Part I Special Relativity

Part I Speial Relativity G. W. Gibbons D.A.M.T.P., Cambridge University, Wilberfore Road, Cambridge CB3 0WA, U.K. February 14, 2008 The views of spae and time whih I wish to lay before you have sprung from the soil of experimental physis, and therein lies their strength. They are radial. Heneforth spae by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union will preserve an independent reality. H Minkowski (1908). Contents 1 The Shedule 5 1.1 Units.................................. 5 2 Einstein s Theory of Speial Relativity 6 3 *Early ideas about light* 6 3.1 Maxwell s equations......................... 10 4 *The Speed of Light* 11 4.1 *Roemer s measurement of *.................... 11 4.2 *Fizeau s measurement of *.................... 12 4.3 *Fouault s rotating mirror*..................... 12 5 Absolute versus Relative motion 12 6 Veloity omposition formulae 14 7 Galilean Priniple of Relativity 14 7.1 Waves and Galilean Transformations................ 16 1

8 Spaetime 16 8.1 Example: uniform motion in 1+1 dimensions........... 17 8.2 Example: uniform motion in 2+1 dimensions........... 17 8.3 Example: non-uniform motion in 1+1 dimensions......... 17 9 Minkowski s Spaetime viewpoint 17 10 Einstein s Priniple of Relativity 18 10.1 Mihelson-Morley Experiment.................... 18 10.2 Derivation of the Lorentz Transformation formulae........ 20 10.3 Relativisti veloity omposition law................ 22 10.4 *Observational for Einstein s seond postulate*.......... 22 10.5 Light in a medium: Fresnel Dragging................ 23 10.6 Composition of Lorentz Transformations.............. 24 10.7 Veloity of light as an upper bound................. 25 10.8 * Super-Luminal Radio soures*................... 25 10.9 The two-dimensional Lorentz and Poinaré groups........ 26 11 The invariant interval 26 11.1 Timelike Separation......................... 27 11.2 Spaelike separation......................... 28 11.3 Time Dilation............................. 28 11.3.1 Muon Deay......................... 28 11.4 Length Contration.......................... 29 11.5 The Twin Paradox: Reverse Triangle Inequality.......... 29 11.5.1 *Hafele -Keating Experiment*............... 30 11.6 Aelerating world lines....................... 30 12 Doppler shift in one spae dimension 31 12.1 *Hubble s Law*............................ 32 13 The Minkowski metri 33 13.1 Composition of Lorentz Transformations.............. 34 14 Lorentz Transformations in 3 + 1 spaetime dimensions 34 14.1 The isotropy of spae......................... 35 14.2 Some properties of Lorentz transformations............ 36 15 Composition of non-aligned veloities 36 15.1 Aberration of Light.......................... 37 15.2 * Aberration of Starlight*...................... 38 15.3 Water filled telesopes........................ 39 15.4 Headlight effet............................ 39 15.5 Solid Angles.............................. 39 15.6 *Celestial Spheres and onformal transformations*........ 40 15.7 *The visual appearane of rapidly moving bodies*........ 40 15.8 Transverse Doppler effet...................... 41 2

15.9 *The Cosmi Mirowave Bakground*............... 41 16 * Kinemati Relativity and the Milne Universe* 42 16.1 *The Foundations of Geometry*.................. 43 16.2 The Milne metri and Hubble s Law................ 46 16.3 *Relativisti omposition of veloities and trigonometry in Lobahevsky spae*................................. 47 16.4 Parallax in Lobahevsky spae................... 49 17 *Rotating referene frames* 50 17.1 Transverse Doppler effet and time dilation............ 50 17.2 *The Sagna Effet*......................... 51 17.3 Length Contration.......................... 52 17.4 Mah s Priniple and the Rotation of the Universe........ 52 18 General 4-vetors and Lorentz-invariants 53 18.1 4-veloity and 4-momentum..................... 53 18.2 4-veloity............................... 54 18.3 4-momentum and Energy...................... 54 18.4 Non-relativisti limit......................... 55 18.5 Justifiation for the name energy.................. 55 18.6 *Hamiltonian and Lagrangian*................... 56 19 Partiles with vanishing rest mass 56 19.1 Equality of photon and neutrino speeds.............. 57 20 Partile deays ollisions and prodution 57 20.1 Radioative Deays.......................... 57 20.2 Impossibility of Deay of massless partiles............ 58 20.3 Some useful Inequalities....................... 59 20.4 Impossibility of emission without reoil............... 60 20.5 Deay of a massive partile into one massive and one massless partile................................ 60 20.6 Deay of a massive partile into two massless partiles...... 60 21 Collisions, entre of mass 61 21.1 Compton sattering......................... 62 21.2 Prodution of pions.......................... 63 21.3 Creation of anti-protons....................... 63 21.4 Head on ollisions.......................... 64 21.5 Example: Relativisti Billiards................... 66 21.6 Mandelstam Variables......................... 66 3

22 Mirrors and Refletions 67 22.1 *The Fermi mehanism*....................... 67 22.2 *Relativisti Mirrors*........................ 68 22.3 *Corner Refletors on the Moon*.................. 69 22.4 Time reversal............................. 70 22.5 Anti-partiles and the CPT Theorem................ 70 23 4-aeleration and 4-fore 70 23.1 Relativisti form of Newton s seond law.............. 70 23.2 Energy and work done........................ 71 23.3 Example: relativisti rokets..................... 71 24 The Lorentz Fore 72 24.1 Example: partile in a uniform magneti field........... 73 24.2 Uniform eletro-magneti field and uniform aeleration..... 74 25 4-vetors, tensors and index notation 75 25.1 Contravariant vetors........................ 75 25.2 Covariant vetors........................... 76 25.3 Example: Wave vetors and Doppler shift............. 77 25.4 Contravariant and ovariant seond rank tensors......... 77 25.5 The musial isomorphism...................... 79 25.6 De Broglie s Wave Partile Duality................. 79 25.7 * Wave and Group Veloity: Legendre Duality*.......... 80 25.8 The Lorentz equation........................ 82 26 Uniformly Aelerating referene frames: Event Horizons 83 27 Causality and The Lorentz Group 84 27.1 Causal Struture........................... 84 27.2 The Alexandrov-Zeeman theorem.................. 84 27.3 Minkowski Spaetime and Hermitian matries........... 85 28 Spinning Partiles and Gyrosopes 86 28.1 Fermi-Walker Transport....................... 86 28.2 Spinning partiles and Thomas preession............. 86 28.3 Bargmann-Mihel-Telegdi Equations................ 87 4

1 The Shedule Read as follows: INTRODUCTION TO SPECIAL RELATIVITY 8 letures, Easter and Lent terms [Leturers should use the signature onvention (+ ).] Spae and time The priniple of relativity. Relativity and simultaneity. The invariant interval. Lorentz transformations in (1 + 1)-dimensional spaetime. Time dilation and muon deay. Length ontration. The Minkowski metri for (1 + 1)-dimensional spaetime.[4] 4 vetors Lorentz transformations in (3 + 1) dimensions. 4 vetors and Lorentz invariants. Proper time. 4 veloity and 4 momentum. Conservation of 4 momentum in radioative deay.[4] BOOKS G.F.R. Ellis and R.M. Williams Flat and Curved Spae-times Oxford University Press 2000 24.95 paperbak W. Rindler Introdution to Speial Relativity Oxford University Press 1991 19.99 paperbak W. Rindler Relativity: speial, general and osmologial OUP 2001 24.95 paperbak E.F. Taylor and J.A. Wheeler Spaetime Physis: introdution to speial relativity Freeman 1992 29.99 paperbak 1.1 Units When quoting the values of physial quantities, units in whih = and h=1, will frequently be used. Thus, at times for example, distanes may be expressed in terms of light year. Astronomers frequently use parses whih is the distane at whih is short for paralax seond. It is the distane at whih the radius of the earth subtends one seond of ar. One parse works out to be 3.0 10 13 Km or 3.3 light years. A frequently used unit of energy, momentum or mass is the eletron volt or ev whih is the work or energy required to move an eletron through a potential differene of one Volt. Physial units, masses and properties of elementary partiles are tabulated by the Partile Data Group and may be looked up at http://pdg.lbl.gov. Although not neessary in order to follow the ourse, it is a frequently illuminating and often amusing exerise to go bak to the original soures. Many of the original papers quoted here may be onsulted on line. For papers in the Physial Review, bak to its ineption in the late nineteenth entury go to http://prola.aps.org 5

. For many others, inluding Siene and Philosophial Transations of the Royal Soiety (going bak its beginning in the to seventeenth entury) go to http://uk.jstor.org. 2 Einstein s Theory of Speial Relativity is onerned with the motion of bodies or partiles whose relative veloities are omparable with that of light = 299, 792, 458 ms 1. (1) In a nutshell, Newton s Seond Law remains unhanged in the form dp dt = F (2) where F the fore ating on a partile of momentum p and mass m 1, but while aording to Newton s Theory p = mv (3) in Einstein s Theory p = mv 1 v2 2. (4) If this were all there is in it, relativity would, perhaps, not be espeially interesting. What makes Relativity important is that it entails a radial revision of our elementary ideas of spae and time and in doing so leads to the even more radial theory of General Relativity whih omes into play when gravity is important. In this ourse we shall ignore gravity and onfine our attention to Speial Relativity. For matters gravitational the reader is direted to[34]. To see why relativity has suh a profound impat on ideas about spae and time, note that we are asserting that there atually is suh a thing as the veloity of light. For the benefit of those who have not studied Physis at A-level, or who did, but have now forgotten all they ever knew, the next setion ontains a review of the elementary physis of light. 3 *Early ideas about light* Experiments with shadows and mirrors lead to the idea that light is a form of energy that propagates along straight lines alled light rays. On refletion at a smooth surfae S at rest, it is found that 1 properly speaking rest-mass 6

(i) The inident ray, the refleted ray and the normal to the surfae at the point of refletion are o-planar (ii) The inident and refleted rays make equal angles with the normal. Hero of Alexander showed that these laws may be summarized by the statement that if A is a point on the inident ray, B on refleted ray and x S the point at whih the refletion takes plae, then x is suh that the distane d(a,x) + d(x, B) (5) is extremized among all paths from AyB, y S to the surfae and from the surfae to B. When light is refrated at a smooth surfae S it is found that (i) The inident ray, the refrated ray and the normal to the surfae at the point of refration are o-planar (ii) The inident and refrated rays make angles θ i and θ r with the normal suh that sin θ i = n r (6) sinθ r n i where the quantities n i and n r are harateristi of the medium and may depend upon the olour of the light and are alled its refrative index. By onvention one sets n = 1 for the vauum. Pierre Fermat showed that these laws, first learly enuniated in about 1621 by the Leyden mathematiian Willebrod Snellius or Snell in work whih was unpublished before his death in 1626, and later by Desartes, although probably known earlier to Thomas Harriot, may be summarized by the statement that if A is a point on the inident ray, B on refrated ray and x the point at whih the refration takes plae, then x is suh that the optial distane n i d(a,x) + n r d(x, B) (7) is extremized among all paths AyB, y S from A to surfae and from the surfae to B. In other words the differential equations for light rays may be obtained by varying the ation funtional nds (8) where ds is the element Eulidean distane. By the time of Galilei its was widely thought that light had a finite speed, and attempts were made to measure it. Broadly speaking there were two views about the signifiane of this speed. The Emission or Ballisti Theory held by Isaa Newton and his followers aording to whih light onsisted of very small partiles or orpusles with mass m speed and momentum p = m, the speed varying depending upon the 7

medium. On this theory, Snell s law is just onservation of momentum parallel to the surfae. p i sin θ i = p r sin θ r, (9) whene, assuming that the mass is independent of the medium sin θ i sin θ r = r i. (10) The Wave Theory proposed by the duth physiist Christian Huygens in 1678, aording to whih light is a wave phenomenon having a speed and suh that eah point on the wave front gives rise to a seondary spherial wave of radius t whose forward envelope gives the wavefront at a time t later. On this theory, Snell s law arises beause the wavelength λ i of the inident wave and the wavelength of the refrated wave λ r differ. Applying Huygen s onstrution gives sin θ i = λ i. (11) sinθ r λ r Sine, for any wave of frequeny f, λf = and sine the frequeny of the wave does not hange on refration, we have aording to the wave theory: sin θ i sin θ r = i r. (12) The two theories gave the opposite predition for the speed of light in a medium. Sine refrative indies are never found to be less than unity, aording to the emission theory the speed of light in a medium is always greater than in vauo, while aording to the wave theory it is always smaller than in vauo. One way to distinguish between the two theories was to measure the speed of light in vauo and in a medium. This was first done by Fouault in 1850, and more aurately by Mihelson in 1883 using the rotating mirror method of the former, whih will be desribed shortly. By interposing a tube filled with water in the path of the light, they showed that the speed of light in water was slower than in vauo 2. It follows that Hero and Fermat s variational properties may be summarized by the statement that the time taken for light to traverse the physial path is extremized. Another way to distinguish the theories is by their ability to aount for the diffration of light by very small obstales as observed by Grimaldi in 1665 or by experiments on slits, suh as were performed by the polymath Thomas Young 3 in 1801. Following a large number of subsequent experiments, notably by Fresnel, by Fouault s time, some form of wave theory was aepted by almost 2 The argument is in fat slightly indiret sine these experiments atually measure the group veloity of light while refration depends on the phase veloity. The distintion is desribed later. Given one, and information about the dispersion, i.e. how the refrative index varies with wavelength, one may alulate the other. 3 Young played an equal role with Champillon in the translation of the Egyptian hieroglyphis on the Rosetta stone. 8

all physiists. In its simplest form, this postulated that in vauo, some quantity satisfies the salar wave equation 1 2 2 φ t 2 = 2 φ, (13) whih, if is onstant, is easily seen to admit wavelike solutions of the form or more generally φ = Asin ( 2π( x λ ft)), (14) φ = f(k ωt), (15) where f() is an arbitrary C 2 funtion of its argument and ω k =. Sine equation (13) is a linear equation, the Priniple of Superposition holds and solutions with arbitrray profiles, moving in arbitrary diretions may be superposed. A fat whih not only explains many opetial phenomena but also led to the idea of Fourier Analysis.Note that solutions (13) are non-dispersive, the speed is independent of the wavelength λ or frequeny f. Until Einstein s work, almost all physiists believed that wave propagation required some form of material medium and that light was no exeption. The medium was alled the luminiferous aether (or ether) and many remarkable properties were asribed to it. Many physiist, inorretly as it turned out, believed ed that it was inextriably linked with the nature of gravitation. Others, like Lord Kelvin, postulated that atoms ould be thought of as knotted vortex rings. This seemed to require that the ether was some sort of fluid. A key question beame: what is the speed of the earth relative to the aether? The properties of the ether beame even harder to understand when it was established that light ould be polarized. This was first notied by Huygens who was studying the refration of light through a rystal of alite also known as Ieland spar. In 1808 Malus disovered that light ould be polarized by in the proess of refletion.these observations led diretly to the idea that light due to some sort of motion transverse to the diretion of propagation, and so the quantity φ should be some sort of vetor rather than a salar. They also suggested to many that the aether should be some sort of solid. The realization that light was an eletromagneti phenomenon and the great ahievement of the Sottish physiist James Clerk Maxwell (1831-1879) in providing in 1873 a omplete, unified and onsistent set of equations to desribe eletromagnetism, whih moreover predited the existene of eletromagneti waves moving at the speed of light and the subsequent experimental verifiation by the German physiist Heinrih Hertz (1857-1894) around 1887 4 did nothing to dispel the wide-spread onfusion about the aether. Elaborate mehanial models of the aether were onstruted and all the while, it and the earth s motion through it, eluded all experimental attempts at detetion. The general frustration at this time is perhaps refleted in the words of the president of the 4 In fat it seems lear that Hertz had been antiipated by the English Eletriian D E Hughes in 1879, but the signifiane of his work was not appreiated until muh later [49]. 9

British Assoiation, Lord Salisbury who is reported to have proposed, at one it its meeting held at Oxford, a definition of the aether as the nominative of the verb to undulate. In a similar vein. disussing the various allegedly physial interpretations, Hertz delared that To the question What is Maxwell s Theory, I know of no shorter or more definite answer than the following: Maxwell s theory is Maxwell s system of equations. Every theory whih leads to the same system of equations, and therefore omprises the same possible phenomena, I would onsider a form of Maxwell s theory. Maxwell s equations have many beautiful and remarkable properties, not the least important of whih is invariane not under Galilei transformations as might have been expeted if the aether theory was orret, but rather under what we now all Lorentz transformations. This fat was notied for the salar wave equation (13) in 1887, long before Einstein s paper of 1905, by Woldemar Voigt(1850-1919) and both Lorentz and Poinaré were aware of the Lorentz invariane of Maxwell s equations but they regarded this as a purely formal property of the equations. As we shall see, Einstein s insight was in effet to see that it is perhaps the single most important mathematial fat about the equations. From it flows all of Speial Relativity and muh of General Relativity. 3.1 Maxwell s equations These split into two sets. The first set always holds, in vauo or in any material medium and independently of whether any eletri harges or urrents are present. They deny the existene of magneti monopoles and asserts the validity of Mihael Faraday s law of indution. divb = 0, urle = B t. (16) The seond set desribe the response of the fields to the pressene of eletri harges, harge density ρ and urrents, urrent density j. At the expense of introduing two additional fields they may also be ast in a form whih is always orret. They assert the validity of Coulomb s law, and Ampére s law, provided it is supplemented by the last, ruial, additional term, alled the displaement urrent due to Maxwell himself. divd = ρ urlh = j + D t. (17) It follows from the identity div url = 0, that eletri harge is onserved ρ + divj = 0. (18) t 10

In order to lose the system one requires onstitutive relations relating D and H to E and B. In vauo these are linear relations D = ǫ 0 E, H = 1 µ 0 B, (19) where µ 0 and ǫ 0 are two universal physial onstants onstants alled respetively the permeability and permittivity of free spae.thus, in vauuo, Maxwell s equations are linear and the priniple of superposition holds for their solutions. Thus, in vauo dive = ρ ǫ 0 url 1 µ 0 B = j + ǫ 0 E t. (20) If there are no harges or urrents present, use of the identity urlurl = graddiv 2 gives ǫ 0 µ 0 2 E t 2 = 2 E, (21) ǫ 0 µ 0 2 B t 2 = 2 B. (22) Thus eah omponent of the eletri and magneti field travels non-dispersiveley with veloity = 1 ǫ0 µ 0. (23) The divergene free onditions imply that solutions of the form E = E 0 f(k ωt), B = B 0 f(k ωt), are transversely (plane) polarized and moreover sine k.e = 0, k.b = 0 (24) B 0 = n E 0, E 0 = B 0 n, (25) with n = k k, the vetors (E 0,B 0,n) form a right handed normal but not orthonormal triad. Physially the diretion of the polarization is usually taken to be that of the eletri field, sine this is easier to detet. Thus for any given propagation diretion n there are two orthogonal polarization states. in the sense that one may hoose the solutions suh that E 1.E 2. = 0 and thus B 1.B 2 = 0. 4 *The Speed of Light* 4.1 *Roemer s measurement of * That light does indeed have a finite speed was first demonstrated, and the speed estimated by the Danish astronomer Olaus Roemer (1614-1710) in 1676 [1]. He observed the phases of Io, the innermost of the four larger satellites or moons 11

of Jupiter (Io, Europa, Ganymede and Callisto in order outward) whih had been disovered in 1610 by Galileo Galilei (1564-1642) using the newly invented telesope 5 and of Io s motion around Jupiter is about 1.77 days an be dedued by observing its phases, when it is elipsed by Jupiter whose orbital period is 11.86 years. In 1688 G D Cassini had published a set of preditions for these but Roemer observed that they were inaurate by about 15 minutes. The periods are shorter when Jupiter is moving toward the earth than when it is moving away from the earth. Roemer explained this and obtained a value for the speed of light by arguing that when Jupiter is moving toward the earth Io the time between elipses is shorter than when Jupiter is moving away from us beause in the former ase light the total distane light has to travel is shorter than in the latter ase. He obtained a value of 192,000 miles per seond or roughly 310,000 Km per se. If we think anahronistially 6, we might say if we think of Io as a lok, its period is Doppler shifted. 4.2 *Fizeau s measurement of * The first aurate terrestrial measurement of the speed of light was by the Frenh physiist Fizeau who, in 1849 [6], passed a beam of light through a rotating toothed wheel with 720 teeth, refleted it off a plane mirror 8.633 Km away and sent the light bak toward the toothed dis. For a rotation speed of 12.6 turns per seond the light was elipsed giving a speed of about 315,000 Km per seond. 4.3 *Fouault s rotating mirror* In 1850 another Frenh physiist, Fouault [7] refleted light off a mirror whih was rotating about an axis parallel to its plane. The refleted light was then sent bak in the same diretion. If the rate of rotation of the mirror was hosen suitably the light arrived bak at its point of departure. From this Fouault dedued a value for the veloity of light of 298, 000 Kms 1. As mentioned above, he was also able to establish that the speed of light in water is less than in vauo. 5 Absolute versus Relative motion Newton based his theory on the assumption that spae was uniform and desribed by the usual laws of Eulidean geometry. There then arose the issue of whether motion with respet to that bakground was observable. If it was, then one would have a notion of absolute as opposed to relative motion. Newton argued, using the idea of a suspended buket of water, that one does have an 5 The true inventor of the telesope is not known. It seems to have been known to the English osmologist Thomas Digges and the Oxford mathematiian and explorer of Virginia, Thomas Harriot(1560-1621 6 The Doppler effet was proposed by the Austrian physiist C.J. Doppler in 1842 12

