291SPECIALRELATIVITYANDFLATSPACETIMEBy definition, the energy density completely specifies the dust. But p only measures theenergy density in the rest frame; what about other frames? We notice that both n andm are 0-components of four-vectors in their rest frame; specifically,Nμ=(n,O,o,O)andp"= (m,0, 0,0).Therefore p is the μ =0, v =0 component of the tensor pN as measuredin its rest frame. We are therefore led to define the energy-momentum tensor for dust:Tast= p"N"=nmU"U"= pUU",(1.106)where p is defined as the energy density in the rest frame.Having mastered dust, more general perfect ffuids are not much more complicated. Re-member that “perfect" can be taken to mean “isotropic in its rest frame." This in turnmeans that Tw is diagonal - there is no net flux of any component of momentum in anorthogonaldirection.Furthermore,thenonzero spacelike componentsmust all be equal.Tll = T22 = T33. The only two independent numbers are therefore T00 and one of the T;we can choose to call the first of these the energy density p, and the second the pressurep. (Sorry that it's the same letter as the momentum.) The energy-momentum tensor of aperfect fluid therefore takes thefollowing form in its rest frame:000p000pTu(1.107)00p000DWe would like, of course, a formula which is good in any frame. For dust we had Tw =pUUv, so we might begin by guessing (p + p)UμUv, which gives000p+p0000(1.108)00000000To get the answer we want we must therefore add000-p000p(1.109)000p0000Fortunately,this has an obvious covariant generalization, namely pn, Thus, the generalform of the energy-momentum tensorfor a perfectfluid is(1.110)TW=(p+p)U"U+pnThis is an important formula forapplications such as stellar structure and cosmology
1 SPECIAL RELATIVITY AND FLAT SPACETIME 29 By definition, the energy density completely specifies the dust. But ρ only measures the energy density in the rest frame; what about other frames? We notice that both n and m are 0-components of four-vectors in their rest frame; specifically, Nµ = (n, 0, 0, 0) and p µ = (m, 0, 0, 0). Therefore ρ is the µ = 0, ν = 0 component of the tensor p⊗N as measured in its rest frame. We are therefore led to define the energy-momentum tensor for dust: T µν dust = p µN ν = nmUµU ν = ρUµU ν , (1.106) where ρ is defined as the energy density in the rest frame. Having mastered dust, more general perfect fluids are not much more complicated. Remember that “perfect” can be taken to mean “isotropic in its rest frame.” This in turn means that T µν is diagonal — there is no net flux of any component of momentum in an orthogonal direction. Furthermore, the nonzero spacelike components must all be equal, T 11 = T 22 = T 33. The only two independent numbers are therefore T 00 and one of the T ii; we can choose to call the first of these the energy density ρ, and the second the pressure p. (Sorry that it’s the same letter as the momentum.) The energy-momentum tensor of a perfect fluid therefore takes the following form in its rest frame: T µν = ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p . (1.107) We would like, of course, a formula which is good in any frame. For dust we had T µν = ρUµU ν , so we might begin by guessing (ρ + p)U µU ν , which gives ρ + p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . (1.108) To get the answer we want we must therefore add −p 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p . (1.109) Fortunately, this has an obvious covariant generalization, namely pηµν. Thus, the general form of the energy-momentum tensor for a perfect fluid is T µν = (ρ + p)U µU ν + pηµν . (1.110) This is an important formula for applications such as stellar structure and cosmology
301 SPECIALRELATIVITYANDFLATSPACETIMEAs further examples, let's consider the energy-momentum tensors of electromagnetismand scalar field theory. Without any explanation at all, these are given by1(FMAF"-nF) ,T+m(1.111)4元and(ma00.0+m02)(1.112)Tmar=muanvaabo.-You can check for yourself that, for example, Too in each case is equal to what you wouldexpect theenergydensitytobe.Besides being symmetric, Tu has the even more important property of being conserved.In this context, conservation is expressed as the vanishing of the “divergence":O,TW=0.(1.113)This is a set of four equations, one for each value of v. The v = O equation corresponds toconservation of energy, while ,Tuk = 0 expresses conservation of the kth component of themomentum. We are not going to prove this in general; the proof follows for any individualsource of matter from the equations of motion obeyed by that kind of matter. In fact, oneway to define Tw would be “a (2, O) tensor with units of energy per volume, which is con-served."You can prove conservation of the energy-momentum tensor for electromagnetism,for example, by taking the divergence of (1.111) and using Maxwell's equations as previouslydiscussed.A final aside: we have already mentioned that in general relativity gravitation does notcount as a “force." As a related point, the gravitational field also does not have an energy-momentum tensor.In fact it is very hard to come up with a sensible local expression for theenergy of a gravitational field; a number of suggestions have been made, but they all havetheir drawbacks.Although there is no"correct"answer, it is an important issue from thepoint of view of asking seemingly reasonable questions such as "What is the energy emittedper second from a binary pulsar as the result of gravitational radiation?
