Lecture Notes on General Relativity

what is general relativity in physics, how do general relativity and quantum mechanics conflict and what is general relativity equation what does general relativity predict
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Lecture Notes on General Relativity Matthias Blau Albert Einstein Center for Fundamental Physics Institut fu¨r Theoretische Physik Universit¨at Bern CH-3012 Bern, Switzerland The latest version of these notes is available from http://www.blau.itp.unibe.ch/Lecturenotes.html Last update August 8, 2016 10 Introduction 1905 was Einstein’s magical year. In that year, he published three articles, on light quanta, onthefoundationsofthetheoryofSpecialRelativity, andonBrownianmotion, each one separately worthy of a Nobel prize. Immediately after his work on Special Relativity, Einstein started thinking about gravity and how to give it a relativistically invariant formulation. He kept on working on this problem during the next ten years, doing little else. This work, after many trials and errors, culminated in his masterpiece, the General Theory of Relativity, presented in 1915/1916. It is widely considered to be one of the greatest scientific and intellectual achievements of all time, a beautiful theory derived from pure thought and physical intuition, capable of explaining, or at least describing, still today, more than 100 years later, every aspect of gravitational physics ever observed. Einstein’s key insightwas whatisnow knownas theEinstein Equivalence Principle, the (local) equivalence of gravitation and inertia. This ultimately led him to the realisation that gravity is best described and understood not as a physical external force like the other forces of nature but rather as a manifestation of the geometry and curvature of space-time itself. This realisation, in its simplicity and beauty, has had a profound impact on theoretical physics as a whole, and Einstein’s vision of a geometrisation of all of physics is still with us today. The aim of these lecture notes is to provide a reasonably self-contained introduction to General Relativity, including a variety of applications of the theory, ranging from the solar system to gravitational waves, black holes and cosmology. These lecture notes for an introductory course on General Relativity are based on a course that I originally gave in the years 1998-2003 in the framework of the Diploma Course of the ICTP (Trieste, Italy). Currently these notes form the basis of a course that I teach as part of the Master in Theoretical Physics curriculum at the University of Bern. Intheinterveningyears, Ihavemade(andkeepmaking)variousadditionstothelecture notes, and they now include much more material than is needed for (or can realistically be covered in) an introductory 1- or even 2-semester course, say, but I hope to have nevertheless preserved (at least in parts)the introductory character and accessible style of the original notes. Invariably, any set of (introductory) lecture notes has its shortcomings, due to lack of space and time, the requirements of the audience and the expertise (or lack thereof) and interests of the lecturer. These lecture notes are, of course, no exception. In particular, the emphasis in these notes is on developing the theory (I am a theoretical physicist), not on experiments or connecting the theory with observation, but stops shortofdoingrealmathematical generalrelativity (i.e. provingtheorems), asthiswould 12require significantly more mathematical sophistication and machinery than I want to assume (or can develop) in these notes. I hope that these lecture notes nevertheless provide the necessary background for studying these or other more advanced topics not covered in these notes. I should also stress that I have written these notes primarily for myself, and for my students. I am making them publicly available just in case somebody else happens to find them useful, and becauseI know that previous versions of these notes have enjoyed somepopularity. However, ifyou donotlikethesenotesormyway of explainingthings, or do not find what you are looking for, please do not complain to me (yes, this has happenedinthepast). Therewilloccasionallybefurtheradditionsandupdatestothese notes, reflecting however my personal preferences and taste rather than any (futile) aim for completeness. Lecturenotesofthislengthunavoidablycontainsomeminormistakessomewhere. How- ever, I hope that these notes are free of major conceptual errors and blunders. I am of course grateful for any constructive criticism and corrections. If you have such com- ments, or also if you just happen to find these notes useful, please let me know (blau at itp.unibe.ch). 