idea of absolute rotational motion. However aording to his laws of motion there is no obvious dynamial way of deteting absolute translational motion. Sine his laws imply that the entre of mass of an isolated system of bodies one ould define an absolute frame of rest as that in whih the entre of mass of for, example the visible stars, is at rest. One later suggestion was that one ould take the entre of the Milky way. Lambert had suggested that it was the loation of a dark regent or massive body, a suggestion also made by Mädler. Interestingly we now know that at the entre of the Milky Way there is a massive Blak Hole of mass around a million times the mass of the sun, 10 6 M. The mass of the Milky way is about 10 12 M. However any suh entre of mass frame an only be determined by astronomial observations. It ould not be found using purely dynamial experiments beneath loudy skies here on earth. Later, physiists, like Ernst Mah[50], began to worry about the logial foundations of Newton s laws. Exatly what was meant by the statement that a partile ontinues in a state of uniform motion if unaffeted by an external fore? Wasn t Newton s seond law in effet a tautology? et. L. Lange in 1885 [52] and others [50] had realized that an operational meaning ould be given to Newton s laws if one introdues the idea of an inertial frame of referene. This Lange thought of as a oordinate system for R 3 whih ould be determined by the free, mutually non-parallel, motion of three partiles. Then the first law ould be formulated as the non-trivial and empirially verifiable statement that any fourth free partile would move in this frame with uniform motion. In effet we are to use use the straight line motion of partiles to build up what is sometimes alled an inertial oordinate system or inertial referene system. In fat this onstrution losely resembles various onstrutions in projetive geometry, espeially if one adds in time as an extra oordinate. We will disuss this in more detail later. Mah pointed out that even if one used astronomial observations to determine a fundamental inertial frame of referene whih is at rest with respet to the fixed stars, that is stars so distant that their proper motions are negligible, this raises a puzzle. For example, in priniple we an define a non-rotating frame in two different ways, (i) Using gyrosopes for example whih, if they are subjet to no external torque will point in a onstant diretion in an inertial frame of referene, in other words using what has ome to be alled the ompass of inertia. (ii) Using the fixed stars. Nowadays we use quasi stellar radio soures quasars. It is then a remarkable oinidene that, as we shall see in detail later, to very high auray these two definitions agree. Mah had some, not very speifi, suggestions about a possible explanation. Mah s ideas strongly influened those of Einstein, espeially when he was formulating his General Theory of Relativity. They really annot be pursued in detail without General Relativity and without some understanding of Relativisti Cosmology. It was against this bakground that the question of the aether beame so important. If it really existed, it would provide an alternative frame of referene, whih might,or might not, oinide with the astronomially determined or 13

dynamially determined frames of referene. It ould, for example, remove the ambiguity about translational motion. The obvious guess was that it all three frames agreed. But if this was true, then the earth should be moving through the ether and this motion should be detetable. 6 Veloity omposition formulae Given that the speed of light is finite and presumably well defined on would ask, on the basis of Newtonian theory, in what frame? If the there is suh a thing as the veloity of light, independent of referene frame then the standard veloity addition formula in Newtonian Theory v v + u (26) annot be right. In fat, as we shall see later, one has a veloity omposition (rather than addition) formula. In Einstein s Theory v v+u 1+ uv 2 (27) so that if v =, v + u 1 + u 2 =. (28) Exerise Show that if u and v then u+v 1+ uv 2. Exerise Using the formula dw dt = F.v, for the rate of doing work W by a fore F ating on a point moving with veloity v, show that the work done in m aelerating a partile of mass m from rest to a final veloity v is 1 2 m 2. v2 The theory is alled the Theory of Relativity beause it deals with 2 relative veloities and what is alled the Priniple of Relativity. This idea began, at least in modern times, with Galileo and we shall begin with his version of it. 7 Galilean Priniple of Relativity Suppose a boat is moving with uniform veloity along a anal and we are looking at it. We are asked the following Question The lookout is in the row s nest and drops a heavy weight onto the dek. Will it hit the aptain below? Answer Yes. Reason We pass to a frame of referene S moving with the boat. The frame at rest with respet to the anal is an inertial frame of referene. Galileo assumed that 14

The laws of dynamis are the same in all frames of referene whih are in uniform motion with respet to an inertial frame of referene Now if we drop something from rest in frame S it will fall vertially down, Therefore if we drop something from rest in frame S it will fall vertially down, The boxed statement is Galilean Relativity follows in Isaa Newton s (1642- aount of dynamis beause But to transform to frame S we set and hene In frame S m d2 x = F(x, t). (29) dt2 x = x ut, (30) m d2 x dt 2 = F( x + ut) in frame S. (31) Note that Galileo assumed that the passengers in the boat would use the same oordinate t. In priniple one might have thought that one would also have to hange the time oordinate to a new oordinate t for this equivalene to work out but both Galileo and Newton agreed that Time is an absolute oordinate (32) that is, it takes the same value in all inertial frames of referene Formulae (30,33) onstitute a t = t (33) Gallean Transformation t = t, x = x ut. (34) We have just shown that Newton s equations of motion are invariant under Galilean Transformations. We shall now use Galilean transformations to dedue the Non-relativisti Veloity Addition Formulae. If a partile moves with respet to a frame S suh that then Thus and hene x = ṽt + x 0 (35) x ut = ṽ + x 0 (36) x = (u + ṽ)t + x 0. (37) v = u = ṽ, (38) gives the veloity with respet to S. Later we will imitate this simple alulation to obtain the veloity addition formula in speial relativity. 15

7.1 Waves and Galilean Transformations If, in a frame S at rest with respet to the aether, we have a wave of the form φ = sin ( k.x ωt ) (39) Its speed is = ω 2π k, its wavelength λ = k and frequeny f = ω 2π. If we submit it to a Galilei transformation it beomes φ = sin ( k. x (ω u.k)t ). (40) In the frame S, the wave has the same wavelength but the frequeny f = ω is hanged f = f(1 u osθ) (41) and the speed is, where = u osθ. (42) The formula for veloity in the moving frame S is very muh what one expets on the basis of a partile viewpoint but note that the angle θ is the angle between the diretion of the wave n = k k and the relative veloity u of the two frames S and S. Both frames agree on this as do they on the diretion of motion of the wave. In other words, Galilei s transformation formulae predit that there is no aberration. Later, we will obtain the physially orret results using the same method as above, but instead of a Galilei transformation we shall substitute using a Lorentz transformation. 8 Spaetime Before proeeding, we will pause to develop a way of thinking about kinematis that in fat goes bak to Lagrange and D Alembert. The latter wrote, in his artile on dimension in the Enylopédie ou Ditionaire raisonée des sienes, des arts et des metiers in 1764 A lever aquaintane of mine believes that it is possible to think of time as a fourth dimension, so that the produt of time and solidity would in some sense be the produt of four dimensions; it seems to me that this idea, while debatable, has ertain merits-at least the merit of novelty. The German pioneer of psyho-physis Gustav Theodor Fehner (1801-1887) wrote a popular artile entitled Der Raum hat vier Dimension whih disusses related ideas. By that time the study of extra spatial dimensions was quiet advaned and the German Astronomer Johann C F Zollner (1834-1882) gained notoriety for laiming that the alleged ability of self-laimed spiritualists to 16

untie knots sealed in glass jars was only expliable if they had been moved into a fourth spatial dimension. By 1880 s the Frenh railway engineer Ibry was using spaetime diagrams in a pratial way to onstrut railway time tables (see illustration on p 55 of [34]) The following examples illustrate the power of the view point in solving this type of mundane problem. 8.1 Example: uniform motion in 1+1 dimensions A ommuter is usually piked up by his/her spouse who drives at onstant speed from their house to meet the ommuter at 5 o lok. One day the ommuter arrives on an earlier train at 4 o lok and deides to walk. After a while the ommuter is piked up by his/her spouse who has driven to meet him as usual. They arrive bak at their house 10 minutes earlier than usual. For how long did the ommuter walk? 8.2 Example: uniform motion in 2+1 dimensions Four ships, A, B, C, D are sailing in a fog with onstant and different speeds and onstant and different ourses. The five pairs A and B, B and C, C and A, B and D, C and D have eah had near ollisions; all them ollisions. Show that A and D neessarily ollide. Hint Consider the triangle in the three-dimensional spaetime diagram formed by the world-lines of A, B and C. 8.3 Example: non-uniform motion in 1+1 dimensions A mountain hiker sets off at 8.00 am one morning and walks up to a hut where he/she stays the night. Next morning he/she sets of at 8.00 am and walks bak down the same trak. Show that, independently of how fast or slowly he/she walks there is at least one time on the two days when he/she is at the same point on the trak. For an interesting history of ideas of the fourth dimension before Einstein in art and popular ulture,inluding H G Wells s ideas about time travel, one may onsult the interesting book [23]. 9 Minkowski s Spaetime viewpoint In what follows we shall initially be onerned with the simplified situation in whih all motion is restrited to one spae dimensions. Thus the position vetors x have just one omponent. In this ase, it is onvenient to adopt a graphial representation, we draw a spaetime diagram onsisting of points we all events with spaetime oordinates (t, x). The two-dimensional spae with these oordinates is alled spaetime. 17

Passing to another frame of referene is like using oblique oordinates in spaetime. However, aording to Galilei, all observers use the same time oordinate. Geometrially while the lines of onstant x have different slopes in different frames, the lines of onstant time are all parallel to eah other. This means that two events (t 1,x 1 ) and (t 2,x 2 ) whih are simultaneous in frame S must be simultaneous in frame S, that is thus ( t 1, x 1 ) = (t 1, 1 ut 1 ), (43) ( t 2, x 2 ) = (t 1,x 2 ut 2 ) (44) t 1 t 2 = 0 t 1 t 2 = 0. (45) We say that in Newtonian theory simultaneity is absolute, that frame independent. Let s summarize (i) The Laws of Newtonian dynamis are invariant under Galilei transformations t = t, x = x ut. (46) (ii) veloities add v = ṽ + u. (47) (iii) Time is absolute. (iv) Simultaneity is absolute. 10 Einstein s Priniple of Relativity We have disovered that no purely dynamial experiment an determine our absolute veloity. If we are in a losed railway arriage moving uniformly we annot tell, by dropping partiles et, how fast we are traveling. The natural question to ask is whether we an tell using experiments involving light. If this has speed relative to some privileged inertial frame S, (identified before Einstein with the mysterious Aether or Ether ), it should, aording to Galileo, have speed u relative to a frame S moving with respet to the aether. By measuring this speed it should be possible to determine u. This was tried in the 10.1 Mihelson-Morley Experiment This is desribed learly and in detail in Mihelson s own words in [2]. Therefore the present desription will be brief. The light travel times T and T of light moving in diretions restively perpendiular and parallel to the motion along 18

arms of an interferometer of lengths L and L are measured. It was argued that in the parallel diretion (working in frame S) T = L [ u + 1 ] T = 2L u 1. (48) 1 u2 2 On the other hand (working in frame S) it was argued that the total distane the perpendiularly moving light has to travel is, by Pythagoras, L 2 + (ut 2 )2 = 2T 2L T = 1 u2 2 (49) Thus, for example, if L = L and T T we should be able to measure u. However in 1887 the experiment arried out by the Amerian Physiist MIhelson and Morley [3] revealed that T = T! Einstein drew the onlusion that no experiment, inluding those using light, should allow one to measure one s absolute veloity, that is he assumed. The Invariane of the Speed of Light The veloity of light is the same in all frames of referene whih are in uniform motion with respet to an inertial frame. 19

In Einstein s own words the same laws of eletrodynamis and optis will be valid for all frames of referene for whih the equations of mehanis hold good. We will raise this onjeture (the purport of whih will hereafter be alled the Priniple of Relativity ) to the status of a postulate and also introdue another postulate, whih is only apparently irreonilable with the former, namely that light is always propagated in empty spae with a definite veloity whih is independent of the state of motion of the emitting body. If Einstein is orret, then Galilei s transformations annot be orret. We need a new transformations alled Lorentz Transformations. They turn out to be (proof shortly) Lorentz Transformations x = x ut, t = 1 u2 2 t u 2 x 1 u2 2. (50) Note that (i) the time t gets transformed to t as well as x to x. (ii) Simultaneity is no longer absolute t 1 t 2 = t 1 t 2 u 1 2 (x 1 x 2 ) (51) u2 2 and hene t 1 t 2 t 1 t 2, if x 1 x 2. (52) (iii) If we take the non-relativisti limit in whih the speed of light is infinite we Lorentz transformations (50) we reover the Galilei transformations (34). 10.2 Derivation of the Lorentz Transformation formulae We assume (i) ( t, x) are linear funtions of (t, x) (ii) 2 t 2 x 2 = 2 t 2 x 2 and hene the speed of light is invariant beause x = t x = ± t. In this first look at the subjet we assume (ii) but in more sophistiated treatments one makes onsiderably weaker assumptions. A preise statement will be made later. Even at this point it should be lear that we are ignoring trivial dilations or homotheties x = λx, t = λt, for λ 0 whih obviously leave the speed of light invariant. However we do not usually inlude these in the set of Galilei transformations. We shall also treat spae and time translations 20

t t + t 0, x x + x 0 as trivial Thus it is suffiient to onsider light rays through the origin of spaetime (t, x) = (0, 0) We shall also regard as trivial spae reversal x = x, t = t and time reversal x = x, t = t. Clearly (50) satisfy (i) and (ii). The onverse is obtained by setting ( t x ) ( A B = C D ) ( t x ), (53) with A > 0, D > 0 beause we are exluding time reversal and spae reversal. Substitution gives (At B) 2 (Ct Dx) 2 2 t 2 x 2 = 0. (54) Thus equating oeffiients of t 2 and x 2 to zero, we get (i) A 2 C 2 = 1 A = oshθ 1, C = sinhθ 1 (55) (ii) D 2 B 2 = 1 D = oshθ 2, B = sinhθ 2 (56) For some θ 1 and θ 2. Now equating the oeffiient of xt to zero gives (iii) AB = CD oshθ 1 sinhθ 2 = oshθ 2 sinhθ 1 θ 1 = θ 2. (57) Thus ( ) t = x ( oshθ sinhθ sinhθ oshθ ) ( ) t. (58) x Setting x = 0 allows us to see that the origin of the S frame satisfies xoshθ = t sinhθ. But if this is to agree with x = ut, where u is the relative veloity, we must have u = tanhθ := β, (59) where θ is alled the rapidity. It follows that and 1 oshθ = := γ (60) 1 u2 2 sinhθ = u 1 u2 2 = βγ. (61) The quantities β and γ do not, as far as I am aware, have individual names, and perhaps for that reason γ is often, rather inelegantly, referred to as the relativisti gamma fator. The use of the symbols β and γ is both traditional and almost universal in the subjet. A Lorentz transformation of the form (50) is often alled a boost whih is analogous to a an ordinary rotation. The analogue of the verb rotating is, unsurprisingly, boosting. A useful relation, partiularly in Tripos questions, is γ 2 (1 β 2 ) = 1. (62) 21

10.3 Relativisti veloity omposition law Of ourse the point is that veloities don t add. Suppose that In frame S x = ṽ t x 0, (63) then using the lorentz transformations (50) we have that In frames x uv 1 u2 2 = ṽ(t ux 2 ) 1 u2 2 + x 0. (64) Thus and hene x(1 + uv 2 ) = (ṽ + u) + t + x 0 1 u2 2, (65) Relativisti veloity omposition law ṽ = u + ṽ 1 + uṽ 2. (66) Thus, for example, ṽ = v =, whih is the invariane of the speed of light. 10.4 *Observational for Einstein s seond postulate* This is that the veloity of light is independent of the veloity of it s soure. Many high preision experiments give indiret evidene for it s validity. In addition, diret observational support for this inludes (i) The light urves of binary stars. De-Sitter [9] pointed out that if, for example, two stars are in orbit around eah other with orbital period T, then if light oming from that portion of the orbit when the star is moving toward us had a larger speed than when it was moving away from us, then light from an earlier part of the motion might even arrive more than half an orbital period before light oming from the intermediate portion of the orbit when it is neither moving toward us or away from us. This would lead to signifiant distortion of the plot of luminosity or of veloity against time. Consider, for example, the ase when we are in the plane a irular orbit of radius R and period P whose entre is a large distane L from us. The relation between time of emission t e and time of observation t o expeted on the basis of Newtonian theory is, sine R << L, t o = t e + L R sin 2πte P + v os 2πte P. (67) In this formula, the quantity v is the extra veloity supposed to be imparted to the photons moving toward us. Aording to the Ballisti theory of light of Newton and Galilei we would apply the usual rules for partiles of speed in 22

the rest frame of the moving stars. Thus we expet v = v orbital = 2πR P, but in general it ould be muh smaller. Sine L >> R and >> v we have t o t e + L R sin 2πt e P The observed Doppler shift is given by + Lv 2 sin 2πt e P. (68) dt 0 = 1 v dt e os 2πt e P + Lv2π 2 P sin 2πt e P. (69) If unless Lv2π P is small there will be signifiant distortion of the light urves. 2 Indeed t o may not be a monotoni funtion of t e, in whih ase, t e will not be a unique funtion of t o. In other words, pulses from different phases of the orbit may arrive on earth at the same time t o. Suh effets have not be seen. De Sitter himself onsidered the binary star β-aurigae. One example sometimes quoted is, the binary star Castor C. It is 45 light years away and has a period of.8 days. The stars have v orbital = 130Kms 1. The effet should be very large, but the light urves of the two stars are quite normal [11]. Using pulsating X-ray soures in binary star systems, Breher [24] was able to onlude that Einstein s seond postulate was true to better than 2 parts in a thousand million v v orbital < 2 10 9. (70) This is ertainly an improvement on Zurhellen, who in (1914) obtained a limit of 10 6 using ordinary binary stars [29]. (ii) The time of travel over equal distanes of gamma rays emitted by a rapidly moving positron annihilating with a stationary eletron an be measured as they are found to be equal [12]. (iii) A similar measurement an be done using the deay of a rapidly moving neutral pion whih deays into two gamma rays [14]. 10.5 Light in a medium: Fresnel Dragging In a medium, the veloity of light is redued to n, where n 1 is alled the refrative index. In general n may depend upon wavelength λ but here we will neglet that effet. Fresnel proposed, in the 1820 s, measuring the speed of light in a stream of water moving with speed u relative to the laboratory. Naive Newtonian theory would give a speed n + u (71) but experiments by Fizeau in 1851 using the toothed wheel method did not agree with this. If we use the relativisti addition formula in the ase that u is small we get instead n + u(1 1 n 2 ) +... (72) 23

whih does agree with Fizeau s experiments. The fator (1 1 n 2 ) is alled Fresnel s dragging oeffiient and had in fat been proposed earlier by the Frenh physiist Fresnel around 1822 using an argument based on wave theory. The experiment was repeated after Einstein had proposed his theory by the 1904? Nobel prize winning Duthman Zeeman ( ). 10.6 Composition of Lorentz Transformations We ould just multiply the matries but there is a useful trik. We define Thus x ± = x ± t. (73) x = 0 we have a rightmovinglightray (74) x + = 0 we have a leftmovinglightray (75) Now Lorentz transformations (50) take the form x + = e θ x + x + = e +θ x +, (76) x = e +θ x + x = e θ x +. (77) We immediately dedue that the inverse Lorentz transformation is given by setting u u, θ θ,i.e. the inverse of (50 ) is Inverse Lorentz Transformations x = x + u t 1 u2 2, t = t + u x 2. 1 u2 2 (78) Now onsider three frames of referene S, S and S suh that we get from S to S by boosting with veloity u 1 and from S to S by boosting with relative veloity u 2. To get from S to S we have to boost with relative veloity u3. If θ 1, θ 2, θ 3 are the assoiated rapidities, we have Thus i.e. x ± = e θ1 x ±, (79) x ± = e θ2 tx ±. (80) x ± = e θ3 x ±, (81) rapidities add θ 3 = θ 1 + θ 2. (82) Using a standard addition formula for hyperboli funtions tanhθ 3 = tanhθ 1 + tanhθ 2 1 + tanhθ 1 tanhθ 2. (83) That is we re-obtain the veloity omposition formula: u 3 + u2 u1 =. (84) 1 + u1u2 2 24

