Lecture Notes on General Relativity pdf

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Lecture Notes on General Relativity Sean M. Carroll Institute for Theoretical Physics University of California Santa Barbara, CA 93106 carrollitp.ucsb.edu December 1997 Abstract These notes represent approximately one semester’s worth of lectures on intro- ductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein’s equations, and three applications: grav- itational radiation, black holes, and cosmology. Individual chapters, and potentially updated versions, can be found at http://itp.ucsb.edu/carroll/notes/. NSF-ITP/97-147 gr-qc/9712019 arXiv:gr-qc/9712019v1 3 Dec 1997i Table of Contents 0. Introduction table of contents — preface — bibliography 1. Special Relativity and Flat Spacetime thespacetime interval—themetric—Lorentztransformations—spacetime diagrams — vectors — the tangent space — dual vectors — tensors — tensor products — the Levi-Civita tensor — index manipulation — electromagnetism — differential forms — Hodge duality — worldlines — proper time — energy-momentum vector — energy- momentum tensor — perfect fluids — energy-momentum conservation 2. Manifolds examples — non-examples — maps — continuity — the chain rule — open sets — charts and atlases — manifolds — examples of charts — differentiation — vectors as derivatives — coordinate bases — the tensor transformation law — partial derivatives arenottensors —themetric again—canonicalformofthemetric —Riemann normal coordinates — tensor densities — volume forms and integration 3. Curvature covariant derivatives and connections — connection coefficients — transformation properties — the Christoffel connection — structures on manifolds — parallel trans- port — the parallel propagator — geodesics — affine parameters — the exponential map — the Riemann curvature tensor — symmetries of the Riemann tensor — the Bianchi identity — Ricci and Einstein tensors — Weyl tensor — simple examples — geodesic deviation — tetrads and non-coordinate bases — the spin connection — Maurer-Cartan structure equations — fiber bundles and gauge transformations 4. Gravitation the Principle of Equivalence — gravitational redshift — gravitation as spacetime cur- vature — the Newtonian limit — physics in curved spacetime — Einstein’s equations — the Hilbert action — the energy-momentum tensor again — the Weak Energy Con- dition — alternative theories — the initial value problem — gauge invariance and harmonic gauge — domains of dependence — causality 5. More Geometry pullbacks and pushforwards — diffeomorphisms — integral curves — Lie derivatives — the energy-momentum tensor one more time — isometries and Killing vectorsii 6. Weak Fields and Gravitational Radiation the weak-field limit defined — gauge transformations — linearized Einstein equations — gravitational plane waves — transverse traceless gauge — polarizations — gravita- tional radiation by sources — energy loss 7. The Schwarzschild Solution and Black Holes spherical symmetry — the Schwarzschild metric — Birkhoff’s theorem — geodesics of Schwarzschild — Newtonian vs. relativistic orbits — perihelion precession — the event horizon — black holes — Kruskal coordinates — formation of black holes — Penrose diagrams — conformal infinity — no hair — charged black holes — cosmic censorship — extremal black holes — rotating black holes — Killing tensors — the Penrose process — irreducible mass — black hole thermodynamics 8. Cosmology homogeneity and isotropy — the Robertson-Walker metric — forms of energy and momentum — Friedmann equations — cosmological parameters — evolution of the scale factor — redshift — Hubble’s lawiii Preface These lectures represent an introductory graduate course in general relativity, both its foun- dations and applications. They are a lightly edited version of notes I handed out while teaching Physics 8.962, the graduate course in GR at MIT, during the Spring of 1996. Al- though they are appropriately called “lecture notes”, the level of detail is fairly high, either includingallnecessarystepsorleavinggapsthatcanreadilybefilledinbythereader. Never- theless, therearevariousways inwhichthesenotesdifferfromatextbook; mostimportantly, they are not organized into short sections that can be approached in various orders, but are meant to be gone through from start to finish. A special effort has been made to maintain a conversational tone, in an attempt to go slightly beyond the bare results themselves and into the context in which they belong. The primary question facing any introductory treatment of general relativity is the level of mathematical rigor at which to operate. There is no uniquely proper solution, as different students will respond with different levels of understanding and enthusiasm to different approaches. Recognizing