VBibliographyThe typical level of difficulty (especiallymathematical)of thebooks is indicated by a numberof asterisks, one meaning mostly introductory and three being advanced. The asterisks arenormalized to these lecture notes, which would be given [**].The first four books werefrequently consulted in the preparation of these notes, the next seven are other relativity textswhich 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 avery nice introductory text. Especially useful if, for example, you aren't quite clear onwhat the energy-momentum tensor really means. S. Weinberg, Gravitation and Cosmology (Wiley, 1972) [**].A really good book atwhat it does, especially strong on astrophysics, cosmology, and experimental tests.However, it takes an unusual non-geometric approach to the material, and doesn'tdiscuss black holes.. C. Misner, K. Thorne and J. Wheeler, Gravitation (Freeman, 1973) [**j. A heavy book,in various senses. Most things you want to know are in here, although you might haveto work hard to get to them (perhaps learning something unexpected in the process).. R. Wald, General Relativity (Chicago, 1984) [***]. Thorough discussions of a numberof advanced topics, including black holes, global structure, and spinors. The approachis more mathematically demanding than the previous books, and the basics are coveredpretty quickly.:E.Taylor and J.Wheeler, SpacetimePhysics (Freeman, 1992)[*].Agood introductionto special relativity.·R. D'Inverno, Introducing Einstein's Relativity (Oxford, 1992) [**].A book I haven'tlooked at very carefully, but it seems as if all the right topics are covered withoutnoticeable 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 allareas of GR, with fully worked solutions, making it all the more difficult for instructorsto invent problems the students can't easilyfind theanswers to.N.Straumann, General Relativity and RelativisticAstrophysics (Springer-Verlag, 1984)[***].Afairly high-level book, which starts out with a good deal of abstract geometryand goes on to detailed discussions of stellar structure and other astrophysical topics
v Bibliography The typical level of difficulty (especially mathematical) of the books is indicated by a number 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 frequently consulted in the preparation of these notes, the next seven are other relativity texts 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. Thorne and J. Wheeler, Gravitation (Freeman, 1973) [**]. A heavy book, 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 is more mathematically demanding than the previous books, and the basics are covered pretty quickly. • E. Taylor and J. Wheeler, Spacetime Physics (Freeman, 1992) [*]. A good introduction 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 Relativity and Gravitation (Princeton, 1975) [**]. A sizeable collection of problems in all areas of GR, with fully worked solutions, making it all the more difficult for instructors to invent problems the students can’t easily find the answers to. • N. Straumann, General Relativity and Relativistic Astrophysics (Springer-Verlag, 1984) [***]. A fairly high-level book, which starts out with a good deal of abstract geometry and goes on to detailed discussions of stellar structure and other astrophysical topics
viF.de Felice and C.Clarke, Relativity on Curved Manifolds (Cambridge, 1990)[***]A mathematical approach, but with an excellent emphasis on physically measurablequantities.·S.Hawking and G.Ellis, The Large-Scale Structure of Space-Time (Cambridge, 1973)[***].An advanced book which emphasizes global techniques and singularity theorems..R. Sachs and H. Wu, General Relativity for Mathematicians (Springer-Verlag, 1977)[***].Just what the title says, although the typically dry mathematics prose styleis here enlivened by frequent opinionated asides about both physics and mathematics(and the state oftheworld).:B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge, 1980)[**]Another good book by Schutz, this one covering some mathematical points that areleft out of the GR book (but at a very accessible level). Included are discussions of Liederivatives,differentialforms,and applicationstophysicsotherthanGR..V.Guillemin and A. Pollack, Differential Topology (Prentice-Hall, 1974) [**j.Anentertaining 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, fiber bundles and Morse theory, with applicationsto 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 asmanifolds and tensor fields as well as more advanced subjects
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) [***]. An advanced book which emphasizes global techniques and singularity theorems. • 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, fiber bundles and Morse theory, with applications to physics; somewhat concise. • F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups (SpringerVerlag, 1983) [***]. The standard text in the field, includes basic topics such as manifolds and tensor fields as well as more advanced subjects