10.7 Veloity of light as an upper bound Suppose that u 1 < and u 2 <, then u 3 <. Proof Sine the hyperboli tangent funtion is a one to one map of the real line onto the open interval ( 1, +1), we have Thus u 1 < < θ 1 <, (85) u 2 < < θ 2 <. (86) < θ 1 + θ 2 < u 3 <. (87) Thus no matter how we try, we annot exeed the veloity of light. 10.8 * Super-Luminal Radio soures* An interesting apparent ase of super-luminal veloities but whih is perfetly expliable without invoking the existene of anything moving faster than light, has been disovered by radio astronomers. What are alled quasars or quasistellar radio soures exhibit jets of matter symmetrially expelled from a dense entral region probably assoiated with a blak hole. For the sake of a simple first look we assume that we an use the geometry of Minkowski spaetime despite the great distanes and that the entral quasar is loated a distane L away from us. We shall also assume that there is a frame in whih both the entral quasar and ourselves are at rest We assume, in the simplest ase possible, that the matter in the jets are expelled at right angles to our line of sight at time t = 0 and therefore at time t = t o the material in the jets has have travelled a distane vt e. Light or radio waves oming from the jets will arrive here at time The angle α subtended is, for small angles t o = t e + 1 L2 + v 2 t 2 e. (88) α = vt e L. (89) The rate of hange with respet to the observation time is We have Thus dα = v dt e. (90) dt 0 L dt o t e = 1 ( t o + t 2 1 v2 o + (L2 2 + t2 v2 ) 0 )(1. (91) 2 2 dt e = 1 ( 1 + dt o 1 v2 v 2 25 t o t 2 o + L 2 ( 1 v 2 1) ). (92)

For large t o we get α = v 1 L 1 v (93) Clearly if v is lose to, then dte dt o an be muh bigger than unity. Thus the size of the effet is muh larger than one s naive Newtonian expetations. If the jet makes an angle with the line of sight we obtain α = v sin θ L 1 1 v os θ (94) The existene of suh apparent superluminal motions was suggested by the present Astronomer Royal in 1966 while a researh student in DAMTP[22]. Just over 4 years later, in 1971, the radio astronomers Irwin Shapiro amd Marshall Cohen and Kenneth Kellerman astronomers found, using very long base line interferometry (VLBI) suh jets, hanging in apparent size over a period of months, in the quasars 3C273 and 3C279. Nowadays the observation of suh apparently super-luminal soures is ommonplae. 10.9 The two-dimensional Lorentz and Poinaré groups Clearly Lorentz transformations, i.e. boosts in one spae and one time dimension, satisfy the axioms for an abelian group (losure under omposition, assoiativity and existene of an inverse) whih is isomorphi to the positive reals under multipliation (one multiplies e θ ) or all the reals under addition (one adds θ). This is ompletely analogous to the group of rotations, SO(2) in two spatial dimensions. The standard notation for the group of boosts is SO(1, 1). If we add in the abelian group of time and spae translation translations t t + t 0, x x + a, (95) we get the analogue of the Eulidean group plane, E(2) whih is alled the Poinaré group and whih may be denoted E(1, 1). 11 The invariant interval Consider two spaetime events (t 1, x 1 ) and (t 2, x 2 ) in spaetime. The invariant interval τ between them is defined by τ 2 = (t 1 t 2 ) 2 (x 1 x 2 ) 2. (96) The name is hosen beause τ 2 is invariant under Lorentz transformations (50). This is beause of the linearity ( ) ( ) ( ) t 1 t 2 oshθ sinhθ t1 t = 2. (97) x 1 x 2 sinhθ oshθ x 1 x 2 26

Now there are three ases: Timelike separation τ 2 > 0 t 1 t 2 > x 1 x 2 In this ase a partile with v < an move between the two events. Lightlike separation τ 2 = 0 t 1 t 2 = x 1 x 2. (98). (99) In this ase a light ray or partile with v = an move between the two events. Spaelike separation τ 2 < 0 t 1 t 2 < x 1 x 2 In this ase no partile with v < an move between the two events.. (100) 11.1 Timelike Separation In this ase there exists a frame S in whih both events have the same spatial position, x 1 = x 2 τ 2 = ( t 1 t 2 ) τ = t 1 t 2, where we have fixed the sign ambiguity to make τ positive. Proof We need to solve for θ the equation ( ) ( ) oshθ sinhθ t1 t 2 = ( t sinhθ oshθ x 1 x 1 t 2 0) tanhθ = x 1 x 2. 2 t 1 t 2 (101) Clearly a real solution for θ exists. Stritly speaking, this is all we an say purely mathematially. However we an say more if we aept the physial lok postulate that a physial lok at rest in frame S would measure an elapsed time t 1 t 2. Then we an identify τ with the time measured by a lok at rest in S and passing between the two events. We all this the proper time between the two events. At this stage it may be helpful to reall the definition of the seond aording to the Bureau International des Poids et Mesures (BIPM) who are responsible for defining and maintaining the International System of Units (SI units). Traditionally 1/86 400 of the mean solar day, it has been sine 1960 had the definition The seond is the duration of 9 192 631 770 periods of the radiation orresponding to the transition between two hyperfine levels of the ground state of the aesium 133 atom. The definition of the metre is formerly defined in 1960 of the wavelength of krypton 86 radiation but in 1983 the BIPM delared that The metre is the length of the path travelled by light in vauum during a time interval of 1/299 792 458 of a seond. Note that not only does the BIPM ompletely aept Einstein s Priniple of the invariane of light but also that the veloity is independent of wavelength. 27

11.2 Spaelike separation In this ase there exist a frame S in whih both events are simultaneous, t 1 = t 2 τ 2 = 1 2 ( x 1 x 2 ) 2 x 1 x 2 = 2 τ 2. Proof This runs along the same lines as above. By analogy with the lok postulate, we assume that τ 2 is the distane measured between the two events in the frame S in whih are both simultaneous. 11.3 Time Dilation Sine (x 1 x 2 ) 2 2 + (t 1 t 2 ) 2 = τ 2, (102) t 1 t 2 = τ 2 + (x 1 x 2 ) 2 2 τ. (103) Thus varying over all frames we see that τ is the least time between the two events as measured in any frame. Moreover t 1 t 2 u ( x 1 x 2 ) t 1 t 2 = + (104) 1 u2 1 u2 2 2 Thus the time between the two events in frame S is τ t 1 t 2 =. (105) 1 u2 2 In other words, moving loks appear to run more slowly than those at rest. 11.3.1 Muon Deay It was first demonstrated by the physiists Rossi and Hall working in the USA in 1941 [36] that one must use time dilation to aount for the properties of elementary partiles alled muons whih arise in osmi ray showers. Cosmi rays, mainly protons, strike the earth s upper atmosphere at a height of about 16Km and reate pions (mass m π = 140 MeV). The pions rapidly deay to muons (mass m µ = 105MeV) and anti-muon neutrinos (mass very nearly zero) π + µ + + ν µ (106) with lifetime τ = 2.6 10 8 s τ = 7.8m. The muons then deay to positrons (mass m e =.5MeV) and eletron and muon anti-neutrinos µ + e + + ν e + ν µ (107) with lifetime τ = 2.1 10 6 s τ = 658.654m. In other words, in this time as measured in the laboratory, a muon, or indeed any other partile, should be able to travel no more than about.66 Km.. How 28

then an it be deteted on the earth s surfae 16 Km. away from where it was produed? The point is that beause of the effet of time dilation the time of deay in the rest frame of the earth is where v is the speed of the muon. It only needs 2.2 10 6 1 s, (108) 1 v2 2 1 > 24, (109) 1 v2 2 i.e. 1 v 1 + v < 1 24 To produe the neessary amount of time dilation. Sine v (110) 1 this requires 11.4 Length Contration Now onsider two spaelike separated events. Sine (1 v ) < 1 1 2 24 2 = 1 1152. (111) (x 1 x 2 ) 2 2 + (t 1 t 2 ) 2 = τ 2, (112) x 1 x 2 = 2 τ 2 + (t 1 t 2 ) 2 2 2 τ 2. (113) Thus the distane between two spaelike separated events is never less than 2 τ 2. In fat if u is the relative veloity of S and S, thus x 1 x 2 = x 1 x 2 + 1 u2 2 x 1 x 2 = u ( t 1 t 2 ), (114) 1 u2 2 2 τ 2 1 u2 2. (115) We all 2 τ 2 the proper distane between the two events. 11.5 The Twin Paradox: Reverse Triangle Inequality Aording to this old hestnut, timorous stay at home Jak remains at rest in frame S for what he thinks is a propertime τ 3, while his adventurous sister Jill takes a trip at high but uniform speed u 1 (with respet S) to the nearest 29

star, Alpha Centauri, (a distane R 4light years aording to Jak) taking propertime τ 1 and then heads a bak, at speed u 2 taking what she thinks is proper time τ 2. Their world lines form a triangle with timelike sides whose proper times are τ 1, τ 2 and τ 3. Jak rekons that the two legs of Jill s journey take times But Evidently we have the t 1 = R u 1, t 2 = R u 2, τ 3 = t 1 + t 2. (116) t 1 = τ 1 1 u2 1 2, t 2 = τ 1 1 u2 2 2. (117) Reverse triangle inequality τ 3 > τ 1 + τ 2. (118) In other words, by simply staying at home Jak has aged relative to Jill. There is no paradox beause the lives of the twins are not stritly symmetrial. This might lead one to suspet that the aelerations suffered by Jill might be responsible for the effet. However this is simply not plausible beause using idential aelerating phases of her trip, she ould have travelled twie as far. This would give twie the amount of time gained. 11.5.1 *Hafele -Keating Experiment* This effet was verified in (1972) [16] in what is alled the Hafele-Keating experiment. Atomi loks were flown around the world in opposite diretions. On their return they had lost, i.e. measured a shorter time, relative to an atomi lok left at rest. The full interpretation of this result is ompliated by the fat that to work this out properly one must also take into aount the Gravitational Redshift effet due to General Relativity. When all is said and done however, a fairly aurate verifiation of the time dilation effet was obtained. Before this experiment, disussion of the twin paradox and assertions that it implied that speial relativity was flawed were quite ommon. Sine The Hafele-Keating experiment, and more reently the widespread use of GPS reeivers, whih depend on the both time dilation and the gravitational redshift, the dispute has somewhat subsided. For an interesting aount of the onfusion that prevailed in some quarters just before the experiment see [15] 11.6 Aelerating world lines If one has a general partile motion x = x(t) with a non-uniform veloity v = dx dt, we get a urve in spaetime with oordinates (t, x(t)). We an work out the proper time dτ elapsed for a short time interval dt by working infinitesimally dτ 2 = dt 2 1 2 dx2, i.e. dτ = dt 1 v2 2. (119) 30

The total proper time measured by a lok moving along the world line is t2 dτ = dt 1 v2 t2 2 dt = t 2 t 1. (120) t 1 From this we dedue t 1 Proposition Among all world lines beginning at t 1 at a fixed spatial position x and ending at t 2 at the same spatial position x, none has shorter proper time than the world line with x onstant. 12 Doppler shift in one spae dimension It was first suggested by Christian Doppler(1803-1853) in 1842 that the olour of light arriving on earth from binary star systems should hange periodially with time as a onsequene of the high stellar veloities [26]. The radial veloities of a star was first measured using this in the winter of 1867 by the English astronomer William Huggins(1824-1910) at his private observatory at Tulse Hill, at that time, outside London. Huggins ompared the F line in the spetrum of light oming from the Dog star, Sirius, and ompared it with the spetrum of Hydrogen in Tulse Hill. The light was shifted toward the red and he dedued that Sirius was reeding from Tulse Hill with a speed of 41.4 miles per seond. Taking into aount the motion of the earth around the sun, he onluded that the radial veloity of Sirius away from the solar system was 29.4 miles per seond. Consider a wave travelling to the right with speed v in frame v. We an represent the wave by funtion Φ(t, x) desribing some physial property of the wave of amplitude A the form where Φ = Asin(ωt kx) = Asin(2π(ft x )), (121) λ the frequeny f = ω 2π and the wavelength λ = 2π k. (122) The speed v of the wave is v = λf = ω k. (123) In frame S ( φ = Asin ω ( t 1 + u2 2 ( x + u t) ) ) k = Asin( ω t k x) (124) 2 1 u2 2 Thus the angular frequeny and wave number in frame S are u x ω = ω ku, k = 1 u2 2 k uω 2 1 u2 2. (125) 31

The quantities ω and k are alled the angular frequeny and wave number respetively. Consider the speial ase of a light wave for whih v = λf = = ω k. One has Thus ω = ω (1 u ) 1 u = ω 1 1 + u u2 2 f = f k = k (1 u ) 1 u2 2 = ω 1 u 1 + u, (126). (127) 1 u 1 + 1 + u, λ u = λ 1 u. (128) Thus if the emitter and reeiver reede from one another the wavelength is inreased and the frequeny is dereased. One says that the signal is redshifted beause red light has a longer wavelength than blue light. If the emitter and reeiver approah one another the signal is blue-shifted. A quantitative measure z alled the red shift is given by 12.1 *Hubble s Law* λ = 1 + z. (129) λ In 1929, the Amerian Astronomer Edwin Hubble disovered that that the universe is in a state of expansion. The light oming from Galaxies, lying outside our own Milky Way with distanes L >.5Mega parses was found to be systematially red shifted rather than blue shifted, moreover the further way they are the greater is their radial veloity v r. Quantitatively, Hubble s law states that z = v r = H 0 L, (130) where H 0 is a onstant of proportionality alled the Hubble onstant. Hubble estimated that H 0 was about 500 Km per seond per Mega parse. Currently H 0 is believed to be rather smaller, about 70 Km per seond per Mega parse. By now, galaxies have been observed moving away from us with redshifts greater than 4. If interpreted in terms of a relative veloity we have 1 + u 1 u = (1 + z) 2 1 u = 2 1 + (1 + z) 2 = 1 13. (131) Note that while a ompletely aurate aount of Hubble s law an only be given using General Relativity, for whih see for example, [34], as long as one is well inside the 7 Hubble radius 6000Mp (132) H 0 7 Often inorretly referred to be astronomers as the horizon sale. 32

and onsiders times muh shorter than the Hubble time 1 H 0 10Giga years (133) then spaetime is suffiiently flat that it is safe to use the standard geometrial ideas of speial relativity. 13 The Minkowski metri In two spaetime dimensions 8 Lorentz transforms leave invariant 2 t 2 x 2 1. Thinking of ( ) t x = (134) as a position vetor in a 2=1+1 dimensional spaetime 9 we an define an indefinite inner produt where what is alled the x 1 x x = 2 t 2 x 2 1 = xt η x (135) Minkowski metri η = is a symmetri matrix 10 Under a Lorentz transformation ( ) +1 0 = η t. (136) 0 1 x Λx = x (137) with ( ) oshθ sinhθ Λ = sinhθ oshθ Now under a Lorentz transformation, i.e. under (137) (138) x x x t Λ t ηλ x = x t η x, x, (139) we must have 11 Λ t ηλ = η. (140) Obviously Λ is analogous to a rotation matrix but it is not the ase that Λ t = Λ 1, as it would be for an orthogonal matrix, rather (in two dimensions) sine Λ 1 (θ) = Λ( θ) Λ t (θ) = Λ(θ). 8 From now on we shall use x 1 for the spae oordinate. This notation is onsistent with the fat that in higher spaetime dimensions there will be more than one spaetime oordinate. 9 From now on an x without a subsript or supersript should be thought of as a olumn vetor. 10 t denotes transpose and we denote this indefinite inner produt with a so-alled entred do. You should distinguished this from a lowered dot. whih will be used for the ordinary dot produt of ordinary vetors in three dimensions as in x.y. 11 Although in two spaetime dimensions Λ = Λ t, this is no longer true in higher dimensions. For that reason we prefer to inlude the transpose symbol t on the first Λ. 33

13.1 Composition of Lorentz Transformations If we do two Lorentz transformations in suession we have where In two spaetime dimensions we have x Λ(θ 1 )x Λ(θ 2 )Λ(θ 1 )x = Λ(θ 3 )x, (141) Λ(θ 3 ) = Λ(θ 2 )Λ(θ 1 ). (142) θ 3 = θ 1 + θ 2 = θ 2 + θ 1, (143) i.e. Lorentz transformations, like rotations, are ommutative. It is not true in general in higher dimensions that either Lorentz transformations or orthogonal transformations ommute. 14 Lorentz Transformations in 3 + 1 spaetime dimensions We now onsider 4-vetors x = ( ) t, (144) x where an ordinary vetor x is onveniently thought of a olumn vetor x = x 1 x 2. (145) x 3 with The Minkowski indefinite inner produt is x x = 2 t 2 x.x = 2 t 2 x 2 = 2 t 2 x 2 = x t η x, (146) 1 0 0 0 0 1 0 0 η =. (147) 0 0 1 0 0 0 0 1 There are differing onventions for the Minkowski metri. Here we have hosen the mainly minus onvention for whih timelike vetors have positive length squred but equally popular for some irumstanes is the mainly plus onvention obtained by hanging η η. A general Lorentz transformation satisfies (137) but in general Λ Λ t. Example Λ = ( ) 1 0, (148) 0 R 34

with R 1 = R t a three-dimensional rotation or orthogonal matrix. The general Lorentz transformation is rather ompliated; it may ontain up to 6 arbitrary onstants. It simplifies if we don t rotate spatial axes. Roughly speaking, three of them orrespond to a general; rotation of the spatial axes and the other three to the three omponents of the relative veloity of the two frames S and S. Example t osh θ sinh θ 0 0 t x 1 sinh θ osh θ 0 0 x = 1. (149) x 2 0 0 1 0 x 2 x 3 0 0 0 1 x 3 In this ase S moves with respet to S along the x 1 axis with veloity u = tanhθ. 14.1 The isotropy of spae The foregoing work impliitly assumes that spae is isotropi and this assumption alls for some omment. The universe we see about us is ertainly not ompletely isotropi. At the level of the solar system and the galaxy, there are gross departures from isotropy. At larger sales, for example the distribution of galaxies, quasars, radio galaxies et there is ertainly good evidene for statistial isotropy but there are signifiant departures from omplete spherial symmetry about us. In partiular, as we shall desribe in greater detail later, the most distant parts of the universe that we have diret optial aess to, the osmi mirowave bakground (CMB) is isotropi only to a part in one hundred thousand or so. This might lead one to postulate that the metri of spaetime should be anisotropi. Physially suh an anisotropy ould manifest itself in at least two, not ompletely unrelated, ways. (i) The speed of light ould depend upon diretion (ii) The dynamis of partiles ould be anisotropi, for example, rather than the masses m of partiles just being salars, they ould be tensors m ij and Newton s seond law might read m ij d 2 x i dt 2 = F i. (150) One learly has to be areful here that one annot eliminate these effets by redefinition of lengths or times. For example if the mass tensor of every partile were proportional then we ould eliminate any interesting effet as far as partiles were onerned by using linear transformations of the spatial oordinates to diagonalize m ij. The same an be said for the motion of light. It only makes sense to say it is anisotropi relative to some hoie of loks, for example aesium loks, otherwise we ould always delare it to be isotropi by hoie of units. We an say the above in a slightly different way and, in doing so, make ontat with some basi ideas in General Relativity. A key ingredient of Speial 35

Relativity is that there is just one metri, and hene just one fundamental speed, whih gives a universal upper bound for the veloity of all types of matter. We say that the Minkowski metri is universal. The ideas whih have just been desribed extend to situations where gravity is important and form the basis of Einstein s Equivalene Priniple 12. Experimentalists have not be slow to test these assumptions and there exist some extremely stringent bounds on departures from isotropy. For example, using rotating interferometers Brillet and Hall [46] found that frational length hanges l = (1.5 ± 2.5) 10 15, (151) l ompletely onsistent with isotropy. Perhaps even more impressive are what are alled Hughes-Drever experiments whih make use of the earth s daily rotation. Thy look for any twenty four hour periodiity in the Zeeman effet. 14.2 Some properties of Lorentz transformations It is lear from the definition (140) that the the omposition Λ 2 Λ 1 of two Lorentz transformations is again a Lorentz transformation. Moreover taking determinants gives det Λ = ±1, (152) Lorentz transformations are (up to a sign) uni-modular. Moreover the inverse is given by 13 Λ 1 = η 1 Λη (153) It follows that Lorentz transformations form a group, alled the Lorentz group, written as O(3, 1) or O(1, 3). If we insist that det Λ = 1, we get the speial Lorentz group SO(3, 1) or SO(1, 3). 15 Composition of non-aligned veloities A partile moves with respet to S with veloity ṽ what are the veloities with respet to S? We have t t t x 1 ṽ = 1 t x 1 = x 2 ṽ 2 x 2 x 3 tṽ 3 x 3 We read off t(osh θ + ṽ1 sinhθ t(v 1 oshθ + sinhθ) ṽ 2 t ṽ 3 t. (154) v 1 = ṽ1 + u 1 + ṽ1u 2. (155) 12 Tehnially, what we are referring to is the Weak Equivalene Priniple 13 We prefer not to use the fat that in a standard orthonormal basis η = η 1 beause this is a basis dependent statement. 36