this, I have tried to provide something for everyone. The lectures do not shy away from detailed formalism (as for example in the introduction to manifolds), but also attempt to include concrete examples andinformal discussion of the concepts under consideration. As these are advertised as lecture notes rather than an original text, at times I have shamelessly stolen from various existing books on the subject (especially those by Schutz, Wald, Weinberg, and Misner, Thorne and Wheeler). My philosophy was never to try to seek originality for its own sake; however, originality sometimes crept in just because I thought I could be more clear than existing treatments. None of the substance of the material in these notes is new; the only reason for reading them is if an individual reader finds the explanations here easier to understand than those elsewhere. Time constraints during the actual semester prevented me from covering some topics in the depth which they deserved, an obvious example being the treatment of cosmology. If the time and motivation come to pass, I may expand and revise the existing notes; updated versions will be available at http://itp.ucsb.edu/carroll/notes/. Of course I will appreciate having my attention drawn to any typographical or scientific errors, as well as suggestions for improvement of all sorts. Numerous people have contributed greatly both to my own understanding of general relativity and to these notes in particular — too many to acknowledge with any hope of completeness. Special thanks are due to Ted Pyne, who learned the subject along with me, taught me a great deal, and collaborated on a predecessor to this course which we taught as a seminar in the astronomy department at Harvard. Nick Warner taught the graduate course at MIT which I took before ever teaching it, and his notes were (as comparison williv reveal) an important influence on these. George Field offered a great deal of advice and encouragement as I learned the subject and struggled to teach it. Tam´as Hauer struggled along with me as the teaching assistant for 8.962, and was an invaluable help. All of the students in 8.962 deserve thanks for tolerating my idiosyncrasies and prodding me to ever higher levels of precision. During the course of writing these notes I was supported by U.S. Dept. of Energy con- tract no. DE-AC02-76ER03069and National Science Foundation grants PHY/92-06867and PHY/94-07195.v Bibliography The typicallevelofdifficulty(especially mathematical)ofthebooksisindicatedbyanumber of asterisks, one meaning mostly introductory and three being advanced. The asterisks are normalized to these lecture notes, which would be given . The first four books were frequentlyconsultedinthepreparationofthesenotes,thenextsevenareotherrelativitytexts which I have found to be useful, and the last four are mathematical background references. • B.F. Schutz, A First Course in General Relativity (Cambridge, 1985) . This is a very nice introductory text. Especially useful if, for example, you aren’t quite clear on what the energy-momentum tensor really means. • S. Weinberg, Gravitation and Cosmology (Wiley, 1972) . A really good book at what it does, especially strong on astrophysics, cosmology, and experimental tests. However, it takes an unusual non-geometric approach to the material, and doesn’t discuss black holes. • C.Misner,K.ThorneandJ.Wheeler,Gravitation(Freeman,1973). Aheavybook, in various senses. Most things you want to know are in here, although you might have to work hard to get to them (perhaps learning something unexpected in the process). • R. Wald, General Relativity (Chicago, 1984) . Thorough discussions of a number of advanced topics, including black holes, global structure, and spinors. The approach ismoremathematicallydemandingthanthepreviousbooks,andthebasicsarecovered pretty quickly. • E.TaylorandJ.Wheeler, SpacetimePhysics (Freeman, 1992). Agoodintroduction to special relativity. • R. D’Inverno, Introducing Einstein’s Relativity (Oxford, 1992) . A book I haven’t looked at very carefully, but it seems as if all the right topics are covered without noticeable ideological distortion. • A.P. Lightman, W.H. Press, R.H. Price, and S.A. Teukolsky, Problem Book in Rela- tivity and Gravitation (Princeton, 1975) . A sizeable collection of problems in all areasofGR,with fullyworked solutions, making itallthe moredifficult forinstructors to invent problems the students can’t easily find the answers to. • N.Straumann,GeneralRelativityandRelativisticAstrophysics(Springer-Verlag,1984) . A fairly high-level book, which starts out with a gooddeal of abstract geometry and goes on to detailed discussions of stellar structure and other astrophysical topics.vi • F. de Felice and C. Clarke, Relativity on Curved Manifolds (Cambridge, 