0.1 Prerequisites General Relativity may appear to be a difficult subject at first, since it requires a certain amount of new mathematics and takes place in an unfamiliar arena. However, this course is meant to be essentially self-contained, requiring only a familiarity with • Special Relativity, • Lagrangian mechanics, n • vector calculus and calculus inR . To be precise, by special relativity I mean the covariant formulation in terms of the Minkowski metric and Lorentz tensors etc.; special relativity is (regardless of what you may have been taught) not fundamentally a theory about people changing trains erratically, runninginto barnswith poles, or doing strange things to their twins; rather, it is a theory of a fundamental symmetry principle of physics, namely that the laws of physics are invariant underLorentz transformations andthat they shouldtherefore also be formulated in a way which makes this symmetry manifest. Litmus Test: does the content of section 1.2 look familiar to you? I will thus attempt to explain every single other thing that is required to understand the basics of Einstein’s theory of gravity. However, this also means that I will not be 13able to discuss some mathematically more advanced and yet equally important aspects of General Relativity. 0.2 Overview Currently, these notes are organised into 7 parts, namely A: Physics in a Gravitational Field and General Covariance B: General Relativity and Geometry C: Dynamics of the Gravitational Field D: General Relativity and the Solar System E: Black Holes F: Cosmology G: Varia I refer to the Table of Contents for rather detailed information about the contents of the individual parts and sections of these notes and want to just provide some remarks here for a first orientation. Part A of the lecture notes is dedicated to explaining and exploring the consequences of Einstein’s insights into the relation between gravity and space-time geometry, and to developing the machinery (of tensor calculus and Riemannian geometry) required to describe physics in a curved space time, i.e. in a gravitational field. From about section 3 onwards, Part A can be read in parallel with other parts of these notes which deal with various applications of General Relativity. In particular, at this point in the course I find it useful to develop in parallel (and suggest to read in parallel) the more formal material on tensor analysis in Part A, and Part D (dealing with solar system tests of general relativity) – cf. the more detailed suggestions at the end of section 2. Not only does this provide an interesting and physically relevant application and illustration of the machinery developed so far, it also serves to provide an appropriate balance between physics and formalism in the lectures. The topics covered in Parts A and D, together with the first section 18 of Part C dealing with the Einstein field equations, probably form the core of most introductory courses on general relativity. This provides (or is meant to provide) the basis for other applications or investigations of general relativity, and other sections of Part C and Parts E-G provide a reasonably large variety of topics to choose from. 14In Part B of the lecture notes I have collected a number of different more mathematical topics that develop the formalism of tensor calculus and differential geometry in one way or another. Stricly speaking, none of these topics are essential for understanding some of themoreelementary aspects of general relativity to betreated later on (so Part B can also be regarded as a mathematical appendix to the notes). However, some of them are required at a later stage to understand, or even formulate, certain somewhat more advanced aspects of general relativity (and it is perhaps best to then go back to this section if and when needed), and others are included simply because they are fun or beautiful (or, usually, both). 0.3 Literature Most of the material covered in these notes, in particular in the introductory parts, is completely standard and can be found in many places. While my way of explaining things is my own, and numerous gratuitous “Remarks” throughout the notes as well as the selection of more advanced topics reflect my own interests, I make no claim to major originality in these notes and have not attempted to reinvent the wheel. In particular, in earlier versions of these notes the presentation of much of the intro- ductory material followed quite closely the treatment in Weinberg’s classic • S. Weinberg, Gravitation and Cosmology and readers familiar with this book may still recognise the similarities in some places. Even though my own way of thinking about general relativity is much more geometric (and this has definitely influenced later