ṽ 2 ṽ2 1 u2 v 2 = = 2. (156) oshθ + v1 sinhθ 1 + uv1 2 v 3 = ṽ 3 oshθ + v1 15.1 Aberration of Light sinhθ = ṽ3 1 u2 2 1 + uv1 2. (157) Suppose that we have light ray making an angle α with the x 1 axis in frame S. What angle α does it make in frame S? By hoie of axes we an suppose that the ray moves in a x 3 = 0 plane. Thus we put ṽ 1 = os α, ṽ 2 = sin α and ṽ 3 = 0. We get os α + u v 1 = 1 + u os α, v 2 = 1 u2 sin α 2 1 + u (158) os α. We hek that v1 2 + v2 2 = 2 as expeted. Thus we may put v 1 = osα, v 2 = sinα. Moreover tanα = 1 u2 sin α 2 os α + u. (159) Example show that (159) may be re-written as tan( α u 2 ) = + u tan( α ). (160) 2 If φ = tan 1 ( x2 x 1 ) and φ = tan 1 ( x2 x 1 ), then we shall refer to the map S 2 S 2 : (α, φ) ( α φ) given by (160) supplemented φ = φ as the aberration map. Stereographi oordinates are defined by so that ζ = e iφ 1 tan α, (161) 2 dα 2 + sin 2 αdφ 2 dζd ζ = 4 (1 + ζ ζ). (162) 2 We may express the aberration map as a simple dilation of the omplex plane + u ζ = ζ, (163) u whih is learly onformal. It belongs to group of Moebius transformations of the sphere, P SL(2C), whih beome frational linear transformations of the omplex ζ plane ζ aζ + b, ab d = 1, (164) ζ + d with a, b,, d C. 37

15.2 * Aberration of Starlight* If α = π 2 then tanα = u. If u << we get α π 2 u 2. This more or less what one expets on the basis of Newtonian Theory. For the ase of the earth moving around the sun we hoose S be a frame at rest with respet to the sun and S to be one at rest with respet to the earth. Thus u 30Kms 1. We dedue that the apparent positions of stars should hange over a 6 month period with an amplitude of about 10 4 radians. Now there are 360 degrees in a full irle and 60 minutes of ar in eah degree and 60 seonds of ar in eah minute of ar. Thus, for example, there are 21, 600 ar minutes in a full irle. James Bradley, Savilian Professor of Astronomy at Oxford 14 and suessor in 1742 of Edmund Halley as Astronomer Royal developed the tehnology to measure positions to better than an ar minute and was thus able to prove for the first time to prove that the earth moves round the sun. In fat he announed in 1728 [4] that the apparent position of the star Eltanin (γ-draonis) and all adjaent stars partake of an osillatory motion of amplitude 20.4 seonds of ar. The plane of the earth s orbit is alled the plane of the elipti. Spherial polar oordinates with respet to the normal of the elipti are alled right asension (analogous to longitude) and delination, (analogous to latitude and measured from the elestial equator). For stars in the elipti, i.e. with zero delination the apparent motion due to abberation is along a straight line. For stars whose diretion is perpendiular to the elipti, i.e with delination π 2, it is irular. At intermediate delinations it is an ellipse. Remark Bradley s observations do not ontradit Einstein s Priniple of Relativity sine in effet he measured the veloity of the earth relative to what are alled the fixed stars. In the early 1930 s shortly after Edwin Hubble s demonstration of the expansion of the universe, the astronomers Strömberg and Biesbroek working in the USA pointed out that, as expeted aording to Speial Relativity, observations of galaxies believed then to be 70 million light years away, in the onstellation of Ursa Major, whih, by virtue of Hubble s law, are moving away from us at speeds of 11,500 Km per se exhibit the same amount of annual aberration [30, 31] as do nearby stars. In fat, we would now assign these galaxies a greater distane beause they were using Hubbles s value for his onstant 500 Km per seond per Mega parse. This is almost a hundred times larger than the urrently aepted value. Similar remarks have been made by Hekmann [32]. In fat, nowadays astronomers use not the fixed stars but rather about 500 distant stellar radio galaxies to provide a fundamental inertial frame of referene alled the International Celestial Referene Frame (ICRF) entred on the baryentre, that is entre of mass or entroid of the solar system. 14 At an annual salary of 138 5s 9d 38

Remark Bradley was also able to establish that the earths axis nods or nutates. Muh later in the 1830 s the German astronomer Bessel established that the apparent positions of nearby stars alter beause over a 6 month period beause we see them from two ends of a baseline given by the diameter of the earths orbit around the sun. This effet is alled stellar parallax. 15.3 Water filled telesopes Bradley s explanation of aberration gave rise to various ontroversies, partly beause it is diffiult to understand on the basis of the wave theory. In partiular Bosovih suggested that one would obtain a different result if the telesope tube is filled with water, sine the speed of light in water is different from that in vauo or what is almost the same, in air. Indeed Bosovih hoped to measure the speed in water in this way. Eventually, in 1871 the then Astronomer Royal George Biddell Airy put the matter to the test [48]. During the spring and autumn of that year he observed the same star γ Draonis as Bradley had. In Marh the orretion for aberration was -19.66 ar ses while in September it was 19.74 ar ses. These results agreed with those of Bradley, within the errors and showed that the presene of the water was irrelevant. From a modern point of view, this is obvious beause the entire effet is due to the passage to a frame moving with the earth. In that frame light from distant stars enters the telesope tube at the aberrated angle and, provided the surfae of the water is perpendiular to the axis of the telesope tube, it will suffer no refration or further hange of diretion. 15.4 Headlight effet We dedue from (159) that lim α = 0, α π. (165) u Thus a photons emitted by a rapidly moving soure a thrown forward and oupy a very small one around the diretion of motion. This effet has been invoked to explain why the two soures seen in many quasars have suh different apparent brightnesses. Typially the quasars seem to emit from a dense entral soure, possibly a blak hole, two blobs of plasma (i.e.highly ionized gas) in moving n opposite diretions. The idea is that one is moving toward us and one away. Light oming from the latter is highly beamed toward us and hene appears muh brighter than the other, the light of whih is beamed away from us. Relativisti gamma fators γ as higher as 10 are quoted in the astrophysial literature. 15.5 Solid Angles In order to quantify the headlight effet, we note that the aberration map preserves angles but not areas. The infinitesimal area element on S 2 is the same 39

as the solid angle Under the aberration map d 2 Ω = sinαdαdφ. (166) d 2 Ω d 2 Ω = 1 v 2 2 (1 v osα)2 d2 Ω. (167) 15.6 *Celestial Spheres and onformal transformations* The set of light rays passing through the origin O of the frame of referene S and any given time t onstitutes the elestial sphere of an observer at O. The elestial sphere may be oordinatized by a spherial polar oordinate system (α, φ) symmetri with respet to the diretion of motion, i.e. the x 3 axis. Thus φ = tan 1 ( x2 x 3 ). A similar oordinate system ( α, φ) exists for the observer situated at the origin Õ of the frame of referene S. The aberration formulae, i.e.(159) or better (160) and φ = φ provide a map from one elestial sphere to the other. This map allows one to relate the visual pereptions of one observer to the other. A short alulation reveals that infinitesimally dα sinα = d α sin α, dφ = d φ. (168) This implies that aberration map preserves angles. To see why, note that, these would be alulated using the metris or infinitesimal line elements ds 2 = dα 2 + sin 2 αdφ 2, and d s 2 = d α 2 + sin 2 αd φ 2. (169) Thus ds 2 = ( sin α sin α )2 d s 2. (170) 15.7 *The visual appearane of rapidly moving bodies* Formula (168) has a striking onsequene whih was only notied 55 years after Einstein s paper of 1905 by Terrell[18] and by Roger Penrose independently[19]. Previously it had been believed that a something seen as a sphere or a ube in frame S say would, beause of length ontration, be seen as an ellipsoid or a uboid in frame S. The truth is more ompliated, beause from (168) it follows that the aberration map is onformal, it preserves angles.this implies that the ube would appear rotated rather than merely squashed in the diretion of motion. It also implies that a sphere always appears as a sphere. Nowadays, there are a number of simulations, using ray-traing tehniques, of what would be seen for example by a relativisti tram passing in front of the patent offie in Berne. See for example http://www.anu.edu.au/physis/searle 40

15.8 Transverse Doppler effet We onsider a wave moving at the speed of light of the form φ = Asin ω ( t x 1 osα + x 2 sin α). (171) Substitution of the Lorentz transformations gives expressions for the angles in frame S whih are equivalent to the aberration formulae derived earlier to together with the relation ω = 1 u os α ω. (172) 1 u2 2 If α = 0, we reover our previous result. If α = π 2 we find, ontrary to what is predited by Newtonian theory, there is a frequeny hange. To interpret what is going on, we think of the a photon emitted in frame S with frequeny ω e = ω and reeived in frame S with frequeny ω o = ω in a diretion (aording to S) perpendiular to the diretion of motion. We have ω 0 = ω e 1 u2 2. (173) The observed frequeny ω 0 is smaller than the emitted frequeny frequeny preisely be a time dilation fator. This effet was verified experimentally with great preision by Ives and Stillwell in 1938 using the light emitted by moving atoms [10]. 15.9 *The Cosmi Mirowave Bakground* One of the most striking appliations of the transverse Doppler effet formula is to the osmi mirowave bakground or CMB. This was first observed, using a ground based radiometer, by the Amerian physiists Arno Penzias and Robert Wilson in 1965 [37] and led to their Nobel Prize in 1978. They had disovered an almost perfetly isotropi bakground of mirowave photons with a thermal or Plankian spetrum with temperature T 3K. This bath of thermal photons is believed to be spatially uniform and to fill the entire universe and to a reli from an earlier and muh hotter phase alled the Hot Big Bang. In a ertain sense it defines an absolute frame of rest, reminisent of the old aether onept and, superfiially, one might think that this ontradits Einstein s Priniple of Relativity. However, as with the fixed stars observed by Bradley, this is not so. Later observations, using satellite and airraft and balloon borne radiometers observations have shown that the solar system is in motion relative to the CMB. At any given time, the temperature distribution observed T o is not exatly 41

isotropi but varies with angle θ to the diretion of motion as T o (θ) = 1 u2 2 1 uos θ T e, (174) where T e is the temperature one would see at rest. We will derive (174) shortly, in the mean time we note that sine u is small we get a small dipole term T = u osθ T e. (175) The observations[28] exhibit a term of magnitude 3m K in the diretion the onstellation of Herules in the souther hemisphere, delination 7 deg, right asension 11 hours12 min, orresponding to a veloity of about 365Kms 1. In fat it is also possible to detet the earth s annual motion around the sun whih at 30Kms 1 is a fator of ten or so smaller. We turn now to the justifiation of (174). If k is Boltzmann s onstant, a Blak Body distribution of photons at temperature T e has 4fe 2hdf ed 2 Ω e dadt e 1 + exp( hfe kt e ) (176) photons of frequeny f e rossing area da in time dt in frequeny interval df e and solid angle d 2 Ω e, where h = 2π is Plank s onstant. Now using the aberration and Doppler shift formulae derived earlier da is the area perpendiular to the motion and f e d 2 Ω e = f o d 2 Ω 0, f e dt e = f 0 dt 0, (177) we see that the moving observer sees a Plankian spetrum in diretion θ with temperature T 0 suh that whih gives (174). f e T e = f o T o, (178) 16 * Kinemati Relativity and the Milne Universe* The Oxford astronomer Edward Arthur Milne (1896-1950) brother of the hildren s writer Alan Alexander Milne (11882-19560 was dissatisfied with Einstein s theory of gravity, General Relativity and the resultant osmologial models it gives rise to and proposed a theory(kinemati Relativity and a osmologial model of his own now alled the Milne universe. While nowadays his theory has largely been rejeted, his simple osmologial model still provides valuable 42

insights. For us it is interesting beause it illustrates that many of the ideas assoiated with the expanding universe are impliit in speial relativity, and moreover Milne s model a be obtained as an approximation to the behaviour of the exat, and highly ompliated, equations of General Relativity in the limit that the universe is expanding very fast and so that gravitational effets an be ignored. The essential points of Milne s ideas were that (i) Spaetime was the same as Minkowski spaetime E 3,1 but a partiular reation event, let s us pik it as the arbitrary origin of Minkoswki spaetime, was the origin of violent explosion suh that the galaxies are now moving away from it in rapid motion with onstant speed 15. (ii) Astronomial and Atomi, i.e. and laboratory, time measurements need not neessarily agree. In other words, he questioned the lok hypothesis. In order to link these ideas we introdue a set of o-moving oordinates τ, χ, α, φ moving with the galaxies. The oordinate τ is just the proper time along eah galaxy s world line. The oordinates χ, α, φ label the individual galaxies. Beause the galaxies move no faster than light the are onfined to the interior of the future light one of the origin: Eah galaxy uts a surfae of onstant proper time t 2 x 2 = 0. (179) τ = t 2 x 2. (180) one and only one. The surfaes of onstant τ are hyperboloids on whih the Minkowski metri indues a positive definite 3- metri whih is learly invariant under the ation of the Lorentz group O(3, 1). In fat the metris on the surfaes τ = onstant are all proportional to a the fixed metri, say that with τ = 1. The urved 3-dimensional spae τ = 1 t 2 x 2 = 1, (181) is alled hyperboli spae H 3 and is the analogue with negative urvature of the unit 3-sphere S 3 whih has positive urvature. 16.1 *The Foundations of Geometry* Hyperboli spae first arose during the investigations of the Hungarian mathematiian Bolyai and the Russian mathematiian Lobahevsky into the foundations of geometry and in partiular into Eulid s fifth axiom about parallel lines in Eulidean geometry. This states that If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produed indefinitely meet on that side on whih the angle is less than two right angles. 15 i.e. along a straight line in Minkowski spaetime 43

After many years of work by many people Bolyai, Lobahevksy and their followers, were finally able to show that Eulid s axiom is genuinely independent of the other axioms of Eulidean geometry and that there exist three onsistent ongruene geometries, E 3 and S 3 and H 3. In important intermediate step was taken by the Swiss mathematiian and osmologist Johann Heinrih Lambert (1728-1777). 16 Lambert foused attention on a quadrilateral with three right angles, and realized that one ould make three hypotheses about the fourth angle. In self-explanatory terms, these he alled the hypothesis of the right angle, the obtuse angle and the aute angle. Clearly the Lambert himself rejeted the third hypothesis, aording to whih the sum A + B + C of the interior angles of a triangle are less than 2π, beause he realized that the defiit 2π A B C is proportional to the area S of the triangle (A + B + C 2π) = KS, (182) where K is now alled the Guass-urvature of the spae 17 But this would mean that we would have an absolute unit of length. Given that there was no natural value to assign it, not surprisingly perhaps the Frenh philosopher Auguste Calinon suggested in 1893? that it might vary with time. Of ourse, this is exatly what an happen aording to Einstein s theory of General Relativity as was first realized by Friedmann in 1922. An interesting ontribution to this debate was made by the German physiologist and physiist Hermann Helmholtz in 1870. His view of The Origin and Meaning of Geometrial Axioms was that they should be based on the idea of free mobility of rigid bodies. In other words, thinking operationally, the properties spae are what an measured using ideal rigid rods whih an be translated to any point in spae, and, moreover, whih remain rigid when rotated about a, and hene every, point. Consider, for example, a rigid body in ordinary Eulidean spae E 3. This may be rotated about any point still keeping its shape and it an similarly be translated to any point. The set of suh motions 18 onstitute what is alled the Eulidean group E(3) whih may be identified with its onfiguration spae. All those onfigurations related by rotation about some point learly orrespond to the same point in Eulidean spae. By isotropy, the ontinuous rotations about single point onstitute the group SO(3) and so we reover ordinary spae as a oset E 3 = E(3)/SO(3). (183) 16 Lambert shares with Thomas Wright and Immanuel Kant the redit for first reognizing that the Milky way is a roughly flat dis made up of stars in Keplerian orbits about some entral body. Our sun having a period about 250,000 years. The entral body is now known to be a blak holes in the diretion of Sagittarius of whose mass is about 1 million times that of the sun. 17 Gauss atually heked, by surveying, the angle sum for the triangle of sides 69Km,85km and 107km whose verties are the three peaks Inselberg, Broken and Hohenhagen in Germany. 18 That is isometries. 44

Helmholtz thus raised the question: what 6-dimensional groups G exist, with an SO(3) subgroup suh that we an regard spae as G/SO(3). (184) If we make various simplifiations, we arrive at the three possibilities E 3 = E(3)/SO(3) S 3 = SO(4)/SO(3) H 3 = SO(3, 1)/SO(3)/,. (185) From a modern perspetive, Helmholtz s assumption of isotropy is well justified by experiments. Atually we an say more, if we are prepared to aept loal Lorentz invariane. We an re-run Helmholtz s reasoning, but replaing spae by spaetime. If E(3, 1) is the Poinaré group, then Minkowski spaetime E 3,1 is the o-set E 3,1 = E(3, 1)/SO(3, 1). (186) There are three possible, so alled maximally symmetri spaetimes. The other two are alled de-sitter ds 4 and Anti-de-Sitter spaetime AdS 4. Their properties may be explored in detail using the methods of General Relativity. For the present we note that orresponding to Helmholtz s list we have E 3,1 = E(3, 1)/SO(3, 1) ds 4 = SO(4, 1)/SO(3, 1) AdS 4 = SO(3, 2)/SO(3, 1). (187) Toward the end of the nineteenth entury there was an inreasing widespread attitude, that whih of these geometries is the orret one is a matter for astronomial observation. Various attempts were made to determine urvature of spae for example in Germany by Karl Friedrih Gauss and later astronomially by Karl Shwarzshild in 1900, at least 15 years before Einstein s Theory of General relativity. In Shwarzshild s stati universe the spatial geometry was taken, like that of Einstein s Stati Universe onstruted 17 later, taken to have positive urvature. However Shwarzshild differed from Einstein in making the antipodal identifiation on S 3 turning it into real projetive spae 19 RP 3. His motivation for doing so was to avoid the feature, present in the ase of S 3, that all light rays in his world, whih orrespond to all straight lines, passing through a point on S 3 are refoused at the antipodal point. In projetive spaes, distint straight lines interset one and only one and in Shwarzshild s universe any two light rays sent off by us in different diretions would eventually return at the same time in the future. The English philosopher, mathematiian and logiian, and winer in 1950 of the Nobel Prize for Literature, Bertrand Russell(1872-1970) wrote of this period My first philosophial book,an Essay on the Foundations of Geometry, whih was an elaboration of my Fellowship dissertation, seems to me now somewhat foolish. I took up Kant s question, how is geometry possible and deided that it was possible, if and only if, spae was one of the three reognized varieties, one of them Eulidean, the 19 The distintion is preisely the same as between SU(2) and SO(3). 45

other two non-eulidean but having the property of preserving a onstant measure of urvature. Einstein s revolution swept away everything resembling this point of view. The geometry of Einstein s Theory of Relativity is suh as I had delared to be impossible. The theory of tensors, upon whih Einstein based himself, would have been useful to me, but I never heard about it until he used it. Apart form details, I do not think there is anything valid in this early book. Russell went on to write many more books, one of them a popular book was on relativity. 16.2 The Milne metri and Hubble s Law We an write down the Minkowski line element in o-moving oordinates τ, χ, θ, φ by setting ds 2 = dt 2 dx 2 (188) t = τ oshχ,x 1 = τ sinhχsinαosφ,x 2 = τ sinhχsin α osφ,x 3 = τ sinhχosα. (189) On substitution, the Minkowski metri beomes ds 2 = dτ 2 τ 2{ dχ 2 + sinh 2 χ(dθ 2 + sin 2 θdφ 2}. 20 (190) The standard metri on Lobahevsky spae is obtained by setting τ = 1 to obtain the expression inside the brae. Exerise Show that the geodesis of Lobahevsky spae may be identified with the intersetions of the hyperboloid τ = 1 with of timelike two-planes through the origin.(a timelike two plane is one ontaining one, and hene many, timelike vetors). In Milne s model, the galaxies move with the oordinates χ, θ, φ onstant. It follows that in Milne s universe the proper distane between two galaxies at the same proper time from the origin inreases in proportion to τ. This is Milne s explanation for Hubble s law. We shall now show that, using the redshift formula we derived earlier that a photon emitted from one galaxy with frequeny ω e at time τ e and reeived at another galaxy with frequeny ω o at time τ e satisfies ω o ω e = 1 + z = τ o τ e. (191). From our previous work on the redshift in one dimension 1 + z = exp θ (192) 20 You should hek that there is a unique value of τ, α, φ for every event inside the future light one of the origin, exept at the obvious oordinate singularities at α = 0, π. 46

