simple alulations yield E p = δe δe + 2E g 2δE + 2E g. Notie that E p < δe. This is beause a fration of δe has been transformed into the kineti energy of the nuleus. When the photon is emitted the nuleus reeives a kik in the opposite diretion (nulear reoil) so that the total momentum of the system nuleus+photon remains zero. The more elegant way: One an rewrite Eq.4.44 as Q 0 P = Q 1. This may seems not muh different but in fat this signifiantly simplifies alulations beause when we square both sides of this equations the result will inlude only the quantities whih are given and the quantity we need to find. Let us show this: means that Q 0 P 2 = Q 1 2 (4.46) Q 0 2 + P 2 2 Q 0 P = Q 1 2. (4.47) Aording to the general results (4.15) and (4.40), for any partile the magnitude squared of its 4-momentum equals to P 2 = E 2 0/ 2, (4.48) where E 0 = m 0 2 is the rest mass-energy of the partile. Thus, we have Q 0 2 = E 2 g/ 2, Q 1 2 = (E g + δe) 2 / 2, and P 2 = 0.

62 CHAPTER 4. RELATIVISTIC PARTICLE MECHANICS Moreover, Substituting these into Eq.4.47 we find Q 0 P = (E g + δe) E p. (E g + δe) 2 + 2(E g + δe)e p = E 2 g, whih inludes only one unknown quantity, E p. Solving this for E p we find the same answer as before E p = δe δe + 2E g 2δE + 2E g. (4.49) 4.5.2 Absorption of neutrons Problem: A neutron of rest mass-energy E n and kineti energy 2E n is absorbed by a stationary B 10 (Boron-10) nuleus of rest mass-energy E 10 = 10E n and beomes B 11 (Boron-11) nuleus. Find the rest mass-energy E 11 of B 11 immediately after the absorption ( and hene before it moves from the exited state to the ground state). Solution: Use the rest frame of B 10 for the alulations. The 4-momentum of B 10 and the neutron before the ollision are ( ) 10En Q 10 =, 0, ( ) 3En Q n =, p, The 4-momentum of B 11 after the ollision denote as Q 11. The energy-momentum onservation reads Q 10 + Q n = Q 11. From this equation we have Q 10 2 + Q n 2 + 2 Q 10 Q n = Q 11 2. (4.50) Next we use the relation between the rest mass-energy of a partile and the magnitude of its 4-momentum vetor: and alulate Q 10 2 = (10E n) 2 2, Q n 2 = (E n) 2 Q 10 Q n = 30 E2 n 2. Finally we substitute these results into Eq.4.50 2, Q 11 2 = (E 11) 2 2, E 2 n 100E 2 n 60E 2 n = E 2 11 and obtain E 11 = 161E n 12.7E n. 4.6 4-aeleration and 4-fore The 4-aeleration of a massive partile is defined as a = d u dτ. (4.51)

4.6. 4-ACCELERATION AND 4-FORCE 63 Figure 4.1: (In this definition d u = u (τ + dτ) u (τ) and thus we subtrat vetors defined at two different points. We an do this only if we an identity of two vetors defined at two different plaes. In Speial Relativity we say that two suh vetors are idential if they have the same omponents in the orresponding oordinate bases of the same system of pseudo-cartesian oordinates. ) It is easy to see that ( a u ) = 0. (4.52) Indeed, ( a u ) = d u 1 d u = dτ 2 dτ ( u u ) = 1 2 In terms of pseudo-cartesian omponents eq.4.51 reads ( Γ dγ Thus, d a = dτ (Γ, Γv) = Γ d (Γ, Γv) = dt dt, ΓdΓ dt d 2 dτ. ) dv v + Γ2. dt a = (Γ dγ ) dt, ΓdΓ dt v + Γ2 a, (4.53) where a = dv/dt is the usual 3-aeleration vetor. The 4-fore vetor is defined as F = d P dτ. (4.54) Sine for massive partile P = m 0 a we obtain that dm 0 F = u + m0 a (4.55) dτ and thus in general the 4-fore is not parallel to the 4-aeleration. following lassifiation of fores is Speial Relativity. A 4-fore is alled pure if dm 0 /dτ = 0. This fat gives rise to the Eq.4.55 tells us that a pure 4-fore is parallel to the 4-aeleration of partile F = m0 a. (4.56) Combining this result with eq.4.52 we also find that F a = 0. (4.57) A 4-fore is alled heat-like if a = 0. Eq.4.55 tells us that a heat-like 4-fore is parallel to the 4-veloity of partile dm 0 F = u. (4.58) dτ

64 CHAPTER 4. RELATIVISTIC PARTICLE MECHANICS Given the splitting (4.33) of 4-momentum we an split the 4-fore vetor of spae-time into a 3-salar and a 3-vetor as well. We obtain d F = dτ (E/, p) = Γ d ( ) Γ de (E/, p) = dt dt, Γdp. dt Using the same definition of 3-fore as in Newtonian mehanis, we an write the above result as F = ( Γ f = dp dt, (4.59) ) de dt, Γf. (4.60) This equation tells us that the 4-fore determines not only the rate of hange of partile momentum but also the hange its energy. Sine u = (ΓmΓv) we have ( F u ) = γ 2 ( de dt + f v). On the other hand, using eqs.(4.52,4.55) we also obtain Combining these two result we find ( F u ) = 2 dm 0 dτ = 2 γ dm 0 dt. de dt = 2 dm 0 + (f v). (4.61) γ dt Similar to Newtonian mehanis eah kind of interation should be desribed by additional laws speifying the 4-fore. This is the subjet for more advaned ourse on Speial Relativity.