versions of and additions to these notes), I have found that the pragmatic approach adopted by Weinberg is ideally suited to introduce general relativity to students with little mathematical background. Asfarasmorerecentandmodernbooksareconcerned,hereisashortpersonalselection of my favourites: 1. At an introductory level, a book that I like and highly recommend is • J. Hartle, Gravity. An Introduction to Einstein’s General Relativity 2. Atan intermediate level (i.e. moreor less atthelevel of thesenotes), my favourite modern book is • S. Carroll, Spacetime and Geometry: An Introduction to General Relativity 3. At a more advanced level, my favourites are • S. Hawking, G. Ellis, The large scale structure of space-time 15• E. Poisson, A Relativist’s Toolkit: the Mathematics of Black Hole Mechanics • R. Wald, General Relativity and I will frequently refer to these books in the body of the notes for discussions of more advanced and/or more mathematical topics. 4. The history of the development of general relativity is an important and complex subject, crucial for a thorough appreciation of general relativity. My remarks on this subject are scarce and possibly even misleading at times and should not be taken as gospel. For an authoritative and informative account, I strongly recommend the scientific biography of Einstein • A. Pais, Subtle is the Lord: the science and life of Albert Einstein 0.4 References and Footnotes As mentioned before, much of the material covered in these notes is quite standard, and can be found in many places, and I have not attempted to provide references or attributions for this. Nevertheless, these lecture notes contain a large number of footnotes, with significantly higher density in the sections of the notes dealing with more advanced and, specifically, more recent developments. For the most part, these are meant as pointers to the literature for further reading and with more information. However, I have also attempted to indicate explicitly in footnotes whenever I have knowinglyusedoradoptedsomethingspecificfromaspecificsourcethatshouldperhaps not be considered common knowledge. If you feel that somewhere in these notes I have written or used something that should not be considered common knowledge and that has not been properly attributed, please let me know. For referencing I have adopted the following procedure: • When referring to textbooks, I usually just refer to them in the form “Author, Title” (as above), without indicating publisher, year, ...If you actually need this information, it will be easy for you to find it. • When referring to articles, if they are available from the preprint server at http://arXiv.org/ I usually just refer to the arXiv number, regardless of whether or not that article has been published elsewhere (this just reflects the by now standard practice that people are more likely to first go there rather than to the library to look for or at an article). 16• References to pre-arXiv articles are given in the traditional complete “Author(s), Title, Journal, ...” form. 0.5 Exercises Exercises are, of course, an indispensable part of any course, in particular a course on general relativity, since it is impossible to familiarise oneself with the formalism (of tensor calculus) without actually doing calculations. Nevertheless, these lecture notes contain no exercises, or at least none that are explicitly labelled as such. This simply reflects my own style of teaching, where exercises are very much integrated intothecourseandmainlyservethepurposeofgettingstudentstolookatwhatwasdone in the course and to perhaps fill in some details that I skipped in class. In particular, I am no fan of exercises that go significantly beyond what is covered in class or in the notes (if it is relevant, I should explain or include it, if it is not then we may as well not bother). However, most (sub-)sections contain numerous “Remarks”, and many of them con- tain supplementary and/or more advanced information and material, and these may be regarded as (annotated) exercises or used as a basis for exercises. If that does not provide enough or satisfactory material, see also • A. Lightman, W. Press, R. Price, S. Teukolsky, Problem Book in Relativity and Gravitation for almost 500 fully solved problems in relativity. 