11 Tensors and Local Symmetries
11.1 Points and Coordinates A point on a curved surface or in a curved space also is a point in a higher- dimensional flat space called an embedding space. For instance, a point on a sphere also is a point in three-dimensional euclidean space and in four- dimensional spacetime. One always can add extra dimensions, but it’s sim- pler to use as few as possible, three in the case of a sphere. On a su ciently small scale, any reasonably smooth space locally looks like n-dimensional euclidean space. Such a space is called a manifold.In- cidentally, according to Whitney’s embedding theorem, every n-dimensional connected, smooth manifold can be embedded in 2n-dimensional euclidean space R2n. So the embedding space for such spaces in general relativity has no more than eight dimensions. We use coordinates to label points. For example, we can choose a polar axis and a meridian and label a point on the sphere by its polar and az- imuthal angles (✓, ) with respect to that axis and meridian. If we use a dif- ferent axis and meridian, then the coordinates (✓0, 0) for the same point will change. Points are physical, coordinates are metaphysical. When we change our system of coordinates, the points don’t change, but their coordinates do. i i Most points p have unique coordinates x (p) and x0 (p) in their coordinate systems. For instance, polar coordinates (✓, ) are unique for all points on a sphere — except the north and south poles which are labeled by ✓ =0 and ✓ = ⇡ and all 0 <2⇡. By using more than one coordinate system, one usually can arrange to label every point uniquely in some coordinate system. In the flat three-dimensional space in which the sphere is a surface, each point of the sphere has unique coordinates, p~ =(x, y, z). 11.2 Scalars 451 We will use coordinate systems that represent the points of a manifold uniquely and smoothly at least in local patches, so that the maps
i i i i x0 = x0 (p)=x0 (p(x)) = x0 (x) (11.1) and i i i i x = x (p)=x (p(x0)) = x (x0) (11.2) are well defined, di↵erentiable, and one to one in the patches. We’ll often group the n coordinates xi together and write them collectively as x without a superscript. Since the coordinates x(p) label the point p, we sometimes will call them “the point x.” But p and x are di↵erent. The point p is unique with infinitely many coordinates x, x0, x00, . . . in infinitely many coordinate systems.
11.2 Scalars A scalar is a quantity B that is the same in all coordinate systems
B0 = B. (11.3) If it also depends upon the coordinates x of the spacetime point p,thenit is a scalar field, and
B0(x0)=B(x). (11.4)
11.3 Contravariant Vectors i By the chain rule, the change in dx0 due to changes in the unprimed coor- dinates is i i @x0 j dx0 = dx . (11.5) @xj Xj This transformation defines contravariant vectors: a quantity Ai is a component of contravariant vector if it transforms like dxi i i @x0 j A0 = A . (11.6) @xj Xj The coordinate di↵erentials dxi form a contravariant vector. A contravariant vector Ai(x) that depends on the coordinates x is a contravariant vector 452 Tensors and Local Symmetries field and transforms as i i @x0 j A0 (x0)= A (x). (11.7) @xj Xj
11.4 Covariant Vectors The chain rule for partial derivatives @ @xj @ i = i j (11.8) @x0 @x0 @x Xj defines covariant vectors: a quantity Ci that transforms as @xj Ci0 = i Cj (11.9) @x0 Xj is a component of a covariant vector. A covariant vector Ci(x) that de- pends on the coordinates x and transforms as @xj Ci0(x0)= i Cj(x) (11.10) @x0 Xj is a covariant vector field.
Example 11.1 (Gradient of a Scalar). The derivatives of a scalar field B0(x0)=B(x) form a covariant vector field because j @B0(x0) @B(x) @x @B(x) i = i = i j , (11.11) @x0 @x0 @x0 @x Xj which shows that the gradient @B(x)/@xj fits the definition (11.10) of a covariant vector field.
11.5 Euclidean Space in Euclidean Coordinates If we use euclidean coordinates to describe points in euclidean space, then covariant and contravariant vectors are the same. Euclidean space has a natural inner product (section 1.6), the usual dot- product, which is real and symmetric. In a euclidean space of n dimensions, 11.5 Euclidean Space in Euclidean Coordinates 453 we may choose any n fixed, orthonormal basis vectors ei
n (e ,e ) e e = ek ek = (11.12) i j ⌘ i · j i j ij Xk=1 and use them to represent any point p as the linear combination
n i p = ei x . (11.13) Xi=1 The dual vectors ei are defined as those vectors whose inner products with the ej are n i i i (e ,ej)= ekejk = j (11.14) Xk=1 zero or one. Here they are the same as the vectors ei, and so we don’t need i i to distinguish e from ei = e , but we will anyway. i The coe cients x are the euclidean coordinates in the ei basis. Since they and the dual vectors ej are orthonormal, each xi is an inner product or dot product
n n xi = ei p = ei e xj = i xj. (11.15) · · j j Xj=1 Xj=1
If we use di↵erent orthonormal vectors ei0 as a basis n i p = ei0 x0 (11.16) Xi=1 then we get new euclidean coordinates x i = e i p for the same point p. 0 0 · These two sets of coordinates are related by the equations
n i i i j x0 = e0 p = e0 e x · · j Xj=1 n (11.17) j j j k x = e p = e e0 x0 . · · k Xk=1 i Because the dual vectors e are the same as the basis vectors ej and are i j independent of the euclidian coordinates x,thecoe cients@x0 /@x and j i @x /@x0 of the transformation laws for contravariant (11.6) and covariant 454 Tensors and Local Symmetries (11.9) vectors are the same
i n n j @x0 i i j j @x j = e0 ej = ek0 ejk = eik0 ek = e ei0 = i . (11.18) @x · · @x0 Xk=1 Xk=1 So contravariant and covariant vectors transform the same way in euclidean space with euclidean coordinates. i j The relations between x0 and x imply that
n i i j k x0 = e0 e e e0 x0 . (11.19) · j · k j,k=1 X i Since this holds for all coordinates x0 ,wehave
n i j e0 e e e0 = . (11.20) · j · k ik j=1 X The coe cients e i e = e ej form an orthogonal matrix, and the linear 0 · j i0 · operator
n n iT i e e0 = e e0 (11.21) i | iih | Xi=1 Xi=1 is an orthogonal (real, unitary) transformation. The change x x is a ! 0 rotation and/or a reflection (exercise 11.2).
Example 11.2 (A Euclidean Space of Two Dimensions). In two-dimensional euclidean space, one can describe the same point by euclidean (x, y) and polar (r, ✓) coordinates. The derivatives
@r x @x @r y @y = = and = = (11.22) @x r @r @y r @r respect the symmetry (11.18), but (exercise 11.1) the derivatives
@✓ y @x @✓ x @y = = = y and = = = x (11.23) @x r2 6 @✓ @y r2 6 @✓ do not because polar coordinates are not euclidean. 11.6 Summation convention 455 11.6 Summation convention An index that appears twice in the same monomial, once as a subscript and once as a superscript, is a dummy index that is summed over as in n A Bi A Bi. (11.24) i ⌘ i Xi=1 The sum is understood to be over the relevant range of indices, usually from 0 or 1 to 3 or n. These summation conventions make tensor notation almost as compact as matrix notation. They make equations easier to read and write.
Example 11.3 (The Kronecker Delta). The summation convention and the chain rule imply that
@x i @xk @x i 1ifi = j 0 = 0 = i = (11.25) @xk @x j @x j j 0ifi = j. 0 0 ⇢ 6 The repeated index k has disappeared in this contraction.
11.7 Minkowski Space Minkowski space has one time dimension, labeled by k = 0, and n space dimensions. In special relativity, n = 3, and the Minkowski metric ⌘ is 1ifk = ` =0 ⌘ = ⌘k` = 1if1k = ` 3 . (11.26) k` 8 0ifk = ` < 6 k It defines an inner product between: points p and q with coordinates xp and ` xq as (p, q)=p q = pk ⌘ q` =(q, p). (11.27) · k` If one time component vanishes, the Minkowski inner product reduces to the euclidean dot-product (1.82). We can use di↵erent sets e and e of n +1Lorentz-orthonormal { i} { i0 } basis vectors
k ` (e ,e )=e e = e ⌘ e = ⌘ = e0 e0 =(e0 ,e0 ) (11.28) i j i · j i k` j ij i · j i j to represent any point p in the space either as a linear combination of the 456 Tensors and Local Symmetries
i vectors ei with coe cients x or as a linear combination of the vectors ei0 i with coe cients x0 i i p = ei x = ei0 x0 . (11.29) The dual vectors, which carry upper indices, are defined as
i ij i ij e = ⌘ ej and e0 = ⌘ ej0 . (11.30)
They are orthonormal to the vectors ei and ei0 because (ei,e )=ei e = ⌘ik e e = ⌘ik ⌘ = i (11.31) j · j k · j kj j i i i and similarly (e0 ,ej0 )=e0 ej0 = j. Since the square of the matrix ⌘ is the ij j · identity matrix ⌘`i⌘ = ` , it follows that j j ei = ⌘ij e and ei0 = ⌘ij e0 . (11.32) The metric ⌘ raises (11.30) and lowers (11.32) the index of a basis vector. i j The component x0 is related to the components x by the linear map i i i j x0 = e0 p = e0 e x . (11.33) · · j Such a map from a 4-vector x to a 4-vector x0 is a Lorentz transformation
i i j i i x0 = L x with matrix L = e0 e . (11.34) j j · j i i k The inner product (p, q) of two points p = ei x = ei0 x0 and q = ek y = k ek0 y0 is physical and so is invariant under Lorentz transformations i k i k i k i k (p, q)=x y e e = ⌘ x y = x0 y0 e0 e0 = ⌘ x0 y0 . (11.35) i · k ik i · k ik i i r k k s With x0 = L r x and y0 = L s y , this invariance is r s i r k s i k ⌘rs x y = ⌘ik L r x L s y = ⌘ik x0 y0 (11.36) or since xr and ys are arbitrary
i k i k ⌘rs = ⌘ik L r L s = L r ⌘ik L s. (11.37) In matrix notation, a left index labels a row, and a right index labels a i T i column. Transposition interchanges rows and columns L r = L r ,so
T i k T ⌘rs = L r ⌘ik L s or ⌘ = L ⌘L (11.38) in matrix notation. In such matrix products, the height of an index— whether it is up or down—determines whether it is contravariant or covariant but does not a↵ect its place in its matrix. 11.7 Minkowski Space 457 Example 11.4 (A Boost). The matrix
2 100 2 1 00 L = 0 p 1 (11.39) 0010 Bp C B 0001C B C @ A where =1/ 1 v2/c2 represents a Lorentz transformation that is a boost in the x-direction. Boosts and rotations are Lorentz transformations. Work- p ing with 4 4 matrices can get tedious, so students are advised to think in ⇥ terms of scalars, like p x = pi⌘ xj = p x Et whenever possible. · ij · If the basis vectors e and e0 are independent of p and of x,thenthe coe cients of the transformation law (11.6) for contravariant vectors are i @x0 i = e0 e . (11.40) @xj · j Similarly, the component xj is xj = ej p = ej e x i, so the coe cients of · · i0 0 the transformation law (11.9) for covariant vectors are j @x j i = e ei0 . (11.41) @x0 · Using ⌘ to raise and lower the indices in the formula (11.40) for the coef- ficients of the transformation law (11.6) for contravariant vectors, we find, since the inner product (11.27) is symmetric, i ` @x0 i ik ` ik @x j = e0 ej = ⌘ ⌘j` ek0 e = ⌘ ⌘j` k (11.42) @x · · @x0 which is @xj/@x i. So if we use coordinates associated with fixed basis ± 0 vectors ei in Minkowski space, then the coe cients for the two kinds of transformation laws di↵er only by occasional minus signs. Thus if Ai is a contravariant vector i @x0 A0i = Aj (11.43) @xj then the relation (11.42) between the two kinds of coe cients implies that @x i @x` @x` @x` 0i 0 j ik j k j j ⌘si A = ⌘si j A = ⌘si ⌘ ⌘j` k A = s k ⌘j` A = s ⌘`j A @x @x0 @x0 @x0 (11.44) j which shows that A` = ⌘`j A transforms covariantly @x` As0 = s A`. (11.45) @x0 458 Tensors and Local Symmetries The metric ⌘ turns a contravariant vector into a covariant one. It also k switches a covariant vector A` back to its contravariant form A k` k` j k j k ⌘ A` = ⌘ ⌘`jA = j A = A . (11.46) In Minkowski space, one uses ⌘ to raise and lower indices j i ij Ai = ⌘ij A and A = ⌘ Aj. (11.47) In general relativity, the spacetime metric g raises and lowers indices.