1 SPECIAL RELATIVITY AND FLAT SPACETIME 30 As further examples, let’s consider the energy-momentum tensors of electromagnetism and scalar field theory. Without any explanation at all, these are given by T µν e+m = −1 4π (F µλF ν λ − 1 4 η µνF λσFλσ) , (1.111) and T µν scalar = η µλη νσ∂λφ∂σφ − 1 2 η µν(η λσ∂λφ∂σφ + m2φ 2 ) . (1.112) You can check for yourself that, for example, T 00 in each case is equal to what you would expect the energy density to be. Besides being symmetric, T µν has the even more important property of being conserved. In this context, conservation is expressed as the vanishing of the “divergence”: ∂µT µν = 0 . (1.113) This is a set of four equations, one for each value of ν. The ν = 0 equation corresponds to conservation of energy, while ∂µT µk = 0 expresses conservation of the k th component of the momentum. We are not going to prove this in general; the proof follows for any individual source of matter from the equations of motion obeyed by that kind of matter. In fact, one way to define T µν would be “a (2, 0) tensor with units of energy per volume, which is conserved.” You can prove conservation of the energy-momentum tensor for electromagnetism, for example, by taking the divergence of (1.111) and using Maxwell’s equations as previously discussed. A final aside: we have already mentioned that in general relativity gravitation does not count as a “force.” As a related point, the gravitational field also does not have an energymomentum tensor. In fact it is very hard to come up with a sensible local expression for the energy of a gravitational field; a number of suggestions have been made, but they all have their drawbacks. Although there is no “correct” answer, it is an important issue from the point of view of asking seemingly reasonable questions such as “What is the energy emitted per second from a binary pulsar as the result of gravitational radiation?
December 1997LectureNotesonGeneralRelativitySean M. Carroll2ManifoldsAfterthe invention of special relativity,Einstein tried for a number of years to invent aLorentz-invariant theory of gravity, without success. His eventual breakthrough was toreplaceMinkowskispacetimewithacurved spacetime,wherethecurvaturewascreated by(and reacted back on)energy and momentum.Before we explore how this happens, we haveto learn a bit about the mathematics of curved spaces.First we will take a look at manifoldsin general, and then in the next section study curvature. In the interest of generality we willusually work in n dimensions, although you are permitted to take n = 4 if you like.A manifold (or sometimes“differentiable manifold")is one of the most fundamentalconcepts inmathematics and physics.We are all aware of the properties of n-dimensionalEuclidean space, Rn, the set of n-tuples (r',..., rn). The notion of a manifold captures theidea of a space which may be curved and have a complicated topology,but in local regionslooks just like Rn.(Hereby"looks like"wedo not mean that the metric is the same, but onlybasic notions of analysis like open sets, functions, and coordinates.) The entire manifold isconstructed by smoothly sewing together these local regions.Examples of manifolds include:· Rn itself, including the line (R), the plane (R2), and so on. This should be obvious,sinceRnlooks likeRn not onlylocallybutglobally.: The n-sphere, Sn. This can be defined as the locus of all points some fixed distancefrom the origin in Rn+1. The circle is of course Si, and the two-sphere S2 will be oneof our favorite examples of a manifold.: The n-torus Tn results from taking an n-dimensional cube and identifying oppositesides.Thus T2 is the traditional surface of a doughnut.identify opposite sides31
December 1997 Lecture Notes on General Relativity Sean M. Carroll 2 Manifolds After the invention of special relativity, Einstein tried for a number of years to invent a Lorentz-invariant theory of gravity, without success. His eventual breakthrough was to replace Minkowski spacetime with a curved spacetime, where the curvature was created by (and reacted back on) energy and momentum. Before we explore how this happens, we have to learn a bit about the mathematics of curved spaces. First we will take a look at manifolds in general, and then in the next section study curvature. In the interest of generality we will usually work in n dimensions, although you are permitted to take n = 4 if you like. A manifold (or sometimes “differentiable manifold”) is one of the most fundamental concepts in mathematics and physics. We are all aware of the properties of n-dimensional Euclidean space, Rn , the set of n-tuples (x 1 , . . ., xn ). The notion of a manifold captures the idea of a space which may be curved and have a complicated topology, but in local regions looks just like Rn . (Here by “looks like” we do not mean that the metric is the same, but only basic notions of analysis like open sets, functions, and coordinates.) The entire manifold is constructed by smoothly sewing together these local regions. Examples of manifolds include: • Rn itself, including the line (R), the plane (R2 ), and so on. This should be obvious, since Rn looks like Rn not only locally but globally. • The n-sphere, S n . This can be defined as the locus of all points some fixed distance from the origin in Rn+1. The circle is of course S 1 , and the two-sphere S 2 will be one of our favorite examples of a manifold. • The n-torus T n results from taking an n-dimensional cube and identifying opposite sides. Thus T 2 is the traditional surface of a doughnut. identify opposite sides 31