171 From the Einstein Equivalence Principle to Geodesics 1.1 Motivation: The Einstein Equivalence Principle The highly successful Newtonian theory of gravity can be succinctly summarised by two sets of differential equations, one describing the dynamics (motion) of particles in a given gravitational field (described by a potential φ), and the other describing the dynamics of the gravitational field itself, namely how φ is to be determined from a given mass configuration. The former takes the standard Newtonian form ¨ mx=F =−m∇φ (1.1) g (but we will come back in some detail below to the question if/why the same mass parametermappearsonbothsidesofthisequation,soastoincorporatetheobservation, going back to Galileo, that “all bodies fall at the same rate in a a gravitational field”). The latter is the Poisson equation 2 Δφ=4πG μ=(4πG /c )ρ , (1.2) N N with G denoting, here and throughout, Newton’s constant, i.e. the gravitational cou- N 2 pling constant, and where μ is the mass density, and ρ =μc the associated rest mass energy density - I will set c=1 in the following and use ρ. Let us start with the field equation. It is immediately evident that this cannot be the final story. Not only is this equation not Lorentz invariant. Because of the absence of time-derivatives in (1.2), it actually describes an “action at a distance” and an in- stantaneous propagation of the gravitational field to every point in space (if you wiggle your mass distribution here now, this will immediately effect the gravitational potential arbitrarily far away). This is something that Einstein had just successfully exorcised from other aspects of physics, and clearly Newtonian gravity had to be revised as well. It is then also immediately clear that what would have to replace Newton’s theory is something rather more complicated. The reason for this is that, according to Special Relativity, massisjustanotherformofenergy. Then,sincegravity couplestomasses, in arelativisticallyinvarianttheorygravitywillalsohavetocoupletoenergy. Inparticular, therefore, gravity would have to couple to gravitational energy, i.e. to itself. As a consequence, the new gravitational field equations will, unlike Newton’s, have to be non-linear: the field of the sum of two masses cannot equal the sum of the gravitational fieldsofthetwomassesbecauseitshouldalsotakeintoaccountthegravitational energy of the two-body system. Now, having realised that Newton’s theory cannot be the final word on the issue, how does one go about finding a better theory? I will first very briefly discuss (and then dismiss) what at first sight may appear to be the most natural and naive approach to formulating a relativistic theory of gravity, 19namely the simple replacement of Newton’s field equation (1.2) by its relativistically covariant version Δφ=4πG ρ −→ φ=4πG ρ , (1.3) N N whereistheLorentzinvariantd’Alembertorwave operator. Whilethislooks promis- ing, somethingcan’t bequiteright aboutthis equation. We already know (from Special Relativity) that ρ is not a scalar but rather the 00-component of a tensor, the energy- momentum tensor, so if actuallyρ appears on the right-hand side,φ cannot bea scalar, while if φ is a scalar something needs to be done to fix the right-hand side. Turning first to the latter possiblility, one option that suggests itself is to replace ρ by α the trace T =T of the energy-momentum tensor. This is by definition / construction α ascalar, anditwill agreewithρinthenon-relativistic limit(whererestmassdominates over other contributions). Thus a first attempt at fixing the above equation might look like φ=4πG T . (1.4) N This is certainly an attractive equation, but it definitely has the drawback that it is too linear. Recall from the discussion above that the universality of gravity (coupling to all forms of matter) and the equivalence of mass and energy lead to the conclusion thatgravity shouldcoupletogravitational energy, invariablypredictingnon-linear(self- interacting) equations for the gravitational field. However, the left hand side could be such that it only reduces to or Δ of the Newtonian potential in the Newtonian limit of weak time-independent fields. Thus a second attempt at fixing the above equation might look like Φ(φ)=4πG T , (1.5) N where Φ(φ)≈φ for weak fields. Such a scalar relativistic theory of gravity and variants thereof were proposed and studiedamongothersbyAbraham,Mie,andNordstrøm. Asitstands,thisfieldequation appears to be perfectly consistent (and it may be interesting to discuss if/how the Einstein equivalence principle, which will put us on our route towards metrics and space-time curvature is realised in such a theory). However, regardless of this, this theory is incorrect simply because it is ruled out experimentally. The easiest way to see this (with hindsight) is to note that the energy-momentum tensor of Maxwell theory (6.47) is traceless (6.121), and thus the above equation would predict no coupling of gravity to the electro-magnetic field, in particular to light, hence in such a theory there 1 would be no deflection of light by the sun etc. The other possibility to render (1.3) consistent is the, a priori perhaps much less com- pelling, option to think of φ and Δφ or φ not as scalars but as (00)-components of 1 For more on the history and properties of scalar theories of gravity see the review by D. Giulini, What is (not) wrong with scalar gravity?, arXiv:gr-qc/0611100. 