《广义相对论》课程教学资源（书籍文献）Lecturer Note on General Relativity [S.M.Carroll 编著].pdf_P26-P30

1 SPECIAL RELATIVITY AND FLAT SPACETIME 19 means that the order of indices doesn’t really matter, which is why we don’t keep track index placement for this one tensor. Given any tensor, we can symmetrize (or antisymmetrize) any number of its upper or lower indices. To symmetrize, we take the sum of all permutations of the relevant indices and divide by the number of terms: T(µ1µ2···µn)ρ σ = 1 n! (Tµ1µ2···µnρ σ + sum over permutations of indices µ1 · · · µn) , (1.67) while antisymmetrization comes from the alternating sum: T[µ1µ2···µn]ρ σ = 1 n! (Tµ1µ2···µnρ σ + alternating sum over permutations of indices µ1 · · · µn) . (1.68) By “alternating sum” we mean that permutations which are the result of an odd number of exchanges are given a minus sign, thus: T[µνρ]σ = 1 6 (Tµνρσ − Tµρνσ + Tρµνσ − Tνµρσ + Tνρµσ − Tρνµσ) . (1.69) Notice that round/square brackets denote symmetrization/antisymmetrization. Furthermore, we may sometimes want to (anti-) symmetrize indices which are not next to each other, in which case we use vertical bars to denote indices not included in the sum: T(µ|ν|ρ) = 1 2 (Tµνρ + Tρνµ) . (1.70) Finally, some people use a convention in which the factor of 1/n! is omitted. The one used here is a good one, since (for example) a symmetric tensor satisfies Sµ1···µn = S(µ1···µn) , (1.71) and likewise for antisymmetric tensors. We have been very careful so far to distinguish clearly between things that are always true (on a manifold with arbitrary metric) and things which are only true in Minkowski space in Cartesian coordinates. One of the most important distinctions arises with partial derivatives. If we are working in flat spacetime with Cartesian coordinates, then the partial derivative of a (k, l) tensor is a (k, l + 1) tensor; that is, Tα µ ν = ∂αR µ ν (1.72) transforms properly under Lorentz transformations. However, this will no longer be true in more general spacetimes, and we will have to define a “covariant derivative” to take the place of the partial derivative. Nevertheless, we can still use the fact that partial derivatives

1 SPECIAL RELATIVITY AND FLAT SPACETIME 20 give us tensor in this special case, as long as we keep our wits about us. (The one exception to this warning is the partial derivative of a scalar, ∂αφ, which is a perfectly good tensor [the gradient] in any spacetime.) We have now accumulated enough tensor know-how to illustrate some of these concepts using actual physics. Specifically, we will examine Maxwell’s equations of electrodynamics. In 19th-century notation, these are ∇ × B − ∂tE = 4πJ ∇ · E = 4πρ ∇ × E + ∂tB = 0 ∇ · B = 0 . (1.73) Here, E and B are the electric and magnetic field 3-vectors, J is the current, ρ is the charge density, and ∇× and ∇· are the conventional curl and divergence. These equations are invariant under Lorentz transformations, of course; that’s how the whole business got started. But they don’t look obviously invariant; our tensor notation can fix that. Let’s begin by writing these equations in just a slightly different notation, ǫ ijk∂jBk − ∂0E i = 4πJi ∂iE i = 4πJ0 ǫ ijk∂jEk + ∂0B i = 0 ∂iB i = 0 . (1.74) In these expressions, spatial indices have been raised and lowered with abandon, without any attempt to keep straight where the metric appears. This is because δij is the metric on flat 3-space, with δ ij its inverse (they are equal as matrices). We can therefore raise and lower indices at will, since the components don’t change. Meanwhile, the three-dimensional Levi-Civita tensor ǫ ijk is defined just as the four-dimensional one, although with one fewer index. We have replaced the charge density by J 0 ; this is legitimate because the density and current together form the current 4-vector, J µ = (ρ, J1 , J2 , J3 ). From these expressions, and the definition (1.58) of the field strength tensor Fµν , it is easy to get a completely tensorial 20th-century version of Maxwell’s equations. Begin by noting that we can express the field strength with upper indices as F 0i = E i F ij = ǫ ijkBk . (1.75) (To check this, note for example that F 01 = η 00η 11F01 and F 12 = ǫ 123B3.) Then the first two equations in (1.74) become ∂jF ij − ∂0F 0i = 4πJi