1990) . A mathematical approach, but with an excellent emphasis on physically measurable quantities. • S. Hawking and G. Ellis, The Large-Scale Structure of Space-Time (Cambridge, 1973) . Anadvancedbookwhichemphasizes globaltechniquesandsingularitytheorems. • R. Sachs and H. Wu, General Relativity for Mathematicians (Springer-Verlag, 1977) . Just what the title says, although the typically dry mathematics prose style is here enlivened by frequent opinionated asides about both physics and mathematics (and the state of the world). • B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge, 1980) . Another good book by Schutz, this one covering some mathematical points that are left out of the GR book (but at a very accessible level). Included are discussions of Lie derivatives, differential forms, and applications to physics other than GR. • V. Guillemin and A. Pollack, Differential Topology (Prentice-Hall, 1974) . An entertaining survey of manifolds, topology, differential forms, and integration theory. • C. Nash and S. Sen, Topology and Geometry for Physicists (Academic Press, 1983) . Includes homotopy, homology, fiberbundles andMorsetheory, withapplications to physics; somewhat concise. • F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups (Springer- Verlag, 1983) . The standard text in the field, includes basic topics such as manifolds and tensor fields as well as more advanced subjects.December 1997 Lecture Notes on General Relativity Sean M. Carroll 1 Special Relativity and Flat Spacetime We will begin with a whirlwind tour of special relativity (SR) and life in flat spacetime. The point will be both to recall what SR is all about, and to introduce tensors and related concepts that will be crucial later on, without the extra complications of curvature on top of everything else. Therefore, for this section we will always be working in flat spacetime, and furthermore we will only use orthonormal (Cartesian-like) coordinates. Needless to say it is possible to do SR in any coordinate system you like, but it turns out that introducing the necessary tools for doing so would take us halfway to curved spaces anyway, so we will put that off for a while. It is often said that special relativity is a theory of 4-dimensional spacetime: three of space, one of time. But of course, the pre-SR world of Newtonian mechanics featured three spatial dimensions and a time parameter. Nevertheless, there was not much temptation to consider these as different aspects of a single 4-dimensional spacetime. Why not? t space at a fixed time x, y, z Consider a garden-variety 2-dimensional plane. It is typically convenient to label the points on such a plane by introducing coordinates, forexample by defining orthogonalx and y axes and projecting each point onto these axes in the usual way. However, it is clear that most of the interesting geometrical facts about the plane are independent of our choice of coordinates. As a simple example, we can consider the distance between two points, given 11 SPECIAL RELATIVITY AND FLAT SPACETIME 2 by 2 2 2 s = (Δx) +(Δy) . (1.1) ′ ′ In a different Cartesian coordinate system, defined byx andy axes which are rotated with respect to the originals, the formula for the distance is unaltered: 2 ′ 2 ′ 2 s =(Δx) +(Δy) . (1.2) We therefore say that the distance is invariant under such changes of coordinates. y’ y Δs Δy Δy’ x’ Δx’ x Δx Thisiswhyitisusefultothinkoftheplaneas2-dimensional: althoughweusetwodistinct numbers to label each point, the numbers are not the essence of the geometry, since we can rotate axes into each other while leaving distances and so forth unchanged. In Newtonian physics this is not the case with space and time; there is no useful notion of rotating space and time into each other. Rather, the notion of “all of space at a single moment in time” has a meaning independent of coordinates. Such is not the case in SR. Let us consider coordinates (t,x,y,z) on spacetime, set up in the following way. The spatial coordinates (x,y,z) comprise a standard Cartesian system, constructed forexample by welding together rigid rods which meet atright angles. The rods mustbemovingfreely, unaccelerated. Thetimecoordinateisdefinedbyasetofclockswhich are not moving with respect to the spatial coordinates. (Since this is a thought experiment, we imagine that the rods are infinitely long and there is one clock at every point in space.) The clocks are synchronized in the following sense: if you travel from one point in space to any other in a straight line at constant speed, the time difference between the clocks at the1 SPECIAL RELATIVITY AND FLAT SPACETIME 3 ends of your journey is the same as if you had made the same trip, at the same speed, in the other direction. The coordinate system thus constructed is an