20some tensor, in which case one could try to salvage (1.3) by promoting it to a tensorial equation Some tensor generalising Δφ ∼4πG T . (1.6) αβ N αβ This is indeed the form of the field equations for gravity (the Einstein equations) we will ultimately be led to (see section 18.4), but Einstein arrived at this in a completely different, and much more insightful, way. Letusnow,verybrieflyandinastreamlinedway,trytoretrace(oneaspectof)Einstein’s thoughts, namely on the relation between inertial and gravitational mass, which, as we will see, will lead us rather quickly to the geometric picture of gravity sketched in the Introduction. To that end we return to the Newtonian equation of motion (1.1). Recall that in this Newtonian theory, there are two a priori completely independent concepts of mass: • inertialmassm (oraccelerationmass),whichaccountsfortheresistanceofabody i or particle against acceleration and appears universally on the left-hand-side of the Newtonian equation of motion ma=F (1.7) i in conjunction with the accelerationa; • gravitational mass m which is the mass the gravitational field couples to, i.e. it g is the gravitational charge of a particle, F =−m∇φ . (1.8) g g Now it is an important empirical fact that the inertial mass of a body is equal to its gravitational mass. This realisation, at least with this clarity, is usually attributed to Newton, although it goes back to experiments and observations by Galileo usually paraphrased as “all bodies fall at the same rate in a gravitational field”. (It is not true, though, that Galileo dropped objects from the leaning tower of Pisa to test this - he used an inclined plane, a water clock and a pendulum). These experiments were later on improved, in various forms, by Huygens, Newton, Bessel and others and reached unprecedented accuracy with the work of Baron von Eo¨tv¨os(1889-...),whowasabletoshowthatinertialandgravitational massofdifferent 9 materials (like wood and platinum) agree to one part in 10 . In the 1950/60’s, this was 11 still further improved by R. Dicke to something like one part in 10 . More recently, rumours of a ‘fifth force’, based on a reanalysis of Eo¨tv¨os’ data (but buried in the meantime) motivated experiments with even higher accuracy and nodifference between m andm was found. i g 21Figure 1: Experimenter and his two stones freely floating somewhere in outer space, i.e. in the absence of forces. Newton’s theory would in principle be perfectly consistent with m 6= m , just as the i g formally analagous equation for an electrically charged particle with charge q in an e electrostatic field E =−∇φ, ¨ mx=−q∇φ , (1.9) i e is perfectly acceptable for any ratio q /m , and Einstein was very impressed with the e i observed equality of m and m . This should, he reasoned, not be a mere coincidence i g but is probably trying to tell us something rather deep about the nature of gravity. With his unequalled talent for discovering profound truths in simple observations, he concluded(callingthis“derglu¨cklichste GedankemeinesLebens”(thehappiestthought of my life)) that the equality of inertial and gravitational mass suggests a close relation between inertia and gravity itself, suggests, in fact, that locally effects of gravity and acceleration are indistinguishable, gravitational mass = inertial mass because (locally) GRAVITY = ACCELERATION He substantiated this with some classical thought experiments, Gedankenexperimente, as he called them, which in the meantime have morphed into and have come to be known as the elevator thought experiments, which we will now discuss. 1. Consider somebody in a small sealed box (elevator) somewhere in outer space. In the absence of any forces, this person will float. Likewise, two stones he has just dropped (see Figure 1) will float with him. 2. Now assume (Figure 2) that somebody on the outside suddenly pulls the box up with a constant acceleration. Then of course, our friend will be pressed to the bottom of the elevator with a constant force and he will also see his stones drop to the floor. 22Figure 2: Constant acceleration upwards mimics the effect of a gravitational field: ex- perimenter and stones drop to the bottom of the box. 3. Now consider (Figure 3) this same box brought into a constant gravitational field. Then again, he will be pressed to the bottom of the elevator with a constant force andhewillseehisstonesdroptothefloor. Withnoexperimentinsidetheelevator can he decide if this is actually due to a gravitational field or due to the fact that somebody is pulling the elevator upwards. Thus our first lesson is that, indeed, locally the effects of acceleration and gravity are indistinguishable. 