11.8 Lorentz Transformations In section 11.7, Lorentz transformations arose as linear maps of the coor- dinates due to a change of basis. They also are linear maps of the basis vectors ei that preserve the inner products
(e ,e )=e e = ⌘ = e0 e0 =(e0 ,e0 ). (11.48) i j i · j ij i · j i j The vectors ei are four linearly independent four-dimensional vectors, and so they span four-dimensional Minkowski space and can represent the vectors ei0 as k ei0 =⇤i ek (11.49) k where the coe cients ⇤i are real numbers. The requirement that the new basis vectors ei0 are Lorentz orthonormal gives k ` k ` k ` ⌘ = e0 e0 =⇤ e ⇤ e =⇤ e e ⇤ =⇤ ⌘ ⇤ (11.50) ij i · j i k · j ` i k · ` j i k` j or in matrix notation ⌘ =⇤⌘ ⇤T (11.51)
T T ` ` T where ⇤ is the transpose (⇤ ) j =⇤j .Evidently⇤ satisfies the definition (11.38) of a Lorentz transformation. What Lorentz transformation is it? The point p must remain invariant, so by (11.34 & 11.49) one has i k i j k j j p = ei0 x0 =⇤i ek L jx = j ek x = ej x (11.52) k i k T T 1 whence ⇤i L j = j or ⇤ L = I.So⇤ = L . By multiplying condition (11.51) by the metric ⌘ first from the left and then from the right and using the fact that ⌘2 = I,wefind 1=⌘2 = ⌘ ⇤ ⌘ ⇤T =⇤⌘ ⇤T ⌘ (11.53) which gives us the inverse matrices
1 T T T 1 ⇤ = ⌘ ⇤ ⌘ = L and (⇤ ) = ⌘ ⇤ ⌘ = L. (11.54) 11.9 Special Relativity 459 In special relativity, contravariant vectors transform as
i i j dx0 = L j dx (11.55)
j 1j i and since x = L i x0 , the covariant ones transform as j @ @x @ 1j @ j @ i = i j = L i j =⇤i j . (11.56) @x0 @x0 @x @x @x By taking the determinant of both sides of (11.51) and using the transpose (1.194) and product (1.207) rules for determinants, we find that det ⇤= 1. ±
11.9 Special Relativity The spacetime of special relativity is flat, four-dimensional Minkowski space. The inner product (p q) (p q) of the interval p q between two points is · physical and independent of the coordinates and therefore invariant. Points p and q that have (p q) (p q) > 0 are spacelike.Ifp and q are spacelike, · then they occur at the same time in some Lorentz frame. Points p and q that have (p q) (p q) < 0 are timelike.Ifp and q are timelike, then · they occur at the same place in some Lorentz frame. If the points p and q are close neighbors with coordinates xi + dxi for p and xi for q, then that invariant inner product is
(p q) (p q)=e dxi e dxj = dxi ⌘ dxj = dx2 (dx0)2 (11.57) · i · j ij with dx0 = cdt. (At some point in what follows, we’ll measure distance in light-seconds so that c = 1.) If the points p and q are on the trajectory of a massive particle moving at velocity v, then this invariant quantity is the square of the invariant distance
ds2 = dx2 c2dt2 = v2 c2 dt2 (11.58) which always is negative since v d⌧ 2 = ds2/c2 = 1 v2/c2 dt2. (11.59) A particle of mass zero moves at the speed of light, and so its element d⌧ of proper time is zero. But for a particle of mass m>0 moving at speed v, the element of proper time d⌧ is smaller than the corresponding element of laboratory time dt by the factor 1 v2/c2. The proper time is the time in the rest frame of the particle, d⌧ = dt when v = 0. So if T (0) is the lifetime p 460 Tensors and Local Symmetries of a particle at rest, then the apparent lifetime T (v) when the particle is moving at speed v is d⌧ T (0) T (v)=dt = = (11.60) 1 v2/c2 1 v2/c2 which is longer — an e↵ect knownp as time dilationp . Example 11.5 (Time Dilation in Muon Decay ). A muon at rest has a mean life of T (0) = 2.2 10 6 seconds. Cosmic rays hitting nitrogen and ⇥ oxygen nuclei make pions high in the Earth’s atmosphere. The pions rapidly decay into muons in 2.6 10 8 s. A muon moving at the speed of light from ⇥ 10 km takes at least t = 10 km/300, 000 (km/sec) = 3.3 10 5 stohitthe ⇥ ground. Were it not for time dilation, the probability P of such a muon reaching the ground as a muon would be t/T (0) 15 7 P = e =exp( 33/2.2) = e =2.6 10 . (11.61) ⇥ The (rest) mass of a muon is 105.66 MeV. So a muon of energy E = 749 MeV has by (11.68) a time-dilation factor of 1 E 749 1 = = =7.089 = . (11.62) 1 v2/c2 mc2 105.7 1 (0.99)2 So a muonp moving at a speed of v =0.99 c hasp an apparent mean life T (v) given by equation (11.60) as 6 E T (0) 2.2 10 s 5 T (v)= T (0) = = ⇥ =1.6 10 s. (11.63) mc2 1 v2/c2 1 (0.99)2 ⇥ The probability of survivalp with timep dilation is t/T (v) P = e =exp( 33/16) = 0.12 (11.64) so that 12% survive. Time dilation increases the chance of survival by a factor of 460,000 — no small e↵ect. 11.10 Kinematics From the scalar d⌧, and the contravariant vector dxi, we can make the 4-vector dxi dt dx0 dx 1 ui = = , = (c, v) (11.65) d⌧ d⌧ dt dt 1 v2/c2 ✓ ◆ p 11.10 Kinematics 461 in which u0 = c dt/d⌧ = c/ 1 v2/c2 and u = u0 v/c.Theproductmui is the energy-momentum 4-vector pi p dxi dt dxi m dxi pi = mui = m = m = d⌧ d⌧ dt 1 v2/c2 dt m E = (c, v)= , p .p (11.66) 1 v2/c2 c ✓ ◆ Its invariant innerp product is a constant characteristic of the particle and proportional to the square of its mass c2 pi p = mc ui mc u = E2 + c2 p 2 = m2 c4. (11.67) i i Note that the time-dilation factor is the ratio of the energy of a particle to its rest energy 1 E = (11.68) 1 v2/c2 mc2 and the velocity of the particlep is its momentum divided by its equivalent mass E/c2 p v = . (11.69) E/c2 The analog of F = m a is d2xi dui dpi m = m = = f i (11.70) d⌧ 2 d⌧ d⌧ in which p0 = E/c, and f i is a 4-vector force. Example 11.6 (Time Dilation and Proper Time). In the frame of a labo- i ratory, a particle of mass m with 4-momentum plab =(E/c, p, 0, 0) travels i a distance L in a time t for a 4-vector displacement of xlab =(ct, L, 0, 0). In its own rest frame, the particle’s 4-momentum and 4-displacement are i i prest =(mc, 0, 0, 0) and xrest =(c⌧, 0, 0, 0). Since the Minkowski inner product of two 4-vectors is Lorentz invariant, we have pix = pix or Et pL = mc2⌧ = mc2t 1 v2/c2 (11.71) i rest i lab i 2 so a massive particle’s phase exp( ip xi/~)isexp(imcp⌧/~). Example 11.7 (p + ⇡ ⌃+K). What is the minimum energy that a ! beam of pions must have to produce a sigma hyperon and a kaon by striking a proton at rest? Conservation of the energy-momentum 4-vector gives 462 Tensors and Local Symmetries pp + p⇡ = p⌃ + pK .Wesetc = 1 and use this equality in the invariant form 2 2 2 (pp+p⇡) =(p⌃+pK ) . We compute (pp+p⇡) in the the pp =(mp, 0) frame 2 and set it equal to (p⌃ + pK ) in the frame in which the spatial momenta of the ⌃and the K cancel and vanish at minimum pion energy: (p + p )2 = p2 + p2 +2p p = m2 m2 2m E p ⇡ p ⇡ p · ⇡ p ⇡ p ⇡ =(p + p )2 = (m + m )2 . (11.72) ⌃ K ⌃ K Thus, since the relevant masses (in MeV) are m⌃+ = 1189.4, mK+ = 493.7, mp = 938.3, and m⇡+ = 139.6, the minimum total energy of the pion is 2 2 2 (m⌃ + mK ) mp m⇡ E⇡ = 1030 MeV (11.73) 2mp ⇡ of which 890 MeV is kinetic. 11.11 Electrodynamics In electrodynamics and in mksa (si) units, the three-dimensional vector potential A and the scalar potential form a covariant 4-vector potential A = , A . (11.74) i c ✓ ◆ The contravariant 4-vector potential is Ai =( /c, A). The magnetic in- duction is B = A or B = ✏ @ A (11.75) r⇥ i ijk j k j in which @j = @/@x , the sum over the repeated indices j and k runs from 1 to 3, and ✏ijk is totally antisymmetric with ✏123 = 1. The electric field is @A @A @ @A E = c 0 i = i (11.76) i @xi @x0 @xi @t ✓ ◆ where x0 = ct. In 3-vector notation, E is given by the gradient of and the time-derivative of A E = A˙. (11.77) r In terms of the second-rank, antisymmetric Faraday field-strength tensor @A @A F = j i = F (11.78) ij @xi @xj ji 11.11 Electrodynamics 463 the electric field is Ei = cFi0 and the magnetic field Bi is 1 1 @A @A B = ✏ F = ✏ k j =( A) (11.79) i 2 ijk jk 2 ijk @xj @xk r⇥ i ✓ ◆ where the sum over repeated indices runs from 1 to 3. The inverse equation Fjk = ✏jkiBi for spatial j and k follows from the Levi-Civita identity (1.455) 1 1 ✏ B = ✏ ✏ F = ✏ ✏ F jki i 2 jki inm nm 2 ijk inm nm 1 1 = ( ) F = (F F )=F . (11.80) 2 jn km jm kn nm 2 jk kj jk In 3-vector notation and mksa = si units, Maxwell’s equations are a ban on magnetic monopoles and Faraday’s law, both homogeneous, B = 0 and E + B˙ = 0 (11.81) r · r ⇥ and Gauss’s law and the Maxwell-Amp`ere law, both inhomogeneous, D = ⇢ and H = j + D˙ . (11.82) r · f r ⇥ f Here ⇢f is the density of free charge and jf is the free current density.By free, we understand charges and currents that do not arise from polarization and are not restrained by chemical bonds. The divergence of H vanishes r⇥ (like that of any curl), and so the Maxwell-Amp`ere law and Gauss’s law imply that free charge is conserved 0= ( H)= j + D˙ = j +˙⇢ . (11.83) r · r ⇥ r · f r · r · f f If we use this continuity equation to replace j with ⇢˙ in its middle r· f f form 0 = j + D˙ , then we see that the Maxwell-Amp`ere law preserves r· f r· the Gauss-law constraint in time @ 0= j + D˙ = ( ⇢ + D) . (11.84) r · f r · @t f r · Similarly, Faraday’s law preserves the constraint B =0 r · @ 0= ( E)= B =0. (11.85) r · r ⇥ @tr · In a linear, isotropic medium, the electric displacement D is related to the electric field E by the permittivity ✏, D = ✏E, and the mag- netic or magnetizing field H di↵ers from the magnetic induction B by the permeability µ, H = B/µ. On a sub-nanometer scale, the microscopic form of Maxwell’s equations 464 Tensors and Local Symmetries applies. On this scale, the homogeneous equations (11.81) are unchanged, but the inhomogeneous ones are ˙ ⇢ ˙ E E = and B = µ0 j + ✏0 µ0 E = µ0 j + 2 (11.86) r · ✏0 r ⇥ c in which ⇢ and j are the total charge and current densities, and ✏0 = 8.854 10 12 F/m and µ =4⇡ 10 7 N/A2 are the electric and magnetic ⇥ 0 ⇥ constants, whose product is the inverse of the square of the speed of light, 2 ✏0µ0 =1/c . Gauss’s law and the Maxwell-Amp`ere law (11.86) imply (ex- ercise 11.6) that the microscopic (total) current 4-vector j =(c⇢, j) obeys the continuity equation⇢ ˙ + j = 0. Electric charge is conserved. r · In vacuum, ⇢ = j = 0, D = ✏0E, and H = B/µ0, and Maxwell’s equations become B = 0 and E + B˙ =0 r · r ⇥ 1 (11.87) E = 0 and B = E˙ . r · r ⇥ c2 Two of these equations B = 0 and E = 0 are constraints. Taking r · r · the curl of the other two equations, we find 1 1 ( E)= E¨ and ( B)= B¨. (11.88) r ⇥ r ⇥ c2 r ⇥ r ⇥ c2 One may use the Levi-Civita identity (1.455) to show (exercise 11.8) that ( E)= ( E) E and ( B)= ( B) B r ⇥ r ⇥ r r· 4 r ⇥ r ⇥ r r· 4 (11.89) in which 2. Since in vacuum the divergence of E vanishes, and 4⌘r since that of B always vanishes, these identities and the curl-curl equations (11.88) tell us that waves of E and B move at the speed of light 1 1 E¨ E = 0 and B¨ B =0. (11.90) c2 4 c2 4 We may write the two homogeneous Maxwell equations (11.81) as @ F + @ F + @ F = @ (@ A @ A )+@ (@ A @ A ) i jk k ij j ki i j k k j k i j j i + @ (@ A @ A )=0 (11.91) j k i i k (exercise 11.9). This relation, known as the Bianchi identity, actually is a generally covariant tensor equation `ijk ✏ @iFjk = 0 (11.92) in which ✏`ijk is totally antisymmetric, as explained in Sec. 11.37. There are four versions of this identity (corresponding to the four ways of choosing 11.12 Tensors 465 three di↵erent indices i, j, k from among four and leaving out one, `). The ` = 0 case gives the scalar equation B = 0, and the three that have r· ` = 0 give the vector equation E + B˙ = 0. 6 r ⇥ In tensor notation, the microscopic form of the two inhomogeneous equa- tions (11.86)—the laws of Gauss and Amp`ere—are ki k @iF = µ0 j (11.93) in which jk is the current 4-vector jk =(c⇢, j) . (11.94) The Lorentz force law for a particle of charge q is d2xi dui dpi dx m = m = = f i = qFij j = qFij u . (11.95) d⌧ 2 d⌧ d⌧ d⌧ j We may cancel a factor of dt/d⌧ from both sides and find for i =1, 2, 3 dpi dp = q F i0 + ✏ B v or = q (E + v B) (11.96) dt ijk k j dt ⇥ and for i =0 dE = q E v (11.97) dt · which shows that only the electric field does work. The only special-relativistic correction needed in Maxwell’s electrodynamics is a factor of 1/ 1 v2/c2 in these equations. That is, we use p = mu = mv/ 1 v2/c2 not p = mv p in (11.96), and we use the total energy E not the kinetic energy in (11.97). p The reason why so little of classical electrodynamics was changed by special relativity is that electric and magnetic e↵ects were accessible to measure- ment during the 1800’s. Classical electrodynamics was almost perfect. Keeping track of factors of the speed of light is a lot of trouble and a distraction; in what follows, we’ll often use units with c = 1. 11.12 Tensors Tensors are structures that transform like products of vectors. A first-rank tensor is a covariant or a contravariant vector. Second-rank tensors also are 466 Tensors and Local Symmetries distiguished by how they transform under changes of coordinates: i j ij @x0 @x0 kl contravariant M 0 = M @xk @xl i l i @x0 @x k mixed Nj0 = k j Nl (11.98) @x @x0 @xk @xl covariant Fij0 = i j Fkl. @x0 @x0 We can define tensors of higher rank by extending the these definitions to quantities with more indices. Example 11.8 (Some Second-Rank Tensors). If Ak and B` are covariant vectors, and Cm and Dn are contravariant vectors, then the product Cm Dn m n is a second-rank contravariant tensor, and all four products Ak C , Ak D , m n m n Bk C , and Bk D are second-rank mixed tensors, while C D as well as Cm Cn and Dm Dn are second-rank contravariant tensors. Since the transformation laws that define tensors are linear, any linear combination of tensors of a given rank and kind is a tensor of that rank and kind. Thus if Fij and Gij are both second-rank covariant tensors, then so is their sum Hij = Fij + Gij. (11.99) A covariant tensor is symmetric if it is independent of the order of its indices. That is, if Sik = Ski,thenS is symmetric. Similarly, a contravariant tensor is symmetric if permutations of its indices leave it unchanged. Thus A is symmetric if Aik = Aki. A covariant or contravariant tensor is antisymmetric if it changes sign ik when any two of its indices are interchanged. So Aik, B , and Cijk are antisymmetric if ik ki Aik = Aki and B = B and (11.100) C =C = C = C = C = C . ijk jki kij jik ikj kji Example 11.9 (Three Important Tensors). The Maxwell field strength Fkl(x) is a second-rank covariant tensor; so is the metric of spacetime gij(x). i The Kronecker delta j is a mixed second-rank tensor; it transforms as i l i k i i @x0 @x k @x0 @x @x0 i j0 = k j l = k j = j = j. (11.101) @x @x0 @x @x0 @x So it is invariant under changes of coordinates. 