211 SPECIALRELATIVITYANDFLATSPACETIME,F0i =4元J0(1.76)Using the antisymmetry of Fu, we see that these maybe combined into the single tensorequationFVμ=4元".(1.77)A similar line of reasoning,which is left as an exercise to you,reveals that the third andfourth equations in (1.74) can be written(1.78)QuFv) = 0 .The four traditional Maxwell equations are thus replaced by two, thus demonstrating theeconomy of tensor notation.More importantly,however, both sides of equations (1.77)and(1.78) manifestly transform as tensors; therefore, if they are true in one inertial frame, theymust be true in any Lorentz-transformed frame. This is why tensors are so useful in relativitywe often want to express relationships without recourse to any reference frame, and it isnecessary that the quantities on each side of an equation transform in the same way underchange of coordinates. As a matter of jargon, we will sometimes refer to quantities whichare written in terms of tensors as covariant (which has nothing to do with “covariant"as opposed to “contravariant"). Thus, we say that (1.77) and (1.78) together serve as thecovariant form of Maxwell's equations, while (1.73) or (1.74)are non-covariant.Let us now introduce a special class of tensors,known as differential forms (orjust"forms"). A differential p-form is a (O,p) tensor which is completely antisymmetric. Thus,scalars are automatically 0-forms, and dual vectors are automatically one-forms (thus ex-plaining this terminology from a while back).We also have the 2-form Fu and the 4-formEμpa:The space of all p-formsisdenoted Ap,and the spaceofall p-form fields over a mani-fold M is denoted Ap(M).A semi-straightforward exercise in combinatorics reveals that thenumber of linearly independent p-forms on an n-dimensional vector space is n!/(p!(n -p)!).So at a point on a 4-dimensional spacetime there is one linearly independent 0-form, four1-forms, six 2-forms, four 3-forms, and one4-form.There are no p-forms for p >n,sinceallof the components will automatically be zero by antisymmetry.Whyshould we care about differential forms?Thisis ahard question to answerwithoutsome more work, but the basic idea is that forms can be both differentiated and integrated,without the help of any additional geometric structure.We will delay integration theoryuntil later, but see how to differentiate forms shortly.Given a p-form A and a q-form B, we can form a (p + q)-form known as the wedgeproduct AAB by taking the antisymmetrized tensor product:_ (p+ q)! A(1.79)(A^B)μ--μp+[-μpBup+1-p+a] :p! q!

1 SPECIAL RELATIVITY AND FLAT SPACETIME 21 ∂iF 0i = 4πJ0 . (1.76) Using the antisymmetry of F µν, we see that these may be combined into the single tensor equation ∂µF νµ = 4πJν . (1.77) A similar line of reasoning, which is left as an exercise to you, reveals that the third and fourth equations in (1.74) can be written ∂[µFνλ] = 0 . (1.78) The four traditional Maxwell equations are thus replaced by two, thus demonstrating the economy of tensor notation. More importantly, however, both sides of equations (1.77) and (1.78) manifestly transform as tensors; therefore, if they are true in one inertial frame, they must be true in any Lorentz-transformed frame. This is why tensors are so useful in relativity — we often want to express relationships without recourse to any reference frame, and it is necessary that the quantities on each side of an equation transform in the same way under change of coordinates. As a matter of jargon, we will sometimes refer to quantities which are written in terms of tensors as covariant (which has nothing to do with “covariant” as opposed to “contravariant”). Thus, we say that (1.77) and (1.78) together serve as the covariant form of Maxwell’s equations, while (1.73) or (1.74) are non-covariant. Let us now introduce a special class of tensors, known as differential forms (or just “forms”). A differential p-form is a (0, p) tensor which is completely antisymmetric. Thus, scalars are automatically 0-forms, and dual vectors are automatically one-forms (thus explaining this terminology from a while back). We also have the 2-form Fµν and the 4-form ǫµνρσ. The space of all p-forms is denoted Λp , and the space of all p-form fields over a manifold M is denoted Λp (M). A semi-straightforward exercise in combinatorics reveals that the number of linearly independent p-forms on an n-dimensional vector space is n!/(p!(n − p)!). So at a point on a 4-dimensional spacetime there is one linearly independent 0-form, four 1-forms, six 2-forms, four 3-forms, and one 4-form. There are no p-forms for p > n, since all of the components will automatically be zero by antisymmetry. Why should we care about differential forms? This is a hard question to answer without some more work, but the basic idea is that forms can be both differentiated and integrated, without the help of any additional geometric structure. We will delay integration theory until later, but see how to differentiate forms shortly. Given a p-form A and a q-form B, we can form a (p + q)-form known as the wedge product A ∧ B by taking the antisymmetrized tensor product: (A ∧ B)µ1···µp+q = (p + q)! p! q! A[µ1···µpBµp+1···µp+q] . (1.79)