where θ is the relative rapidity between the two frames. From the embedding equations (189), taking one the emitting galaxy to have τ = τ e, χ = χ e and the observing galaxy to τ e, χ = χ e and for both to have the same angular oordinates, we have Thus relative rapidity is and so x e x o = τ e τ o osh(χ o χ e ). (193) θ = χ o χ e. (194) Now, from the metri form (190), a radial light ray satisfies dτ = ±τdχ, (195) τ o τ e = exp θ = 1 + z. (196) In fat (196) is idential to what one would obtain for this metri using the standard rules of General Relativity. For nearby soures, z s small and we get the simple form of Hubble s Law z = θ, τ o τ e = θ. (197) For large redshifts however there are substantial differenes from this simple linear relation. The observational data also exhibit departures from the linear law at high redshifts. At present the onsensus among astronomers is that this departure is inonsistent with the preditions of the Milne model 21. However it should be borne in mind that not many years ago that the onsensus among the same astronomers was that the observations did support the Milne model! 16.3 *Relativisti omposition of veloities and trigonometry in Lobahevsky spae* It was pointed out by Variak in 1911 [38] that the omposition of veloity law, whih reads in vetor notation (v,u) 1 ( γ ) u 1 + u.v u + v (u (u v)), (198) 2 1 + γ 2 u with γ u = 1 1 u2 2, whih, onsidered as a map from B 3 B 3 B 3, where B 3 is the ball of radius in three dimensional Eulidean spae, is in general neither ommutive nor assoiative, has an elegant geometrial desription in terms of the trigonometry of hyperboli spae. Reall that, in onventional spherial trigonometry, one onsiders triangles bounded by portions of great irles. The great irles an be thought of as intersetions of planes through the entre of a sphere of unit radius. It is standard 21 Known in this ontext as a k = 1 Friedmann-Lemaitre-Robertson- Walker low density universe. 47

notation to write a, b, for the lengths of the sides and A, B, C for the three angles, angle A being opposite side a et. All relevant formulae an be derived from the basi relations osa = osbos + sin b sinosa, et, (199) first apparently derived by the Arab prine and astronomer Mohammad ben Gebir al Batani (830-929) 22 known in Latin as Albategnius. A onise derivation of Albategnius s formulae is provided by ontemplating the vetor identity (r s).(t u) = (r.t)(s.u) (r.u)(s.t), (200) whih is equivalent to the tensor identity ǫ irs ǫ ibu = δ rt δ su δ ru δ st. (201) Now let r = n a, s = n b, t = n, u = n a, where n a is a unit vetor from the origin to the vertex A et. We now return to relativity. Consider three frames of referene S a,s b and S. Assoiate with eah a pseudo-orthonormal basis, also known as a tetrad or vierbein or repère mobile whose time like legs are a e 0 et. By the results of the previous setion these three timelike legs may be thought of the three verties of a triangle in the two-dimensional hypberboli plane. The sides of the triangles are geodesis orresponding for example to the two-plane spanned by a e 0 and be 0. The lengths of the sides, labeled a, b, of the triangle are related given by the relative rapidities: osha = b e 0 e 0 et. (202) Thus the formulae for omposition of boost may be interpreted as giving the lengths of the sides of a hyperboli triangle. In order to obtain the formula, we need only modify the proof given above. In three dimensional Minkowski spae we an still define a ross produt: We an also see that Thus for Minkowski 3-vetors (u v) µ = ǫ µνλ u ν v λ. (203) η µν ǫ µαβ ǫ νστ = η ασ η βτ η ατ η βσ. (204) (r s) (t u) = (r t)(s u) (r u)(s t). (205) If we substitute r = t s = u and both r and s are timelike and r s is spaelike we see that r s 2 = (r r)(s s)(osh 2 θ 1) (206) 22 All dates in these notes are CE, i.e. Christian era 48

that is Our desired formula is thus r s = r s sinh θ. (207) osha = oshb osh sinhbsinhosa (208) Note that formally, one may obtain (208) from the Albategnius s formula by analyti ontinuation a ia whih may be interpreted as passing to the ase of imaginary radius. In fat, one may ask whether hyperboli geometry, like spherial geometry, an be obtained by onsidering a surfae embedded in ordinary Eulidean spae E 3. The answer turns out be no. However, as we have seen there is no diffiulty in obtaining it form a surfae in three-dimensional Minkowski spaetime E 2,1. 16.4 Parallax in Lobahevsky spae Consider an equilateral hyperboli triangle ABC whose sides AC and AB are equal and whose side AB has length 2r.As the point C reedes to infinity the angles CAB = CBA tend to a onstant value Π(r) alled the angle of paralleism whih depends on the distane r. The formulae of the previous setion may be used to derive the relation sinπ = 1 oshr. (209) Thus for small r we obtain the Eulidean value π 2 but in general the angle of parallism is less than π 2. Now suppose that A and B are the positions of the earth on its orbit around the sun at times whih differ by six months. If a star is situated at S somewhere on the line OC, where O is the midpoint of the side AB, and if at C there is some muh more distant star, then the angles CAS,SAB and CBS and SBA may be measured. In the Eulidean ase, one has CAS + SAB = π 2. (210) The angle π 2 SAB = is alled the parallax and it may be used to estimate the distane of the star OS in terms of the radius of the earth s orbit. This was first done by Bessel in 1838 for the star 61 Cygni and he found a parallax of 0.45 seonds of ar. Astronomers say that 61 Cygni is situated at a distane of.45 parses. Lobahevsky, not believing that spae was Eulidean, attempted unsusefully to measure the urvature of spae by measuring the angle of parallism Π = CAS + SAB (211) for the star Sirius some time before Bessel. Later Shwarshild repeated this attempt for other stars and obtained the lower bound of 64 light years. In fat, if Π = π 2 δ and δ is small, then δ r, (212) 49

where r is the ratio of the earth s radius to the radius of urvature of Lobahevsky spae. The International Celestial Referene System is aurate to no better than.05 milli-ar-seonds. Thus one ould expet to get a lower bound for the urvature of spae in this way using present day tehnology, no better than about 10 5 parses. 17 *Rotating referene frames* We know both from elementary experiene and from Newtonian dynamis that we an tell by means of loal experiments if the referene frame we are using is rotating with respet to an inertial frame of referene, for example that determined by the fixed stars. As Newton himself observed, the water in a rotating buket at rest on a rigidly rotating platform rises up the sides due to apparent entrifugal fores. No trip to Paris is omplete without a visit to the Panthéon to view Fouault s pendulum demonstrating the rotation of the earth, even on those days when the skies are overed by loud and astronomial methods are not available. Thus we do not expet the Minkowski metri of flat spaetime, when written in o-rotating oordinates t, z, ρ, φ to take the same form ds 2 = dt 2 dz 2 dρ 2 ρdφ 2, (213) as it does in non-rotating ylindrial polars t, z, ρ, φ. For partiles in uniform rigid rotation about the z-axis, φ = φ + ωt, where ω is the rate of rotation in radians per se, and φ, ρ, z labels the partiles. Their veloity, relative to an inertial frame is v(ρ) = ωρ. Substitution and ompletion of a square gives the so-alled Langevin form of the flat Minkowski metri ds 2 = (1 ω 2 ρ 2 ) ( dt ρ2 d φ ) 2 dz 2 1 ω 2 ρ 2 dρ 2 ρ 2 1 ω 2 ρ 2 d φ 2. (214) Clearly the Langevin form of the metri breaks down at the veloity of light ylinder ρ = ω 1. For ρ > ω 1 the partiles on the platform would have to travel faster than light. Any physial platform must have smaller proper radius than ω 1. Inside the veloity of light ylinder, ρ = ω 1, the metri is independent of time. Thus all distanes are independent of time. In other words, the system really is in a state of rigid rotation. 17.1 Transverse Doppler effet and time dilation We an read off immediately that for a partile at rest on the platform, dτ = 1 ω 2 ρ 2 dt. (215) Thus means that a signal of duration dt onsisting of n pulses sent from ρ = ρ e with frequeny f e, so that f e = n dτ e will be reeived at ρ = ρ e with frequeny 50

f o = n dτ o. Thus f 0 f e = 1 ω 2 ρ e 1 ve 1 ω 2 = 2 ρ o 1 vo 2. (216) Thus, for example, photon emitted from somewhere on the platform and absorbed at the entre will be redshifted. In fat this is just the transverse Doppler effet in a different guise. The effet was first demonstrated experimentally in 1960 by Hay, Shiffer, Cranshaw and Egelstaff making use of the Mössbauer effet for a 14-4 KeV γ rays emitted by a Co 57 soure with an Fe 57 absorber [33]. 17.2 *The Sagna Effet* The Langevin form of the flat Minkowski metri is invariant neither under time reversal nor reversal of the o-moving angle φ but it is invariant under simultaneous reversal of both. This gives rise to a differene between the behaviour of light moving in the diretion of rotation ompared with that moving in the opposite diretion. The effet is really rather elementary but it has given rise to onsiderable disussion. A light ray, passing along a light pipe for example, satisfies where dt = ρ2 ωd φ 1 ω 2 + dl (217) ρ2 dl 2 = dr2 + dz 2 1 ω 2 ρ 2 + ρ 2 d φ 2 (1 ω 2 ρ 2 ) 2. (218) If the light ray exeutes a losed urve C, as judged on the platform in the pro-grade (in the diretion of rotation d φ > 0) it will take a longer t = than the time taken t if it traverses the urve in the retro-grade sense (d φ < 0). The time differene between these times is t + t = 2ω C ρ 2 d φ 1 ω 2 ρ 2 = 2ω D ρdρd φ (1 ω 2 ρ) 2 (219) with D = C. If the urve C is well inside the veloity of light ylinder and so ω 2 ρ 2 << 1, we find t + t = 4ωA, (220) where A is the area of the domain D enlosed by the urve C. The differene between the two travel times really has nothing muh to do with Einstein s theory of Speial Relativity; it may be asribed to the simple fat that light has to travel further in one diretion than in the other. The effet is now usually named after Sagna. If C is taken to be around the equator, at sea level t + t = 414.8ns. This is substantial and must be taken into aount in the alibration of GPS reeivers. 51

The Sagna effet was first demonstrated using interferometry by Harres in 1911 and Sagna in 1913 [35]. The method was later used to measure the absolute rotation rate of the earth by Mihelson and Gale in 1925. Using an optial loop 2/5 miles wide and 2/5 mile long, they verified the shift of 236/1000 of a fringe predited. Current laser tehnology allows the measurements of 0.00001 degh 1. Of the experiment, Mihelson said Well gentlemen, we will undertake this, although my onvition is strong that we shall prove only that the earth rotates on its axis, a onlusion whih I think we may be said to be sure of already 17.3 Length Contration It is a striking mathematial fat is that if we use dl 2 in (218) as a spatial line element in the dis, i.e. set dz = 0 we find that dl 2 = 1 4ω 2 ( dχ 2 + sinh 2 χd φ 2), (221) where ωρ = tanh( χ 2 ). The urved metri in brakets in (221) is that of the unit pseudo-sphere, i.e. two-dimensional hyperboli spae with onstant Gauss urvature = 1. However the physial line element or proper distane on the rotating dis, is not (218) but rather ( from (214) ds 2 = dρ 2 + ρ2 d φ 2 1 ωρ 2. (222) The metri (222) is also urved. Note that radial diretions, orthogonal to the motion agree with those in an inertial frame, but, as expeted, irumferential distanes are inreased, relative to those in an inertial frame by a fator 1 1 v2 (ρ). (223) Beause of this inrease, it is not possible to embed isometrially the surfae with oordinates ρ, φ and metri (222) as a surfae of revolution in ordinary Eulidean 3-spae E 3. 17.4 Mah s Priniple and the Rotation of the Universe The Austrian physiist and philosopher Ernst Mah was muh exerised by ideas of absolute motion and absolute rest. In effet he pointed out that inertial frames of referene in whih Newton s laws hold, and those at rest with the fixed stars need not neessarily oinide. In fat they do, to high auray, and this requires some sort of explanation. Nowadays this an be provided using the theory of Inflation. This requires developing some General Relativity, but it is possible to use Speial Relativity to quantify the extent of the agreement. 52

The English physiist Stephen William Hawking, 17th Luasian professor pointed out that if the universe were rotating about us, then distant light reeived here should suffer a diretion dependent transverse Doppler shift. The photons whih have travelled furthest in their journey toward us are part of the Cosmi Mirowave Bakground (CMB). If the universe were rotating with angular veloity ω we would expet a temperature variation with angle of the form T 0 T =, (224) 1 ω2 r 2 os 2 θ 2 where θ is the angle made by the line of sight with the axis of rotation. Roughly speaking we may take r as the age of the universe and so the angle φ turned through in that time, in radians, is related to the maximum variation of temperature aross the sky by φ = 2 T T. (225) Given that the measured temperature flutuations are about one part in 10 5, the universe annot have rotated by more than one hundredth of a turn sine its beginning. 18 General 4-vetors and Lorentz-invariants In general we set ( ) v0 v =, et, and v v u = v t ηu = u t ηv = v 0 u 0 v.u. (226) One may hek that v w is a Lorentz invariant using (137) but it may also be seen from the following elementary but illuminating Proposition If v and w are 4-vetors then v u = u v is a Lorentz invariant. Proof Evidently (v + u) (v + u) = v v + v u + u v + u u = v v + u u = 2u v. (227) The left hand side is Lorentz-invariant and the first two terms on the left hand side are Lorentz. Thus u v is Lorentz-invariant. In other words we an a alulate with the Minkowski inner produt in the same way we would for any quadrati form. 18.1 4-veloity and 4-momentum The world line of a partile in spaetime is a urve f : λ x(λ) and is speified by giving its spaetime oordinates t = t(λ) and x = x(λ) as a funtion of some parameter 23 λ along the urve. One might thing it more natural to 23 In this setion λ has nothing to do with wavelength. 53

desribe the motion by giving x as a funtion of t, and indeed this is possible for the partiular hoie λ = t but this is distinguishes the time oordinate from the spatial oordinates but as we have disovered this is against the spirit of Relativity. Moreover, as we shall see, there are advantages in not making that hoie. 18.2 4-veloity We an then define the tangent 4-vetor of the urve f by T(λ) = dx dλ = ( dt dλ dx dλ ). (228) If we insist that under a Lorentz transformation the parameter λ is unhanged λ λ, the T will transform under Lorentz transformations (137) as a 4-vetor. Note that this would not be true if we made the hoie λ = t. Suppose that T T > 0, then the urve f is said to be timelike and we an make the hoie λ = τ where τ is proper time along the world line. We then define the veloity 4-vetor often alled the 4-veloity by U = dx dτ = It follows from the definition of proper time that i.e. ( dt ) dτ dx. (229) dτ U U = 2 ( dt dτ )2 ( dx dτ )2 = 2 ( dτ dτ )2, (230) U U = 2. (231) In units in whih = 1, U is a unit timelike vetor. We shall always assume that t is a stritly monotonially inreasing funtion of λ, i.e. T 0 > 0, whih in the timelike ase means that U 0 > 0. Suh a timelike tangent vetor is alled future direted 18.3 4-momentum and Energy This is defined by p = m U, (232) where m is a positive onstant alled the rest mass of the partile. If U is future direted then 4-momentum will also future direted, p 0 > 0. We have p p = m 2 2. (233) Now p 0 = mdt dτ = m E := 1, v2 2 p = mdx dτ = mv. (234) 1 v2 2 54

In other words p = ( E p ). (235) The quantity E is the energy of the partile, as will be justified shortly. Now thus and moreover p p = E2 2 p2 = m 2 2, (236) E = m 2 4 + p 2 2. (237) p = E v. (238) 18.4 Non-relativisti limit For small v2 2 we have up to terms of O(v 3 ), p = mv +..., E = m 2 + 1 2 mv2 +.... (239) We all m 2 the rest mass energy. Note that E is non-zero even if the partile is at rest. Therefore it is reasonable to define the kineti energy by 18.5 Justifiation for the name energy If we suppose a general equation of motion of the form then the rate of doing work W on the partile is That is dw dt In the speial ase that we set T = E m 2. (240) dp dt = F (241) = F.v = v. dp dt = d (v.p) pdv dt dt. (242) dw = d(v.p) p.dv. (243) p = mv 1 v2 2, (244) we find dw dt = d dt ( mv 2 ) mv. v (245) 1 v2 1 v2 2 2 55

That is = d ( mv 2 + m 2 1 v2 ) de = dt 1 2 dt. (246) v2 2 F.v = de dt. (247) It is reasonable therefore to regard E or T = E m 2 as the energy of the partile. In fat we usually inlude the rest mass energy m 2 in the energy beause in energeti nulear proesses in whih partiles deay into other partiles of different rest masses, for instane, the rest mass term must be inluded in the total energy budget. That E really is the type of energy you might pay your eletriity bill to aquire gas been demonstrated by timing rapidly eletrons to find their veloity and then absorbing them into a alorimeter to measure their energy in alories. Sadly for those who dream of perpetual motion, Einstein s formula (237) was verified [13]. 18.6 *Hamiltonian and Lagrangian* Quite generally in dynamis, for example when onsidering exitations in ondensed matter systems, we define what is alled the Hamiltonian funtion H(p) of a free partile whose spatial momentum is p by dh = v.dp,. (248) If the system is to be onservative, H must be an exat differential and we have H(p) = W. (249) We also define the Lagrangian funtion L(v) of a partile with spatial veloity v as the Legendre transform of the Hamiltonian, i.e., so that from (248, 243) L + H = p.v, (250) v = H p, In the speial ase of a relativisti partile one has H = m 2 4 + 2 p 2, p = L v. (251) L = m 2 19 Partiles with vanishing rest mass 1 v2 2. (252) Einstein s theory allows for the possibility of partiles whih annot be treated in Newton s mehanis, those whose speed is stritly onstant.the onstant value 56

of the speed an only be, by Einstein s Priniple of Relativity, exatly that of light. It turns out that by using only momentum p and energy E as the basi variables the basi equations still make sense if we set m = 0. We have in general v = E v and so if v =, we have Now sine we have E = p (253) p = ( E p ), (254) p p = 0. (255) Partiles of this type inlude the photon whih is responsible, aording to quantum eletro-dynamis, for eletromagneti phenomena and more speulatively the graviton whih, aording to quantum gravity, is responsible for gravitational phenomena. In addition there are three types of neutrinos, ν e, ν µ, ν τ assoiated with the eletron, muon and tau partile respetively. 19.1 Equality of photon and neutrino speeds At 7:35:40 UT 24 on 23 February 1987 eletron neutrinos ν e from the Large Magelllani Cloud arrived in Japan and were deteted using the KAMIOKANDE neutrino telesope. By 10:38 UT the same day, the first optial brightening of what is now known as the supernova SN1987A were seen. Thus the travel time for neutrinos and photons (160,000 years) differed by less than 3 hours. It follows that their speeds differ by less than two parts in an Amerian billion (10 9 ) [17]. For most purposes therefore one may regard neutrinos as being massless,like the photon. Other experiments however based on neutrinos arriving here on earth from the sun indiate that they do have a very small mass, of the order 10 4 ev. Example If the neutrino atually has a mass m and energy E, and the SN1987 is at a distane L from us, then if T γ = L is the transit time of the photon and T ν = L v that of a neutrino, we have m 2 E = (T ν T γ )(T ν + T γ ) Tν 2. (256) 20 Partile deays ollisions and prodution 20.1 Radioative Deays Perhaps the simplest proess one may onsider is the deay of a partile of rest mass m 1 into partiles of rests mass m 2 and m 3. To get a Lorentz-invariant law 24 Universal time, i.e. almost exatly Greenwih Mean Time, GMT 57

of deay we express it in terms of 4-vetors. The simplest possibility would be p 1 = p 2 + p 3. (257) The omponents of (257) give four equations: the onservation of Energy E 1 = E2 + E3, and Momentum p 1 = p 2 + p 3. (258) Suppose the deaying partile is at rest in frame S. Then p 1 = (m 1, 0) p 2 + p 3 = 0, (259) and hene p 2 = p 3 = p. The two partiles produed move off with equal and opposite momentum. We also have E 1 = m 1 2 = m 2 2 4 + p 2 2 + m 2 3 4 + p 2 2 m 1 m 2 + m 3. (260) Partile 1 an only deay into partile 2 plus partile 3 if its rest mass exeeds the sum of the rest masses of the produts. If this is true, then a solution for p always exists. Put another way, the kineti energy liberated T = T 2 + T 2 = (m 1 m 2 m 3 ) 2 and this must be positive and this must ome from the original rest mass energy. In general one expets that unless there is some reason, for example a onservation law like that of eletri harge, that heavy partiles will always be able to deay into lighter partiles. Only the partile with least rest mass an be stable. This is the eletron. It ould, in priniple deay into two photons, but photons arry no eletri harge and hene this is impossible. 20.2 Impossibility of Deay of massless partiles Suppose that with p 1 massless, i.e. It follows that p 1 = p 2 + p 3, (261) p 1 p 1 = 0. (262) p 2 2 + p 2 3 + 2p 2 p 3 = 0 = m 2 2 2 + m 2 3 + 2p 2 p 3. (263) But this is impossible beause if for p 2 and p 3 future direted timelike or lightlike To prove (264) note that p 2 p 3 0. (264) p 2 p 3 = E 2E 3 2 p 2.p 3 = m 2 1 2 + p 2 2 m 2 3 2 + p 2 3 p 2.p 3 (265) 58

but the left hand side is bounded below by p 2 p 3 p 2.p 3 0, (266) sine, by the usual Shwarz inequality, v.u v u, for any pair of 3-vetors v and u. Note that equality an only be attained if p 2 and p 3 are two parallel lightlike vetors. However the deay of a photon for example into two ollinear photons should perhaps be better thought of as superposition. Moreover in quantum mehanial terms it is a proess with vanishing small phase spae volume. 20.3 Some useful Inequalities For a timelike or light 4-vetor v we define v = v v. The working in the previous subsetion an be re-arranged to show Proposition(Reverse Shwarz Inequality) If p 2 and p 3 are two future direted timelike or lightlike vetors, then Similarly p 2 p 3 p 2 p 3. (267) Proposition(Reverse triangle Inequality) if p 1, p 2 and p 3 are all timelike or light like sides of a triangle then p 1 p 2 + p 3. (268) The reverse triangle inequality an be extended to give Proposition If p 1, p 2,...p n+1 are future direted timelike or null and p 1 = p 2 +... + p n+1 (269) then p 1 p 1 +... + p n+1,. (270) A slight extension of the same working yields PropositionIf p q 0 for all future direted timelike and lightlike 4-vetors p, then q is future direted timelike or lightlike. Finally Proposition(Convexity of the future light one) If p and q are future direted and timelike then so is any positive linear ombination ap + bq, a 0, b 0. 59

