SPECIAL RELATIVITY. MATH2410 KOMISSAROV S.S

Transcription

1 SPECIAL RELATIVITY. MATH2410 KOMISSAROV S.S 2012

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3 Contents Contents 2 1 Spae and Time in Newtonian Physis Spae Einstein summation rule Time Galilean relativity Newtonian Mehanis Galilean transformation The lak of speed limit Light Advaned material: Maxwell equations, eletromagneti waves, and Galilean invariane Maxwell equations Some relevant results from vetor alulus Wave equation in eletromagnetism Plane waves Wave equation is not Galilean invariant Basi Speial Relativity Einstein s postulates Einstein s thought experiments Experiment 1. Relativity of simultaneity Experiment 2. Time dilation Experiment 3. Length ontration Synronization of loks Lorentz transformation Derivation Newtonian limit Relativisti veloity addition One-dimensional veloity addition Three-dimensional veloity addition Aberration of light Doppler effet Transverse Doppler effet Radial Doppler effet General ase Spae-time Minkowski diagrams Spae-time Light one Causal struture of spae-time Types of spae-time intervals

4 4 CONTENTS 3.6 Vetors Definition Operations of addition and multipliation Coordinate transformation Infinitesimal displaement vetors Tensors Definition Components of tensors Coordinate transformation Metri tensor Definition Classifiation of spae-time vetors and other results speifi to spae-time Relativisti partile mehanis Tensor equations and the Priniple of Relativity veloity and 4-momentum Energy-momentum onservation Photons Partile ollisions Nulear reoil Absorption of neutrons aeleration and 4-fore

5 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 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.

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

8 8 CONTENTS

9 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 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

11 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 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 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

13 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 ( ) 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 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.

15 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 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 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.

17 1.8. ADVANCED MATERIAL: MAXWELL EQUATIONS, ELECTROMAGNETIC WAVES, AND GALILEAN INV 1.8 Advaned material: Maxwell equations, eletromagneti waves, and Galilean invariane Maxwell equations Maxwell ( ) 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( ) and Einstein( ) 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 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 ) f (x 2 ) f (x 3 ) 2. (1.31)

18 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) 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 B t 2 2 B = 0. (1.35) 1 2 ψ 2 t 2 2 ψ = 0, (1.36) 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 = 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.

19 1.8. ADVANCED MATERIAL: MAXWELL EQUATIONS, ELECTROMAGNETIC WAVES, AND GALILEAN INV 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 Denoting x 1 as simply x we have ψ t 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 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 20 CHAPTER 1. SPACE AND TIME IN NEWTONIAN PHYSICS

21 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 ( ) 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 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 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 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.

23 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 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 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 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.

25 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 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 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

27 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 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 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.

29 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( ) and Larmor( ) 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 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 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? 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 =