4. Now consider somebody cutting the cable of the elevator (Figure 4). Then the elevator will fall freely downwards but, as in Figure 1, our experimenter and his stones will float as in the absence of gravity. Thus lesson number two is that, locally the effect of gravity can be eliminated by going to a freely falling reference frame (or coordinate system). This should not come as a surprise. In the Newtonian theory, if the free fall in a constant gravitational field is described by the equation x¨ =g (+ other forces) , (1.10) then in the accelerated coordinate system 2 ξ(x,t)=x−gt /2 (1.11) the same physics is described by the equation ¨ ξ =0 (+ other forces) , (1.12) and the effect of gravity has been eliminated by going to the freely falling coordi- natesystemξ. Thecrucialpointhereisthatinsuchareferenceframenotonlyour 23Figure 3: Effect of a constant gravitational field: indistinguishablefor our experimenter from that of a constant acceleration in Figure 2. Figure 4: Free fall in a gravitational field has the same effect as no gravitational field (Figure 1): experimenter and stones float. 24Figure 5: Experimenter and his stones in a non-uniform gravitational field: the stones will approach each other slightly as they fall to the bottom of the elevator. observer will float freely, but because of the equality of inertial and gravitational mass he will also observe all other objects obeying the usual laws of motion in the absence of gravity. 5. In the above discussion, I have put the emphasis on constant accelerations and on ‘locally’. To see the significance of this, consider our experimenter with his elevator in the gravitational field of the earth (Figure 5). This gravitational field is not constant but spherically symmetric, pointing towards the center of the earth. Therefore the stones will slightly approach each other as they fall towards the bottom of the elevator, in the direction of the center of the gravitational field. 6. Thus, if somebodycuts the cable now and the elevator is again in free fall (Figure 6), our experimenter will float again, so will the stones, but our experimenter will also notice that the stones move closer together for some reason. He will have to conclude that there is some force responsible for this. This is lesson number three: in a non-uniform gravitational field the effects of gravity cannot be eliminated by going to a freely falling coordinate system. This isonlypossiblelocally, onsuchscales onwhichthegravitational fieldisessentially constant. Einstein formalised the outcome of these thought experiments in what is now known as the Einstein Equivalence Principle which roughly states that physics in a freely falling 25Figure 6: Experimentator and stones freely falling in a non-uniform gravitational field. The experimenter floats, so do the stones, butthey move closer together, indicating the presence of some external force. frame in a gravitational field is the same as physics in an inertial frame in Minkowski space in the absence of gravitation. Two formulations are At every space-time point in an arbitrary gravitational field it is possible to choose a locally inertial (or freely falling) coordinate system such that, within a sufficiently small region of this point, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence 2 of gravitation. and Experimentsinasufficientlysmallfreelyfallinglaboratory,overasufficiently short time, give results that are indistinguishable from those of the same 3 experiments in an inertial frame in empty space. There are different versions of this principle depending on what precisely one means by ‘thelawsofnature’. Ifonejustmeansthelaws ofNewtonian(or relativistic) mechanics, then this priciple essentially reduces to the statement that inertial and gravitational mass are equal. Usually, however, this statement is taken to imply also Maxwell’s 2 S. Weinberg, Gravitation and Cosmology. 3 J. Hartle, Gravity. An Introduction to Einstein’s General Relativity. 