inertial frame. An event is defined as a single moment in space and time, characterized uniquely by (t,x,y,z). Then, without any motivation for the moment, let us introduce the spacetime interval between two events: 2 2 2 2 2 s =−(cΔt) +(Δx) +(Δy) +(Δz) . (1.3) (Notice that it can be positive, negative, or zero even for two nonidentical points.) Here, c is some fixed conversion factor between space and time; that is, a fixed velocity. Of course it will turn out to be the speed of light; the important thing, however, is not that photons happen to travel at that speed, but that there exists a c such that the spacetime interval is invariant under changes of coordinates. In other words, if we set up a new inertial frame ′ ′ ′ ′ (t,x,y,z ) by repeating our earlier procedure, but allowing for an offset in initial position, angle, and velocity between the new rods and the old, the interval is unchanged: 2 ′ 2 ′ 2 ′ 2 ′ 2 s =−(cΔt) +(Δx) +(Δy) +(Δz ) . (1.4) This is why it makes sense to think of SR as a theory of 4-dimensional spacetime, known as Minkowski space. (This is a special case of a 4-dimensional manifold, which we will deal with in detail later.) As we shall see, the coordinate transformations which we have implicitly defined do, in a sense, rotate space and time into each other. There is no absolute notion of “simultaneous events”; whether two things occur at the same time depends on the coordinates used. Therefore the division of Minkowski space into space and time is a choice we make for our own purposes, not something intrinsic to the situation. Almost all of the “paradoxes” associated with SR result from a stubborn persistence of the Newtonian notions of a unique time coordinate and the existence of “space at a single moment in time.” By thinking in terms of spacetime rather than space and time together, these paradoxes tend to disappear. Let’s introduce some convenient notation. Coordinates on spacetime will be denoted by letters with Greek superscript indices running from 0 to 3, with 0 generally denoting the time coordinate. Thus, 0 x =ct 1 x =x μ x : (1.5) 2 x =y 3 x =z (Don’t start thinking of the superscripts as exponents.) Furthermore, for the sake of sim- plicity we will choose units in which c =1 ; (1.6)1 SPECIAL RELATIVITY AND FLAT SPACETIME 4 we will therefore leave out factors ofc in all subsequent formulae. Empirically we know that 8 cisthespeedoflight,3×10 meterspersecond; thus, weareworkinginunitswhere1second 8 equals 3×10 meters. Sometimes it will be useful to refer to the space and time components μ of x separately, so we will use Latin superscripts to stand for the space components alone: 1 x =x i 2 x : x =y (1.7) 3 x =z Itisalsoconvenient towritethespacetimeintervalinamorecompactform. Wetherefore introduce a 4×4 matrix, the metric, which we write using two lower indices:   −1 0 0 0   0 1 0 0   η =  . (1.8) μν   0 0 1 0 0 0 0 1 (Some references, especially field theory books, define the metric with the opposite sign, so be careful.) We then have the nice formula 2 μ ν s =η Δx Δx . (1.9) μν Notice that we use the summation convention, in which indices which appear both as superscripts and subscripts are summed over. The content of (1.9) is therefore just the same as (1.3). Now we can consider coordinate transformations in spacetime at a somewhat more ab- stract level than before. What kind of transformations leave the interval (1.9) invariant? One simple variety are the translations, which merely shift the coordinates: ′ μ μ μ μ x →x =x +a , (1.10) μ where a is a set of four fixed numbers. (Notice that we put the prime on the index, not on μ the x.) Translations leave the differences Δx unchanged, so it is not remarkable that the μ interval is unchanged. The only other kind of linear transformation is to multiply x by a (spacetime-independent) matrix: ′ ′ μ μ ν x =Λ x , (1.11) ν or, in more conventional matrix notation, ′ x = Λx. (1.12) μ These transformations do not leave the differences Δx unchanged, but multiply them also by the matrix Λ. What kind of matrices will leave the interval invariant? Sticking with the matrix notation, what we would like is 2 T ′ T ′ s = (Δx) η(Δx) = (Δx) η(Δx) T T = (Δx) Λ ηΛ(Δx) , (1.13)1 SPECIAL RELATIVITY AND FLAT SPACETIME 5 and therefore T η =Λ ηΛ , (1.14) or ′ ′ μ ν η = Λ Λ η ′ ′ . (1.15) ρσ ρ σ μ ν ′ μ ′ ′ We want to find the matrices Λ such that the components of the matrix η are the ν μ ν same as those of η ; that is what it means for the interval to be invariant under these ρσ transformations. The matrices which satisfy (1.14) are known as the Lorentz transformations; the set of them forms a group under matrix multiplication, known as theLorentz group. There is a close analogy between this group and O(3), the rotation group in three-dimensional space. The