December1997LectureNotesonGeneralRelativitySean M. Carroll1Special Relativity and Flat SpacetimeWe 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 relatedconcepts that will be crucial later on, without the extra complications of curvature on topof everything else. Therefore, for this section we will always be working in fat spacetime,and furthermore we will only use orthonormal (Cartesian-like) coordinates. Needless to sayit is possible to do SR in any coordinate system you like, but it turns out that introducingthe necessary tools for doing so would take us halfway to curved spaces anyway, so we willputthatoff forawhile.It is often said that special relativity is a theory of 4-dimensional spacetime: three ofspace, one of time.But of course, the pre-SR world of Newtonian mechanics featured threespatial dimensions and a time parameter. Nevertheless, there was not much temptation toconsider these as different aspects of a single 4-dimensional spacetime.Why not?space at afixedtimex, y,zConsider a garden-variety 2-dimensional plane. It is typically convenient to label thepoints on such a plane by introducing coordinates, for example by defining orthogonal andy axes and projecting each point onto these axes in the usual way.However,it is clear thatmost of the interesting geometrical facts about the plane are independent of our choice ofcoordinates. As a simple example, we can consider the distance between two points, given1
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? space at a fixed time t 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, for example by defining orthogonal x 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 1
21 SPECIALRELATIVITYANDFLATSPACETIMEby2 = (Ar)2 + (Ay)?.(1.1)In a different Cartesian coordinate system, defined by r' and y' axes which are rotated withrespect to the originals, the formula for the distance is unaltered:s2= (△r)2 +(△y)?.(1.2)We therefore say that the distance is invariant under such changes of coordinates.1yAAsAyAyAxAxThis is why it is useful to think of the plane as 2-dimensional: although we use two distinctnumbers to label each point, the numbers are not the essence of the geometry, since we canrotate axes into each other while leaving distances and so forth unchanged.In Newtonianphysics this is not the case with space and time; there is no useful notion of rotating spaceand 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, r, y, z) on spacetime, set up inthefollowing way. The spatial coordinates (,y,z)comprise a standard Cartesian system.constructed for example by welding together rigid rods which meet at right angles. The rodsmustbemovingfreely,unaccelerated.Thetimecoordinateisdefinedbyasetofclockswhicharenotmovingwithrespecttothespatialcoordinates.(Sincethisisathoughtexperimentwe 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 toany other in a straight line at constant speed, the time difference between the clocks at the
1 SPECIAL RELATIVITY AND FLAT SPACETIME 2 by s 2 = (∆x) 2 + (∆y) 2 . (1.1) In a different Cartesian coordinate system, defined by x ′ and y ′ axes which are rotated with respect to the originals, the formula for the distance is unaltered: s 2 = (∆x ′ ) 2 + (∆y ′ ) 2 . (1.2) We therefore say that the distance is invariant under such changes of coordinates. ∆ ∆ ∆ y x’ x y y’ x x’ s y’ ∆ ∆ This is why it is useful to think of the plane as 2-dimensional: although we use two distinct 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 for example by welding together rigid rods which meet at right angles. The rods must be moving freely, unaccelerated. The time coordinate is defined by a set of clocks which 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 the
31 SPECIALRELATIVITYANDFLATSPACETIMEends of your journey is the same as if you had made the same trip, at the same speed, in theother 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, r,y, z). Then, without any motivation for the moment, let us introduce the spacetimeinterval betweentwo events:2 = -(c△t)? + (△r)? + (△y)? +(△z)?(1.3)(Notice that it can be positive, negative, or zero even for two nonidentical points.) Here, cis some fixed conversion factor between space and time; that is, a fixed velocity. Of courseit will turn out to be the speed of light; the important thing, however, is not that photonshappen to travel at that speed, but that there exists a c such that the spacetime intervalis invariant under changes of coordinates. In other words, if we set up a new inertial frame(t',r',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:s? = -(cAt)? + (Ar')? + (Ay)? + (Az).(1.4)This is why it makes sense to think of sR as a theory of 4-dimensional spacetime, knownas Minkowski space. (This is a special case of a 4-dimensional manifold, which we willdeal with in detail later.) As we shall see, the coordinate transformations which we haveimplicitly defined do, in a sense, rotate space and time into each other. There is no absolutenotion of "simultaneous events"; whether two things occur at the same time depends on thecoordinates used. Therefore the division of Minkowski space into space and time is a choicewe make for our own purposes, not something intrinsic to the situation.Almost all of the“paradoxes"associated with SR result from a stubborn persistence ofthe Newtonian notions of a unique time coordinate and the existence of “space at a singlemoment in time." By thinking in terms of spacetime rather than space and time together,theseparadoxes tend todisappear.Let's introduce some convenient notation. Coordinates on spacetime will be denoted byletterswithGreek superscript indices runningfrom0 to3,with0generallydenoting thetime coordinate. Thus,ro = ctl =a(1.5)rh :r?=y23=2(Don't start thinking of the superscripts as exponents.) Furthermore, for the sake of sim-plicity we will choose units in which(1.6)c=1;
1 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: s 2 = −(c∆t) 2 + (∆x) 2 + (∆y) 2 + (∆z) 2 . (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: s 2 = −(c∆t ′ ) 2 + (∆x ′ ) 2 + (∆y ′ ) 2 + (∆z ′ ) 2 . (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, x µ : x 0 = ct x 1 = x x 2 = y x 3 = z (1.5) (Don’t start thinking of the superscripts as exponents.) Furthermore, for the sake of simplicity we will choose units in which c = 1 ; (1.6)