264 theory, quantum mechanics etc. What it pragmatically asserts in one of its stronger forms is that ... thereisnoexperimentthatcandistinguishauniformacceleration from a uniform gravitational field. (J. Hartle, loc. cit.) The power of the above principle, which we will regard as a heuristic guideline, rather than trying to (prematurely) give it a mathematically precise formulation, lies in the fact that we can combine it with our understanding of physics in accelerated reference systems to gain insight into the physics in a gravitational field. Two immediate conse- quences of this (which cannot be derived on the basis of Newtonian physics or Special Relativity alone) are • light is deflected by a gravitational field just like material objects; • clocks run slower in a gravitational field than in the absence of gravity. To see the inevitability of the first assertion, imagine a light ray entering the rocket / elevator in Figure 1 horizontally through a window on the left hand side and exiting again at the same height through a window on the right. Now imagine, as in Figure 2, accelerating the elevator upwards. Then clearly the light ray that enters on the left will exit at a lower point of the elevator on theright because the elevator is accelerating upwards. By the equivalence principle one should observe exactly the same thing in a constant gravitational field (Figure 3). It follows that in a gravitational field the light ray is bent downwards, i.e. it experiences a downward acceleration with the (locally constant) gravitational acceleration g. To understand the second assertion, one can e.g. simply appeal to the so-called “twin- paradox” of Special Relativity: the accelerated twin is younger than his unaccelerated inertial sibling. Hence accelerated clocks run slower than inertial clocks. Hence, by the equivalence principle, clocks in a gravitational field run slower than clocks in the absence of gravity. Alternatively, one can imagine two observers at the top and bottom of the elevator, having identical clocks and sending light signals to each other at regular intervals as determined by their clocks. Once the elevator accelerates upwards, the observer at the bottom will receive the signals at a higher rate than he emits them (because he is accelerating towards the signals he receives), and he will interpret this as his clock running more slowly than that of the observer at the top. By the equivalence principle, the same conclusion now applies to two observers at different heights in a gravitational 4 For a discussion of different formulations of the equivalence principle and the logical relations among them, see E. di Casola, S. Liberati, S. Sonego, Nonequivalence of equivalence principles, arXiv:1310.7426 gr-qc. 27field. This can also be interpreted in terms of a gravitational redshift or blueshift (photons losing or gaining energy by climbing or falling in a gravitational field), and we will return to a more quantitative discussion of this effect in section 2.9. 1.2 Lorentz-Covariant Formulation of Special Relativity (Review) What the equivalence principle tells us is that we can expect to learn something about the effects of gravitation by transforming the laws of nature (equations of motion) from an inertial Cartesian coordinate system to other (accelerated, curvilinear) coordinates. As a first step, we will, in section 1.3 below, discuss the above example of an observer undergoing constant acceleration in the context of special relativity. As a preparation for this, and the remainder of the course, this section will provide a lightning review of the Lorentz-covariant formulation of special relativity, mainly to set the notation and conventions that will be used throughout, and only to the extent that it will be used in the following. 1. Minkowski space(-time) (a) ThearenaofspecialrelativityisMinkowskispace-timehenceforthMinkowski space for short, the union of space and time is implied by uttering the word “Minkowski”. It is the space of events, labelled by 3 Cartesian spatial coor- k dinates x and a time-coordinate t or, more usefully, by the coordinates a 0 k k (ξ )=(ξ =ct,ξ =x ) , (1.13) a where c is the speed of light. Typically in these notes ξ will indicate such a (locally) inertial coordinate system, whereas generic coordinates will be μ called x etc. We will almost always work in units in which c=1. (b) Minkowski space is equipped with a prescription for measuring distances, encoded in a line-element which, in these coordinates, takes the form X 2 0 2 k 2 ds =−(dξ ) + (dξ ) . (1.14) k (c) This can be written as X 2 0 2 k 2 a b ds =−(dξ ) + (dξ ) ≡η dξ dξ . (1.15) ab k with metric (η )=diag(−1,+1,+1,+1) or, more explicitly, ab   −1 0 0 0   0 +1 0 0   η =  (1.16) ab   0 0 +1 0 0 0 0 +1 (thus we are using the “mostly plus” convention). 