11.13 Di↵erential Forms 467 ` Example 11.10 (Contractions). Although the product Ak C is a mixed k second-rank tensor, the product Ak C transforms as a scalar because ` k ` k @x @x0 m @x m ` m ` Ak0 C0 = k m A` C = m A` C = mA` C = A` C . (11.102) @x0 @x @x A sum in which an index is repeated once covariantly and once contravari- antly is a contraction as in the Kronecker-delta equation (11.25). In gen- eral, the rank of a tensor is the number of uncontracted indices. 11.13 Di↵erential Forms By (11.10 & 11.5), a covariant vector field contracted with contravariant co- ordinate di↵erentials is invariant under arbitrary coordinate transformations j i i @x @x0 k j k k A0 = Ai0 dx0 = i Aj k dx = k Aj dx = Ak dx = A. (11.103) @x0 @x k This invariant quantity A = Ak dx is a called a 1-form in the language of di↵erential forms introduced about a century ago by Elie´ Cartan (1869– 1951, son of a blacksmith). The wedge product dx dy of two coordinate di↵erentials is the directed ^ area spanned by the two di↵erentials and is defined to be antisymmetric dx dy = dy dx and dx dx = dy dy = 0 (11.104) ^ ^ ^ ^ so as to transform correctly under a change of coordinates. In terms of the coordinates u = u(x, y) and v = v(x, y), the new element of area is @u @u @v @v du dv = dx + dy dx + dy . (11.105) ^ @x @y ^ @x @y ✓ ◆ ✓ ◆ Labeling partial derivatives by subscripts (6.20) and using the antisymmetry (11.104), we see that the new element of area du dv is the old area dx dy ^ ^ multiplied by the Jacobian J(u, v; x, y) of the transformation x, y u, v ! du dv =(u dx + u dy) (v dx + v dy) ^ x y ^ x y = u v dx dx + u v dx dy + u v dy dx + u v dy dy x x ^ x y ^ y x ^ y y ^ =(u v u v ) dx dy x y y x ^ u u = x y dx dy = J(u, v; x, y) dx dy. (11.106) v v ^ ^ x y 468 Tensors and Local Symmetries A contraction H = 1 H dxi dxk of a second-rank covariant tensor with 2 ik ^ a wedge product of two di↵erentials is a 2-form. A p -form is a rank-p covariant tensor contracted with a wedge product of p di↵erentials 1 K = K dxi1 ...dxip . (11.107) p! i1...ip ^ The exterior derivative d di↵erentiates and adds a di↵erential; it turns a p-form into a (p + 1)-form. It converts a function or a 0-form f into a 1-form @f df = dxi (11.108) @xi and a 1-form A = A dxj into a 2-form dA = d(A dxj)=(@ A ) dxi dxj. j j i j ^ Example 11.11 (The Curl). The exterior derivative of the 1-form A = Ax dx + Ay dy + Az dz (11.109) is a 2-form that contains the curl (6.39) of A dA = @ A dy dx + @ A dz dx y x ^ z x ^ + @ A dx dy + @ A dz dy x y ^ z y ^ + @ A dx dz + @ A dy dz x z ^ y z ^ =(@ A @ A ) dy dz (11.110) y z z y ^ +(@ A @ A ) dz dx z x x z ^ +(@ A @ A ) dx dy x y y x ^ =( A) dy dz +( A) dz dx +( A) dx dy. r⇥ x ^ r⇥ y ^ r⇥ z ^ j The exterior derivative of the 1-form A = Aj dx is the 2-form dA = d A dxj = @ A dxi dxj = 1 F dxi dxj = F (11.111) j ^ i j ^ 2 ij ^ i in which @i = @/@x .So d turns the electromagnetic 1-form A—the 4-vector potential or gauge field Aj—into the Faraday 2-form—the tensor Fij. The square dd of the exterior derivative vanishes in the sense that dd applied to any p-form Q is zero i r i d d Qi...dx ... = d (@rQi...) dx dx ... ^ ^ ^ (11.112) =(@ @ Q ) dxs dxr dxi ...=0 ⇥ ⇤ ⇥s r i... ^ ^ ⇤^ because @ @ Q is symmetric in r and s while dxs dxr is anti-symmetric. s r ^ Some writers drop the wedges and write dxi dxj as dxidxj while keeping ^ 11.13 Di↵erential Forms 469 the rules of antisymmetry dxidxj = dxjdxi and dxi 2 = 0. But this econ- omy prevents one from using invariant quantities like S = 1 S dxidxk in 2 ik which Sik is a second-rank covariant symmetric tensor. If Mik is a covariant second-rank tensor with no particular symmetry, then (exercise 11.7) only i k its antisymmetric part contributes to the 2-form Mik dx dx and only its i k ^ symmetric part contributes to the quantity Mik dx dx . The exterior derivative d applied to the Faraday 2-form F = dA gives dF = ddA = 0 (11.113) which is the Bianchi identity (11.92). A p-form H is closed if dH = 0. By (11.113), the Faraday 2-form is closed, dF = 0. A p-form H is exact if there is a (p 1)-form K whose di↵erential is H = dK. The identity (11.112) or dd = 0 implies that every exact form is closed. The lemma of Poincar´eshows that every closed form is locally exact. i If the Ai in the 1-form A = Aidx commute with each other, then the 2-form A A = 0. But if the A don’t commute because they are matrices ^ i or operators or Grassmann variables, then A A = 1 [A ,A ] dxi dxj need ^ 2 i j ^ not vanish. Example 11.12 (If B˙ = 0, the electric field is closed and locally exact). If B˙ = 0, then by Faraday’s law (11.81) the curl of the electric field vanishes, E = 0. In terms of the 1-form E = E dxi for i = 1, 2, 3, the vanishing r⇥ i of its curl E is r⇥ 1 dE = @ E dxj dxi = (@ E @ E ) dxj dxi =0. (11.114) j i ^ 2 j i i j ^ So E is closed. We can define a quantity VP (x) as a line integral of the 1-form E along a path P to x from some starting point x0 x V (x)= E dxi = E. (11.115) P i ZP, x0 ZP In general, VP (x) will depend on the path P . But the path dependence V (x) V (x) is the line integral of E around a closed loop which by P 0 P Stokes’s theorem (6.44) is an integral of the vanishing curl E over any r⇥ surface S whose boundary @S is the closed curve P P 0 i VP 0 (x) VP (x)= Ei dx = ( E) da = 0 (11.116) P P S r⇥ · I 0 Z as long as E = 0 on and near the surface S. In the notation of forms, r⇥ 470 Tensors and Local Symmetries Stokes’s theorem is V (x) V (x)= E = dE = 0 (11.117) P 0 P Z@S ZS (George Stokes, 1819–1903). Thus the potential VP (x)=V (x)isindepen- dent of the path, E = V (x), and the 1-form E = E dxi = @ Vdxi = r i i dV is locally exact. The general form of Stokes’s theorem is that the integral of any p-form H over the boundary @R of any (p + 1)-dimensional, simply connected, orientable region R is equal to the integral of the (p + 1)-form dH over R H = dH (11.118) Z@R ZR which for p = 1 gives (6.44 & 11.117). Example 11.13 (Stokes’s Theorem for 0-forms). Here p = 0, the region R =[a, b] is 1-dimensional, H is a 0-form, and Stokes’s theorem is b b H(b) H(a)= H = dH = dH(x)= H0(x) dx (11.119) Z@R ZR Za Za familiar from elementary calculus. Example 11.14 (Exterior Derivatives Anticommute with Di↵erentials). i j The exterior derivative acting on two one-forms A = Aidx and B = Bjdx is d(A B)=d(A dxi B dxj)=@ (A B ) dxk dxi dxj (11.120) ^ i ^ j k i j ^ ^ =(@ A ) B dxk dxi dxj + A (@ B ) dxk dxi dxj k i j ^ ^ i k j ^ ^ =(@ A ) B dxk dxi dxj A (@ B ) dxi dxk dxj k i j ^ ^ i k j ^ ^ =(@ A ) dxk dxi B dxj A dxi (@ B ) dxk dxj k i ^ ^ j i ^ k j ^ = dA B A dB. ^ ^ If A is a p-form, then d(A B)=dA B +( 1)pA dB (exercise 11.10). ^ ^ ^ 11.14 Tensor Equations Maxwell’s homogeneous equations (11.92) relate the derivatives of the field- strength tensor to each other as 0=@iFjk + @kFij + @jFki. (11.121) 11.15 The Quotient Theorem 471 They are generally covariant tensor equations (sections 11.35 & 11.37). They follow from the Bianchi identity (11.113) dF = ddA =0. (11.122) Maxwell’s inhomegneous equations (11.93) relate the derivatives of the field- strength tensor to the current density ji and to the square root of the mod- ulus g of the determinant of the metric tensor gij (section 11.18) @(pgFik) = µ pgji. (11.123) @xk 0 They are generally covariant tensor equations. We’ll write them as invari- ant forms in section 11.29 and derive them from an action principle in sec- tion 11.43. If we can write a physical law in one coordinate system as a tensor equation Gkl = 0 (11.124) then in any other coordinate system, the corresponding tensor equation ij G0 = 0 (11.125) also is valid since i j ij @x0 @x0 kl G0 = G =0. (11.126) @xk @xl Similarly, physical laws remain the same when expressed in terms of invari- ant forms. Thus by writing a theory in terms of tensors or forms, one gets a theory that is true in all coordinate systems if it is true in any. Only such generally covariant theories have a chance at being right in our coordinate system, which is not special. One way to make a generally covariant theory is to start with an action that is invariant under all coordinate transformations. 11.15 The Quotient Theorem Suppose that the product BAof a quantity B (with unknown transforma- tion properties) with an arbitrary tensor A (of a given rank and kind) is a i tensor. Then B is itself a tensor. The simplest example is when BiA is a scalar for all contravariant vectors Ai i j Bi0A0 = BjA . (11.127) 472 Tensors and Local Symmetries Then since Ai is a contravariant vector i i @x0 j j B0A0 = B0 A = B A (11.128) i i @xj j or i @x0 j B0 B A =0. (11.129) i @xj j ✓ ◆ Since this equation holds for all vectors A, we may promote it to the level of a vector equation i @x0 B0 B =0. (11.130) i @xj j j k Multiplying both sides by @x /@x0 and summing over j, we get i j j @x0 @x @x Bi0 j k = Bj k (11.131) @x @x0 @x0 which shows that the unknown quantity Bi transforms as a covariant vector @xj Bk0 = k Bj. (11.132) @x0 The quotient rule works for unknowns B and tensors A of arbitrary rank and kind. The proof in each case is similar to the one given here. 11.16 The embedding space The change in a point p(x) on a smooth manifold due to an infinitesimal change dxi(p) in the coordinates xi(p) of the point is @p(x) dp↵(x)= dxi = e ↵(x) dxi (11.133) @xi i in which the superscript ↵ labels the n coordinates of the embedding space Rn and the index i labels the coordinates on the manifold. The embedding n space R has its own metric I↵ which determines the inner products of basis vectors as n e e = e ↵ I e . (11.134) i · k i ↵ j ↵X, =1 The inner metric I usually is a diagonal matrix with eigenvalues 1. ↵ ± 11.17 Tangent vectors 473 Example 11.15 (A sphere). For the surface of a 2-sphere in R3, one might have ↵ =1, 2, 3 and i = ✓, . One then could use the 3 3 identity matrix ⇥ I to form inner products e e = e ↵ I e = e ↵ e ↵. (11.135) i · j i ↵ j i j Example 11.16 (Spacetime). If we use four coordinates to describe ordi- nary spacetime as a curved surface in a flat embedding space R8,thenwe would have ↵ =0, 1,...,7 and i =0, 1, 2, 3. We then could use an 8 8 ⇥ diagonal matrix I with entries I = = 1 to form inner products ↵ ± ↵ ± 7 e e = e ↵ I e . (11.136) i · j i ↵ j ↵X, =0 11.17 Tangent vectors Vectors defined by (11.133) as derivatives @p↵(x) e ↵(x) (11.137) i ⌘ @xi of physical points p with respect to the coordinates xi are tangent to the manifold at p. They are local coordinate basis vectors.Thesetan- gent vectors span a tangent space of as many dimensions as there are coordinates xi. In a di↵erent system of coordinates x0, the same displacement (11.133) is i dp = ei0 (x0) dx0 . The basis vectors ei(x) and ei0 (x0) @p @p ei(x)= i and ei0 (x0)= i . (11.138) @x @x0 are linearly related to each other and transform as covariant vectors @p @xj @p @xj ei0 (x0)= i = i j = i ej(x). (11.139) @x0 @x0 @x @x0 474 Tensors and Local Symmetries 11.18 The metric tensor Points are physical, coordinate systems metaphysical. So p, q, p q, and (p q) (p q) are all invariant quantities. When p and q = p + dp lie · infinitesimally close to each other on the spacetime manifold, the vector i dp = ei dx is the sum of the basis vectors multiplied by the changes in the coordinates xi. Both dp and the inner product dp dp are physical and so are · independent of the coordinates. The inner product e e of two coordinate i · j basis vectors is defined in terms of the metric the flat embedding space Rn, which is a diagonal matrix I with eigenvalues 1; it is a sum over the n ± values of ↵ and that label the coordinates of Rn n 1 e e = e ↵ I e . (11.140) i · j i ↵ j ↵X, =0 The squared separation dp2 is n 1 dp2 dp dp =(e dxi) (e dxj)= e↵ I e dxidxj. (11.141) ⌘ · i · j i ↵ j ↵X, =0 It is the same in any other coordinate system n 1 2 i j ↵ i j dp dp dp =(e0 dx0 ) (e0 dx0 )= e0 I e0 dx0 dx0 . (11.142) ⌘ · i · j i ↵ j ↵X, =0 This invariant squared separation dp2 defines a 4 4metricg on spacetime ⇥ ij dp2 = e e dxidxj = g dxidxj (11.143) i · j ij as the inner products n 1 g = e e = e ↵ I e . (11.144) ij i · j i ↵ j ↵X, =0 We normally will apply the summation convention (section 11.6) to these inner indices ↵ and . Since the basis vectors ei transform (11.139) as covariant vectors, the metric gij transforms as a covariant tensor k ` k ` ↵ ↵ @x @x @x @x 0 gij0 = ei0 I↵ ej = ei I↵ ej i j = gk` i j , (11.145) @x0 @x0 @x0 @x0 11.18 The metric tensor 475 as it must be since the squared separation dp2 is independent of our coordi- nates i j i j @x0 k @x0 ` k ` g0 dx0 dx0 = g0 dx dx = g dx dx . (11.146) ij ij @xk @x` k` Thus the metric tensor gij is a rank-2 covariant tensor. By its construc- tion (11.144), it also is a 4 4 real, symmetric matrix. ⇥ Example 11.17 (Graph paper). Imagine a piece of slightly crumpled graph paper with horizontal and vertical lines. The lines give us a two-dimensional coordinate system (x1,x2) that labels each point p(x) on the paper. The vectors e1(x)=@1p(x) and e2(x)=@2p(x) define how a point moves dp(x)= i 1 2 ei(x) dx when we change its coordinates by dx and dx . The vectors e1(x) and e2(x) span a di↵erent tangent space at the intersection of every horizon- tal line with every vertical line. Each tangent space is like the tiny square of the graph paper at that intersection. We can think of the two vectors ei(x) as three-component vectors in the three-dimensional embedding space we live in. The squared distance between any two nearby points separated by dp(x)isds2 dp2(x)=e2(x)(dx1)2 +2e (x) e (x) dx1dx2 + e2(x)(dx2)2 ⌘ 1 1 · 2 2 in which the inner products g = e (x) e (x) are defined by the euclidian ij i · j metric of the embedding euclidian space R3. Example 11.18 (The sphere S2). Let the point p be a euclidean 3-vector representing a point on the two-dimensional surface of a sphere of radius R. Spherical coordinates label the point p = R(sin ✓ cos , sin ✓ sin , cos ✓), the coordinate basis vectors are @p @p e = = R ✓ˆ and e = = R sin ✓ ˆ, (11.147) ✓ @✓ @ and the embedding metric is the 3 3 identity matrix I = .Theinner ⇥ ↵ ↵ products of the basis vectors are the components (11.144) of the sphere’s metric tensor g g e e e e R2 0 ✓✓ ✓ = ✓ · ✓ ✓ · = (11.148) g g e e e e 0 R2 sin2 ✓ ✓ ✓ ◆ ✓ · ✓ · ◆ ✓ ◆ which has determinant R4 sin2 ✓.Sincee e = 0, the squared distance is ✓ · ds2 = e e d✓2 + e e d 2 = R2d✓2 + R2 sin2 ✓d 2. (11.149) ✓ · ✓ · If we adopt a dimensionless scale factor a and set ar = R sin ✓, so that adr = R cos ✓d✓,then a2dr2 a2dr2 ds2 = + a2r2d 2 = + a2r2d 2, (11.150) cos2 ✓ 1 (a/R)2r2 476 Tensors and Local Symmetries and in terms of k = a2/R2 the r, metric is g g 1/(1 kr2)0 rr r = a2 . (11.151) g g 0 r2 ✓ r ◆ ✓ ◆ Example 11.19 (The hyperboloid H2). The hyperboloid H2 in R3 is the surface defined by R2 = x2 + y2 z2. If we label its points as p = R(sinh ✓ cos , sinh ✓ sin , cosh ✓), then its coordinate basis vectors are @p e = = R (cosh ✓ cos , cosh ✓ sin , sinh ✓) ✓ @✓ (11.152) @p e = = R ( sinh ✓ sin , sinh ✓ cos , 0). @ The embedding metric is I = diag(1, 1, 1), so z is a time coordinate. Since e e = 0, the squared distance between nearby points is ✓ · ds2 = e e d✓2 + e e d 2 = R2 d✓2 + R2 sinh2 ✓d 2. (11.153) ✓ · ✓ · In terms of the dimensionless scale factor a, the parameter k =(a/R)2, and the radial variable r = R sinh ✓/a, the squared distance ds2 is dr2 dr2 ds2 = a2 + a2r2d 2 = a2 + r2 d 2 . (11.154) cosh2 ✓ 1+kr2 ✓ ◆ Example 11.20 (The sphere S3). The sphere S3 is a 3-dimensional space in R4 defined by R2 = x2 + y2 + z2 + w2. If we label its points as p( ,✓, )=R(sin sin ✓ cos , sin sin ✓ sin , sin cos ✓, cos ), (11.155) then its coordinate basis vectors are @p e = = R(cos sin ✓ cos , cos sin ✓ sin , cos cos ✓, sin ) @ @p e = = R(sin cos ✓ cos , sin cos ✓ sin , sin sin ✓, 0) (11.156) ✓ @✓ @p e = = R( sin sin ✓ sin , sin sin ✓ cos , 0, 0). @ The inner metric of the embedding space is I =(1, 1, 1, 1), and in this metric the basis vectors are orthogonal. In terms of the dimensionless scale factor 11.18 The metric tensor 477 a, the parameter k =(a/R)2, and the radial variable r = R sin /a,the squared distance ds2 between two nearby points is ds2 = e e d 2 + e e d✓2 + e e d 2 · ✓ · ✓ · = R2 d 2 +sin2 d✓2 +sin2 sin2 ✓d 2 dr2 = a2 + r2d✓2 + r2 sin2 ✓d 2 (11.157) 1 sin2 ✓ ◆ dr2 = a2 + r2d✓2 + r2 sin2 ✓d 2 . 1 kr2 ✓ ◆ Example 11.21 (The hyperboloid H3). The hyperboloid H3 is a 3-space in R4 defined by R2 = x2 + y2 + z2 w2. If we label its points as p( ,✓, )=R (sinh sin ✓ cos , sinh sin ✓ sin , sinh cos ✓, cosh ), (11.158) then its coordinate basis vectors are @p e = = R(cosh sin ✓ cos , cosh sin ✓ sin , cosh cos ✓, sinh ) @ @p e = = R(sinh cos ✓ cos , sinh cos ✓ sin , sinh sin ✓, 0) ✓ @✓ @p e = = R( sinh sin ✓ sin , sinh sin ✓ cos , 0, 0). @ (11.159) The inner metric is I =(1, 1, 1, 1), and w is a time coordinate. The basis vectors are orthogonal. In terms of the dimensionless scale factor a,the parameter k =(a/R)2, and the radial variable r = R sinh /a, the squared distance ds2 between two nearby points is ds2 = e e d 2 + e e d✓2 + e e d 2 · ✓ · ✓ · = R2 d 2 +sinh2 d✓2 +sinh2 sin2 ✓d 2 dr2 = a2 + r2d✓2 + r2 sin2 ✓d 2 (11.160) 1+sinh2 ✓ ◆ dr2 = a2 + r2d✓2 + r2 sin2 ✓d 2 . 1+kr2 ✓ ◆ The squared distances (11.157 & 11.160) respectively are those of the closed and open Robinson-Walker metrics (11.454) for dx0 = 0. 478 Tensors and Local Symmetries 11.19 The principle of equivalence Since the metric tensor gij(x) is real and symmetric, it can be diagonalized at any point p(x)bya4 4 orthogonal matrix O(x) ⇥ e0 000 T k ` 0 e1 00 O i gk` O j = 0 1 (11.161) 00e2 0 B 000e C B 3C @ A which arranges the four real eigenvalues ei of the matrix gij(x) in the order e e e e . Thus the coordinate transformation 0 1 2 3 k T k @x O i i = (11.162) @x0 e | i| takes any spacetime metric gk`(x) with onep negative and three positive eigen- values into the Minkowski metric ⌘ij of flat spacetime 1000 @xk @x` 0100 gk`(x) i j = gij0 (x0)=⌘ij = 0 1 (11.163) @x0 @x0 0010 B 0001C B C @ A at the point p(x)=p(x0). The principle of equivalence says that in these free-fall coordinates x0, the physical laws of gravity-free special relativity apply in a suitably small region about the point p(x)=p(x0). It follows from this principle that the metric gij of spacetime accounts for all the e↵ects of gravity. 2 In the x0 coordinates, the invariant squared separation dp is that of spe- cial relativity 2 i j i j dp = g0 dx0 dx0 = e0 (x0) e0 (x0) dx0 dx0 ij i · j a b i j a b i j = ei0 (x0)⌘abej0 (x0) dx0 dx0 = i ⌘ab j dx0 dx0 (11.164) i j 2 0 2 2 = ⌘ dx0 dx0 =(dx0) (dx0 ) = ds . ij The x0 coordinates are not unique because every Lorentz transformation leaves the metric ⌘ invariant. Coordinate systems in which gij(x0)=⌘ij are called inertial coordinate systems. The congruency transformation (1.312 & 11.161–11.163) preserves the signs of the eigenvalues e which are the signature ( 1, 1, 1, 1) of the metric i tensor. 11.20 Tetrads 479 11.20 Tetrads We defined metric tensor (11.144) in terms of four n-vectors ei in the em- n T bedding space R as gij = ei ej = ei Iej in which I is the metric of the · embedding space which is a diagonal matrix (11.136) with n eigenvalues 1. ± If we instead invert the equation (11.163) that relates the metric tensor to the flat metric and write @x a @x b g = 0 ⌘ 0 , (11.165) ij @xi ab @xj then we will have expressed the metric in terms of four 4-vectors @x a c a(x)= 0 = ca (x)=ca(x) (11.166) i @xi i i as a b gij(x)=ci (x) ⌘ab c j(x) (11.167) in which ⌘ab is the 4 4 metric (11.163) of flat Minkowski space. Whether ⇥ a ´ the fundamental variables are the four 4-vectors ci (x)introducedbyElie Cartan (1869–1951) or the metric tensor gij is an open question. Cartan’s a four 4-vectors ci (x) are called a moving frame,atetrad, or a vierbein. 11.21 The Contravariant Metric Tensor ik The inverse g of the covariant metric tensor gkj satisfies ik i ik g0 gkj0 = j = g gkj (11.168) in all coordinate systems. To see how it transforms, we use the transforma- tion law (11.145) of gkj t u i ik ik @x @x j = g0 gkj0 = g0 k gtu j . (11.169) @x0 @x0 1 T 1 1T 1 1 In matrix notation, this is I = g0 HgH which implies g0 = H g H or in tensor notation i ` i` @x0 @x0 vw g0 = g . (11.170) @xv @xw 480 Tensors and Local Symmetries Thus the inverse gik of the covariant metric tensor is a second-rank con- travariant tensor called the contravariant metric tensor. 11.22 Dual vectors and the raising and lowering of indices The contraction of a contravariant vector Ai with any rank-2 covariant tensor gives a covariant vector, but we reserve the symbol Ai for the covariant vector that is the contraction of Aj with the metric tensor j Ai = gijA . (11.171) This operation is called lowering the index on Aj. Similarly, the contraction of a covariant vector Bj with any rank-2 con- travariant tensor is a contravariant vector, and we reserve the symbol Bi for contravariant vector that is the contraction i ij B = g Bj (11.172) of Bj with the inverse of the metric tensor. This is called raising the index on Bj. The vectors ei, for instance, are given by i ij e = g ej. (11.173) They are therefore orthonormal or dual to the basis vectors ei e ej = e gjke = gjke e = gjkg = gjkg = j (11.174) i · i · k i · k ik ki i with respect to the metric ⌘ab of the flat space tangent to the manifold. 11.23 Orthogonal coordinates in Rn In flat n-dimensional euclidian space, it is convenient to use orthogonal basis vectors and orthogonal coordinates. A change dxi in the coordi- nates moves the point p by (11.133) i dp = ei dx . (11.175) The metric gij is the inner product (11.144) g = e e . (11.176) ij i · j Since the vectors ei are orthogonal, the metric is diagonal g = e e = h2 . (11.177) ij i · j i ij 11.24 Polar Coordinates 481 The inverse metric ij 2 g = hi ij (11.178) raises indices. For instance, the dual vectors i ij 2 i i e = g e = h e satisfy e e = . (11.179) j i i · k k The invariant squared distance dp2 between nearby points (11.141) is dp2 = dp dp = g dxi dxj = h2 (dxi)2 (11.180) · ij i and the invariant volume element is dV = dnp = h ...h dx1 ... dxn = gdx1 ... dxn = gdnx (11.181) 1 n ^ ^ ^ ^ in which g = det gij is the square root of the positive determinant of gij. The important special case in which all the scale factors h are unity is p i cartesian coordinates in euclidean space (section 11.5). We also can use basis vectorse ˆi that are orthonormal. By (11.177 & 11.179), these vectors eˆ = e /h = h e i satisfye ˆ eˆ = . (11.182) i i i i i · j ij In terms of them, a physical and invariant vector V takes the form i i i 1 V = ei V = hi eˆi V = e Vi = hi eˆi Vi =ˆei V i (11.183) where i 1 V h V = h V (no sum). (11.184) i ⌘ i i i The dot-product is then V U = g V i U j = V U . (11.185) · ij i i In euclidian n-space, we even can choose coordinates xi so that the vectors i ei defined by dp = ei dx are orthonormal. The metric tensor is then the n n identity matrix g = e e = I = . But since this is euclidian ⇥ ik i · k ik ik n-space, we also can expand the n fixed orthonormal cartesian unit vectors `ˆ in terms of the e (x) which vary with the coordinates as `ˆ= e (x)(e (x) `ˆ). i i i · 11.24 Polar Coordinates In polar coordinates in flat 2-space, the change dp in a point p due to a change in its coordinates is dp =ˆrdr+ ✓ˆrd✓ so dp = er dr + e✓ d✓ with er =ˆer =ˆr and e✓ = r eˆ✓ = r ✓ˆ. The metric tensor for polar coordinates is 10 (g )=(e e )= . (11.186) ij i · j 0 r2 ✓ ◆ 482 Tensors and Local Symmetries r ✓ The contravariant basis vectors are e =ˆr and e =ˆe✓/r. A physical vector i i V is V = V ei = Vi e = V r rˆ + V ✓ ✓ˆ. 11.25 Cylindrical Coordinates For cylindrical coordinates in flat 3-space, the change dp in a point p due to a change in its coordinates is dp = ⇢ˆd⇢ + ˆ ⇢d + zˆdz = e⇢ d⇢ + e d + ez dz (11.187) with e⇢ = eˆ⇢ = ⇢ˆ, e = ⇢ eˆ = ⇢ ˆ, and ez = eˆz = zˆ. The metric tensor for cylindrical coordinates is 100 (g )=(e e )= 0 ⇢2 0 (11.188) ij i · j 0 1 001 @ A with determinant det g g = ⇢2. The invariant volume element is ij ⌘ dV = ⇢dx1 dx2 dx3 = pgd⇢d dz= ⇢d⇢d dz. (11.189) ^ ^ ⇢ z The contravariant basis vectors are e = ⇢ˆ, e = eˆ /⇢, and e = zˆ.A physical vector V is i i V = V ei = Vi e = V ⇢ ⇢ˆ + V ˆ + V z zˆ. (11.190) Incidentally, since p =(⇢ cos ,⇢ sin , z) , (11.191) the formulas for the basis vectors of cylindrical coordinates in terms of those of rectangular coordinates are (exercise 11.14) ⇢ˆ = cos xˆ +sin yˆ ˆ = sin xˆ + cos yˆ zˆ = zˆ. (11.192) 11.26 Spherical Coordinates For spherical coordinates in flat 3-space, the change dp in a point p due to a change in its coordinates is dp = rˆdr + ✓ˆrd✓+ ˆ r sin ✓d = er dr + e✓ d✓ + e d (11.193) 11.27 The Gradient of a Scalar Field 483 so er = rˆ, e✓ = r ✓ˆ, and e = r sin ✓ ˆ or er =(sin✓ cos , sin ✓ sin , cos ✓) e = r (cos ✓ cos , cos ✓ sin , sin ✓) (11.194) ✓ e = r sin ✓ ( sin , cos , 0) . The metric tensor for spherical coordinates is 10 0 (g )=(e e )= 0 r2 0 (11.195) ij i · j 0 1 00r2 sin2 ✓ @ A with determinant det g g = r4 sin2 ✓. The invariant volume element is ij ⌘ dV = r2 sin2 ✓dx1 dx2 dx3 = pgdrd✓d = r2 sin ✓drd✓d . (11.196) ^ ^ The orthonormal basis vectors are eˆr = rˆ, eˆ✓ = ✓ˆ, and eˆ = ˆ.The contravariant basis vectors are e r = rˆ, e ✓ = ✓ˆ/r, e = ˆ/r sin ✓. A physical vector V is i i V = V ei = Vi e = V r rˆ + V ✓ ✓ˆ + V ˆ. (11.197) Incidentally, since p =(r sin ✓ cos , r sin ✓ sin , r cos ✓) , (11.198) the formulas for the basis vectors of spherical coordinates in terms of those of rectangular coordinates are (exercise 11.15) er =sin✓ cos xˆ +sin✓ sin yˆ + cos ✓ zˆ e = r (cos ✓ cos xˆ + cos ✓ sin yˆ sin ✓ zˆ) (11.199) ✓ e = r sin ✓ ( sin xˆ + cos yˆ) . 11.27 The Gradient of a Scalar Field If f(x) is a scalar field, then the di↵erence between it and f(x + dx)defines the gradient f as (6.26) r @f(x) df (x)=f(x + dx) f(x)= dxi = f(x) dp. (11.200) @xi r · j Since dp = ej dx ,theinvariant form i @f eˆi @f f = e i = i (11.201) r @x hi @x 484 Tensors and Local Symmetries satisfies this definition (11.200) of the gradient @f @f @f f dp = ei e dxj = i dxj = dxi = df . (11.202) r · @xi · j @xi j @xi In two polar coordinates, the gradient is ˆ i @f eˆi @f @f ✓ @f f = e i = i =ˆr + . (11.203) r @x hi @x @r r @✓ In three cylindrical coordinates, it is (6.27) i @f eˆi @f @f 1 @f ˆ @f f = e i = i = ⇢ˆ + + zˆ (11.204) r @x hi @x @⇢ ⇢ @ @z and in three spherical coordinates it is (6.28) @f i eˆi @f @f 1 @f ˆ 1 @f ˆ f = i e = i = rˆ + ✓ + . (11.205) r @x hi @x @r r @✓ r sin ✓ @ 11.28 Levi-Civita’s Tensor In 3 dimensions, Levi-Civita’s symbol ✏ ✏ijk is totally antisymmetric ijk ⌘ with ✏123 = 1 in all coordinate systems. We can turn his symbol into something that transforms as a tensor by multiplying it by the square root of the determinant of a rank-2 covariant tensor. A natural choice is the metric tensor. Thus the Levi-Civita tensor ⌘ijk is the totally antisymmetric rank-3 covariant (pseudo)tensor ⌘ijk = pg✏ijk (11.206) in which g = det g is the absolute value of the determinant of the metric | mn| tensor gmn. The determinant’s definition (1.184) and product rule (1.207) imply that Levi-Civita’s tensor ⌘ijk transforms as @xt @xu ⌘0 = g ✏0 = g ✏ = det g ✏ ijk 0 ijk 0 ijk @x m @x n tu ijk s ✓ 0 0 ◆ p p @xt @x u = det det det (g ) ✏ @x m @x n tu ijk s ✓ 0 ◆ ✓ 0 ◆ @x @x = det pg✏ = det p g✏ @x ijk @x ijk ✓ 0 ◆ ✓ 0 ◆ t u v t u v @x @x @x @x @x @x = i j k pg✏tuv = i j k ⌘tuv (11.207) @x0 @x0 @x0 @x0 @x0 @x0 11.29 The Hodge Star 485 in which is the sign of the Jacobian det(@x/@x0). Levi-Civita’s tensor is a pseudotensor because it doesn’t change sign under the parity transforma- tion x i = xi. 0 We get ⌘ with upper indices by using the inverse gnm of the metric tensor ijk it ju kv it ju kv mn ⌘ = g g g ⌘tuv = g g g pg✏tuv = pg✏ijk det(g ) (11.208) ijk = pg✏ijk/ det(gmn)=s✏ijk/pg = s✏ /pg in which s is the sign of the determinant det gij = sg. Similarly in 4 dimensions, Levi-Civita’s symbol ✏ ✏ijk` is totally ijk` ⌘ antisymmetric with ✏0123 = 1 in all coordinate systems. No meaning attaches to whether the indices of the Levi-Civita symbol are up or down; some authors even use the notation ✏(ijk`) or ✏[ijk`] to emphasize this fact. In 4 dimensions, the Levi-Civita pseudotensor is ⌘ijk` = pg✏ijk` (11.209) where again g = det g . The determinant’s definition (1.184) and product | ij| rule (1.207) imply that it transforms as @x @x ⌘0 = g ✏ = det pg✏ = det pg✏ ijk` 0 ijk` @x ijk` @x ijk` ✓ 0 ◆ ✓ 0 ◆ p t u v w t u v w @x @x @x @x @x @x @x @x = i j k ` p g✏tuvw = i j k ` ⌘tuvw @x0 @x0 @x0 @x0 @x0 @x0 @x0 @x0 (11.210) where is the sign of the Jacobian det(@x/@x0). Raising the indices on ⌘ with det gij = sg we have ijk` it ju kv `w it ju kv `w mn ⌘ = g g g g ⌘tuvw = g g g g pg✏tuvw = pg✏ijk` det(g ) = pg✏ / det(g )=s✏ /pg s✏ijk`/pg. ijk` mn ijk` ⌘ (11.211) In n dimensions, one may define Levi-Civita’s symbol ✏(i1 ...in) as to- tally antisymmetric with ✏(1 ...n) = 1 and his pseudotensor as ⌘i1...in = pg✏(i1 ...in). 