rotation group can be thought of as 3×3 matrices R which satisfy T 1=R 1R , (1.16) where 1 is the 3×3 identity matrix. The similarity with (1.14) should be clear; the only difference is the minus sign in the first term of the metricη, signifying the timelike direction. The Lorentz group is therefore often referred to as O(3,1). (The 3×3 identity matrix is simply the metric for ordinary flat space. Such a metric, in which all of the eigenvalues are positive, is calledEuclidean, while those such as (1.8) which feature a single minus sign are called Lorentzian.) Lorentz transformations fall into a number ofcategories. First there are the conventional rotations, such as a rotation in the x-y plane:   1 0 0 0   ′ 0 cosθ sinθ 0   μ Λ =  . (1.17) ν   0 −sinθ cosθ 0 0 0 0 1 The rotation angle θ is a periodic variable with period 2π. There are also boosts, which may be thought of as “rotations between space and time directions.” An example is given by   coshφ −sinhφ 0 0   ′ −sinhφ coshφ 0 0   μ Λ =  . (1.18) ν   0 0 1 0 0 0 0 1 The boost parameter φ, unlike the rotation angle, is defined from −∞ to ∞. There are also discrete transformations which reverse the time direction or one or more of the spa- tial directions. (When these are excluded we have the proper Lorentz group, SO(3,1).) A general transformation can be obtained by multiplying the individual transformations; the1 SPECIAL RELATIVITY AND FLAT SPACETIME 6 explicit expression for this six-parameter matrix (three boosts, three rotations) is not suffi- ciently pretty or useful to bother writing down. In general Lorentz transformations will not commute, so the Lorentz group is non-abelian. The set of both translations and Lorentz transformations is a ten-parameter non-abelian group, the Poincar´e group. You should not be surprised to learn that the boosts correspond to changing coordinates by moving to a frame which travels at a constant velocity, but let’s see it more explicitly. ′ ′ For the transformation given by (1.18), the transformed coordinates t and x will be given by ′ t = tcoshφ−xsinhφ ′ x = −tsinhφ+xcoshφ . (1.19) ′ From this we see that the point defined by x =0 is moving; it has a velocity x sinhφ v = = = tanhφ . (1.20) t coshφ −1 To translate into more pedestrian notation, we can replace φ =tanh v to obtain ′ t = γ(t−vx) ′ x = γ(x−vt) (1.21) √ 2 where γ = 1/ 1−v . So indeed, our abstract approach has recovered the conventional expressions for Lorentz transformations. Applying these formulae leads to time dilation, length contraction, and so forth. An extremely useful tool is the spacetime diagram, so let’s consider Minkowski space from this point of view. We can begin by portraying the initial t and x axes at (what are conventionally thoughtofas)rightangles, andsuppressing they andz axes. Then according ′ ′ to (1.19), under a boost in the x-t plane the x axis (t = 0) is given by t =xtanhφ, while ′ ′ thet axis (x = 0) is given by t =x/tanhφ. We therefore see that the space and time axes are rotated into each other, although they scissor together instead of remaining orthogonal in the traditional Euclidean sense. (As we shall see, the axes do in fact remain orthogonal in the Lorentzian sense.) This should come as no surprise, since if spacetime behaved just like a four-dimensional version of space the world would be a very different place. It is also enlightening to consider the paths corresponding to travel at the speed c = 1. These aregiven inthe originalcoordinate system byx =±t. In the new system, amoment’s ′ ′ thought reveals that the paths defined by x = ±t are precisely the same as those defined by x = ±t; these trajectories are left invariant under Lorentz transformations. Of course we know that light travels at this speed; we have therefore found that the speed of light is the same in any inertial frame. A set of points which are all connected to a single event by1 SPECIAL RELATIVITY AND FLAT SPACETIME 7 t t’ x = -t x = t x’ = -t’ x’ = t’ x’ x straight lines moving at the speed of light is called a light cone; this entire set is invariant under Lorentz transformations. Light cones are naturally divided into future and past; the set of all points inside the future and past light cones of a point p are called timelike separated from p, while those outside the light cones are spacelike separated and those ontheconesarelightlikeornullseparatedfromp. Referringbackto(1.3),weseethatthe interval between timelike separated points is negative, between spacelike separated points is 2 positive, and between null separated points is zero. (The interval is defined to bes , not the square root of this quantity.) Notice the distinction between this situation and that in the Newtonian world; here, it is impossible to say (in a coordinate-independent way) whether a point that is