6CCP3630 General Relativity - Kcl.ac.uk

that are not as well explained as you would have liked, ask me or read some of the references provided.I will be teaching the course in the order the lectures are presented. But I suspect we will not havesufficient time to cover all the topics. So sometimes I will ask you to read sections that I don’t have timeto cover – so do try to keep up with your reading. There are additional material in these lecture noteswhich are non-examinable, and is marked with two asterisks **. Material marked with a single asterisk* is examinable, but is review material which will not be lectured on in class – so read ahead!There are 5 homework sets to help you along, and the notes often refer you to these problems. The bestway is to work through the problems when asked by the notes. GR, like most physics subjects, rewardsthose who practice, practice and practice. Fortunately, there are plenty of examples and problems thatyou can easily find on the internet. Unfortunately, that means that any problem I can come up withprobably has a solution somewhere posted online. So, while you are encouraged to use the internet tohelp you study, try to resist checking the answers!Finally, a word on notation. As you will soon see, there is going to be a lot of new notation introduced,and they can be quite daunting at first, but it is really quite intuitive once you get used to it. A wordof warning when looking through other references (and the internet) though – everybody seems to havetheir own favorite notation, especially when it comes to representing abstract tensors and componenttensors (if you don’t know what these are, don’t worry!), so it is easy to get confused. My advice is topick a reference book or two, get used to their notation, and stick to them (and these lecture notes) anddon’t google too much on the internet.6

Recommended Books* is highly recommended for this course.Online resources will be posted on the class webpage. B. Schutz, *A First Course in General Relativity, 2nd ed., Cambridge University Press 2009. Thisis an excellent book, one of the first books on GR aimed at undergraduates and still the best in myopinion! B. Schutz, *Geometrical Methods of Mathematical Physics, Cambridge University Press 1985.Schutz’s book on differential geometry is the best book on differential geometry aimed at a physicist. It has the right amount of rigour, and great explanations that is a hallmark of Schutz’s books.Best of all – it spends very little time on GR, so it provides a nice counter balance to any GR booktrying to teach differential geometry. Aimed at advanced undergraduates and beginning PhDs. B. Wald, General Relativity, The University of Chicago Press 1984. Aimed at PhDs and researchers,Wald’s GR book has been around for 3 decades but it still one of the best. Wald’s mathematical approach can be quite dense sometimes, but it is accurate, precise and complete. The WaldInfallibility Principle : When in doubt, go Wald. Charles Misner, Kip Thorne and John Wheeler, Gravitation, Freeman 1973. The first modern bookon GR, and what a tome. It tries to be super complete and goes a long way in achieving that goal.Personally, I don’t use it as much though that’s probably because I was not taught using it. It’s amassive book, and good not just for GR, but also to kill pesky cockcroaches that infest dirty PhDoffices. S. Carroll, *Spacetime and Geometry, Addison-Wesley 2004. This is a relatively new addition tothe many graduate level GR books, but has since gained a loyal following and is now one of themost popular books for instructors to teach GR from. It strikes a nice balance between the intuitivephysics explanation of Schulz with the mathematical rigour of Wald. It is aimed at a beginningPhD student, but a motivated undergraduate will enjoy this. S. Weinberg, Gravitation and Cosmology, John Wiley 1972. This book by God, I mean, Weinberg,is the one book that bucks the trend of tying GR with differential geometry. Written by one of themost remarkable physicists of our generation who is the authority on field theory, this book teachesGR from a physically intuitive viewpoint with a generous dose of field theory. This may not bethe one GR book you learn GR from, but it is full of deep insight into the theory that only God, Imean Weinberg, can provide. J. Hartle, *An Introduction to Einstein’s General Relativity, Pearson 2003. This book is aimed atthe undergraduate level, approaching GR in a highly physically intuitive way. It deliberately avoidsa lot of fancy mathematics. An excellent reference book for this course. A. Lightman, W. Press, R. Price, S. Teukolsky, *Problem Book in General Relativity, PrincetonUniversity Press, 1975. This is a collection of GR problems with full solutions, causing muchgrief to instructors desperately trying to find new problems. Anyhow, it’s a great resource, andyou are encouraged to try some of the problems. In fact, it is now available online for free athttp://apps.nrbook.com/relativity/index.html.7