20.4 Impossibility of emission without reoil A partile of rest mass m annot emit one or more partiles keeping its rest mass onstant. Thus free eletron annot emit one or more photons. To see why not, suppose p 1 = p 2 + p 3, (271) with p = p 2 and p 3 future direted and timelike. We get p 2 1 = p2 2 + p2 3 + 2p p 3, p 2 3 + 2p 2 p 3 = 0, (272) whih is impossible. This has an appliation to so alled bremsstrahlung radiation emitted by an aelerated eletron. This an only our if there is some other body or partile to take up the reoil. 20.5 Deay of a massive partile into one massive and one massless partile For example one ould onsider pion( m π = 140 Mev) deaying into a muon (m µ = 105 Mev) and an anti-muon-neutrino. We have p 2 2 = 0, thus This gives π µ + ν µ (273) p 1 = p 2 + p 3, (p 1 p 3 ) 2 = 0 = p 2 1 + p 2 3 2p 1 p 3. (274) m 2 1 2 + m 2 3 2 = 2( E 1E 2 2 p 1.p 3 ). (275) Suppose we are in the rest frame of partile 1 so p 1 = (m 1, 0) t. We get m 2 1 2 = m 2 3 = 2m 1E 3, E 3 = m2 1 + m2 3 2m 1 2. (276) The relativisti gamma fator of the third partile is given by γ = 1 E 3 = 1 m v2 3 2 = 1 2 (m 1 + m 3 ). (277) m 3 m 1 2 20.6 Deay of a massive partile into two massless partiles For example one might onsider the deay of a neutral pion (m pi 0 = 135MeV) into two photons ( symbol γ) with lifetime τ = 8.4 10 17 s. π 0 2γ (278) 60

We have (beause p 2 2 = p2 3 = 0) Thus p 1 = p 2 + p 3, p 2 1 = 2p 2 p 3. (279) m 2 1 2 = 2( E 2E 3 2 p 2.p 3 ). (280) Now for a massless partile E 2 p 2 = n 2 (281) where n 3 is a unit vetor in the diretion of motion. Thus if n 2.n 3 = osθ, where θ is the angle (not rapidity!) between the diretions of the two massless partiles, we have Thus There are two simple ases m 2 1 4 = (1 n 2.n 3 ) = (1 osθ) = 2 sin 2 ( θ ). (282) 2E 2 E 3 2 sin 2 ( θ 2 ) = m2 1 4E 2 E 3. (283) (i) Partile 1 deays from rest. We have E 2 = E 3 = 1 2 m2 1 2. This implies θ = π. (ii) Partile 1 is moving along say the 1-axis with relativisti gamma fator 1 γ = and the two photons are emitted symmetrially making an angle 1 θ v2 2 2 to the 1-axis. We have E 2 = E 3 = 1 2 m 1 2 γ, whene sin 2 ( θ 2 ) = 1 γ 2. (284) If partile 1 is moving very fast, γ >> 1 then θ will be very small. This is the headlight effet. 21 Collisions, entre of mass We have in general p 1 + p 2 = p 3 + p 4. (285) The 4-vetor p + p = m om U, with U U = 2 defines the entre of mass energy, that is m om 2 is the total energy in the entre of mass frame, i.e. in the frame in whih p 1 + p 2 = 0. Th relation (p 1 + p 2 ) 2 = (p 3 + p 4 ) 2 = m 2 om 2 (286) often brings simplifiations to the algebra. 61

As an example, if a partile of mass m and 3-momentum p ollides with a partile of mass m whih is at rest, then m om = 2 E 2 M + M2 + m 2, (287) where E = m 2 4 + p 2 2 is the energy of the inident partile. The quantity m om 2 is usually taken as a measure of the available energy in suh ollisions. For large E it is muh smaller than E, rising as E 1 2 rather than linearly with energy. This is beause so muh energy must go into the reoil. An extreme ase is provided by Ultra-High Energy Cosmi rays. These hit protons (mass 938MeV) in the upper atmosphere with energies up to 10 21 ev = 10 9 TeV. They are by far the fastest partiles known to us. However energy available for nulear reations is no more than about 10 3 TeV. 21.1 Compton sattering In this proess, for the disovery of whih Compton was awarded the Nobel prize in 1927, an X-ray photon is sattered of an eletron whih is initially at rest. We have p 2 1 = p 2 3 = 0, p 2 2 = p 2 4 = m 2 e 2, p 2 = (m e, 0) t. Now This gives (p 1 p 3 ) = p 4 p 1, (p 1 p 3 ) 2 = (p 4 p 2 ) 2. (288) 2p 1 p 3 = 2m 2 e 2 2p 4 p 2. (289) This gives, using the method for photons used earlier But energy onservation gives Substitution and simplifiation gives 2 E 1E 3 2 (1 osθ) = 2m 2 e 2 2E 4m e 2 (290) E 4 = E 1 + m e 2 E 3. (291) (1 osθ) = m e 2( 1 E 3 1 E 1 ). (292) Aording to quantum theory E 1 = hf 1 = h λ 3, where h is Plank s onstant, f 1 is the frequeny and λ 1 the wavelength of the inident photon and f 3 and λ 3 that of the sattered photon. We get (1 osθ) = m e h (λ 3 λ 1 ). (293) Clearly the wavelength of the sattered photon is longer than that of the inident photon beause kineti energy has been imparted to the eletron. 62

21.2 Prodution of pions Protons in osmi rays striking the upper atmosphere may produe either neutral (π 0 ) or positively harged (π + ) pions aording to the reations p + p p + p + π 0 or p + p p + n + π + (294) respetively, where n is the neutron. Sine the mass of the proton is 1836.1 times the eletron mass and that of the neutron 1838.6, whih is why the latter an deay to the former aording to the reation n p + e + ν e (295) in about 13 minutes, we ignore the differene and all the ommon mass M. Numerially it is about 938MeV. Despite the fat that the mass of the neutral pion π o is 264 times the eletron mass and that the harged pions π ± are both 273.2 times the eletron mass, the latter annot deay into the former by onservation of eletri harge. In either ase, we all the mass m. Its value is about 140Mev. If T is the kineti energy of the inident proton and p is its momentum, then equating the invariant (p 1 + p 2 ) 2 = (p 3 + p 4 + p 5 ) 2 and using the inequality p 3 + p 4 + p 5 (m 3 + m 4 + m 5 ), (296) we get (T + 2M 2 ) 2 2 p 2 (2M + m) 2 4. (297) Using what is sometimes alled the on-shell ondition ( E + M)2 p 2 = M 2 2, (298) one finds that the T 2 terms anel and one obtains the threshold This is about 290Mev in the present ase. 21.3 Creation of anti-protons T 2m 2 (1 + m ). (299) 4M If a proton p ollides at suffiiently high speed against a stationary proton a proton-anti-proton pair an be reated as in the following reation p + p p + p + (p + p) (300) ( p denotes an anti-proton). One might have thought that the least kineti energy T required for this proess is 2m p 2, but this is not orret. Most of the inident kineti energy goes into the kineti energy of the reoiling proton. In fat the threshold, i.e., the minimum energy required, is 6m p 2 = 5.6MeV. To see why, note that 4-momentum onservation gives p 1 + p 2 = p 1 + p 2 + p 3 + p 4 (301) 63

thus p 1 + p 2 = p 1 + p 2 + p 3 + p 4 p 1 + p 2 + p 3 + p 4 = 4m p. (302) (we use the fat that m p = m p.) Now p 1 = (m p, 0) and p 2 = ( E,p), where E is the total energy, inluding rest mass energy of the inident proton and p is 3-momentum. Thus But Simplifying gives p 1 + p 2 = ( E + m p) 2 p 2 4m p 2. (303) p 2 = E2 2 m2 p 2. (304) E 7m p 2, T 6m p 2. (305) The first prodution of anti-protons on earth was ahieved by Chamberlain and Segré at the Berkley Bevatron in California. This linear aelerator was built to be apable of aelerating protons up to energies of 6.6 BeV 25 Chamberlain and Segré reeived the Nobel Prize in 1959 for this work. 21.4 Head on ollisions In this ase we have p 1 = (M, 0) t p 2 = ( E, p, 0.0)t (306) with By head on we mean E 2 2 p 2 = m 2 4. (307) with p 3 = ( E 2, p M, 0, 0) t p 4 = ( E 4, p M, 0, 0) t (308) E 3 2 2 p m 2 = m 2 2, and E 4 2 2 p M 2 = M 2 2. (309) The two onservation equations are E + M = 2 p 2 m + 4 p 2 m + 2 p 2 M + 4 M 2, p = p m + p M. (310) Our strategy is to eliminate p m and solve for p M in terms of E. This leads to some moderately heavy algebra so we will go through all the steps, if only to illustrate the superiority of 4-vetor methods. We set = 1 during the intermediate stages of the alulation. 25 A BeV is nowadays alled a Gev. 64

Using momentum onservation, the energy onservation equation gives E + M p 2 M + M2 = (p p M ) 2 + m 2. (311) Squaring gives (E + M) 2 2(E + M) p 2 M + M2 + p 2 M + M2 = (p p M ) 2 + m 2. (312) Thus E 2 + 2ME + 2M 2 + p 2 M 2(E + M) p 2 M + M2 = p 2 2p m p + m 2. (313) But E 2 = p 2 + m 2, and hene, 2EM + 2pp M + 2M 2 = 2(E + M) p 2 M + M2. (314) Dividing by two and squaring one more gives M 2 (E +M) 2 +2M(M +E)pp M +p 2 p 2 M = (E +M)2 M 2 +(E +M) 2 p 2 M. (315) Thus, taking out a fator of p M, with Finally we get, restoring units, 2M(E + M)p = ((E + M) 2 p 2 )p M (316) p M = 2Mp( E 2 + M) 2M E 2 + M 2 + m 2, p m = p(m 2 M 2 ) 2M E 2 + M 2 + m 2, (317) E = m 2 4 + 2 p 2. (318) You should hek that in the non-relativisti limit,, one reovers the usual Newtonian formulae. Just as in that ase, if the inident partile is more massive than the partile it hits (i.e.m > M ) it moves forward after the ollision, while if it is less massive it reverse its diretion. If the two partiles are perfetly mathed, (i.e. m 2 = M 2 ) all the inident energy will be transferred to the target partile. By ontrast if, for example, the target is very massive, (i.e. M ), the inident partile is refleted bak with the same speed it arrives with. If the inident energy is very large ompared with the rest masses of both itself and the target then p M p, p m (m2 M 2 ). (319) 2M All the inident momentum is transferred to the target. 65

21.5 Example: Relativisti Billiards It is well known to players of billiards that if ball is struk and the proess is elasti, i.e. no energy is lost, and the ollision is not exatly head-on, then the two balls move off at an angle θ = 90 deg between eah others diretion, as seen in the frame of referene of the table. In relativisti billiards this is not so. The angle θ depends on the ratio of the energies imparted to the two balls and the inident kineti energy T. If the balls have mass m and eah emerges form the ollision arrying the same energy, then one finds 2 sin 2 θ 2 = 4m2 T + 4m2. (320) If T << m 2 we reover the non-relativisti result. By ontrast, if T >> m 2 we find θ 0. This is another manifestation of the headlight effet. 21.6 Mandelstam Variables. Consider a four-body sattering a + b + d, (321) with partiles of masses m a, m b, m, m d. Conservation of 4-momentum gives p a + p b = p + p d, (322) where p a, p b, p, p d are all taken to be future direted. One has p a p a = m 2 a, et. (323) The energy momentum in the entre of mass frame is given by p a + p b = p + p d (324) and thus the energy available for any reation, i.e. the entre of mass energy is s, where s = (pa + pb) 2 = (p + pd) 2. (325) Beause p a +p b is non-spaelike s is non-negative, s 0. The energy momentum transferred from partile a to partile t = (p a p ) 2 = (p b p d ) 2. (326) By the reverse Shwartz inequality t (m a m ) 2. In partiular if m a = m, the momentum transfer p a p is spaelike. Now the four vetors p a, p b, p, p d are not linearly dependent, they lie in a timelike three-plane. Their endpoints thus define a tetrahedron. In a tetrahedron, the lengths of opposite sides are equal and s and t give the lengths squared of two of the three possible pairs of opposite sides. The remaining length squared is given by u = (p a p d ) 2 = (p b p ) 2. (327) 66

From three vetors linearly independent vetors with 12 omponents one expets to be able to form only 6 = 12 6 independent Lorentz salars and hene the seven quantities m a, m b m, m d, s, t, an not be independent. A simple alulation shows that s + t + u = m 2 a + m 2 b + m 2 + m 2 d. (328) If the masses m a, m b, m, m d are fixed s, t, u may be thought of as a set over three over omplete oordinates on a two-dimensional spae of sattering states sine they are onstrained by the relation (328). Example all four masses equal. In the entre of mass frame the ingoing 3-momenta are equal and opposite as are the outgoing momenta. Their ommon magnitude is p 2 and the sattering angle is θ, one has s = 4(m 2 + p 2 ) t = 2p 2 (1 osθ), u = 2p 2 (1 + osθ). (329) The allowed range of s is thus s m 2 and of t 0 t 4p 2. It is onvenient to regard s, t, u as triangular oordinates in the plane. In partiular we regard them as giving the oriented perpendiular distanes from the sides of an equilateral triangle of height m 2 a + m 2 b + m2 + m 2 d. The sides of the triangle are thus given by s = 0,t = 0 and u = 0. Not all points in the plane orrespond to physially allowed values of s, t, u. For example, in the ase of four equal masses, the physial region lies outside the equilateral triangle and oupies an infinitely large 60 deg setor starting from the vertex s = 4m 2 and bounded by two half lines given by t = 0 and u = 0 obtained by produing the two sides adjaent to that vertex. It is possible to give physial meaning but this requires the idea of antipartiles. 22 Mirrors and Refletions 22.1 *The Fermi mehanism* Fermi proposed a mehanism for aelerating osmi rays. The details have something in ommon with a well known thought experiment in whih photons are onfined within a ylinder and work done on them by means of a slowly (i.e. adiabatially) moving piston. What Fermi had in mind was a large loud of mass M moving slowly with veloity u <<. Inident on it is a relativisti partile with momentum p and energy E = p. The partile is sattered bak with momentum p and energy E = E + δe = p. The veloity energy of the loud beomes u = u + du but it;s rest mass is unhanged. In this approximation, energy and momentum onservation beome 1 2 Mu2 + E = 1 2 M(u + δu)2 + E, Mu p = M(u + δu) + p. (330) 67

We have not inlude the rest mass energy M 2 beause it anels on both sides of the energy onservation equation. One gets Thus Muδu = E E, and Mδu = (p p) = E + E. (331) Mδu 2E, δe = 2Eu. (332) The first equation of (332) tells us by how muh the loud slows down, and the seond that the energy of the partiles is multiplied by a fator (1 + 2 u ) whih is greater than unity as long as u is positive. Note that this fator depends only on u. It does not depend on the mass of the loud. Fermi imagined partiles bouning bakwards and forward between two louds whih were slowly approahing eah another. The energy of the trapped partile would go up like a geometrial progression and it would seem that very high energies ould be ahieved. In pratie, while it is easy to see why partiles might satter off suh louds, it is not so easy to see how they would get trapped between two louds and so Fermi s theory has fallen out of favour. However it is interesting as an illustration of sattering. In fat the loud behaves like a mirror and the effet may be understood heuristially as a manifestation of the Doppler effet. The inoming partile has energy E(1 + u ) with respet to a rest frame sitting on the loud. on the loud with frequeny E(1 + u ) In the rest frame of the loud it is re-emitted with this frequeny in the opposite diretion and this is seen in the original rest frame as having energy (1+ u )(1+ u )E (1+2u )E. In fat refletion problems of this type an also be solved by omposing Lorentz transformations. Now if L is the distane between the two louds, then the time between bounes is 2L and in this time the distane has diminished by an amount δl = ul. Thus during an adiabati hange δl L = δe, EL = onstant. (333) E If, for example, we think of a photon with frequeny f or wavelength λ find from (333) f 1 L or λ L. 22.2 *Relativisti Mirrors* Suppose the mirror oupies the region of spaetime e 1 x > d, (334) where e 1 is a unit spaelike vetor, orresponding geometrially to the normal to the timelike hyperplane, e 1 x = d, (335) 68

and d a onstant giving the distane of the plane from the origin. If a partile, for example a light ray, with 4-momentum p is inident on the mirror and elastially or speularly refleted off the mirror the refleted partile, has 4-momentum R 1 (p) = p + 2e i (e p). (336) The possibly unfamiliar sign in (336) is beause the normal satisfies e 1 e 1 = 1 (337) Note that the refletion operator R 1 (p) leaves the rest mass unhanged sine R 1 (p) R 1 (p) = p p. (338) In other words R 1 (p) is an isometry, it leaves Minkowski lengths unhanged. In omponents, if e 1 = (βγ, γ, 0, 0) t, p = (E, p 1, p 2, p 3 ) t, (339) where β and γ have there usual meanings in terms of the veloity v of the mirror, and we take p 1 to be positive so that the inident partile is moving from right to left. Then refleted partile has momentum R 1 (p) = (E 1 + v2 1 v 2 + 2v 1 v 2 p 1, +p 1 1 + v 2 1 v 2 + 2v 1 v 2 E, p 2, p 3 ) t. (340) As an example, onsider the mirror at rest, v = 0. R 1 (p) = (E, p 1, p 2, p 3 ). (341) The energy E is unhanged and only the omponent of momentum perpendiular to the mirror is reversed. Another interesting ase is of a light ray or photon moving perpendiular to the mirror. Thus p = (E, E, 0, 0) t, and R 1 (p) = ( 1 + v ) 2(E, E, 0, 0) t 1 v (342) We see that the light ray is refleted bakward with two fators of the Doppler shift as desribed in the previous setion. Note that if the mirror is moving and the inident photon is not moving exatly perpendiularly to the mirror then the angle of refletion will not equal the angle of inidene. 22.3 *Corner Refletors on the Moon* During the first Apollo landing on the moon in 1969 a orner refletor was left on the lunar surfae. Within weeks laser photon pulses sent from the Lik Observatory in California refleted off the refletor and reeived bak in California. Over the past 30 years or so the number of refletors and the preision has been 69

inreased so that at any given time, the distane to the moon an be determined to better than 1m. A orner refletor effet three suessive refletions in there mutually perpendiular mirrors, the walls of an orthant in the rest frame of the refletor. If the walls have spaelike normals e 1, e 2, e 3, then the effet of three refletions is given by R 1 R 2 R 3 (p) = P(p) = e 0 (p )e 0 + e 1 (e 1 p) + e 2 (e 2 p) + e 3 (e 3 p). (343) The operator P(p) is alled spatial parity and reverses the spatial omponents of any 4-vetor it ats on. Thus, aording to an observer in its rest frame, a the orner refletor send bak a photo in preisely the diretion it omes from. 22.4 Time reversal 22.5 Anti-partiles and the CPT Theorem 23 4-aeleration and 4-fore Given a timelike urve x = x(τ), where τ is propertime along the urve,, we define its aeleration 4-vetor by a = du dτ = d2 x dτ 2. (344) But U U = 2, du dτ U + U du du = 0, 2U dτ dτ = 0. (345) Thus 4-aeleration and 4-veloity are orthogonal a U = 0. (346) Thus, sine U is timelike, a must be spaelike. Geometrially, U is the unit tangent vetor of the world line and a is its urvature vetor. 23.1 Relativisti form of Newton s seond law This may be written as m d2 U = G, or ma = G, (347) dτ2 where the 4-fore G is not an arbitrary 4-vetor but must be orthogonal to U, a G = 0. (348) 70