322MANIFOLDS. A Riemann surface of genus g is essentially a two-torus with g holes instead of justone. S? may be thought of as a Riemann surface of genus zero.For those of you whoknowwhat thewords mean, every“compact orientable boundaryless"two-dimensionalmanifold is a Riemann surface of some genus.genus 2genus 1genus o More abstractly, a set of continuous transformations such as rotations in Rn forms amanifold. Lie groups are manifolds which also have a group structure.. The direct product of two manifolds is a manifold. That is, given manifolds M andM' of dimension n and n', we can construct a manifold M x M', of dimension n + n',consisting of ordered pairs (p,p')for all p e M and p' e M'.With all of these examples, the notion of a manifold may seem vacuous; what isn't amanifold? There are plenty of things which are not manifolds, because somewhere theydo not look locally likeRn.Examples includea one-dimensional line running intoa two-dimensional plane, and two cones stuck together at their vertices. (A single cone is okay;you can imagine smoothing out the vertex.)We will now approach the rigorous definition of this simple idea, which requires a numberof preliminary definitions.Many of them arepretty clear anyway,but it's nice to be complete
2 MANIFOLDS 32 • A Riemann surface of genus g is essentially a two-torus with g holes instead of just one. S 2 may be thought of as a Riemann surface of genus zero. For those of you who know what the words mean, every “compact orientable boundaryless” two-dimensional manifold is a Riemann surface of some genus. genus 0 genus 1 genus 2 • More abstractly, a set of continuous transformations such as rotations in Rn forms a manifold. Lie groups are manifolds which also have a group structure. • The direct product of two manifolds is a manifold. That is, given manifolds M and M′ of dimension n and n ′ , we can construct a manifold M × M′ , of dimension n + n ′ , consisting of ordered pairs (p, p′ ) for all p ∈ M and p ′ ∈ M′ . With all of these examples, the notion of a manifold may seem vacuous; what isn’t a manifold? There are plenty of things which are not manifolds, because somewhere they do not look locally like Rn . Examples include a one-dimensional line running into a twodimensional plane, and two cones stuck together at their vertices. (A single cone is okay; you can imagine smoothing out the vertex.) We will now approach the rigorous definition of this simple idea, which requires a number of preliminary definitions. Many of them are pretty clear anyway, but it’s nice to be complete
332MANIFOLDSThe most elementary notion is that of a map between two sets. (We assume you knowwhat a set is.) Given two sets M and N, a map @: M → N is a relationship which assigns, toeach element of M, exactly oneelement of N.Amap is therefore justa simplegeneralizationof afunction.The canonical picture of amaplooks likethis:MeNGiven two maps @: A- B and : B-→ C, we define the composition o@: A - Cby the operation (bo)(a) = w((a)). So a E A, d(a) e B, and thus (o d)(a) e C. Theorder inwhich themaps are written makes sense, sincethe one on the right acts first.Inpictures:opCAeYBA map is called one-to-one (or"injective")if each element of N has at most oneelement of M mapped into it,and onto(or"surjective")if eachelement of Nhasatleastone element of M mapped into it. (If you think about it, a better name for "one-to-one"would be“two-to-two".) Consider a function :R-→ R. Then ()= e" is one-to-one, butnot onto; Φ(r) = r3 - r is onto, but not one-to-one; Φ(r) = r3 is both; and p(r) = a? isneither.The set M is known as the domain of the map , and the set of points in N which Mgets mapped into is called the image of . For some subset U C N, the set of elements ofM which get mapped to U is called the preimage of U under p, or -1(U). A map which is
2 MANIFOLDS 33 The most elementary notion is that of a map between two sets. (We assume you know what a set is.) Given two sets M and N, a map φ : M → N is a relationship which assigns, to each element of M, exactly one element of N. A map is therefore just a simple generalization of a function. The canonical picture of a map looks like this: ϕ M N Given two maps φ : A → B and ψ : B → C, we define the composition ψ ◦ φ : A → C by the operation (ψ ◦ φ)(a) = ψ(φ(a)). So a ∈ A, φ(a) ∈ B, and thus (ψ ◦ φ)(a) ∈ C. The order in which the maps are written makes sense, since the one on the right acts first. In pictures: ψ ϕ A B C ϕ ψ A map φ is called one-to-one (or “injective”) if each element of N has at most one element of M mapped into it, and onto (or “surjective”) if each element of N has at least one element of M mapped into it. (If you think about it, a better name for “one-to-one” would be “two-to-two”.) Consider a function φ : R → R. Then φ(x) = e x is one-to-one, but not onto; φ(x) = x 3 − x is onto, but not one-to-one; φ(x) = x 3 is both; and φ(x) = x 2 is neither. The set M is known as the domain of the map φ, and the set of points in N which M gets mapped into is called the image of φ. For some subset U ⊂ N, the set of elements of M which get mapped to U is called the preimage of U under φ, or φ −1 (U). A map which is