Chapter 1IntroductionFrom Newton to EinsteinA high-brow is someone who looksat a sausage and thinks ofPicasso.A.P. Herbert1.1The Newtonian WorldviewI’ve got bad news. For many years in your life, you have been told several lies. The first lie is that aparticle of mass m and position x, under a force F moves under the Newton’s Second Law of MotionF md2 xdt2(1.1)where t is some absolute time that everyone in the entire Universe can agree upon. The second lie isthat two particles of masses m and M will feel an attractive force towards each other whose amplitudeis given by Newton’s Law of GravitationGM mF (1.2)r2where r is the distance between these two masses, and G 6.673 10 11 Nm2 kg 2 s 2 is Newton’sconstant. Equivalently, one can express the gravitational potential Φ of any mass m at point x0 asΦ(x) Gm, x x0 (1.3)and then the gravitational acceleration g is simply the gradient of the potentialg Φ.(1.4)We usually call the potential “the gravitational field”, whose gradient is the acceleration, although as wewill see later it also makes sense to think of the acceleration as a “field”. Equivalently, one can derivethe potential Φ of any mass distribution specified by the density ρ(x) via the Poisson Equation 2 Φ 4πGρ(x).The Poisson Equation is exactly equivalent to Eq. (1.2).8(1.5)

You, and most people, bought these lies. The reason is that, like all successful lies, these lies arevery good lies. These equations, combined with Newton’s first and third laws of motion, allow us tocompute the dynamics of the very small such as the curve of a football being struck by Roberto Carlosto the very large such as the motion of the planets around the Sun. We use these equations to accuratelycompute the trajectory our spacecraft must take to reach Pluto more than 4 billion km away and modelthe aerodynamics of an airplane. It unifies seemingly standalone concepts such as Galileo’s Theory ofInertia and Kepler’s Law of Planetary Motion, and asserts that the same laws apply to physics at allscales from the atomic scale to the galactic scale. By all measures, they are incredibly successful lies.Newton’s Laws are not just a bunch of equations where you plug in some initial conditions (usuallya particle’s initial position and velocity) to compute the result (either some future/past position andvelocity), they actively promote the idea that space and time are separate entities, to be treated differently.In the Newtonian view of the Universe, we live in a 3-dimensional world. This 3D world then dynamicallyevolves, with the evolution govern by some quantity we call “time”. Every being in this Universe, mustagree on this time. Furthermore, all forces are instantaneous across infinite distances. The Laws, andtheir underlying ideas about how space and time are subdivided into their separate domains, are usuallyknown collectively as Newtonian Mechanics.Nevertheless, despite their successes, there are clues to why they are not quite the right picture of theUniverse. These clues can be divided into two kinds : observational and theoretical. The (more obviousand commonly stated) observational clue is that the perihelion (the point of minimum distance fromthe Sun) of Mercury is precessing, i.e. the perihelion is changing by 43 arcseconds per century. WhileNewton’s laws correctly predicted Mercury’s orbital period and the fact that the orbit is an ellipse, itdoes not predict this precession. This precession was discovered by Urbain Le Verrier in 1859 (where helooked at data from 1697 to 1848), and recognized by him to be a problem for Newtonian mechanics. Atotally Newtonian Mechanical solution to this problem is to postulate the existence of a yet unseen planet(named Vulcan) in order to provide the necessary gravitational tug to generate this effect – this is theNewtonian version of the Dark Matter problem. This ultimately failed hypothesis is not without merit:non-conformance of Uranus’ orbit to Newton’s Laws led to the prediction and subsequent discovery ofNeptune in 1846.Less well known and more subtly, the Newtonian picture of the Universe has theoretical issues aswell. Maxwell had discovered his eponymous equations by 1861. These equations unify the theories ofelectric and magnetic fields, and predict that these fields propagate as waves at a finite speed of light.These equations are B µ0 J µ0 0 t E,(1.6) E t B,ρc ·E , 0 · B 0.(1.7)(1.8)(1.9)Here 0 8.854 10 12 kg 1 m 3 s4 A2 is the permittivity of free space and µ0 4π 10 7 kgms 2 A 2 isthe permeability of free space, and that we have used ρc for charge density. Note that we have used thenotation t / t – if you are not familiar with this you should probably start to get used to it as youwill see it will save you a lot of writing in the future.In electrostatics (i.e. the study of electromagnetism when all time derivatives are zero), Coulomb’slaw (discovered in 1781) states that the magnitude of the force between two particles of electric chargesq1 and q2 separate by distance r, is given byF kq1 q2,r29(1.10)