23.2 Energy and work done We have This beomes Thus p = mu, dp dτ = G, 1 1 v2 2 de dt = 1 v2 2 G 0, ( d E ) = dt p ( ) G0. (349) G dp dt = 1 v2 2 G. (350) 1 v2 G = F, (351) 2 where F is the old fashioned Newtonian fore. Now G.v G U = G 0 = 0. (352) 1 v2 1 v2 2 2 Thus whih gives G 0 = 1 de dt whih is preisely the result we used earlier. 23.3 Example: relativisti rokets F.v 1 v2 2 (353) = F.v, (354) These have variable rest-mass, m = m(τ). The equation of motion is d(mu) dτ = J, (355) where J is the rate of emission of 4-momentum of the ejeta. Physially J must be timelike, J J > 0, whih leads to the inequality ṁ > a. (356) m Thus to obtain a ertain aeleration, as in the Twin-Paradox set-up over a ertain proper time requires a lower bound on the total mass of the fuel used ln( m final ) < a dτ. (357) m initial 71

In two dimensional Minkowski spaetime E 1,1 U a = (oshθ, sinhθ) a a = dθ dτ, (358) where θ is the rapidity.we find m final 1 + vinitial 1 vfinal < = 1 (359) m initial 1 v initial 1 + v final 1 + z Consider for example two observers, one of whom is at rest and and engaged in heking Goldbah s onjeture that every even number is the sum of two primes using a omputer. The seond observer, initialy at rest with respet to the first observer v initial = 0, deides to use time dilation to find out faster by aelerating toward the stationary observer thus aquiring a veloity v final and blue shift fator 1+z. The inrease in the rate of gain of information is bounded by the energy or mass of the fuel expended. 24 The Lorentz Fore We will illustrate the general theory in the previous setion by means of the simplest way of solving the onstraint G U = 0. We set G = eηfu, (360) where e is a onstant and F is a 4 4 matrix. (the inlusion of η is for later onveniene. Sine η is invertible, and in fat η 2 = 1, we ould absorb it into the definition of F). Now, using the fat that η 2 = 1, U G = u t ηηfu = U t FU. (361) Thus we may satisfy our onstraint by demanding that F = F t. (362) We all F the Faraday tensor. The word tensor will not be explained here sine we won t need at this point. We give the omponents of F suggestive names. E 0 1 E 2 E 3 F = E1 0 B 3 B 2 E2 B 3 0 B 1. (363) E3 B 2 B 1 0 Thus 1 0 0 0 0 1 0 0 ηf = 0 0 1 0 0 0 0 1 E 0 1 E 2 E 3 E 0 1 E 2 E 3 0 B 3 B 2 B 3 0 B 1 = E 1 0 B 3 B 2 E 2 B 3 0 B 1 E B 2 B 1 0 3 B 2 B 1 0 (364) E1 E2 E3 72

0 E 1 G = eηfu = e or in 3-vetor notation E 1 E 2 E 3 0 B 3 B 2 E 2 B 3 0 B 1 E 3 B 2 B 1 0 dp dt = e(e + v B), de dt γ eγv. E v 1 γ eγ(e = 1 + v 2 B 3 v 3 B 2 ) v 2 γ eγ(e 2 + v 3 B 1 v 1 B 3 ) v 3 γ eγ(e 3 + v 1 B 2 v 2 B 1 ) (365) = ee.v. (366) These are just the Lorentz fore equations for a partile arrying an eletri harge e in an eletri field E and magneti field B. 24.1 Example: partile in a uniform magneti field In a vanishing eletri field, E = 0, the energy E and hene the speed v is onstant. Thus p = mγv, where the relativisti gamma fator γ is onstant. If the magneti field is uniform, independent of time, and aligned, for example, along the x 3 diretion, we have p 3 = onstant x 3 (t) = v 3 t + x 3 (0), where v 3 is the onstant omponent of the veloity in the 3-diretion. Thus v1 2 + v2 2 := v 2 = onstant. Now we have v 1 = e B mγ v 2, Thus, with a hoie of origin for time with and v 2 = e B mγ v 1 (367) ẋ 1 = v 1 = v osωt, ẋ 2 = v 2 = v sinωt, (368) ω L = e B m ω = ω L γ, (369) is the Larmor Frequeny. (370) The projetion of the motion in the x 1, x 2 plane is irular. Up to a translation x 1 = R sinωt, x 2 = R osωt, (371) with Thus if p = mγv, p = p 2 1 + p2 3, v = Rω. (372) p = e B R. (373) This result is used by osmi ray physiist, who measure the radius R of partiles the traks of partiles, to obtain their momentum. Numerially, (373) p = 300 B R, (374) 73

with p in ev, B in Gauss and R in m. The radius of the earth is 6, 400 Km and its magneti moment 8 10 25 Gaussm 2. Thus only partiles of 59.5 GeV or more an be expeted to reah the surfae of the earth. Example The relation (373) was used by Buherer in 1909 [20] to hek the relativisti formula relating energy and momentum. Buherer produed eletrons of known kineti energy by sending through a known potential differene V and then sent them through known magneti fields and measured the radii of their orbits. He found agreement with the relation ev = m 2 e 4 + e 2 2 B 2 R 2 m e 2. (375) 24.2 Uniform eletro-magneti field and uniform aeleration If F is a onstant matrix we an integrate the equation of motion rather easily. In this denote we shall denote d 26 dτ by a dot. The equation of motion is a = ẍ = e m η 1 Fẋ. (376) Now d τ (a a) = 2a (ȧ) = 2at ηȧ. (377) But ȧ = e m ηf U = e ηfa. m (378) Thus a t ηȧ = e m at ηη 1 Fa = e m at Fa = 0, Beause F is antisymmetri, F = F t. Thus the magnitude of the aeleration is onstant. Of ourse its diretion hanges. Now the first integral of (377) is ẋ = e m η 1 Fx + U 0. (379) Let s set U 0 = 0 and onsider the ase of a purely eletri field along the x 1 axis. η 1 F = The equations beomes, with dt dτ = e E m x 1, ( 0 E E 0 Thus, with a hoie of origin of proper time τ, t = Asinh ( e E τ m ). (380) dx 1 dτ = e E t. (381) m ), x1 = Aosh ( e E τ ), (382) m 26 The inverse is expliitly inluded to make ontat with the index notation we will introdue later 74

where A is a onstant of integration. The world line is a hyperbola The magnitude of the aeleration is 2 t 2 x 2 1 = A2. (383) a = AE m. (384) 25 4-vetors, tensors and index notation In advaned work, partiularly when passing to Einstein s theory of General Relativity, it is helpful to adopt a notation whih is a natural extension of elementary Cartesian tensor analysis (see e.g. [21]). The notation is universally used in physis and engineers throughout the world and despite the initial impression that it is rather ompliated, rather more so than the matrix notation we have been using so far, experiene shows that when the basi onventions have been absorbed, it provides both a very ompat notation and one whih allows for very effiient alulations. All legal expressions are automatially ovariant,, i.e have well defined transformation rules under Lorentz transformations, and, from mathematial point of view, it allows one to write down mathematially well defined formulae and make well defined mathematial onstrutions without needing expertise in abstrat algebra or needing to be familiar with the ompliated basis independent definitions introdued in books on multi-linear algebra. What, for example pure mathematiians all funtoriality is almost guaranteed. In fat the notation was introdued and widely adopted by pure mathematiians during the first half of the twentieth entury and then abandoned by them, in favour of oordinate free notations. Suh notations have many merits but they often require detailed explanations to unpak them. The wise words of Arthur Cayley(1821-1895), inventor of matries and explorer of higher dimensions, invariants and o-variants speaking in a related ontext seem appropriate: My own view is that quaternions are merely a partiular method, or say a theory, in oordinates. I have the highest admiration for the notion of a quaternion;but,... as I onsider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its appliations. As another illustration... I ompare a quaternion formula as a poket-map - a apital thing to put in one s poket, but for use must be unfolded:the formula, to be understood, must be translated into oordinates. 25.1 Contravariant vetors One labels the omponents of a 4-vetor in some basis with indies whih take values 0, 1, 2, 4 and whih are plaed upstairs. In ommon with most modern books these indies will be denoted by letters from the Greek alphabet. Instead 75

of 0 one sometimes uses 4. Thus the following notation should be thought of as onveying the same information. x x µ x 0 x 1 x 2 x 3 ( ) x 0. (385) The usual time oordinate is given by x 0 = t but in advaned work we usually set = 1 so we will always think of x 0 as the time oordinate. Of ourse, mathematially, the first is the abstrat 4-vetor, the seond the set of its omponents in a basis (i.e. in a partiular frame of referene) and the third its representation as a olumn vetor. The real 4-dimensional spae of 4- vetors will be alled V. The Lorentzian inner produt is written as x i x y = x t ηx = x µ η µν y ν. (386) Evidently η µν are the omponents of a the quadrati form η in the basis. Now, as with any vetor spae, Lorentz transformation may be be viewed passively: as a hange of basis or atively as a linear map Λ : V V. In either ase we have x µ x µ = Λ µ νx ν. (387) Thus Λ µ ν are the omponents of the linear map or endomorphism Λ. The first, upper, so-alled ontravariant index labels rows and the seond, so alled ovariant index labels the olumns of the assoiated matrix. Note that the Einstein summation onvention applies in the modified form that ontrations are allowed only between a ovariant index and a ontravariant index, i.e. between an upstairs and a downstairs index. 25.2 Covariant vetors Now what about row vetors, e.g z = y t, where y is a olumn vetor?. We write the omponents of z with indies downstairs and so all of the following should onvey the same information Thus z z µ (z 0 z 1 z 2 x 3 ) (z 0 z i ). (388) zx = z µ x µ = z 0 x 0 + z 1 x 1 + z 2 x 2 + z 3 x 3 = z 0 x 0 + z i x i. (389) We would like zx to be invariant Lorentz transformations and hene it must transform like z z = zλ 1, z ν = z µ (Λ 1 ) µ ν z µ Λ µ ν = z ν. (390) Clearly olumn vetors and row vetors transform in the opposite way, one with Λ and the other with (Λ 1 ) t. We say they transform ontragrediently. 76

Alternatively we refer to olumn vetors as ontravariant vetors and row vetors as ovariant vetors. Another way to say this is that x µ are the omponents of an element of the four-dimensional vetor spae V of 4-vetors, and z µ, the omponents of the four-dimensional dual vetor spae V, i.e. the spae of linear maps from V to R. 25.3 Example: Wave vetors and Doppler shift If one looks bak at our derivation of the Doppler effet, we wrote Φ = Asin(kx ωt) = Asin( k x ω t) (391) and dedued the transformation rules for the angular frequeny ω and wave vetor k (125 using the invariane of the phase k x ω t = k ωt. (392) In our present language we see that we an think of t, x as a ontravariant vetor x µ and ω, k as a ovariant vetor k µ. Thus Φ = Asin(k µ x µ ). (393) Now we see that our Lorentz transformation rule (50) is that for a ontravariant vetor and our Doppler shift rule (125) is that for a ovariant vetor. The invariane of the phase is the statement that k µ x µ = k µ x µ. (394) Abbreviating the term ovariant vetor to ovetor and ontravariant vetors to ontravetors,, we an say that a wave ovetor k µ belongs to the vetor spae V dual to the vetor spae V of ontravetors. Geometrially, the surfaes of onstant phase are hyperplanes in Minkowski spaetime E 3,1 φ = k µ x µ = onstant. (395) The wave ovetor k µ orresponds to the o-normal to the 3-dimensional hypersurfaes of of onstant phase. k µ = φ x µ (396) 25.4 Contravariant and ovariant seond rank tensors Now onsider how a quadrati form given by hanges under a Lorentz transformation x t Qx = x µ Q µν y ν (397) x x = Λx, i.e. x µ = Λ µ νx ν, ỹ µ = Λ µ νy ν. (398) 77

If we define the transformed quadrati form by then x t Qy = x t Qỹ, (399) Q = (Λ 1 ) t QΛ 1 Λ t QΛ = Q or Λ α µ Qαβ Λ β ν = Q µν. (400) The Lorentz invariane ondition (137) reads Λ α µ η αβ Λ β ν = η µν. (401) (Λ t ) µ α = Λ α µ and η = η. (402) We say that η µν are the omponents of a (symmetri) seond rank ovariant tensor η sine they transform in the same fashion as the tensor produt or outer produt x µ y ν of two ovariant vetors x µ and y µ. The omponents of the inverse of the metri are (η 1 ) µν = η µν = η νµ, (403) and satisfy η µα η αν = δ µ ν, (404) where δ ν µ is the Kroneker delta, i.e. the unit matrix, whose trae or ontration is δ µ µ = 4. We say that η µν are the omponents of a (symmetri) seond rank ontravariant tensor η sine they transform in the same fashion as the tensor produt or outer produt of two ontravariant vetors x µ and y ν. The Minkowski metri η µν = η 27 νµ an be thought of as a symmetri seond tensor, i.e. mathematially speaking a symmetri bilinear map V V R. The Faraday tensor F µν = F νµ is an example of an antisymmetri seond tensor, i.e. mathematially speaking an antisymmetri bilinear map V V R. Under a Lorentz transformation its omponents hange F µν F µν s.t. F µν = Λ α µ F αβ Λ β ν = F αβ Λ α µλ β ν. (405) Note that the transformation rule is exatly the same as for the metri η µν. The same rule holds for an arbitrary seond rank tensor Q αβ, symmetri, antisymmetri or with no speial symmetry. The omponents of an n-th rank ovariant tensor transform, i.e. a tensor with n indies downstairs or mathematially speaking, a multi-linear real valued map from the n-fold Cartesian produt V... R transform analogously. The symmetry or anti-symmetry of a tensor is a Lorentz-invariant. In the ase of rank the symmetri and antisymmetri parts Q αβ = Q (αβ) + Q [αβ], (406) 27 Note that from now on we are will be indulging in the standard abuse of language whih refers to an objet by its omponents. 78

with Q (αβ) = 1 2 (Q αβ + Q βα ), Q [αβ] = 1 2 (Q αβ Q βα ) (407) transform separately into themselves. The proof proeeds by index shuffling. For example in the anti-symmetri ase Q [µν] = Q αβ Λ α [µλ β ν] = Q αβ Λ [α [µλ β] ν = Q [αβ] Λ α µλ β ν. (408) An idential argument with square brakets replaing round brakets applies in the symmetri ase. 25.5 The musial isomorphism Note that if v µ transforms like a ontravariant vetor and Q µν is a seond rank ovariant tensor the Q µν x ν, and Q µν x µ (409) are ovariant vetors whih oinide or oinide up to a sign of q is symmetri or antisymmetri respetively. Thus, in the ase of Minkowski spae, the distintion between ontravariant and ontravariant vetors is more apparent than real, beause one may pass from one to the other by index lowering and index raising using the metri η µν or inverse metri η µν respetively. We use a notation in whih the same kernel letter is used for vetors and tensors whih are identified using index raising or lowering. Thus we write, for example x µ = η µν x ν x µ = η µν x µ. (410) In other words the metri η effets an isomorphism between the vetor spae V of ontravariant 4-vetors and its dual vetors spae V of ovariant 4-vetors. Thus p q = q p = η µν p µ q ν = p µ q ν = p µ q ν. (411) Note that the order of indies is still important. F µ ν = η µα F αν and F n u µ = F µβ η βµ should be distinguished. One sometimes uses the musial symbols and to denote index raising and lowering respetively and so the isomorphism is referred to as the musial isomorphism. 25.6 De Broglie s Wave Partile Duality Tn In 1924 the Frenh aristorat Louis-Vitor 7 e du de Broglie(1892-1987) proposed, in his dotoral dissertation that just as light, believed sine the interferene experiments of Thomas Young, to be a wave phenomenon, has, aording to Albert Einstein s photon hypothesis (for whih he was awarded the Nobel 79

prize in 1922) some of the properties of partiles, so should ordinary partiles, like eletrons, and indeed all forms of matter of waves, aording to the universal sheme, Energy and frequeny E = hf Wavelength and momentum p = h λ, (412) where h is Plank s onstant. The Amerian physiist Clinton Josephson Davidson(1881-1958) and the English physiist George Paget Thomson(1892-19750) were awarded the Nobel prize in 1937 for the experimental demonstration of the diffration of eletrons. George Thomson was the son of the 1906 Nobelist Joseph John Thomson (1856-1940) who established the existene of the eletron. It was said of the pair that the father reeived the prize for proving that eletrons are partiles and the son for proving that they waves. No parallel ase appears to be known, and indeed may not be possible, in the ase of the mathematiians equivalent of the Nobel prize, the Fields medal. Note that de Broglie s proposal allows us to reonile the two opposing theories of refration, the emission and the wave theory desribed earlier. One may indeed think of Snell s law as expressing onservation of momentum parallel to the refrating surfae as long as one uses de Broglie s relation p = h λ for the momentum rather than Newtons formula p = m. An important part of de Broglie s preposterous proposal, for whih he was awarded the Nobel prize in 1929, was that he ould show that it is ovariant with respet to Lorentz transformations. With the formalism we have just developed this is simple. Defining = h 2π, his proposal beomes p µ = k µ = η µν k µ, (413) In other words his wave-partile duality is equivalent to the musial isomorphism. 25.7 * Wave and Group Veloity: Legendre Duality* I order to reonile de Broglie s proposal with our usual ideas it is neessary to reall some fats about wave motion. In fat, these fats are also relevant for some of the optial experiments mentioned earlier. In general, monohromati wave motion, that is waves of a single fixed wavelength have a single well defined frequeny and onversely 28 Thus the phase travels with the phase veloity v p = fλ = ω k. (414) In general the motion is dispersive, whih means that the phase veloity v p depends on wavelength λ. For example, for light, we define the refrative index by v p = n and a little familiarity with prisms and the rainbow soon onvines 28 In some situations, suh as in ondensed matter physis, it may be that the frequeny is a multivalued funtion of wavelength. In what follows, we exlude this possibility. 80

one that refrative index depends upon wavelength, n = n(λ). In other words the dispersion relation ω = ω(k) is not, in general ω = k, but more general. Now pure monohromati waves never exist in pratie. The best one an arrange is a superposition of a group or wave paket of waves with almost the same frequeny ( ) Φ(x, t) = A(k )e i k.x ω(k )t d 3 k (415) where A(k ) is peaked near k = k. We set k = k + s ω(k ) = ω(k) + v g.s + O( s 2 ) (416) where the group veloity v g = ω k.. (417) One now performs a stationary phase or saddle point evaluation of the integral. This amounts to assuming that One finds that A(s) = e a2 2 s 2. (418) ( ) Φ e i k.x ω(k)t e 1 2a x vgt 2. (419) One sees that the peak of the wave paket moves with the group veloity, not the phase veloity. Note that de Broglie s proposal is ompatible with Hamiltonian mehanis. If we set H = ω, p = k, (420) then (417) and (251 ) beome idential. Now let s turn to the speial ase of a relativisti partile. Using units in whih = = 1, the dispersion relation is Thus v p = ω = m 2 + k 2. (421) 1 + ( m k k )2, v g = m2 + k. (422) 2 The group veloity v g oinides with what we have been thinking of the veloity v of the relativisti partile and is never greater in magnitude than the speed of light. By ontrast the phase veloity is always greater than that of light. If v g = v g, then and we have the strikingly simple relation v p v g = 2. (423) In Hamiltonian mehanis, the passage between momentum and veloity is via the Legendre transform. The Legendre transform is a duality or involution beause the Legendre transform of a Legendre transform gets you bak to 81

yourself. The musial isomorphism is also an involution. These fats are of ourse related. We an onsider an arbitrary a ovariant Lagrangian L(v µ ) and ovariant or super Hamiltonian H(p µ ) suh that and p µ = L v µ, H + L = p µ v µ, (424) vµ = H p µ. (425) To obtain the standard, experimentally well verified Lorentz-invariant relation between energy and momentum, we hoose we have L = m 2 η µνv µ v ν 1 2m ηµν p µ p ν, (426) p µ = mη µν v ν v µ = 1 m ηµν p ν. (427) It is illuminating to look at this from the Galilean perspetive. 29 Unlike the ase with the Lorentz group, Galilean boosts form a three-dimensional invariant subgroup subgroup of the full Galilei group. Under its ation, the four quantities x, t transform linearly as ( ) ( ) ( ) x 1 u x. (428) t 0 1 t whih gives a reduible but not fully reduible representation sine the subspaes t = onstant are left invariant. The phase k.x ωt is left invariant and so the wave vetor k and frequeny ω transform under the ontragredient representation (i.e. under the transpose of the inverse) ( k ω ) ( 1 0 u t 1 ) ( k ω ). (429) These two representations are not equivalent, essentially beause no nondegenerate metri is available to raise and lower indies. This is one way of understanding the differene between the preditions about aberration made aording to the partile and wave viewpoint in Galilean physis. 25.8 The Lorentz equation Having set up the notation, we are now in a position to write down the equation of a relativisti partile of mass m and harge e moving in an eletro-magneti field F µν = F νµ m d2 x µ = ef µ dx ν ν dτ dτ,. (430) 29 In what follows we shall use some standard group-theoreti terminology whih will not be defined here. An understanding of the rest of this setion is not neessary for the rest of the letures. 82