282. Lorentz Transformations (a) Lorentz transformations are by definition those linear transformations a a a b ¯ ξ 7→ξ =L ξ (1.17) β that leave the Minkowski line-element invariant, 2 a b a b 2 a b ¯ ¯ ds¯ ≡η dξ dξ =η dξ dξ =ds ⇔ η L L =η . (1.18) ab ab ab cd c d In matrix notation this can also be written as t ¯ ξ =Lξ , LηL =η (1.19) t where L is the transpose of L. This is the defining condition for Lorentz t transformations,andtheLorentzsignatureanalogueoftheconditionR1R = 1foranorthogonaltransformation(rotationorreflection)inEuclideanspace, with metric η →1 =δ . ab ik ik Alternative notation: a¯ a¯ b a¯ a¯ b ¯ ξ =L ξ or ξ =L ξ (1.20) b b a a¯ ¯ Strictly speakingξ andξ may be considered to refer to two different quan- ¯ tities, to the cooordinates of the new point ξ in the old coordinate system versusthecoordinates of theold pointξ in thenewcoordinate system. How- ever, for many elementary purposes this difference between what is known as the active (moving points) versus the passive (relabelling points) point of view is not crucial, and one should not be hung-up on notation: coordinates arefundamentally justbookkeepingdevices sousewhatever is convenient for current bookkeeping or other purposes. (b) Infinitesimal Lorentz rotations, i.e. Lorentz transformations with L of the formL =1+ω, ω infinitesimal, are characterised by t t (1+ω) η(1+ω)=η ⇒ (ηω)+(ηω) =0 . (1.21) Thus the matrix ηω is anti-symmetric. In components, an infinitesimal Lorentz transformation therefore has the form a a b c δξ =ω ξ with ω ≡η ω =−ω . (1.22) ab ac ba b b (c) Poincar´etransformationsarethoseaffinetransformationsthatleavetheMin- kowskiline-elementinvariant. TheyarecomposedofLorentztransformations and arbitrary constant translations and thus have the form a a a b a ¯ ξ 7→ξ =L ξ +ζ , (1.23) β 29infinitesimally a a b a δξ =ω ξ +ǫ . (1.24) b Any two inertial systems in the sense of the equivalence principle of special relativity are related by a Poincar´e transformation. 3. Distance & Causal Structure (a) The Minkowski metric defines the Lorentz (and Poincar´e) invariant distance 2 a a b b (Δξ) =η (ξ −ξ )(ξ −ξ ) (1.25) ab P Q P Q a a betwen two events P and Q with coordinates ξ and ξ respectively. P Q 2 (b) Depending on the sign of (Δξ) , the two events P,Q are called, spacelike, lightlike (null) or timelike separated,   0 spacelike separated  2 (Δξ) = (1.26) =0 lightlike separated   0 timelike separated (c) The set of events that are lightlike separated from P define the lightcone at P. It consists of two components (joined at P), the future and the past 0 0 lightcone, distinguished by the sign of ξ −ξ (positive for Q on the future Q P 0 0 lightcone, ξ ξ , negative for Q on the past lightcone). Q P 4. Curves and Tangent Vectors a (a) A parametrised curve is given by a map λ7→ ξ (λ). The tangent vector to the curve at the point ξ(λ ) has components 0 d ′a a ξ (λ )= ξ (λ) . (1.27) 0 λ=λ 0 dλ It is called spacelike, lightlike (null) or timelike, depending on the sign of ′a ′b η ξ ξ , ab   0 spacelike  ′a ′b η ξ ξ =0 lightlike (1.28) ab   0 timelike This sign (and hence this classification) depends only on the image of the curve, not its parametrisation. (b) Acurvewhosetangentvectoriseverywheretimelikeiscalledatimelikecurve (and likewise for lightlike and spacelike curves). A curve whose tangent vector is everywhere timelike or null (i.e. non-spacelike) is called a causal curve. Worldlines of massive particles are timelike curves, those of massless particles (light) are null curves. 30(c) AnaturalLorentz-invariant parametrisation oftimelike curvesisprovidedby the Lorentz-invariant proper time τ along the curves, a a ξ =ξ (τ) , (1.29) with p p p 2 a b ′a ′b cdτ = −ds = −η dξ dξ = −η ξ ξ dλ ab ab (1.30) a b dξ (τ)dξ (τ) 2 ⇒ η =−c . ab dτ dτ Likewise spacelike curves are naturally parametrised by proper distance ds. The derivative with respect to proper time will be denoted by an overdot, d a a ˙ ξ (τ)= ξ (τ) . (1.31) dτ a ˙ Because τ is Lorentz-invariant, τ¯ =τ, tangent vectors ξ of τ-parametrised curves transform linearly under Lorentz transformations, a ¯ d ∂ξ d a a b a b ˙ ¯ ¯ ˙ ξ (τ)= ξ (τ)= ξ (τ)=L ξ (τ) . (1.32) b b dτ ∂ξ dτ These are the prototypes of what are called Lorentz vectors or, more gener- ally, Lorentz tensors. 5. Lorentz Vectors a (a) Lorentz vectors (or 4-vectors) are objects with components v which trans- a form under Lorentz transformations with the matrixL (to bethought of as b a a ¯ the Jacobian of the transformation relating ξ and ξ ), a a b v¯ =L v . (1.33) b (b) η can be regarded as defining an indefinite scalar product on the space of ab a b Lorentz vectors, and the Minkowski norm η v v and the Minkowski scalar ab a b product η v w of Lorentz vectors are invariant under Lorentz transforma- ab tions, a b a b a b a b η v¯ v¯ =η v v , η v¯ w¯ =η v w . (1.34) ab ab ab ab Avector iscalled, spacelike, lightlike (null)or timelike dependingonthesign of its Minkowski norm. (c) One can identify Minkowski space with its “tangent space”, i.e. with the 1,3 vector space V = R of 4-vectors equipped with the quadratic form or scalar productη with signature (1,3). ab 6. Other Lorentz Tensors 31

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