where k 8.99 1010 Nm2 C 2 is the Coulomb’s constant. This force is attractive if the charges are ofopposite sign, and repulsive otherwise. In analogy to the gravitational potential, we can also write theCoulomb’s potential of a particle of charge q at x0V (x) kq. x x0 (1.11)The gradient of the potential in electrostatics is then simply the electric field E that you know and loveE V (x).(1.12)Comparing the above equation to Eq. (1.4), and Eq. (1.3) to Eq. (1.11) (the minus sign difference is dueto the convention we chose for our potential V ). Similarly, taking the gradient of the electric field1 givesus one of Maxwell equations Eq. (1.8), which is the electromagnetic version of the Poisson Equation Eq.(1.5). The electromagnetic analog of Newton’s 2nd law of motion is the Lorentz forceF q(E v B)(1.13)where v is the velocity of the charge. This law was discovered experimentally, but have you ever wonderedwhy this is so – in particular why is the force proportional to the electric field but proportional to thecross product of the particle velocity with the magnetic field B? We will answer this question soonenough, but let’s plow on.Despite all these tantalizing analogies, the point of departure of Newton’s gravity and electromagnetism is the presence of waves. The reason is that Maxwell equations go beyond electrostatics, butdescribe electrodynamics, i.e. where charges and fields evolve with time2 . In the absence of anycharges, Maxwell’s equations Eq. (1.6) to Eq. (1.9) can be recast as a pair of wave equations t2 E 11 2 E 0 , t2 B 2 B 0, 0 µ0 0 µ0(1.14)with solutionsE E0 ei(k·x kct) , B B0 ei(k·x kct)p(1.15)where k is the wavevector and k k , which is simply waves propagating at velocity c 1/( 0 µ0 ) 3 1010 cm s 1 , i.e. we (or more accurately, Maxwell) have discovered that electromagnetic waves travelat light speed. The question is then: why isn’t there an equivalent “gravitational field” (or accelerationfield) waves predicted in the Newton’s Law of gravity too? The short answer is that Newton’s Law ofgravity Eq. (1.2) has no time derivatives in it, while the Coulomb Force Eq. (1.10) is supplemented bythe other equations of Maxwell. This is our first purely theoretical hint that perhaps Newton’s gravity isnot the complete theory of gravity – maybe there exist similar “dynamical” equations for gravity too?Now you might argue that gravity is fundamentally different from electromagnetism, so even thoughthere are analogies between the two forces, Eq. (1.2) is really all there is to it. However, wave solutionsalso imply something important: since they propagate at some finite speed (namely c), their effects taketime to propagate from one point to another. If the Sun suddenly goes out, it will take eight minutes orso for us to find out about it. Newton’s law simply states that the effects of gravity propagate at infinitespeed – along with its difficult philosophical consequences.1.2From Space and Time to SpacetimeIn 1905, Einstein formulated his Theory of Special Relativity. You have studied special relativityin your previous modules, so this won’t be new to you. However, let’s review it again, but now in a1 Onecan also take the gradient of the magnetic field to give the equation · B 0, where the magnetic field is notsourced – a consequence of the fact that we have no magnetic monopoles in nature.2 For example, Eq. (1.12) is really E · V (x) A – we will discuss this in further detail in a moment.t10