26 Uniformly Aelerating referene frames: Event Horizons Uniform translational motion is, aording to Speial Relativity, unobservable. Uniformly aelerated motion however is observable. To reveal some of its effets, we may pass to an aelerated system of oordinates, often alled Rindler oordinates x 0 = ρ sinht, x 3 = ρ osht, (431) whih, if 0 < ρ <, < t <, over only one quarter of two-dimensional Minkowski spaetime E 1,1, the so-alled Rindler wedge x 1 > x 0. (432) In this wedge the flat Minkowski metri takes the stati form ds 2 = ρ 2 dt 2 dρ 2. (433) From our previous work, we see that The urves ρ = onstant have onstant aeleration a = 1 ρ. We shall refer to these urves as Rindler observers. They are in fat the orbits of a one parameter family of Lorentz boosts, t t + t 0 x ± = x 1 ± x 0 e ±t0 x ±. Note that the propertime τ Rindler along a Rindler observer is given by τ Rindler = ρt. The aeleration of the set of Rindler observers goes to infinity on the boundary of the Rindler wedge, i.e. on the pair of null hypersurfaes surfaes x 0 = ±x 1. These surfaes are alled the future and past horizons of the Rindler observers. That is beause the past, respetively future, light ones of all the points on the worldline of a Rindler observer, and thus neessarily their interiors, lie to the past, respetively future of these null hypersurfaes. In other words the future horizon is the boundary of the set of events that an ever ausally influene a Rindler observer and the past horizon the boundary of the set of events whih a Rindler horizon may ausally influene. Thus the nature of all events for whih X 0 > x 1 an never be known to Rindler observers. On the other hand, there is no boundary to the past of an inertial observer, i.e. a timelike geodesi. For example a timelike observer with say x 1 = onstant > 0 will simply pass through the future horizon and out of the Rindler wedge in finite propertime τ Inertial = x 1. A simple alulation shows that a light ray emitted from the event (x 0, x 1 ) will be reeived by a Rindler observer at a propertime τ Rindler = 1 ( x 1 ) ρ ln τ Inertial. (434) ρ Aording to the Rindler observer, the light oming from the Inertial observer is inreasing redshifted. The motion appears to be slower and slower. So muh so, that the redshift beomes infinite as the Inertial observer is on the point of passing through the future event horizon and aording to the Rindler observer the Inertial observer never atually passes through in finite time. 83

The rather ounter-intuitive phenomena desribed above have a very preise parallel in the behaviour of the event horizons of blak holes. The fat that they may our in suh as simple situation as that two-dimensional Minkowski spaetime shows that although apparently paradoxial, there is nothing logially inonsistent about them. 27 Causality and The Lorentz Group. 27.1 Causal Struture We may endow Minkowski spaetime with a ausal struture, that is a partially ordering, alled a ausal relation whih is reflexive and transitive. In a general, time orientable, spaetime, one says that x ausally preedes y and writes x y (435) if the event x an be joined to the event y by a future direted timelike or null urve. Thus (i) x y and y z x z and (ii) x x. (436) There is an obvious dual relation, written as x y in whih past and future are interhanged. A stronger relation, alled hronology an also be introdued. We say that x hronologially preedes y if there is a future direted timelike urve joining x to y and write x y. (437) In Minkowski spaetime the urves may be taken to be straight lines, i.e. geodesis. We write x y x 0 y 0 (x 1 y 1 ) 2 + (x 2 y 2 ) 2 + (x 3 y 3 ) 2. (438) 27.2 The Alexandrov-Zeeman theorem Our derivation of the Lorentz group earlier depended upon the assumption of linearity. In fat this may be removed. Alexandrov[54] and independently Zeeman[53] have shown that any ontinuous map of Minkowski spaetime into Minkowski spaetime, as long as it is higher than 1+1 dimensional whih preserves the light one of the origin must in fat be linear. It follows that suh a transformation is the produt of a dilation and a Lorentz transformation. In other words, in four spaetime dimensions, one may haraterize the eleven dimensional group onsisting of the Poinaré group semi-diret produt dilatations as the automorphism group of the ausal struture of Minkowski spaetime. Sine the proof entails speial tehniques we will not give it here. 84

In 1 + 1 dimensions, things are very different. In light one oordinates the metri is ds 2 = dx + dx. (439) The light one and ausal struture is learly left unhanged under x ± x ± = f ± (x ± ), (440) where the two funtions f ± are arbitrary monotoni C 1 funtions of their argument. Thus the group of ausal automorphisms of two-dimensional Minkowski spaetime is infinite dimensional. It is the produt of two opies of the infinite dimensional group Diff(R) of invertible and differentiable maps of the real line into itself. This fat plays an important role in what is alled String Theory. 27.3 Minkowski Spaetime and Hermitian matries We may identify four-dimensional Minkowski spaetime with the spae of 2 2 Hermitian matries X = X aording to the sheme ( ) x X = 0 + x 3 x 1 + ix 2 x 1 ix 2 x 0 x 3. (441) Now if X and Y are lightlike separated, then One an say more, det(x Y ) = 0 (442) X Y X Y is non negative definite (443) The Minkowski metri may be written as ds 2 = det dx. (444) Now onsider the group GL(2, C) ats on Hermitian matries by onjugation Moreover if we insist that X SXS = X, X = X. (445) dets = 1, (446) we obtain the group, SL(2,C) of 2 2 uni-modular omplex valued matries, whih is 6 dimensional. We have exhibited a homomorphism from SL(2,C) to SO(3, 1) 30 The kernel of this homomorphism is easily seen to be the group Z 2 given by S = ±1. Thus the homomorphism is a double overing, S and S give the same element of the identity omponent SO 0 (3, 1). We ould pursue this homomorphism further, but at this point we prefer to return to the ausal struture (443) on Hermitian matries. Orderings of this type were studied by Hua. They have other appliations, inluding to providing a natural ordering, orresponding to purity on density matries in quantum mehanis. A speial ase of Hua s formalism is the ase of 2 2 matries. In tis ase he is able to re-obtain the Alexandrov-Zeeman result [55]. 30 Stritly speaking sine SL(2,C) is onneted it is onto the onneted omponent S 0 (3, 1). 85

28 Spinning Partiles and Gyrosopes 28.1 Fermi-Walker Transport We begin by onsidering a timelike urve C with unit tangent vetor u = dx dτ and a vetor e defined along the urve satisfying where u = du dτ de dτ + u(e u) = 0, (447) is the aeleration of the urve C. One has d d (e u) = 0, dτ dτ (e s) = 1 (e u)(e u). (448) 2 Thus if s satisfies (447) along C, we say that it is Fermi-Walker transported along C. From (448) it follows that if s is orthogonal to u, (u s) = 0, at one point on the urve C then it is orthogonal to u at all points of C. Moreover its length e = e e will be onstant along e. If e 1 and e 2 are Fermi-Walker transported along C, then d dτ (e 1 e 2 ) = (e 1 u)(e 2 u) (e 2 u)(e 1 u). (449) Thus, if e 1 and e 2 are initially orthogonal to u and eah other they will remain orthogonal to u and to eah other. Introduing a third vetor e 3 we an arrange that e 0 = u, e 1, e 2, e 3 we may onstrit in this way a pseudo-orthonormal frame along the urve C. Physially we an think of e a, a = 0, 1, 2, 3 as a loally non-rotating frame defined along the aelerating worldline C. 28.2 Spinning partiles and Thomas preession Let s be the spin vetor of a partile whose 4-veloity is u. In a loal rest frame, s should be purely spatial, so s u = 0. (450) In the absene of an external torque, we postulate that its omponents are onstant in a Fermi-Walker transported frame, along the world line i.e. ds dτ + u(s u) = 0. (451) Note that if the world line of the partile is aelerating, even in the absene of an external torque, the spin, while staying onstant in magnitude, will hange in diretion. This is alled Thomas preession. Its existene was pointed out in 1927[39]. If s u = 0, then we an write (451) as ds µ dτ + Uµ νs ν = 0, (452) 86

with U µν = u µ ( u) ν ( u) µ u ν. (453) One may think of U µν as an infinitesimal rotation. Thomas regards this as the result of the ommutator of two suessive, non parallel boosts. If an external torque H is applied the equation beomes ds dτ + u(s u) = H, where u H = 0. (454) 28.3 Bargmann-Mihel-Telegdi Equations In (1926) Goudsmit and Uhlenbek, studying the fine struture of atomi spetral lines and their behaviour in external magneti fields, the Zeeman effet, realized that the eletron at rest has both an intrinsi spin s of magnitude s = 2, and an intrinsi magneti moment µ, so that immersed in a magneti field B the spin hanges as ds = µ B. (455) dt In fat Goudsmit and Uhlenbek argued on the grounds of atomi spetra that s = g e s, (456) 2m with the gyromagneti ratio g = 2. Thus the spin preesses aording to the equation ds dt = g e 2m s B. (457) The reason for the apparently odd normalization is that for ordinary orbital motion for whih the spin oinides with the orbital angular momentum, s = L, g takes the value 1. In fat, a little later in 1927? Paul Adrien Maurie Dira (1902-1984), the 15th Luasian professor) proposed that relativisti eletrons satisfy what we now all the Dira equation, rather than it s non-relativisti approximation the Shrödinger equation. Dira showed that the value g = 2 follows naturally from his equation. His work was reognized by the ward of the Nobel prize in (1933). Later, in the 1940 s advanes in radio engineering allowed more preise measurements in atomi spetral lines, and revealed a level of hyper-fine struture beyond that predited from the Dira equation. In partiular, there is shift or splitting in the lowest lines of hydrogen, due to a differene in the energy between an orbiting eletron spinning up or spinning down, relative to the diretion of the orbital angular momentum. The differene in energy, alled the Lamb shift after the man who measured it is extremely small and a transition between the two levels gives rise to radio waves of 21m wavelength. It was suggested in 1944 by the Duth astronomer Henk van der Hulst that radiation of this wavelength should be emitted by interstellar louds of neutral Hydrogen and its detetion was ahieved by various groups in 1951. Today radio-astronomy using the 21m 87

line allows is an important area of researh, not least beause the preise frequeny allows the measurement of the veloity of louds of neutral Hydrogen using the Doppler effet. To aount for the Lamb shift it is neessary to assign an anomalous gyromagneti moment to the eletron, in other words g 2 = 0. The value of g 2 an be alulated using the relativisti quantum mehanial theory of photons interating with eletrons known as quantum eletrodynamis QED. At present the agreement between theory and experiment is better than To measure g 2 one aelerates eletrons in known eletromagneti fields and measures the preession of the spin. A relativistially ovariant set of equations desribing this, properly taking into aount the effets of Thomas preession, was set up by Bargmann-Mihel and Telegdi, then at Prineton. Basially one needs a ovariant expression for the torque whih will reprodue (??) in the rest frame of the eletron. One s first guess might be H = g e 2m Fs, H µ = g e 2m F µνs ν? (458) but this does not satisfy H µ u µ = 0. (459) In order to remedy this defet we introdue a projetion operator h µ ν = δ µ ν u µ u ν. (460) whih thought of as an endomorphism h projets an arbitrary vetor orthogonal to u. One may also think of h µν = h νµ as the restrition of the spaetime metri η µν to a loal 3-plane orthogonal to the tangent of the world line C. The projetion operator satisfies h 2 = h, h µ λ hλ ν = hµ ν h µ ν uν = 0. (461) Now using the projetion operator we are able to propose or H = g e 2m hfs = g e ( ) Fs u(u Fs) 2m (462) H µ = g e 2m hλ µ F λτs τ = g e ( Fµτ s τ u µ u α F αβ s β). (463) 2m Note that s H = (u s)(u Fs) and so if u s = 0, the length s of the spin-vetor s is onstant. In the presene of an eletromagneti field one has thus u = e Fu, (464) m ds dτ = g e e Fs + (2 g) u(u Fs). (465) 2m 2m Clearly the ase g = 2 is very speial. The spin vetor s and the 4-veloity u obey the same equation, and thus they move rigidly together. By ontrast, if g 2, this is not the ase, the spin preesses in the moving frame, allowing a measurement to be made. 88

Referenes [1] O Roemer, Phil Trans 12 (1677) 893 [2] A A Mihelson Studies in Optis(1927) University of Chiago Press [3] A A Mihelson and Morley Phil Mag 24 (1887) 449 [4] Bradley Phil Trans 35(1728)637 [5] A Einstein et al. The Priniple of Relativity (Dover) [6] H Fizeau, Compte Rendu Aaadamie des Sienes de Paris 29 (1849) 90 [7] M L FouaultCompte Rendu Aaadamie des Sienes de Paris 30 (1850) 551 Compte Rendu Aaadamie des Sienes de Paris 55 (1862) 501 Compte Rendu Aaadamie des Sienes de Paris 55 (1862) 792 [8] D F Comstok Phys Rev 30 (1920) 267 [9] W De-Sitter,Pro Amsterdam Aad16 (1913) 395 [10] H E Ives and CR Stillwell J opt So Amer28(1939 215;31(1941)369 [11] A H Joy and R F Sanford Ap J 64 (1926) 250 [12] D Sadeh Phys Rev Lett Experimental Evidene for the onstany of the veloity of of gamma rays using annihilation in flight10 (1963) 271-273 [13] W Bertozzi Amer J Phys 55(1964)1 [14] Alväger, Farley, Kjellamn and Wallin Phys Lett 12(1964) 260 [15] L Marder, Time and the spae traveller (1971) George Allen and Unwin [16] J C Hafele and R E Keating Siene 177 (1972) 166-168,168-170 [17] M J Longo, Phys Rev36D (1987) 3276-3277 [18] J Terrell Phys Rev116(1959) 1041 [19] R Penrose, Pro Camb Phil So 55 (1959) 137 [20] A H Buherer Ann Physik 28(1909) 513 [21] H Jeffreys Cartesian tensors [22] M J Rees, Nature 211 (1966) 468-470 [23] L Dalrymple Henderson The Fourth Dimension in Non-Eulidean geometry in Modern Art Prineton University Press [24] K Breher Phys Rev Lett 39(1977) 1051 89

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Index 4-fore, 70 4-veloity, 54 aberration map, 37 absolute unit of length, 44 aeleration 4-vetor, 70 adiabatially, 67 amplitude, 31 angle of paralellism, 49 angular frequeny, 32 anomalous gyromagneti moment, 88 antipodal identifiation, 45 available energy, 62 Ballisti theory of light, 22 baryentre, 38 blue-shifted, 32 boost, 21 boosting, 21 bremsstrahlung radiation, 60 Bureau International des Poids et Mesures, 27 ausal relation, 84 ausal struture, 84 elestial equator, 38 elestial sphere, 40 entre of mass, 38 entre of mass energy, 61, 66 entre of mass frame, 61 entroid, 38 hronology, 84 lok hypothesis, 43 CMB, 41 CMB dipole, 42 o-moving oordinates, 43 o-rotating oordinates, 50 ompass of inertia, 13 onfiguration spae, 44 onstitutive relations, 11 ontragrediently, 76 ontravariant, 76, 77 ontravetors, 77 Cosmi Mirowave Bakground, 53 osmi mirowave bakground, 35 osmi mirowave bakground, 41 ovariant, 75 77 ovetor, 77 urvature vetor, 70 delination, 38 dilations, 20 Dira equation, 87 dispersion, 80 dispersion relation, 81 displaement urrent, 10 duality, 81 Einstein s Stati Universe, 45 Einsteins Equivalene Priniple, 36 elasti, 66 elasti or speular refletion, 69 eletron and muon anti-neutrinos, 28 eletron volt, 5 Emission or Ballisti theory of light, 7 Enylopédie, 16 events, 17 Faraday tensor, 72 Fermi-Walker transport, 86 fixed stars, 38 fixed stars, 13, 38 foundations of geometry, 43 Fourier Analysis, 9 frational linear transformations, 37 frame of referene, 14 Fresnel s dragging oeffiient, 24 future direted, 54 future horizon, 83 Gauss-urvature, 44 General Relativity, 6 Gravitational Redshift, 30 graviton, 57 Greenwih Mean Time, 57 92

Hafele-Keating experiment, 30 Hamiltonian funtion, 56 homotheties, 20 horizon sale, 32 Hot Big Bang, 41 Hubble onstant, 32 Hubble radius, 32 Hubble time, 33 Hubble s law, 32 Hyperboli spae, 43 hyperfine struture, 87 hyperplanes, 77 hypersurfaes, 77 index lowering, 79 index raising, 79 index shuffling, 79 inertial referene system, 13 inertial oordinate system, 13 inertial frame of referene, 13, 14 infinitesimal line elements, 40 infinitesimal area element, 39 Inflation, 52 International. Celestial Referene Frame, 38 International System of Units (SI units), 27 intrinsi spin, 87 invariant interval, 26 involution, 81 isometry, 69 kernel letter, 79 Kinemati Relativity, 42 kineti energy, 55 Kroneker delta, 78 Lagrangian funtion, 56 Lamb shift, 87 Langevin, 50 Larmor frequeny, 73 Legendre transform, 56 light rays, 6 light years, 5 Lorentz group, 36 Lorentz Transformations, 20 magneti moment, 74, 87 maximally symmetri spaetimes, 45 Milne universe, 42 minutes of ar, 38 Moebius transformations, 37 momentum transfer, 66 monohromati, 80 muons, 28 n-th rank ovariant tensor, 78 neutrinos, 57 non-dispersive, 9 non-relativisti limit, 20 normal, 68 nutates, 39 oblique oordinates in spaetime, 18 on-shell ondition, 63 optial distane, 7 orthant, 70 orthogonal poariztaion states, 11 outer produt, 78 past horizon, 83 permeability, 11 permittivity, 11 phase veloity, 80 photon, 57 pions, 28 Plankian spetrum, 41 plane of the elipti, 38 plasma, 39 Poinaré group, 26 Priniple of Relativity, 14 priniple of superposition, 9 pro-grade, 51 projetion operator, 88 proper distane, 29 proper motions, 13 proper time, 27 pseudo-orthonormal basis, 48 pseudo-sphere, 52 QED, 88 quantum eletrodynamis, 88 quasars, 13, 25 93

quasi-stellar radio soures, 13, 25 railway time tables, 17 rapidity, 21 red shift, 32 redshifted, 32 refletion operator, 69 refrative index, 23 relative veloities, 14 relativisti gamma fator, 21, 39 repère mobile, 48 rest mass, 54 rest mass energy, 55 retro-grade, 51 right asension, 38 Rindler oordinates, 83 Rindler observers, 83 Rindler wedge, 83 rotating buket, 50 uni-modular, 36 unit tangent vetor, 70 Universal Time, 57 veloity 4-vetor, 54 very long base line interferometry, 26 vierbein, 48 VLBI, 26 wave ovetor, 77 wave number, 32 wave paket, 81 Wave theory of light, 8 salar wave equation, 9 Shrödinger equation, 87 Shwarzshild s stati universe, 45 seonds of ar, 38 simultaneity is absolute, 18 spae, 6 spae reversal, 21 spaetime, 17 spaetime diagram, 17 spatial parity, 70 speial Lorentz group, 36 stellar parallax, 39 stereographi oordinates, 37 symmetri seond rank tensor, 78 tangent 4-vetor, 54 tensor produt, 78 tetrad, 48 thermal, 41 Thomas preession, 86 threshold, 63 time, 6 time reversal, 21 timelike, 54 transverse Doppler shift, 53 triangular oordinates, 67 94