spacelike separated from p is in the future of p, the past of p, or “at the same time”. To probe the structure of Minkowski space in more detail, it is necessary to introduce the concepts of vectors and tensors. We will start with vectors, which should be familiar. Of course, inspacetime vectors arefour-dimensional, andareoftenreferred toasfour-vectors. This turns out to make quite a bit of difference; for example, there is no such thing as a cross product between two four-vectors. Beyond the simple fact of dimensionality, the most important thing to emphasize is that each vector is located at a given point in spacetime. You may be used to thinking of vectors as stretching from one point to another in space, and even of “free” vectors which you can slide carelessly from point to point. These are not useful concepts in relativity. Rather, to each point p in spacetime we associate the set of all possible vectors located at that point; this set is known as the tangent space atp, orT . The name is inspired by thinking of the p set of vectors attached to a point on a simple curved two-dimensional space as comprising a1 SPECIAL RELATIVITY AND FLAT SPACETIME 8 plane which is tangent to the point. But inspiration aside, it is important to think of these vectors as being located at a single point, rather than stretching from one point to another. (Although this won’t stop us from drawing them as arrows on spacetime diagrams.) T p p manifold M Later we will relate the tangent space at each point to things we can construct from the spacetime itself. For right now, just think of T as an abstract vector space for each point p in spacetime. A (real) vector space is a collection of objects (“vectors”) which, roughly speaking, can be added together and multiplied by real numbers in a linear way. Thus, for any two vectors V and W and real numbers a and b, we have (a+b)(V +W) =aV +bV +aW +bW . (1.22) Every vector space has an origin, i.e. a zero vector which functions as an identity element under vector addition. In many vector spaces there are additional operations such as taking an inner (dot) product, but this is extra structure over and above the elementary concept of a vector space. A vector is a perfectly well-defined geometric object, as is a vector field, defined as a set of vectors with exactly one at each point in spacetime. (The set of all the tangent spaces of a manifold M is called the tangent bundle, T(M).) Nevertheless it is often useful for concrete purposes to decompose vectors into components with respect to some set of basis vectors. A basis is any set of vectors which both spans the vector space (any vector is a linear combination of basis vectors) and is linearly independent (no vector in the basis is a linear combination of other basis vectors). For any given vector space, there will be an infinite number of legitimate bases, but each basis will consist of the same number of1 SPECIAL RELATIVITY AND FLAT SPACETIME 9 vectors, known as the dimension of the space. (For a tangent space associated with a point in Minkowski space, the dimension is of course four.) Let us imagine that at each tangent space we set up a basis of four vectors eˆ , with (μ) μ μ∈0,1,2,3 as usual. In fact let us say that each basis is adapted to the coordinates x ; that is, the basis vector eˆ is what we would normally think of pointing along the x-axis, (1) etc. It is by no means necessary that we choose a basis which is adapted to any coordinate system at all, although it is often convenient. (We really could be more precise here, but lateron we will repeat thediscussion atanexcruciating level ofprecision, sosome sloppiness now is forgivable.) Then any abstract vector A can be written as a linear combination of basis vectors: μ A =A eˆ . (1.23) (μ) μ The coefficientsA arethecomponents ofthe vectorA. More oftenthan notwe will forget μ the basis entirely and refer somewhat loosely to “the vector A ”, but keep in mind that this is shorthand. The real vector is an abstract geometrical entity, while the components are just the coefficients of the basis vectors in some convenient basis. (Since we will usually suppress the explicit basis vectors, the indices will usually label components of vectors and tensors. This is why there are parentheses around the indices on the basis vectors, toremind us that this is a collection of vectors, not components of a single vector.) A standard example of a vector in spacetime is the tangent vector to a curve. A param- eterized curve or path through spacetime is specified by the coordinates as a function of the μ parameter, e.g. x (λ). The tangent vector V(λ) has components μ dx μ V = . (1.24) dλ μ The entire vector is thus V = V eˆ . Under a Lorentz transformation the coordinates (μ) μ x change according to (1.11), while