deliberately high-brow language that you might not be familiar with. In fact, let’s start with Newton’sequations in high brow language. What follows is a completely unhistorical description – instead we’llfocus on the modern view of things.1.2.1Newton’s 2nd Law of Motion : Galilean Transformations, Space andTimeWe begin by considering physics in 3 space and 1 time dimensions (usually shortened to “3 1D”). Wewant to define some frame of reference so that we can describe some physical situations, e.g. say “thisparticle is moving at velocity v with respect to X ” where X can be some other particle, or more usually, asome frame of reference. The usual way to do this is to lay down some coordinates on this frame so that wecan do some calculations on it. You can be quite creative with labeling your coordinates, but we want toimpose additional restrictions: we want the coordinates to label the space homogenously in space and timeand isotropically. What this means is that your coordinate system should look “the same” everywhere,and has no special directions. Such a coordinate frame is called an inertial frame. A simple choice ofcoordinates which satisfies these considerations are the usual Cartesian coordinates (t, x, y, z), wheret labels time and (x, y, z) the 3 spatial coordinates. On the other hand, the spherical coordinate system(t, r, θ, φ) has a special point (the origin), so is not an inertial frame. Usually, non-inertial frames usuallylead to “fictitious” forces such as the centripetal force or the Coriolis force.Given such coordinates, we can “do physics”, i.e. solve equations using it. For example, you can solveNewton’s equations d2 x/dt2 F/m. For constant F and m, the solution, as you have done countlesstime, isF 2x t vt x0(1.16)2mwhere v ẋ(0) is the initial velocity at time t 0 and x0 x(0) is the position at time t 0.Suppose now I give you another inertial frame with a new coordinate system, say (t0 , x0 , y 0 , z 0 ) which isrelated to (t, x, y, z) in some undefined way (for now). The question is: can you describe the same physicsusing this new coordinate system? Let’s do a specific example, suppose the two coordinate systems arerelated byt0 t , x0 x ut.(1.17)In words, the primed frame is moving by some constant velocity u with respect to the unprimed frame.Such a coordinate transform is called a boost. It is easy to describe the physical quantities of velocityand acceleration in the new frame in terms of the quantities of the old frame, vizv0 dx0 t d(x ut) 0 v u0dt tdt(1.18)anddv0 t d(v u) 0 a.(1.19)dte tdtThus the velocity in the primed frame is shifted by the constant u while the acceleration remains thesame – by the “same” we mean that the value remains the same in both frames. This is all very familiar.In fact, from Eq. (1.19), we can see immediately that Newton’s equation of motion Eq. (1.1) underthe transformation Eq. (1.17), returns to its exact same functional forma0 F md2 xd2 x 0 m.dt2dt02(1.20)In high-brow language, we say that “Newton’s 2nd Law of motion remains invariant3 under the transformation Eq. (1.17).” This means that, it doesn’t matter which of the inertial frames (primed or unprimed)3 Forthose who have taken the 5CCP2332 Symmetry in Physics module, this would be very familiar to you.11

you are on, you can happily use Newton’s 2nd Law to compute motion of particles and you will get thecorrect answer relative to your frame.In addition to Eq. (1.17), what are the set of all possible linear4 coordinate transformations whichleave Newton’s 2nd Law Eq. (1.1) invariant? We can shift the coordinates by constants, or a translation,x0 x x0 , t0 t s,(1.22)it is easy to check that sinced t0 d, dx0 dx dt t dt0Newton’s 2nd Law Eq. (1.1) remains invariant.The other set of possible coordinate transformations are Rotationsx0 Rx(1.23)(1.24)where R is some 3 3 square matrix called a rotation matrix. For example, a rotation of angle θaround the z axis would yield a rotation matrix cos θ sin θ 0 (1.25)R sin θcos θ 0 .001You can imagine some more complicated rotations. Under rotation, Newton’s 2nd law transforms asF0 md2 (Rx)dt2(1.26)so as long as we rotate the force F0 RF, we recover the Newton’s 2nd L

First and foremost, I would like to thank my former PhD advisor Sean Carroll who, years ago, took a hard look at this completely unprepared ex-mechanical engineer and decided that I can be taught General Relativity while simultaneously trying to modif