221 SPECIALRELATIVITYANDFLATSPACETIMEThus, for example, the wedge product of two 1-forms is(1.80)(A^B)= 2AμB = A,B,- A,Bμ Note that(1.81)A^B=(-1)PB^A,so you can alter the order of a wedge product if you are careful with signs.The exterior derivative“d"allows us to differentiate p-form fields to obtain (p+1)-formfields. It is defined as an appropriately normalized antisymmetric partial derivative:(1.82)(dA)-p+ = (p + 1)O Aμ2--p+1] :The simplest example is the gradient, which is the exterior derivative of a 1-form:(1.83)(do)μ=0μΦ:The reason why the exterior derivative deserves special attention is that it is a tensor,even incurved spacetimes, unlike its cousin the partial derivative. Since we haven't studied curvedspaces yet, we cannot prove this, but (1.82) defines an honest tensor no matter what themetric and coordinates are.Another interesting fact about exterior differentiation is that, for any form A,(1.84)d(dA) = 0 ,which is often written d? = o. This identity is a consequence of the definition of d and thefact that partial derivatives commute, .g = OgO (acting on anything). This leads us tothe following mathematical aside, just for fun. We define a p-form A to be closed if dA= 0and exact if A =dB for some (p-1)-form B.Obviously,all exact forms are closed, but theconverse isnot necessarily true.Ona manifold M, closed p-forms comprise a vector spaceZp(M), and exact forms comprise a vector space Bp(M). Define a new vector space as theclosed forms modulothe exact forms:ZP(M)HP(M) =(1.85)BP(M)This is known as the pth de Rham cohomology vector space, and depends only on thetopology of the manifold M.(Minkowski space is topologically equivalent to R4, which isuninteresting, so that all of the HP(M) vanish for p > O; for p = 0 we have Ho(M) =RTherefore in Minkowski space all closed forms are exact except for zero-forms; zero-formscan't be exact since there are no -1-forms for them to be the exterior derivative of.) It isstriking that information about the topology can be extracted in this way, which essentiallyinvolves the solutions to differential equations:The dimension bp of the space Hp(M) is

1 SPECIAL RELATIVITY AND FLAT SPACETIME 22 Thus, for example, the wedge product of two 1-forms is (A ∧ B)µν = 2A[µBν] = AµBν − AνBµ . (1.80) Note that A ∧ B = (−1)pqB ∧ A , (1.81) so you can alter the order of a wedge product if you are careful with signs. The exterior derivative “d” allows us to differentiate p-form fields to obtain (p+1)-form fields. It is defined as an appropriately normalized antisymmetric partial derivative: (dA)µ1···µp+1 = (p + 1)∂[µ1Aµ2···µp+1] . (1.82) The simplest example is the gradient, which is the exterior derivative of a 1-form: (dφ)µ = ∂µφ . (1.83) The reason why the exterior derivative deserves special attention is that it is a tensor, even in curved spacetimes, unlike its cousin the partial derivative. Since we haven’t studied curved spaces yet, we cannot prove this, but (1.82) defines an honest tensor no matter what the metric and coordinates are. Another interesting fact about exterior differentiation is that, for any form A, d(dA) = 0 , (1.84) which is often written d2 = 0. This identity is a consequence of the definition of d and the fact that partial derivatives commute, ∂α∂β = ∂β∂α (acting on anything). This leads us to the following mathematical aside, just for fun. We define a p-form A to be closed if dA = 0, and exact if A = dB for some (p−1)-form B. Obviously, all exact forms are closed, but the converse is not necessarily true. On a manifold M, closed p-forms comprise a vector space Z p (M), and exact forms comprise a vector space Bp (M). Define a new vector space as the closed forms modulo the exact forms: H p (M) = Z p (M) Bp (M) . (1.85) This is known as the pth de Rham cohomology vector space, and depends only on the topology of the manifold M. (Minkowski space is topologically equivalent to R4 , which is uninteresting, so that all of the Hp (M) vanish for p > 0; for p = 0 we have H0 (M) = R. Therefore in Minkowski space all closed forms are exact except for zero-forms; zero-forms can’t be exact since there are no −1-forms for them to be the exterior derivative of.) It is striking that information about the topology can be extracted in this way, which essentially involves the solutions to differential equations. The dimension bp of the space Hp (M) is