the parameterization λ is unaltered; we can therefore deduce that the components of the tangent vector must change as ′ ′ μ μ μ ν V →V = Λ V . (1.25) ν However, the vector itself (as opposed to its components in some coordinate system) is invariant under Lorentz transformations. We can use this fact to derive the transformation properties of the basis vectors. Let us refer to the set of basis vectors in the transformed ′ coordinate system as eˆ . Since the vector is invariant, we have (ν ) ′ μ ν V =V eˆ = V eˆ ′ (μ) (ν ) ′ ν μ ′ = Λ V eˆ . (1.26) μ (ν ) μ But this relation must hold no matter what the numerical values of the componentsV are. Therefore we can say ′ ν eˆ =Λ eˆ ′ . (1.27) (μ) μ (ν )1 SPECIAL RELATIVITY AND FLAT SPACETIME 10 To get the new basis eˆ ′ in terms of the old one eˆ we should multiply by the inverse (ν ) (μ) ′ ν of the Lorentz transformation Λ . But the inverse of a Lorentz transformation from the μ unprimed to the primed coordinates is also a Lorentz transformation, this time from the primed to the unprimed systems. We will therefore introduce a somewhat subtle notation, by writing using the same symbol for both matrices, just with primed and unprimed indices adjusted. That is, ′ −1 ν μ ′ (Λ ) = Λ , (1.28) μ ν or ′ ′ ′ μ σ σ μ ν μ Λ ′ Λ =δ , Λ ′ Λ =δ , (1.29) ′ ν μ ν ρ ν ρ μ whereδ isthetraditionalKroneckerdeltasymbolinfourdimensions. (NotethatSchutzuses ρ a different convention, always arranging the two indices northwest/southeast; the important thing is where the primes go.) From (1.27) we then obtain the transformation rule for basis vectors: μ eˆ ′ =Λ ′ eˆ . (1.30) (ν ) ν (μ) Therefore the set of basis vectors transforms via the inverse Lorentz transformation of the coordinates or vector components. It is worth pausing a moment to take all this in. We introduced coordinates labeled by upper indices, which transformed in a certain way under Lorentz transformations. We then considered vector components which also were written with upper indices, which made sense since they transformed in the same way as the coordinate functions. (In a fixed coordinate μ system, each of the four coordinates x can be thought of as a function on spacetime, as can each of the four components of a vector field.) The basis vectors associated with the coordinate system transformed via the inverse matrix, and were labeled by a lower index. Thisnotationensuredthattheinvariantobjectconstructedbysummingoverthecomponents and basis vectors was left unchanged by the transformation, just as we would wish. It’s probably not giving too much away to say that this will continue to be the case for more complicated objects with multiple indices (tensors). Once we have set up a vector space, there is an associated vector space (of equal dimen- sion) which we can immediately define, known as the dual vector space. The dual space is usually denoted by an asterisk, so that the dual space to the tangent space T is called p ∗ the cotangent space and denoted T . The dual space is the space of all linear maps from p ∗ the original vector space to the real numbers; in math lingo, ifω∈T is a dual vector, then p it acts as a map such that: ω(aV +bW) =aω(V)+bω(W)∈R , (1.31) whereV,W are vectors anda, b are real numbers. The nice thing about these maps is that they form a vector space themselves; thus, if ω and η are dual vectors, we have (aω+bη)(V) =aω(V)+bη(V) . (1.32)1 SPECIAL RELATIVITY AND FLAT SPACETIME 11 To make this construction somewhat more concrete, we can introduce a set of basis dual (ν) ˆ vectors θ by demanding (ν) ν ˆ θ (eˆ ) =δ . (1.33) (μ) μ Then every dual vector can be written in terms ofits components, which we label with lower indices: (μ) ˆ ω =ω θ . (1.34) μ In perfect analogy with vectors, we will usually simply writeω to stand for the entire dual μ vector. Infact,youwillsometimeseeelementsofT (whatwehavecalledvectors)referredto p ∗ as contravariant vectors, and elements of T (what we have called dual vectors) referred p toascovariantvectors. Actually,ifyoujustrefertoordinaryvectorsasvectorswithupper indices and dual vectors as vectors with lower indices, nobody should be offended. Another name for dual vectors is one-forms, a somewhat mysterious designation which will become clearer soon. The component notation leads to a simple way of writing the action of a dual vector on a vector: ν (μ) ˆ ω(V) = ω V θ (eˆ ) μ (ν) ν μ = ω V δ μ ν μ = ω V ∈R . (1.35) μ This is why it is rarely necessary to write the basis vectors (and dual vectors) explicitly; the components do all of the work. The form of (1.35) also suggests that we can think of vectors as linear maps on dual vectors, by defining μ V(ω)≡ω(V)=ω V . (1.36) μ Therefore, the dual space to the dual vector space is the original vector space itself. Of course in spacetime