11.29 The Hodge Star This section is optional on a first reading. In 3 cartesian coordinates, the Hodge dual turns 1-forms into 2-forms dx = dy dz dy = dz dx dz = dx dy (11.212) ⇤ ^ ⇤ ^ ⇤ ^ 486 Tensors and Local Symmetries and 2-forms into 1-forms (dx dy)=dz (dy dz)=dx (dz dx)=dy. (11.213) ⇤ ^ ⇤ ^ ⇤ ^ It also maps the 0-form 1 and the volume 3-form into each other 1=dx dy dz (dx dy dz) = 1 (11.214) ⇤ ^ ^ ⇤ ^ ^ (William Vallance Douglas Hodge, 1903–1975). More generally in 3-space, we define the Hodge dual, also called the Hodge star, as 1 ` j k ` j k `t ju kv 1= ⌘`jkdx dx dx (dx dx dx )=g g g ⌘tuv ⇤ 3! ^ ^ ⇤ ^ ^ (11.215) 1 dxi = gi` ⌘ dxj dxk (dxi dxj)=gik gj` ⌘ dxm ⇤ 2 `jk ^ ⇤ ^ k`m and so if the sign of det g is s = +1, then 1 = 1, dxi = dxi, ij ⇤⇤ ⇤⇤ (dxi dxk)=dxi dxk, and (dxi dxj dxk)=dxi dxj dxk. ⇤⇤ ^ ^ ⇤⇤ ^ ^ ^ ^ Example 11.22 (Divergence and Laplacian). The dual of the 1-form @f @f @f df = dx + dy + dz (11.216) @x @y @z is the 2-form @f @f @f df = dy dz + dz dx + dx dy (11.217) ⇤ @x ^ @y ^ @z ^ and its exterior derivative is the laplacian @2f @2f @2f d df = + + dx dy dz (11.218) ⇤ @x2 @y2 @z2 ^ ^ ✓ ◆ multiplied by the volume 3-form. Similarly, the dual of the one form A = Ax dx + Ay dy + Az dz (11.219) is the 2-form A = A dy dz + A dz dx + A dx dy (11.220) ⇤ x ^ y ^ z ^ and its exterior derivative is the divergence @A @A @A d A = x + y + z dx dy dz (11.221) ⇤ @x @y @z ^ ^ ✓ ◆ times dx dy dz. ^ ^ 11.29 The Hodge Star 487 In flat Minkowski 4-space with c = 1, the Hodge dual turns 1-forms into 3-forms dt = dx dy dz dx = dy dz dt ⇤ ^ ^ ⇤ ^ ^ (11.222) dy = dz dx dt dz = dx dy dt ⇤ ^ ^ ⇤ ^ ^ 2-forms into 2-forms (dx dt)=dy dz (dx dy)= dz dt ⇤ ^ ^ ⇤ ^ ^ (dy dt)=dz dx (dy dz)= dx dt (11.223) ⇤ ^ ^ ⇤ ^ ^ (dz dt)=dx dy (dz dx)= dy dt ⇤ ^ ^ ⇤ ^ ^ 3-forms into 1-forms (dx dy dz)= dt (dy dz dt)= dx ⇤ ^ ^ ⇤ ^ ^ (11.224) (dz dx dt)= dy (dx dy dt)= dz ⇤ ^ ^ ⇤ ^ ^ and interchanges 0-forms and 4-forms 1=dt dx dy dz (dt dx dy dz)= 1. (11.225) ⇤ ^ ^ ^ ⇤ ^ ^ ^ More generally in 4 dimensions, we define the Hodge star as 1 1= ⌘ dxk dx` dxm dxn ⇤ 4! k`mn ^ ^ ^ 1 dxi = gik ⌘ dx` dxm dxn ⇤ 3! k`mn ^ ^ 1 (dxi dxj)= gik gj` ⌘ dxm dxn (11.226) ⇤ ^ 2 k`mn ^ (dxi dxj dxk)=git gju gkv ⌘ dxw ⇤ ^ ^ tuvw dxi dxj dxk dx` = git gju gkv g`w⌘ = ⌘ijk`. ⇤ ^ ^ ^ tuvw ⇣ ⌘ Thus (exercise 11.17) if the determinant det gij of the metric is negative, then dxi = dxi (dxi dxj)= dxi dxj ⇤⇤ ⇤⇤ ^ ^ (11.227) (dxi dxj dxk)=dxi dxj dxk 1= 1. ⇤⇤ ^ ^ ^ ^ ⇤⇤ In n dimensions, the Hodge star turns p-forms into n p-forms ⌘ i1 ip i1k1 ipkp k1...kp`1...`n p `1 `n p dx ... dx = g ...g dx ... dx . ⇤ ^ ^ (n p)! ^ ^ (11.228) 488 Tensors and Local Symmetries Example 11.23 (The Inhomogeneous Maxwell Equations). The homoge- neous Maxwell equations are dF = ddA =0. (11.229) To get the inhomogeneous Maxwell equations, we first form the dual F = ⇤ dA ⇤ F = 1 F dxi dxj = 1 F gikgj`⌘ dxm dxn = 1 F k`⌘ dxm dxn, ⇤ 2 ij ⇤ ^ 4 ij k`mn ^ 4 k`mn ^ and then apply the exterior derivative d F = 1 d F k`⌘ dxm dxn = 1 @ F k`⌘ dxp dxm dxn. ⇤ 4 k`mn ^ 4 p k`mn ^ ^ ⇣ ⌘ k ⇣ ⌘ To get back to a 1-form like j = jk dx , we apply a second Hodge star d F = 1 @ F k` ⌘ (dxp dxm dxn) ⇤ ⇤ 4 p k`mn ⇤ ^ ^ 1 ⇣ k` ⌘ ps mt nu v = 4 @p F ⌘k`mn g g g ⌘stuv dx 1 ⇣ k` ⌘ ps mt nu v = 4 @p pgF ✏k`mn g g g pg✏stuv dx (11.230) 1 ⇣ k`⌘ ps mt nu wv = 4 @p pgF ✏k`mn g g g g ✏stuv pgdxw ⇣ ⌘ 1 k` pg = 4 @p pgF ✏k`mn ✏pmnw dxw det gij ⇣ ⌘ s k` = @p pgF ✏k`mn ✏pmnw dxw 4pg ⇣ ⌘ in which we used the definition (1.184) of the determinant. Levi-Civita’s 4-symbol obeys the identity (exercise 11.18) ✏ ✏pwmn =2 p w w p . (11.231) k`mn k ` k ` Applying it to d F , we get ⇤ ⇤ s k` p w w p s kp d F = @p pgF dxw = @p pgF dxk. ⇤ ⇤ 2pg k ` k ` pg ⇣ ⌘ ⇣ ⌘ In our spacetime s = 1. Setting d F equal to j = j dxk = jk dx ⇤ ⇤ k k multiplied by the permeability µ0 of the vacuum, we arrive at expressions for the microscopic inhomogeneous Maxwell equations in terms of both tensors and forms 1 kp k @p pgF = µ0 j and d F = µ0 j. (11.232) pg ⇤ ⇤ ⇣ ⌘ They and the homogeneous Bianchi identity (11.92, 11.113, & 11.294) ijk` ✏ @`Fjk = dF = ddA= 0 (11.233) 11.30 Notations for Derivatives 489 are invariant under general coordinate transformations. 11.30 Notations for Derivatives We have various notations for derivatives. We can use the variables x, y, and so forth as subscripts to label derivatives @f @f f = @ f = and f = @ f = . (11.234) x x @x y y @y If we use indices to label variables, then we can use commas @f @2f f = @ f = and f = @ @ f = (11.235) ,i i @xi ,ik k i @xk@xi k and f,k0 = @f/@x0 . In the next section, we will use a semicolon to mean a covariant derivative. 11.31 Covariant derivatives and a ne connections If F (x) is a vector field, then its invariant description in terms of spacetime- dependent basis vectors ei(x)is i F (x)=F (x) ei(x). (11.236) Since the basis vectors ei(x) vary with x, the derivative of F (x) contains two terms @F @Fi @e = e + F i i . (11.237) @x` @x` i @x` In general, the derivative of a vector ei is not a linear combination of the basis vectors ek. For instance, on the 2-dimensional surface of a sphere in 3-dimensions, the derivative @e ✓ = rˆ (11.238) @✓ points to the sphere’s center and isn’t a linear combination of e✓ and e . ` k The inner product of a derivative @ei/@x with a dual basis vector e is the Levi-Civita a ne connection @e k = ek e = ek i (11.239) i` · i,` · @x` 490 Tensors and Local Symmetries which relates spaces that are tangent to the manifold at infinitesimally sep- arated points. It is called an a ne connection because the di↵erent tangent spaces lack a common origin. In terms of the a ne connection (11.239 ), the inner product of the deriva- tive (11.237) with ek is @F @Fi @e @Fk ek = ek e + F i ek i = + F i k (11.240) · @x` · @x` i · @x` @x` i` a combination that is called a covariant derivative (section 11.34) @Fk @F D F k F k + F i k F k = ek . (11.241) ` ⌘r` ⌘ @x` i` ⌘ ;` · @x` It is a second-rank mixed tensor. The covariant derivative of the scalar F is @Fk D F = e ek @ F = + F i k e = F k e . (11.242) ` k · ` @x` i` k ;` k ✓ ◆ k Some physicists write the a ne connection i` as k = k (11.243) i` i` ⇢ and call it a Christo↵el symbol of the second kind. The coordinate basis vectors ei are the spacetime derivatives (11.138) of the point p, and so the a ne connection (11.239) is a double derivative of p @e @2p @2p @e k = ek i = ek = ek = ek ` = k (11.244) i` · @x` · @x`@xi · @xi@x` · @xi `i and thus is symmetric in its two lower indices k k i` = `i. (11.245) A ne connections are not tensors. Tensors transform homogeneously; k connections transform inhomogeneously. The connection i` transforms as k m n k k @ei0 @x0 p @x @ @x 0 = e0 = e e i` · @x ` @xp · @x ` @xm @x i n 0 0 ✓ 0 ◆ k m n k 2 n @x0 @x @x p @en @x0 @ x = p ` i e m + n ` i (11.246) @x @x0 @x0 · @x @x @x0 @x0 k m n k 2 n @x0 @x @x p @x0 @ x = p ` i nm + n ` i . @x @x0 @x0 @x @x0 @x0 11.32 Torsion 491 k Although the connection i` is not a tensor, its variation k = k (g + g ) k (g ) (11.247) i` i` nm nm i` nm is a tensor because the inhomogeneous term in this equation (11.246) cancels k in the di↵erence i`. The electromagnetic field Ai(x) and other gauge fields are connections. k Since the Levi-Civita connection i` is symmetric in i and `, in four- dimensional spacetime, there are 10 of them for each k, or 40 in all. The 10 correspond to 3 rotations, 3 boosts, and 4 translations. 11.32 Torsion We can use Cartan’s tetrads to define a more general covariant derivative. k The quantity F = F ck is a scalar under general coordinate transformations. a k a It has one flat-space index F = F ck which participates under Lorentz transformations, but we’ll ignore that for the moment. Its k-derivative is a k a k a F,i = F,i ck + F ck,i. (11.248) a k The four 4-vectors ck have four dual vectors ca that obey the rules a ` ` a k a ck ca = k and ck cb = b . (11.249) The inner product of this equation with the dual vector c` a ` k a ` k ` a k ` k ` a F,i ca = F,i ck ca + F ca ck,i = F,i k + F ca ck,i (11.250) = F ` + F k c` ca F ` + F k !` F ` ,i a k,i ⌘ ,i ki ⌘ ;i ` k is Cartan’s covariant derivative F;i of the contravariant vector F .Itisa ` mixed tensor F;i defined in terms of a more general connection ` ` a ! ki = ca ck,i (11.251) ` ` that, unlike the Levi-Civita connection ki = ik, is not neccessarily sym- metric in its two lower indices k and i. The antisymmetric part of the Cartan connection is the torsion tensor 1 T ` !` !` (11.252) ki ⌘ 2 ki ik ⇣ ⌘ which is a tensor. 492 Tensors and Local Symmetries 11.33 Parallel Transport The movement of a vector along a curve on a manifold so that its length and direction in successive tangent spaces do not change is called parallel transport. If the vector is F = F ie , then we want the inner product ek dF i · of dF with all dual tangent vectors ek to vanish along the curve. But this is just the condition that the covariant derivative (11.241) of F should vanish along the curve @F @Fk ek = + F i k = D F k =0. (11.253) · @x` @x` i` ` Example 11.24 (Parallel Transport on a Sphere). We parallel-transport the vector v =(0, 1, 0) up from the equator along the line of longitude = 0. Along this path, the vector v =(0, 1, 0) is constant, so @✓v = 0 and k D✓v = 0. Then we parallel-transport it down from the north pole along the line of longitude = ⇡/2 to the equator. Along this path, = ⇡/2, and the vector v = e /r =(0, cos ✓, sin ✓) obeys the parallel-transport condition ✓ (11.253) because its ✓-derivative @v 1 @e @ = ✓ = (0, cos ✓, sin ✓)=(0, sin ✓, cos ✓)= rˆ @✓ r @✓ @✓ | =⇡/2 (11.254) is perpendicular to the tangent vectors e✓ and e along the curve = ⇡/2. We then parallel-transport v along the equator back to the starting point = 0. Along this path, the vector v =(0, 0, 1) is constant, so @ v =0 and D vk = 0. The change from v =(0, 1, 0) to v =(0, 0, 1) is due to the curvature of the sphere. 11.34 Covariant Derivatives i In comma notation, the derivative of a contravariant vector field F = F ei is i i F,` = F,` ei + F ei,` (11.255) which in general lies outside the space spanned by the basis vectors ei.So we use the a ne connections (11.239) to form the inner product ek F = ek F i e + F ie = F i k + F i k = F k + F i k . (11.256) · ,` · ,` i i,` ,` i i` ,` i` This covariant derivative of a contravariant vector field often is writ- ten with a semicolon F k = ek F = F k + F i k . (11.257) ;` · ,` ,` i` 11.34 Covariant Derivatives 493 It transforms as a mixed second-rank tensor. The invariant change dF projected onto ek is ek dF = ek F dx` = F k dx`. (11.258) · · ,` ;` In terms of its covariant components, the derivative of a vector V is k k k V,` =(Vk e ),` = Vk,` e + Vk e,`. (11.259) i k To relate the derivatives of the vectors e to the a ne connections i`,we di↵erentiate the orthonormality relation k = ek e (11.260) i · i which gives us 0=ek e + ek e or ek e = ek e = k . (11.261) ,` · i · i,` ,` · i · i,` i` Since e ek = k , the inner product of e with the derivative of V is i · ,` i` i e V = e V ek + V ek = V V k . (11.262) i · ,` i · k,` k ,` i,` k i` This covariant derivative⇣ of a covariant⌘ vector field often is written with a semicolon V = e V = V V k . (11.263) i;` i · ,` i,` k i` It transforms as a rank-2 covariant tensor. Note the minus sign in Vi;` and the plus sign in F k. The change e dV is ;` i · e dV = e V dx` = V dx`. (11.264) i · i · ,` i;` ` Since dV is invariant, ei covariant, and dx contravariant, the quotient rule (section 11.15) confirms that the covariant derivative Vi;` of a covariant vector Vi is a rank-2 covariant tensor. We used similar logic to derive Cartan’s covariant derivative (11.250) of a contravariant vector k k i k F;` = F,` + F ! i`. (11.265) To find his formula for the covariant derivative of a covariant vector, we first k di↵erentiate the scalar V = Vk c k k Va,` = Vk,` ca + Vk c a,` (11.266) a and then use (11.249) to contract it with the dual vector ci a k a a k k a k ci Va,` = Vk,` ca ci + Vk ci c a,` = Vk,` i + Vk ci c a,` (11.267) a k = Vi,` + Vk ci c a,`. 