231 SPECIALRELATIVITYANDFLATSPACETIMEcalled the pth Betti number of M, and the Euler characteristic is given by the alternatingsumx(M) =Z(-1)Pbp :(1.86)=Cohomology theory is the basis for much of modern differential topology.Moving back to reality, the final operation on differential forms we will introduce isHodge duality. We define the “Hodge star operator" on an n-dimensional manifold as amap from p-forms to (n-p)-forms,11"Vp(1.87)(*A)μi--n-pμ-npArp,Dmapping A to “A dual". Unlike our other operations on forms, the Hodge dual does dependonthemetricofthemanifold (whichshouldbeobvious,sincewehad toraisesomeindiceson the Levi-Civita tensor in order to define (1.87)).Applying the Hodge star twice returnseither plus or minus the original form:**A = (-1)$+p(n-P)A ,(1.88)where s is the number of minus signs in the eigenvalues of the metric (for Minkowski space,s = 1).Two facts on the Hodge dual: First,"duality"in the sense of Hodge is different than therelationship between vectors and dual vectors, although both can be thought of as the spaceof linear maps from the original space to R. Notice that the dimensionality of the space of(n - p)-forms is equal to that of the space of p-forms, so this has at least a chance of beingtrue. In the case of forms, the linear map defined by an (n -p)-form acting on a p-form isgiven by the dual of the wedge product of the two forms. Thus, if A(n-p) is an (n - p)-formand B(p) is a p-form at some point in spacetime, we have* (A(n-p) ΛB(p) ER :(1.89)The second fact concerns differential forms in 3-dimensional Euclidean space. The Hodgedual of the wedge product of two 1-forms gives another 1-form:(1.90)* (U ^ V);=EkU,V.(All of the prefactors cancel.) Since 1-forms in Euclidean space are just like vectors, we havea map from two vectors to a single vector.You should convince yourself that this is just theconventional cross product, and that the appearance of the Levi-Civita tensor explains whythe cross product changes sign under parity (interchange of two coordinates, or equivalentlybasis vectors). This is why the cross product only exists in three dimensions - because only

1 SPECIAL RELATIVITY AND FLAT SPACETIME 23 called the pth Betti number of M, and the Euler characteristic is given by the alternating sum χ(M) = Xn p=0 (−1)p bp . (1.86) Cohomology theory is the basis for much of modern differential topology. Moving back to reality, the final operation on differential forms we will introduce is Hodge duality. We define the “Hodge star operator” on an n-dimensional manifold as a map from p-forms to (n − p)-forms, (∗A)µ1···µn−p = 1 p! ǫ ν1···νp µ1···µn−pAν1···νp , (1.87) mapping A to “A dual”. Unlike our other operations on forms, the Hodge dual does depend on the metric of the manifold (which should be obvious, since we had to raise some indices on the Levi-Civita tensor in order to define (1.87)). Applying the Hodge star twice returns either plus or minus the original form: ∗ ∗A = (−1)s+p(n−p)A , (1.88) where s is the number of minus signs in the eigenvalues of the metric (for Minkowski space, s = 1). Two facts on the Hodge dual: First, “duality” in the sense of Hodge is different than the relationship between vectors and dual vectors, although both can be thought of as the space of linear maps from the original space to R. Notice that the dimensionality of the space of (n − p)-forms is equal to that of the space of p-forms, so this has at least a chance of being true. In the case of forms, the linear map defined by an (n − p)-form acting on a p-form is given by the dual of the wedge product of the two forms. Thus, if A (n−p) is an (n − p)-form and B(p) is a p-form at some point in spacetime, we have ∗ (A (n−p) ∧ B (p) ) ∈ R . (1.89) The second fact concerns differential forms in 3-dimensional Euclidean space. The Hodge dual of the wedge product of two 1-forms gives another 1-form: ∗ (U ∧ V )i = ǫi jkUjVk . (1.90) (All of the prefactors cancel.) Since 1-forms in Euclidean space are just like vectors, we have a map from two vectors to a single vector. You should convince yourself that this is just the conventional cross product, and that the appearance of the Levi-Civita tensor explains why the cross product changes sign under parity (interchange of two coordinates, or equivalently basis vectors). This is why the cross product only exists in three dimensions — because only