we will be interested not in a single vector space, but in fields of vectors and dualvectors. (The set ofallcotangent spaces overM is thecotangent bundle, ∗ T (M).) Inthatcasethe actionofadualvector fieldonavector field isnotasinglenumber, but ascalar (orjust “function”)onspacetime. Ascalar is aquantity without indices, which is unchanged under Lorentz transformations. We can use the same arguments that we earlier used for vectors to derive the transfor- mation properties of dual vectors. The answers are, for the components, ν ω ′ =Λ ′ ω , (1.37) μ μ ν and for basis dual vectors, ′ ′ (ρ ) ρ (σ) ˆ ˆ θ =Λ θ . (1.38) σ1 SPECIAL RELATIVITY AND FLAT SPACETIME 12 This is just what we would expect from index placement; the components of a dual vector transform under the inverse transformation of those of a vector. Note that this ensures that the scalar (1.35) is invariant under Lorentz transformations, just as it should be. Let’sconsidersomeexamplesofdualvectors,firstinothercontextsandtheninMinkowski space. Imagine the space ofn-component column vectors, for some integern. Then the dual space is that of n-component row vectors, and the action is ordinary matrix multiplication:   1 V   2 V       ·   V = , ω =(ω ω ··· ω ) , 1 2 n   ·      ·  n V   1 V   2 V      · i   ω(V) = (ω ω ··· ω ) =ωV . (1.39) 1 2 n i   ·      ·  n V Another familiar example occurs in quantum mechanics, where vectors in the Hilbert space are represented by kets, ψi. In this case the dual space is the space of bras, hφ, and the action gives the number hφψi. (This is a complex number in quantum mechanics, but the idea is precisely the same.) In spacetime the simplest example of a dual vector is the gradient of a scalar function, the set of partial derivatives with respect to the spacetime coordinates, which we denote by “d”: ∂φ (μ) ˆ dφ = θ . (1.40) μ ∂x The conventional chain rule used to transform partial derivatives amounts in this case to the transformation rule of components of dual vectors: μ ∂φ ∂x ∂φ = ′ ′ μ μ μ ∂x ∂x ∂x ∂φ μ = Λ ′ , (1.41) μ μ ∂x wherewehaveused(1.11)and(1.28)torelatetheLorentztransformationtothecoordinates. The fact that the gradient is a dual vector leads to the following shorthand notations for partial derivatives: ∂φ =∂ φ =φ, . (1.42) μ μ μ ∂x1 SPECIAL RELATIVITY AND FLAT SPACETIME 13 μ (Very roughly speaking, “x has an upper index, but when it is in the denominator of a derivative it implies a lower index on the resulting object.”) I’m not a big fan of the comma notation, but we will use∂ all the time. Note that the gradient does in fact act in a natural μ way on the example we gave above of a vector, the tangent vector to a curve. The result is ordinary derivative of the function along the curve: μ ∂x dφ ∂ φ = . (1.43) μ ∂λ dλ As a final note on dual vectors, there is a way to represent them as pictures which is consistent with the picture of vectors as arrows. See the discussion in Schutz, or in MTW (where it is taken to dizzying extremes). A straightforward generalization of vectors and dual vectors is the notion of a tensor. Just as a dual vector is a linear map from vectors to R, a tensor T of type (or rank) (k,l) is a multilinear map from a collection of dual vectors and vectors to R: ∗ ∗ T : T ×···×T ×T ×···×T →R p p p p (k times) (l times) (1.44) Here, “×” denotes the Cartesian product, so thatfor exampleT ×T is the space ofordered p p pairs of vectors. Multilinearity means that the tensor acts linearly in each of its arguments; for instance, for a tensor of type (1,1), we have T(aω+bη,cV +dW)=acT(ω,V)+adT(ω,W)+bcT(η,V)+bdT(η,W). (1.45) From this point of view, a scalar is a type (0,0) tensor, a vector is a type (1,0) tensor, and a dual vector is a type (0,1) tensor. The space of all tensors of a fixed type (k,l) forms a vector space; they can be added together and multiplied by real numbers. To construct a basis for this space, we need to define a new operation known as the tensor product, denoted by⊗. If T is a (k,l) tensor and S is a (m,n) tensor, we define a (k+m,l+n) tensor T ⊗S by (1) (k) (k+m) (1) (l) (l+n) T ⊗S(ω ,...,ω ,...,ω ,V ,...,V ,...,V ) (1) (k) (1) (l) (k+1) (k+m) (l+1) (l+n) =T(ω ,...,ω ,V ,...,V )S(ω ,...,ω ,V ,...,V ) . (1.46) (i) (i) (Note that the ω and V are distinct dual vectors and vectors, not components thereof.) In other words, first act T on the appropriate set of dual vectors and vectors, and then act S on the remainder, and then multiply the answers. Note that, in general, T ⊗S = 6 S⊗T. It is now straightforward to construct a basis for the space of all (k,l) tensors, by taking tensor products of basis vectors and dual vectors; this basis will consist of all tensors of the form (ν ) (ν ) 1 ˆ ˆ l eˆ ⊗···⊗eˆ ⊗θ ⊗···⊗θ . (1.47) (μ ) (μ ) 1 k

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