494 Tensors and Local Symmetries a k To evaluate ci c a,`, we di↵erentiate the first of the dual-vector relations (11.249) to show that ca ck = ck ca . (11.268) i a,` a i,` Cartan’s covariant derivative of a covariant vector then is V = V V ck ca = V V !k . (11.269) i;` i,` k a i,` i,` k i` 11.35 The Covariant Curl k Because the connection i` is symmetric (11.245) in its lower indices, the covariant curl of a covariant vector Vi is simply its ordinary curl V V = V V k V + V k = V V . (11.270) `;i i;` `,i k `i i,` k i` `,i i,` Thus the Faraday field-strength tensor Fi` which is defined as the curl of the covariant vector field Ai F = A A (11.271) i` `,i i,` is a generally covariant second-rank tensor. If however we must use Cartan’s covariant derivative (11.250), then we would define the Faraday field-strength tensor Fi` as F = A A = A A + A !k !k = A A +2A T k i` `;i i;` `,i i,` k i` `i `,i i,` k i` ⇣ ⌘ (11.272) k in which T i` is the torsion tensor (11.252). Example 11.25 (Orthogonal coordinates). In orthogonal coordinates, the curl is defined (6.39, 11.110) in terms of the totally antisymmetric Levi- ijk 123 Civita symbol ✏ (with ✏123 = ✏ = 1), as 3 3 1 ijk V = ( V )i eˆi = ei ✏ Vk;j (11.273) r⇥ r⇥ h1h2h3 Xi=1 ijkX=1 which, in view of (11.270) and the antisymmetry of ✏ijk,is 3 3 1 ijk V = ( V )i eˆi = ei ✏ Vk,j (11.274) r⇥ r⇥ hihjhk Xi=1 ijkX=1 or by (11.182 & 11.184) 3 3 1 ijk 1 ijk V = hieˆi ✏ Vk,j = hieˆi ✏ (hkV k),j . r⇥ hihjhk hihjhk ijkX=1 ijkX=1 (11.275) 11.36 Covariant derivative of a tensor 495 Often one writes this as a determinant e e e h eˆ h eˆ h eˆ 1 1 2 3 1 1 1 2 2 3 3 V = @ @ @ = @ @ @ . r⇥ h h h 1 2 3 h h h 1 2 3 1 2 3 V V V 1 2 3 h V h V h V 1 2 3 1 1 2 2 3 3 (11.276) In cylindrical coordinates, the curl is ⇢⇢ˆ ˆ zˆ 1 V = @ @ @ . (11.277) r⇥ ⇢ ⇢ z V ⇢V V ⇢ z In spherical coordinates, it is rrˆ ✓ˆ rsin ✓ ˆ 1 V = @ @ @ . (11.278) r⇥ r2 sin ✓ r ✓ V rV r sin ✓ V r ✓ In more formal language, the curl is 1 dV = d V dxk = V dxi dxk = (V V ) dxi dxk. (11.279) k k,i ^ 2 k,i i,k ^ ⇣ ⌘ 11.36 Covariant derivative of a tensor Let’s write a second-rank contravariant tensor in a coordinate-invariant way as a sum of direct products of tangent vectors ei T = T jm e e . (11.280) j ⌦ m Its derivative is T, = T jm, e e + T jm e e + T jm e e . (11.281) k k j ⌦ m j,k ⌦ m j ⌦ m,k i` The covariant derivative T ;k of the tensor is the inner product of this derivative with the contravariant or dual tangent vectors ei e` ⌦ T i`; = ei e` T, k ⌦ · k = T jm, ei e e` e + T jm ei e e` e + T jm ei e e` e k · j · m · j,k · m · j · m,k = T jm, i ` + T jm ei e ` + T jm i e` e (11.282) k j m · j,k m j · m,k = T i`, +T j` ei e + T im e` e . k · j,k · m,k 496 Tensors and Local Symmetries In it we recognize two instances of our formula i = ei e for the a ne jk · j,k connection (11.239). Replacing these inner products with a ne connections, i` we may write the covariant derivative T;k of the tensor as i` i` j` i im ` T;k = T,k + T jk + T mk. (11.283) In particular, when T i` = F i G`, this formula says that covariant derivatives, like ordinary derivatives, obey the Leibniz rule i ` i ` i ` (F G );k = F;k G + F G;k. (11.284) Covariant derivatives, like ordinary derivatives, are derivations (AB);k = A;k B + AB;k. (11.285) A similar argument leads to a similar formula for the covariant derivative of a covariant tensor. A coordinate-independent form of a covariant tensor Tjm is T = T ej em. (11.286) jm ⌦ Its derivative is T, = T , ej em + T ej, em + T ej em, . (11.287) k jm k ⌦ jm k ⌦ jm ⌦ k The covariant derivative Ti`;k of the tensor is the inner product of this deriva- tive with the covariant tangent vectors e e i ⌦ ` Ti`;k = ei e` T,k ⌦ · (11.288) = T + T e ej, +T e em, . i`,k j` i · k im ` · k In it we see two instances of our other formula e ek = k for the i · ,` i` a ne connection (11.261). Replacing them, we may write we may write the covariant derivative Ti`;k of the tensor as T = T T j T m . (11.289) i`;k i`,k j` ik im `k The rule for a general tensor is to treat every contravariant index as in (11.283) and every covariant index as in (11.289). The covariant derivative of a mixed rank-4 tensor, for instance, is T ab = T ab + T jb a + T am b T ab j T ab m . (11.290) xy;k xy,k xy jk xy mk jy xk xm yk 11.37 Covariant Derivatives and Antisymmetry 497 11.37 Covariant Derivatives and Antisymmetry Let us apply our rule (11.289) for the covariant derivative of a second-rank tensor Ai` A = A A m A m (11.291) i`;k i`,k m` ik im `k to an antisymmetric tensor A = A . (11.292) i` `i Then by adding together the three cyclic permutations of the indices i`k we find that the antisymmetry of the tensor and the symmetry (11.245) of the m m a ne connection ik = ki conspire to cancel the quadratic terms leaving us with A + A + A = A A m A m i`;k ki;` `k;i i`,k m` ik im `k + A A m A m ki,` mi k` km i` + A A m A m `k,i mk `i `m ki = Ai`,k + Aki,` + A`k,i (11.293) an identity named after Luigi Bianchi (1856–1928). The Maxwell field-strength tensor Fi` is antisymmetric by construction (F = A A ), and so the Maxwell’s homogeneous equations i` `,i i,` 1 ijk` ✏ Fjk,` = Fjk,` + Fk`,j + F`j,k 2 (11.294) = A A + A A + A A =0 k,j` j,k` `,kj k,`j j,`k `,jk are tensor equations valid in all coordinate systems. This remains true even if we use Cartan’s covariant derivative (11.250) to define the Cartan-Faraday tensor (11.272). It is amazing how right Maxwell was in the middle of the nineteenth century. 11.38 A ne Connection and Metric Tensor m To relate the a ne connection `i to the derivatives of the metric tensor gk`, we lower the contravariant index m to get m m k`i = gkm `i = gkm i` = ki` (11.295) which is symmetric in its last two indices and which some call a Christo↵el symbol of the first kind,written[`i, k]. One can raise the index k back up by using the inverse of the metric tensor mk mk n m n m g k`i = g gkn `i = n `i = `i . (11.296) 498 Tensors and Local Symmetries m Although we can raise and lower these indices, the connections `i and k`i are not tensors. The definition (11.239) of the a ne connection tells us that = g m = g em e = e e = = e e . (11.297) k`i km `i km · `,i k · `,i ki` k · i,` By di↵erentiating the definition g = e e of the metric tensor, we find i` i · ` g = e e + e e = e e + e e = + . (11.298) i`,k i,k · ` i · `,k ` · i,k i · `,k `ik i`k Permuting the indices cyclicly, we have gki,` = ik` + ki` (11.299) g`k,i = k`i + `ki. If we now subtract relation (11.298) from the sum of the two formulas (11.299) keeping in mind the symmetry abc = acb, then we find that four of the six terms cancel g + g g = + + + = 2 (11.300) ki,` `k,i i`,k ik` ki` k`i `ki `ik i`k k`i leaving a formula for k`i = 1 (g + g g ) . (11.301) k`i 2 ki,` `k,i i`,k Thus the connection is three derivatives of the metric tensor s = gsk = 1 gsk (g + g g ) . (11.302) i` k`i 2 ki,` `k,i i`,k 11.39 Covariant Derivative of the Metric Tensor Let us apply our formula (11.289) for the covariant derivative of a covariant tensor to the metric tensor gi` g g g m g n . (11.303) i`;k ⌘ i`,k m` ik in `k l n If we now substitute our formula (11.302) for the connections ik and `k g = g g 1 gms (g + g g ) g 1 gns (g + g g ) i`;k i`,k m` 2 is,k sk,i ik,s in 2 s`,k ks,` `k,s (11.304) `k and use the fact (11.168) that the metric tensors gi` and g are mutually inverse and symmetric, then we find 1 s 1 s gi`;k = gi`,k (gis,k + gsk,i gik,s) (gs`,k + gks,` g`k,s) 2 ` 2 i (11.305) = g 1 (g + g g ) 1 (g + g g )=0. i`,k 2 i`,k `k,i ik,` 2 i`,k ki,` `k,i 11.40 Divergence of a contravariant vector 499 The covariant derivative of the metric tensor vanishes. This result follows from our choice of the Levi-Civita connection (11.239); it is not true for some other connections. Covariant derivatives obey the Leibniz rule (11.284), so a ac ac ac ac b;` =(g gcb);` = g;` gcb + g gcb;` = g;` gcb (11.306) = a i g k a g = a g a g =0. i` b cb b` k cb b` cb b` cb ac The covariant derivative of the inverse metric tensor vanishes, g;` = 0. 11.40 Divergence of a contravariant vector The contracted covariant derivative of a contravariant vector is a scalar known as the divergence, V = V i = V i + V k i . (11.307) r· ;i ,i ki Because gik = gki, in the sum (11.302) over i i = 1 gi` (g + g g ) (11.308) ki 2 i`,k `k,i ki,` the last two terms cancel because they di↵er only by the interchange of the dummy indices i and ` i` `i i` g g`k,i = g gik,` = g gki,`. (11.309) So the contracted connection collapses to i 1 i` ki = 2 g gi`,k. (11.310) There is a nice formula for this last expression. To derive it, let g g be ⌘ i` the 4 4 matrix whose elements are those of the covariant metric tensor g . ⇥ i` Its determinant, like that of any matrix, is the cofactor sum (1.195) along any row or column, that is, over ` for fixed i or over i for fixed ` det(g)= gi` Ci` (11.311) iXor ` in which the cofactor C is ( 1)i+` times the determinant of the reduced i` matrix consisting of the matrix g with row i and column ` omitted. Thus the partial derivative of det g with respect to the i`th element gi` is @ det(g) = Ci` (11.312) @gi` in which we allow gi` and g`i to be independent variables for the purposes of this di↵erentiation. The inverse gi` of the metric tensor g,liketheinverse 500 Tensors and Local Symmetries (1.197) of any matrix, is the transpose of the cofactor matrix divided by its determinant det(g) C 1 @ det(g) gi` = `i = . (11.313) det(g) det(g) @g`i Using this formula and the chain rule, we may write the derivative of the determinant det(g) as @ det(g) `i det(g),k = gi`,k =det(g) g gi`,k (11.314) @gi` and so since gi` = g`i, the contracted connection (11.310) is i 1 i` det(g),k det(g) ,k g,k (pg),k = g gi`,k = = | | = = (11.315) ki 2 2det(g) 2 det(g) 2g pg | | in which g det(g) is the absolute value of the determinant of the metric ⌘ tensor. Thus from (11.307 & 11.315), we arrive at our formula for the covariant divergence of a contravariant vector: (pg) (pgVk) V = V i = V i + i V k = V k + ,k V k = ,k . (11.316) r· ;i ,i ki ,k pg pg Example 11.26 (Maxwell’s inhomogeneous equations). An important ap- plication of this divergence formula (11.316) is the generally covariant form (11.232) of Maxwell’s inhomogeneous equations 1 k` k pgF = µ0j . (11.317) pg ,` ⇣ ⌘ Example 11.27 (Energy-momentum tensor). Another application is to the divergence of the symmetric energy-momentum tensor T ij = T ji ij ij i kj j im T;i = T,i + ki T + mi T (pgT kj) (11.318) = k + j T im. pg mi 11.40 Divergence of a contravariant vector 501 i More formally, the Hodge dual (11.226) of the 1-form V = Vi dx is i 1 ik ` m n V = Vi dx = Vi g ⌘k`mn dx dx dx ⇤ ⇤ 3! ^ ^ (11.319) 1 = pgVk ✏ dx` dxm dxn 3! k`mn ^ ^ in which g is the absolute value of the determinant of the metric tensor gij. The exterior derivative now gives 1 k p ` m n d V = pgV ✏k`mn dx dx dx dx . (11.320) ⇤ 3! ,p ^ ^ ^ ⇣ ⌘ So using (11.226) to apply a second Hodge star, we get (exercise 11.20) 1 k p ` m n d V = pgV ✏k`mn dx dx dx dx ⇤ ⇤ 3! ,p ⇤ ^ ^ ^ ⇣ ⌘ ⇣ ⌘ 1 k pt `u mv nw = pgV ✏k`mn g g g g ⌘tuvw 3! ,p ⇣ ⌘ 1 k pt `u mv nw = pgV ✏k`mn g g g g ✏tuvwpg 3! ,p ⇣ ⌘ 1 k pg p`mn = pgV ✏k`mn ✏ 3! ,p det gij ⇣ ⌘ s k p s k = pgV k = pgV (11.321) pg ,p pg ,k ⇣ ⌘ ⇣ ⌘ So in our spacetime with det g = g ij 1 d V = pgVk . (11.322) ⇤ ⇤ pg ,k ⇣ ⌘ i In 3-space the Hodge star (11.215) of a 1-form V = Vi dx is 1 1 V = V dxi = V gi` ⌘ dxj dxk = pgV` ✏ dxj dxk. (11.323) ⇤ i ⇤ i 2 `jk ^ 2 `jk ^ Applying the exterior derivative, we get the invariant form 1 ` p j k d V = pgV ✏`jk dx dx dx . (11.324) ⇤ 2 ,p ^ ^ ⇣ ⌘ We add a star by using the definition (11.215) of the Hodge dual in a 3-space in which the determinant det gij is positive and the identity (exercise 11.19) pjk p ✏`jk ✏ =2 ` (11.325) 502 Tensors and Local Symmetries as well as the definition (1.184) of the determinant 1 ` p j k d V = pgV ✏`jk dx dx dx ⇤ ⇤ 2 ,p ⇤ ^ ^ ⇣ ⌘ ⇣ ⌘ 1 ` pt ju kv = pgV ✏`jk g g g ⌘tuv 2 ,p ⇣ ⌘ 1 ` pt ju kv = pgV ✏`jk g g g ✏tuvpg 2 ,p ⇣ ⌘ 1 ` pjk pg = pgV ✏`jk ✏ 2 ,p det gij ⇣ ⌘ 1 ` p 1 p = pgV ` = (pgV ) . (11.326) pg ,p pg ,p ⇣ ⌘ Example 11.28 (Divergence in Orthogonal Coordinates). In two orthog- onal coordinates, equations (11.177 & 11.184) imply that pg = h1h2 and k V = V k/hk, and so the divergence (11.316) of a vector V is 2 1 h1h2 V = V k (11.327) r· h1h2 hk ,k Xk=1 ✓ ◆ which in polar coordinates (section 11.24) with hr = 1 and h✓ = r,is 1 1 V = r V + V = r V + V . (11.328) r· r r ,r ✓ ,✓ r r ,r ✓,✓ h i h i In three orthogonal coordinates, equations (11.177 & 11.184) give pg = k h1h2h3 and V = V k/hk, and so the divergence (11.316) of a vector V is (6.29) 3 1 h1h2h3 V = V k . (11.329) r· h1h2h3 hk ,k Xk=1 ✓ ◆ In cylindrical coordinates (section 11.25), h⇢ = 1, h = ⇢, and hz = 1; so 1 V = ⇢ V + V + ⇢ V r· ⇢ ⇢ ,⇢ , z ,z 1 h i = ⇢ V + V + ⇢ V . (11.330) ⇢ ⇢ ,⇢ , z,z h i In spherical coordinates (section 11.26), hr = 1, h✓ = r, h = r sin ✓, g = det g = r4 sin2 ✓ and the inverse gij of the metric tensor is | | 10 0 ij 2 (g )= 0 r 0 . (11.331) 0 2 2 1 00r sin ✓ @ A 11.41 The Covariant Laplacian 503 So our formula (11.327) gives us 1 V = r2 sin ✓ V + r sin ✓ V + r V r· r2 sin ✓ r ,r ✓ ,✓ , 1 h i = sin ✓ r2V + r sin ✓ V + rV (11.332) r2 sin ✓ r ,r ✓ ,✓ , h i as the divergence V . r· 11.41 The Covariant Laplacian In flat 3-space, we write the laplacian as = 2 or as . In euclidian 2 2 2 r·r r 4 coordinates, both mean @x + @y + @z . In flat minkowski space, one often 2 turns the triangle into a square and writes the 4-laplacian as 2 = @0 . ,i4 ik The gradient f,k of a scalar field f is a covariant vector, and f = g f,k is its contravariant form. The invariant laplacian 2f of a scalar field f ,i is the covariant divergence f;i . We may use our formula (11.316) for the divergence of a contravariant vector to write it in these equivalent ways ,i ik ,i ik (pgf ),i (pgg f,k),i 2f = f =(g f,k);i = = . (11.333) ;i pg pg To find the laplacian 2f in terms of forms, we apply the exterior derivative i to the Hodge dual (11.226) of the 1-form df = f,idx 1 d df = d f dxi = d f gik ⌘ dx` dxm dxn ⇤ ,i ⇤ 3! ,i k`mn ^ ^ ✓ ◆ (11.334) 1 ,k p ` m n = f pg ✏k`mn dx dx dx dx 3! ,p ^ ^ ^ ⇣ ⌘ and then add a star using (11.226) 1 ,k p ` m n d df = f pg ✏k`mn dx dx dx dx ⇤ ⇤ 3! ,p ⇤ ^ ^ ^ ⇣ ⌘ ⇣ ⌘ (11.335) 1 ,k pt `u mv nw = f pg ✏k`mn g g g g pg✏tuvw. 3! ,p ⇣ ⌘ The definition (1.184) of the determinant now gives (exercise 11.20) 1 ,k p`mn pg d df = f pg ✏k`mn ✏ ⇤ ⇤ 3! ,p det g (11.336) ⇣,k ⌘ p s s ,k = f pg k = f pg . ,p pg pg ,k ⇣ ⌘ ⇣ ⌘ 504 Tensors and Local Symmetries In our spacetime det g = sg = g, and so the laplacian is ij 1 2f = d df = f ,k pg . (11.337) ⇤ ⇤ pg ,k ⇣ ⌘ Example 11.29 (Invariant Laplacians). In two orthogonal coordinates, equations (11.177 & 11.178) imply that pg = det(gij) = h1h2 and that | | f ,i = gik f = h 2 f , and so the laplacian (11.333) of a scalar f is ,k i ,i p 2 1 h1h2 f = 2 f,i ,i. (11.338) 4 h1h2 hi ! Xi=1 2 In polar coordinates, where h1 = 1, h2 = r, and g = r , the laplacian is 1 1 1 2 f = (rf ) + r f = f + r f + r f . (11.339) 4 r ,r ,r ,✓ ,✓ ,rr ,r ,✓✓ In three orthogonalh coordinates, i equations (11.177 & 11.178) imply that ,i ik 2 pg = det(gij) = h1h2h3 and that f = g f = h f,i, and so the | | ,k i laplacian (11.333) of a scalar f is (6.33) p 3 1 h1h2h3 f = 2 f,i ,i. (11.340) 4 h1h2h3 hi ! Xi=1 2 In cylindrical coordinates (section 11.25), h⇢ = 1, h = ⇢, hz = 1, g = ⇢ , and the laplacian is 1 1 1 1 f = (⇢f ) + f + ⇢f = f + f + f + f . (11.341) 4 ⇢ ,⇢ ,⇢ ⇢ , ,zz ,⇢⇢ ⇢ ,⇢ ⇢2 , ,zz In spherical coordinates (section 11.26), hr = 1, h✓ = r, h = r sin ✓, and g = det g = r4 sin2 ✓. So (11.340) gives us the laplacian of f as (6.35) | | 2 r sin ✓f,r ,r +(sin✓f,✓),✓ +(f, / sin ✓), f = 2 4 r sin ✓ 2 r f,r ,r (sin ✓f,✓),✓ f, = 2 + 2 + 2 . (11.342) r r sin ✓ r2 sin ✓ If the function f is a function only of the radial variable r, then the laplacian is simply 1 2 1 2 f(r)= r f 0(r) 0 = [rf(r)]00 = f 00(r)+ f 0(r) (11.343) 4 r2 r r in which the primes⇥ denote ⇤r-derivatives. 11.42 The Principle of Stationary Action 505 11.42 The Principle of Stationary Action It follows from a path-integral formulation of quantum mechanics that the classical motion of a particle is given by the principle of stationary action S = 0. In the simplest case of a free non-relativistic particle, the lagrangian is L = mx˙ 2/2 and the action is t2 m S = x˙ 2 dt. (11.344) 2 Zt1 The classical trajectory is the one that when varied slightly by x (with x(t1)= x(t2)=0) does not change the action to first order in x.We first note that the change x˙ in the velocity is the time derivative of the change in the path d d x˙ = x˙ 0 x˙ = (x0 x)= x. (11.345) dt dt So since x(t1)= x(t2)=0, the stationary path satisfies t2 t2 d x 0= S = mx˙ x˙ dt = mx˙ dt · · dt Zt1 Zt1 t2 d = m (x˙ x) mx¨ x dt dt · · Zt1 t2 t2 = m [x˙ x]t2 m x¨ x dt = m x¨ x dt. (11.346) · t1 · · Zt1 Zt1 If the first-order change in the action is to vanish for arbitrary small varia- tions x in the path, then the acceleration must vanish x¨ = 0 (11.347) which is the classical equation of motion for a free particle. If the particle is moving under the influence of a potential V (x), then the action is t2 m S = x˙ 2 V (x) dt. (11.348) t 2 Z 1 ⇣ ⌘ Since V (x)= V (x) x, the principle of stationary action requires that r · t2 0= S = ( mx¨ V ) x dt (11.349) r · Zt1 or mx¨ = V (11.350) r 506 Tensors and Local Symmetries which is the classical equation of motion for a particle of mass m in a po- tential V . The action for a free particle of mass m in special relativity is ⌧2 t2 S = m d⌧ = m 1 x˙ 2 dt (11.351) ⌧1 t1 Z Z p where c = 1 and x˙ = dx/dt. The requirement of stationary action is t2 t2 x˙ x˙ 0= S = m 1 x˙ 2 dt = m · dt (11.352) 2 t1 t1 1 x˙ Z p Z But 1/ 1 x˙ 2 = dt/d⌧ and so p p t2 dx d x dt ⌧2 dx d x dt dt 0= S = m dt = m d⌧ dt · dt d⌧ dt · dt d⌧ d⌧ Zt1 Z⌧1 ⌧2 dx d x = m d⌧. (11.353) d⌧ · d⌧ Z⌧1 So integrating by parts, keeping in mind that x(⌧2)= x(⌧1)=0,wehave ⌧2 d dx d2x ⌧2 d2x 0= S = m x x d⌧ = m x d⌧. d⌧ d⌧ · d⌧ 2 · d⌧ 2 · Z⌧1 ✓ ◆ Z⌧1 (11.354) To have this hold for arbitrary x,weneed d2x = 0 (11.355) d⌧ 2 which is the equation of motion for a free particle in special relativity. What about a charged particle in an electromagnetic field Ai? Its action is ⌧2 x2 ⌧2 dxi S = m d⌧ + q A (x) dxi = m + qA (x) d⌧. (11.356) i i d⌧ Z⌧1 Zx1 Z⌧1 ✓ ◆ We now treat the first term in a four-dimensional manner i k i k ⌘ikdx dx k k d⌧ = ⌘ikdx dx = = uk dx = ukd x (11.357) i k ⌘ikdx dx p in which uk = dxk/d⌧ is thep 4-velocity (11.65) and ⌘ is the Minkowski metric (11.26) of flat spacetime. The variation of the other term is i i i k i i Ai dx =( Ai) dx + Ai dx = Ai,k x dx + Ai d x (11.358) 11.43 A Particle in a Gravitational Field 507 Putting them together, we get for S ⌧2 d xk dxi d xi S = mu + qA xk + qA d⌧. (11.359) k d⌧ i,k d⌧ i d⌧ Z⌧1 ✓ ◆ After integrating by parts the last term, dropping the boundary terms, and changing a dummy index, we get ⌧2 du dxi dA S = m k xk + qA xk q k xk d⌧ d⌧ i,k d⌧ d⌧ Z⌧1 ✓ ◆ ⌧2 du dxi = m k + q (A A ) xk d⌧. (11.360) d⌧ i,k k,i d⌧ Z⌧1 If this first-order variation of the action is to vanish for arbitrary xk,then the particle must follow the path du dxi dpk 0= m k + q (A A ) or = qF kiu (11.361) d⌧ i,k k,i d⌧ d⌧ i which is the Lorentz force law (11.95). 11.43 A Particle in a Gravitational Field The invariant action for a particle of mass m moving along a path xi(t)is ⌧2 1 i ` 2 S = m d⌧ = m gi`dx dx . (11.362) ⌧ Z 1 Z ⇣ ⌘ Proceeding as in Eq.(11.357), we compute the variation d⌧ as i ` i ` i ` (gi`)dx dx 2gi`dx dx d⌧ = gi`dx dx = i ` 2 gi`dx dx p = 1 g xkuiu`d⌧ g ui dx` 2 i`,k i` p = 1 g xkuiu`d⌧ g uid x` (11.363) 2 i`,k i` in which u` = dx`/d⌧ is the 4-velocity (11.65). The condition of stationary action then is ⌧2 ⌧2 d x` 0= S = m d⌧ = m 1 g xkuiu` + g ui d⌧ (11.364) 2 i`,k i` d⌧ Z⌧1 Z⌧1 ✓ ◆ 508 Tensors and Local Symmetries ` ` which we integrate by parts keeping in mind that x (⌧2)= x (⌧1)=0 ⌧2 d(g ui) 0=m 1 g xkuiu` i` x` d⌧ 2 i`,k d⌧ Z⌧1 ✓ ◆ ⌧2 dui = m 1 g xkuiu` g uiuk x` g x` d⌧. (11.365) 2 i`,k i`,k i` d⌧ Z⌧1 ✓ ◆ Now interchanging the dummy indices ` and k on the second and third terms, we have ⌧2 dui 0=m 1 g uiu` g uiu` g xkd⌧ (11.366) 2 i`,k ik,` ik d⌧ Z⌧1 ✓ ◆ or since xk is arbitrary dui 0= 1 g uiu` g uiu` g . (11.367) 2 i`,k ik,` ik d⌧ rk i ` If we multiply this equation of motion by g and note that gik,`u u = i ` g`k,iu u ,thenwefind dur 0= + 1 grk (g + g g ) uiu`. (11.368) d⌧ 2 ik,` `k,i i`,k r So using the symmetry gi` = g`i and the formula (11.302) for i`, we get dur d2xr dxi dx` 0= + r uiu` or 0 = + r (11.369) d⌧ i` d⌧ 2 i` d⌧ d⌧ which is the geodesic equation. In empty space, particles fall along geodesics independently of their masses. The right-hand side of the geodesic equation (11.369) is a contravariant vector because (Weinberg, 1972) under general coordinate transformations, r r i ` the inhomogeneous terms arising fromx ¨ cancel those from i`x˙ x˙ . Here and often in what follows we’ll use dots to mean proper-time derivatives. The action for a particle of mass m and charge q in a gravitational field r i` and an electromagnetic field Ai is 1 ⌧2 i ` 2 i S = m gi`dx dx + q Ai(x) dx (11.370) ⌧ Z ⇣ ⌘ Z 1 i because the interaction q Aidx is invariant under general coordinate trans- formations. By (11.360 & 11.366), the first-order change in S is R ⌧2 dui S = m 1 g uiu` g uiu` g + q (A A ) ui xkd⌧ 2 i`,k ik,` ik d⌧ i,k k,i Z⌧1 (11.371) 11.44 Equivalence principle and geodesic equation 509 and so by combining the Lorentz force law (11.361) and the geodesic equation ri r i (11.369) and by writing F x˙ i as F i x˙ ,wehave d2xr dxi dx` q dxi 0= + r F r (11.372) d⌧ 2 i` d⌧ d⌧ m i d⌧ as the equation of motion of a particle of mass m and change q.Itisstriking how nearly perfect the electromagnetism of Faraday and Maxwell is. The action of the electromagnetic field interacting with an electric current jk in a gravitational field is S = 1 F F k` + µ A jk pgd4x (11.373) 4 k` 0 k Z h i in which pgd4x is the invariant volume element. After an integration by parts, the first-order change in the action is @ S = F k` pg + µ jk pg A d4x, (11.374) @x` 0 k Z ⇣ ⌘ and so the inhomogeneous Maxwell equations in a gravitational field are @ pgFk` = µ pgjk. (11.375) @x` 0 ⇣ ⌘ 11.44 Equivalence principle and geodesic equation The principle of equivalence (section 11.19) says that in any gravitational field, one may choose free-fall coordinates in which all physical laws take the same form as in special relativity without acceleration or gravitation—at least over a suitably small volume of spacetime. Within this volume and in these coordinates, things behave as they would at rest deep in empty space far from any matter or energy. The volume must be small enough so that the gravitational field is constant throughout it. Example 11.30 (Elevators). When a modern elevator starts going down from a high floor, it accelerates downward at something less than the lo- cal acceleration of gravity. One feels less pressure on one’s feet; one feels lighter. After accelerating downward for a few seconds, the elevator assumes a constant downward speed, and then one feels the normal pressure of one’s weight on one’s feet. The elevator seems to be slowing down for a stop, but actually it has just stopped accelerating downward. What if the cable snapped, and a frightened passenger dropped his laptop? 510 Tensors and Local Symmetries He could catch it very easily as it would not seem to fall because the elevator, the passenger, and the laptop would all fall at the same rate. The physics in the falling elevator would be the same as if the elevator were at rest in empty space far from any gravitational field. The laptop’s clock would tick as fast as it would at rest in the absence of gravity. The transformation from arbitrary coordinates xk to free-fall coordinates i y changes the metric gj` to the diagonal metric ⌘ik of flat spacetime ⌘ = diag( 1, 1, 1, 1), which has two indices and is not a Levi-Civita tensor. Al- gebraically, this transformation is a congruence (1.314) @xj @x` ⌘ = g . (11.376) ik @yi j` @yk The geodesic equation (11.369) follows from the principle of equiva- lence (Weinberg, 1972; Hobson et al., 2006). Suppose a particle is moving under the influence of gravitation alone. Then one may choose free-fall co- ordinates y(x) so that the particle obeys the force-free equation of motion d2yi = 0 (11.377) d⌧ 2 with d⌧ the proper time d⌧ 2 = ⌘ dyidyk. The chain rule applied to yi(x) ik in (11.377) gives d @yi dxk 0= d⌧ @xk d⌧ ✓ ◆ @yi d2xk @2yi dxk dx` = + . (11.378) @xk d⌧ 2 @xk@x` d⌧ d⌧ We multiply by @xm/@yi and use the identity @xm @yi = m (11.379) @yi @xk k to write the equation of motion (11.377) in the x-coordinates d2xm dxk dx` + m =0. (11.380) d⌧ 2 k` d⌧ d⌧ This is the geodesic equation (11.369) in which the a ne connection is @xm @2yi m = . (11.381) k` @yi @xk@x` 11.45 Weak, Static Gravitational Fields 511 11.45 Weak, Static Gravitational Fields Newton’s equations describe slow motion in a weak, static gravitational. Because the motion is slow, we neglect ui compared to u0 and simplify the geodesic equation (11.369) to dur 0= + r (u0)2. (11.382) d⌧ 00 Because the gravitational field is static, we neglect the time derivatives gk0,0 r and g0k,0 in the connection formula (11.302) and find for 00 r = 1 grk (g + g g )= 1 grk g (11.383) 00 2 0k,0 0k,0 00,k 2 00,k 0 with 00 = 0. Because the field is weak, the metric can di↵er from ⌘ij by only a tiny tensor g = ⌘ + h so that to first order in h 1we ij ij ij | ij|⌧ have r = 1 h for r =1, 2, 3. With these simplifications, the geodesic 00 2 00,r equation (11.369) reduces to d2xr d2xr 1 dx0 2 = 1 (u0)2 h or = h . (11.384) d⌧ 2 2 00,r d⌧ 2 2 d⌧ 00,r ✓ ◆ So for slow motion, the ordinary acceleration is described by Newton’s law d2x c2 = h . (11.385) dt2 2 r 00 If is his potential, then for slow motion in weak static fields g = 1+h = 1 2 /c2 and so h = 2 /c2. (11.386) 00 00 00 Thus, if the particle is at a distance r from a mass M, then = GM/r and h = 2 /c2 =2GM/rc2 and so 00 d2x GM r = = = GM . (11.387) dt2 r r r r3 How weak are the static gravitational fields we know about? The dimen- 2 39 9 sionless ratio /c is 10 on the surface of a proton, 10 on the Earth, 6 4 10 on the surface of the sun, and 10 on the surface of a white dwarf. 11.46 Gravitational Time Dilation i Suppose we have a system of coordinates x with a metric gik and a clock at rest in this system. Then the proper time d⌧ between ticks of the clock is d⌧ =(1/c) g dxi dxj = p g dt (11.388) ij 00 q 512 Tensors and Local Symmetries where dt is the time between ticks in the xi coordinates, which is the labo- ratory frame in the gravitational field g00. By the principle of equivalence (section ??), the proper time d⌧ between ticks is the same as the time be- tween ticks when the same clock is at rest deep in empty space. If the clock is in a weak static gravitational field due to a mass M at a distance r,then g =1+2 /c2 =1 2GM/c2r (11.389) 00 is a little less than unity, and the interval of proper time between ticks d⌧ = p g dt = 1 2GM/c2rdt (11.390) 00 is slightly less than the interval dt betweenp ticks in the coordinate system of an observer at x in the rest frame of the clock and the mass, and in its gravitational field. Since dt > d⌧, the laboratory time dt between ticks is greater than the proper or intrinsic time d⌧ between ticks of the clock una↵ected by any gravitational field. Clocks near big masses run slow. Now suppose we have two identical clocks at di↵erent heights above sea level. The time T` for the lower clock to make N ticks will be longer than the time Tu for the upper clock to make N ticks. The ratio of the clock times will be 2 T` 1 2GM/c (r + h) gh = 1+ 2 . (11.391) Tu 1 2GM/c2r ⇡ c p Now imagine that a photonp going down passes the upper clock which mea- sures its frequency as ⌫u and then passes the lower clock which measures its frequency as ⌫`. The slower clock will measure a higher frequency. The ratio of the two frequencies will be the same as the ratio of the clock times ⌫` gh =1+ 2 . (11.392) ⌫u c As measured by the lower, slower clock, the photon is blue shifted. Example 11.31 (Pound, Rebka, and M¨ossbauer). Pound and Rebka in 1960 used the M¨ossbauer e↵ect to measure the blue shift of light falling down a 22.6 m shaft. They found ⌫` ⌫u gh 15 = =2.46 10 (11.393) ⌫ c2 ⇥ (Robert Pound 1919–2010, Glen Rebka 1931–, Rudolf M¨ossbauer 1929– 2011). 11.47 Curvature 513 Example 11.32 (Redshift of the Sun). A photon emitted with frequency ⌫0 at a distance r from a mass M would be observed at spatial infinity to have frequency ⌫ ⌫ = ⌫ p g = ⌫ 1 2MG/c2r (11.394) 0 00 0 2 for a redshift of ⌫ = ⌫0 ⌫. Since thep Sun’s dimensionless potential /c is MG/c2r = 2.12 10 6 at its surface, sunlight is shifted to the red by ⇥ 2 parts per million. 11.47 Curvature The curvature tensor or Riemann tensor is Ri = i i + i j i j (11.395) mnk mn,k mk,n kj nm nj km which we may write as the commutator i i i R =(Rnk) =[@k + k,@n + n] m mnk m (11.396) =( + )i n,k k,n k n n k m in which the ’s are treated as matrices 0 0 0 0 k 0 k 1 k 2 k 3 1 1 1 1 k 0 k 1 k 2 k 3 k = 0 2 2 2 2 1 (11.397) k 0 k 1 k 2 k 3 B 3 3 3 3 C B k 0 k 1 k 2 k 3C @ A i i j with ( k n) m = kj nm and so forth. Just as there are two conventions for the Faraday tensor Fik which di↵er by a minus sign, so too there are two i conventions for the curvature tensor Rmnk. Weinberg (Weinberg, 1972) uses the definition (11.395); Carroll (Carroll, 2003), Padmanabhan (Padmanab- han, 2010), Schutz (Schutz, 2009), and Zee (Zee, 2013) use an extra minus sign. The Ricci tensor is a contraction of the curvature tensor n Rmk = Rmnk (11.398) and the scalar curvature is a further contraction mk R = g Rmk. (11.399) 514 Tensors and Local Symmetries Example 11.33 (Curvature of a Sphere). While in four-dimensional space- time indices run from 0 to 3, on the sphere they are just ✓ and . There are only eight possible a ne connections, and because of the symmetry (11.245) i i in their lower indices ✓ = ✓, only six are independent. The point p on a sphere of radius r has cartesian coordinates p = r (sin ✓ cos , sin ✓ sin , cos ✓) (11.400) so the two 3-vectors are @p e = = r (cos ✓ cos , cos ✓ sin , sin ✓)=r ✓ˆ ✓ @✓ (11.401) @p e = = r sin ✓ ( sin , cos , 0) = r sin ✓ ˆ. @ The embedding metric is the 3 3 identity matrix, so the metric g made ⇥ ij of the dot products g = e e is ij i · j r2 0 (g )= . (11.402) ij 0 r2 sin2 ✓ ✓ ◆ Di↵erentiating the vectors e✓ and e ,wefind e = r (sin ✓ cos , sin ✓ sin , cos ✓)= r rˆ (11.403) ✓,✓ e = r cos ✓ ( sin , cos , 0) = r cos ✓ ˆ (11.404) ✓, e ,✓ = e✓, (11.405) e = r sin ✓ (cos , sin , 0) . (11.406) , ij The metric with upper indices (g ) is the inverse of the metric (gij) r 2 0 (gij)= (11.407) 0 r 2 sin 2 ✓ ✓ ◆ so the dual vectors ei are ✓ 1 1 e = r (cos ✓ cos , cos ✓ sin , sin ✓)=r ✓ˆ 1 1 e == ( sin , cos , 0) = ˆ. (11.408) r sin ✓ r sin ✓ The a ne connections are given by (11.239) as i = i = e i e . (11.409) jk kj · j,k ✓ ✓ Since both e and e are perpendicular tor ˆ, the a ne connections ✓✓ and ˆ ✓✓ both vanish. Also, e , is orthogonal to ,so = 0 as well. Similarly, ˆ ✓ ✓ e✓, is perpendicular to ✓,so ✓ = ✓ also vanishes. 11.47 Curvature 515 The two nonzero a ne connections are 1 1 = e e = r sin ✓ ˆ r cos ✓ ˆ = cot ✓ (11.410) ✓ · ✓, · and ✓ = e ✓ e · , = sin ✓ (cos ✓ cos , cos ✓ sin , sin ✓) (cos , sin , 0) · = sin ✓ cos ✓. (11.411) In terms of the two non-zero a ne connections ✓ = ✓ = cot ✓ and ✓ = sin ✓ cos ✓, the two Christo↵el matrices (11.397) are 00 00 = = (11.412) ✓ 0 0 cot ✓ ✓ ! ✓ ◆ and 0 ✓ 0 sin ✓ cos ✓ = = . (11.413) 0 cot ✓ 0 ✓ ! ✓ ◆ Their commutator is 0 cos2 ✓ [ , ]= = [ , ] (11.414) ✓ cot2 ✓ 0 ✓ ✓ ◆ and both [ ✓, ✓] and [ , ] vanish. So the commutator formula (11.396) gives for Riemann’s curvature tensor ✓ ✓ R✓✓✓ =[@✓ + ✓,@✓ + ✓] ✓ =0 R✓ ✓ =[@✓ + ✓,@ + ] ✓ =( ,✓) ✓ +[ ✓, ] ✓ = (cot ✓) + cot2 ✓ = 1 ,✓ R✓ =[@ + ,@ + ]✓ = ( )✓ +[ , ]✓ ✓ ✓ ✓ ,✓ ✓ = cos2 ✓ sin2 ✓ cos2 ✓ = sin2 ✓ R =[@ + ,@ + ] =0. (11.415) n The Ricci tensor (11.398) is the contraction Rmk = Rmnk, and so ✓ R✓✓ = R✓✓✓ + R✓ ✓ = 1 (11.416) R = R✓ + R = sin2 ✓. ✓ km The curvature scalar (11.399) is the contraction R = g Rmk, and so since ✓✓ 2 2 2 g = r and g = r sin ✓,itis ✓✓ 2 2 2 2 2 R = g R + g R = r sin ✓r sin ✓ = (11.417) ✓✓ r2 516 Tensors and Local Symmetries for a 2-sphere of radius r. Gauss invented a formula for the curvature K of a surface; for all two- dimensional surfaces, his K = R/2. Example 11.34 (Curvature of a cylindrical hyperboloid). The points of a cylindrical hyperboloid in 3-space satisfy z2 = x2 + y2 r2 and may be parameterized as p = r(cosh ✓ cos , cosh ✓ sin , sinh ✓). The (orthogonal) coordinate basis vectors are e✓ = p,✓ = r(sinh ✓ cos , sinh ✓ sin , cosh ✓) (11.418) e = p = r( cosh ✓ sin , cosh ✓ cos , 0). , The embedding metric is the 3 3 identity matrix, so the metric g = e e ⇥ ij i · j is cosh2 ✓ +sinh2 ✓ 0 (g )=r2 . (11.419) ij 0 cosh2 ✓ ✓ ◆ Tristan Hubsch’s Mathematica package great.m can compute the scalar curvature for us. One enters x = {theta, phi}; (met = {{r^2*(Cosh[theta]^2 + Sinh[theta]^2), 0}, {0, r^2*Cosh[theta]^2}}) // MatrixForm; SCurvature[met, x] and gets R =2/(r cosh(2✓))2 after adjusting for Hubsch’s sign convention. Python has sympy and gravipy. 11.48 Einstein’s equations If we make an action that is a scalar, invariant under general coordinate transformations, and then apply to it the principle of stationary action, we will get tensor field equations that are invariant under general coordinate transformations. If the metric of spacetime is among the fields of the action, then the resulting theory will be a possible theory of gravity. If we make the action as simple as possible, it will be Einstein’s theory. To make the action of the gravitational field, we need a scalar. Apart from the volume 4-form 1= det g d4x = pgd4x = pgcdtd3x,the ⇤ | ik| simplest scalar we can form from the metric tensor and its first and second p derivatives is the scalar curvature R which gives us the Einstein-Hilbert action c3 c3 S = R pgd4x = gik R pgd4x (11.420) EH 16⇡G 16⇡G ik Z Z 11.48 Einstein’s equations 517 39 2 2 11 3 1 2 in which G =6.7087 10 ~c (GeV/c ) =6.6742 10 m kg s is ⇥ ⇥ Newton’s constant. If gik(x) is a tiny local change in the inverse metric, then the rule 1 det A =detA Tr(A A) (1.41, valid for any nonsingular, nondefective ik ik ik matrix A) together with the identity 0 = (g gk`)= g gk` + g gk` imply that ik det g (det g)g gik 1 ik pg = = = pggik g . (11.421) 2pg 2pg 2 So the first-order change in the action density is ik ik ik ik g Rik pg = Rik pg g + g Rik pg + g pg Rik ⇣ ⌘ 1 (11.422) = R pgg gik + gik pg R . ik 2 ik ik ✓ ◆ ik The product g Rik is a scalar, so we can evaluate it in any coordinate a system. In a local inertial frame, where bc = 0 and gde is constant, this invariant variation of the Ricci tensor (11.398) is ik ik n n ik n n g Rik = g in,k ik,n = g (@k in @n ik) (11.423) = gik @ n gin @ k = @ gik n gin k . k in k in k in in n k ⇣ ⌘ By (11.247), the variations in and in are tensors although the connec- tions themselves aren’t. Thus, we can evaluate this invariant variation of the Ricci tensor in any coordinate system by replacing the derivatives with covariant ones getting ik ik n in k g Rik = g in g in (11.424) ;k ⇣ ⌘ which we recognize as the covariant divergence (11.316) of a contravariant vector. The last term in the first-order change (11.422) in the action density is therefore a surface term whose variation vanishes for tiny local changes gik of the metric ik ik n in k pgg Rik = pg g in g in . (11.425) ,k h ⇣ ⌘i Hence the variation of SEH is simply c3 1 S = R g R pg gik d4x. (11.426) EH 16⇡G ik 2 ik Z ✓ ◆ The principle of least action SEH = 0 now gives us Einstein’s equations 518 Tensors and Local Symmetries for empty space: 1 R g R =0. (11.427) ik 2 ik The tensor G = R 1 g R is Einstein’s tensor. ik ik 2 ik Taking the trace of Einstein’s equations (11.427), we find that the scalar curvature R and the Ricci tensor Rik are zero in empty space: R = 0 and Rik =0. (11.428) The energy-momentum tensor Tik is the source of the gravitational field. It is defined so that the change in the action of the matter fields due to a tiny local change gik(x)inthemetricis 1 1 S = T pg gik d4x = T ik pg g d4x (11.429) m 2c ik 2c ik Z Z in which the identity gik = gijg`k g explains the sign change. Now the j` principle of least action S = SEH + Sm = 0 yields Einstein’s equations in the presence of matter and energy 1 8⇡G R g R = T . (11.430) ik 2 ik c4 ik Taking the trace of both sides, we get 8⇡G 8⇡G T R = T and R = T g . (11.431) c4 ik c4 ik 2 ik ✓ ◆ 11.49 Energy-momentum tensor The action Sm of the matter fields is a scalar that is invariant under general coordinate transformations. In particular, a tiny local general coordinate a a a transformation x0 = x + ✏ (x)leavesSm invariant 4 0= Sm = L( i(x)) g(x) d x. (11.432) Z ⇣ p ⌘ The vanishing change Sm = Sm + Smg has a part Sm due to the changes in the fields i(x) and a part Smg due to the change in the metric ik g . The principle of stationary action tells us that the change Sm is zero as long as the fields obey the classical equations of motion. The definition (11.429) of the energy-momentum tensor now tells us that 1 0= S = S = T ik pg g d4x. (11.433) m mg 2c ik Z 11.50 Perfect fluids 519 We take the change in Sm to be 4 4 S = L0( 0 (x0)) g (x ) d x0 L( (x)) g(x) d x m i 0 0 i Z Z (11.434) p 4 p 4 = L0( 0 (x)) g (x) d x L( (x)) g(x) d x. i 0 i Z p Z p So using the identity gik g = gik g , the definition (11.263) of the k` k` covariant derivative of a covariant vector, and the formula (11.302) for the connection in terms of the metric, we find that to lowest order in ✏a(x), the change in the metric is g = g0 (x) g (x)=g0 (x0) g (x) (g0 (x0) g0 (x)) ik ik ik ik ik ik ik =( a ✏a )( b ✏b )g ✏cg i ,i k ,k ab ik,c = g ✏b g ✏a ✏c g ib ,k ak ,i ik,c = g (gbc ✏ ) g (gac ✏ ) ✏c g ib c ,k ak c ,i ik,c (11.435) = ✏ ✏ ✏ g gbc ✏ g gac ✏c g i,k k,i c ib ,k c ak ,i ik,c = ✏ ✏ + ✏ gbc g + ✏ gac g ✏c g i,k k,i c ib,k c ak,i ik,c = ✏ ✏ + ✏ gac (g + g g ) i,k k,i c ia,k ak,i ik,a = ✏ ✏ + ✏ c + ✏ c = ✏ ✏ . i,k k,i c ik c ki i;k k;i Combining this result (11.435) with the vanishing (11.433) of the change Smg,wehave ik 4 0= T pg (✏i;k + ✏k;i) d x. (11.436) Z Since the energy-momentum tensor is symmetric, we may combine the two terms, integrate by parts, divide by pg, and so find that the covariant divergence of the energy-momentum tensor is zero ik ik k ia i ak 1 ik i ak 0=T = T + T + T = (pgT ),k + T (11.437) ;k ,k ak ak pg ak when the fields obey their equations of motion. In a given inertial frame, only the total energy, momentum, and angular momentum of both the matter and the gravitational field are conserved. 11.50 Perfect fluids In many astrophysical and most cosmological models, the energy-momentum tensor is assumed to be that of a perfect fluid, which is isotropic in its rest frame, does not conduct heat, and has zero viscosity. For a perfect fluid of 520 Tensors and Local Symmetries pressure p and density ⇢ with 4-velocity ui (defined by (11.65)), the energy- momentum or energy-momentum tensor Tij is p T = pg +( + ⇢) u u (11.438) ij ij c2 i j in which gij is the spacetime metric. An important special case is the energy-momentum tensor due to a nonzero value of the energy density of the vacuum. In this case p = c2⇢ and the energy-momentum tensor is T = pg = c2⇢g (11.439) ij ij ij 2 in which T00 = c ⇢ is the (presumably constant) value of the energy density of the ground state of the theory. This energy density ⇢ is a plausible candidate for the dark-energy density. It is equivalent to a cosmological constant ⇤=8⇡G⇢. On small scales, such as that of our solar system, one may neglect mat- ter and dark energy. So in empty space and on small scales, the energy- momentum tensor vanishes Tij = 0 along with its trace and the scalar cur- vature T =0=R, and Einstein’s equations (11.431) are Rij =0. (11.440) 11.51 Standard Form Tensor equations are independent of the choice of coordinates, so it’s wise to choose coordinates that simplify one’s work. For a static and isotropic gravitational field, this choice is the standard form (Weinberg, 1972, ch. 8) ds2 = B(r) c2 dt2 + A(r) dr2 + r2 d✓2 +sin2 ✓d 2 (11.441) in which B(r) and A(r) are functions that one may find by solving the 2 2 2 i j field equations (11.430). Since ds = c d⌧ = gij dx dx , the nonzero 2 2 2 components of the metric tensor are grr = A(r), g✓✓ = r , g = r sin ✓, rr 1 ✓✓ 2 and g00 = B(r), and those of its inverse are g = A (r), g = r , 2 2 00 1 g = r sin ✓, and g = B (r). By di↵erentiating the metric tensor i and using (11.302), one gets the components of the connection k`, such as ✓ = sin ✓ cos ✓, and the components (11.398) of the Ricci tensor R , ij 11.52 Schwarzschild’s Solution 521 such as (Weinberg, 1972, ch. 8) B (r) 1 B (r) A (r) B (r) 1 A (r) R = 00 0 0 + 0 0 (11.442) rr 2B(r) 4 B(r) A(r) B(r) r A(r) ✓ ◆✓ ◆ ✓ ◆ in which the primes mean d/dr. 11.52 Schwarzschild’s Solution If one ignores the small dark-energy parameter ⇤, one may solve Einstein’s field equations (11.440) in empty space Rij = 0 (11.443) outside a mass M for the standard form of the Ricci tensor. One finds (Wein- berg, 1972) that A(r) B(r) = 1 and that rB(r)=r plus a constant, and one determines the constant by invoking the Newtonian limit g = B 00 ! 1+2MG/c2r as r . In 1916, Schwarzschild found the solution !1 1 2MG 2MG ds2 = 1 c2dt2 + 1 dr2 + r2 d✓2 +sin2 ✓d 2 c2r c2r ✓ ◆ ✓ ◆ (11.444) which one can use to analyze orbits around a star. The singularity in 1 2MG g = 1 (11.445) rr c2r ✓ ◆ 2 at the Schwarzschild radius rS =2MG/c is an artifact of the coordinates; the scalar curvature R and other invariant curvatures are not singular at the Schwarzschild radius. Moreover, for the Sun, the Schwarzschild radius 2 rS =2M G/c is only 2.95 km, far less than the radius of the Sun, which is 5 2 6.955 10 km. So the surface at rS =2M G/c is far from the empty space ⇥ in which Schwarzschild’s metric applies (Karl Schwarzschild, 1873–1916). 11.53 Black Holes Suppose an uncharged, spherically symmetric star of mass M has collapsed 2 within a sphere of radius rb less than its Schwarzschild radius rS =2MG/c . Then for r>rb, the Schwarzschild metric (11.444) is correct. By Eq.(11.388), the apparent time dt of a process of proper time d⌧ at r 2MG/c2 is 2MG p dt = d⌧/ g00 = d⌧/ 1 2 . (11.446) r c r 522 Tensors and Local Symmetries The apparent time dt becomes infinite as r 2MG/c2. To outside ob- ! servers, the star seems frozen in time. Due to the gravitational redshift (11.394), light of frequency ⌫p emitted at r 2MG/c2 will have frequency ⌫ 2MG p ⌫ = ⌫p g00 = ⌫p 1 2 (11.447) r c r when observed at great distances. Light coming from the surface at rS = 2MG/c2 is redshifted to zero frequency ⌫ = 0. The star is black. It is a black 2 hole with a surface or horizon at its Schwarzschild radius rS =2MG/c , although there is no singularity there. If the radius of the Sun were less than its Schwarzschild radius of 2.95 km, then the Sun would be a black hole. The radius of the Sun is 6.955 105 km. ⇥ Black holes are not really black. Stephen Hawking (1942–) has shown that the intense gravitational field of a black hole of mass M radiates at temperature ~ c3 T = (11.448) 8⇡kGM 5 1 in which k =8.617343 10 eV K is Boltzmann’s constant, and ~ is ⇥ 34 Planck’s constant h =6.6260693 10 Js dividedby2⇡, ~ = h/(2⇡). ⇥ The black hole is entirely converted into radiation after a time 5120 ⇡G2 t = M 3 (11.449) ~ c4 proportional to the cube of its mass. 11.54 Cosmology Astrophysical observations tell us that on the largest observable scales, space is flat or very nearly flat; that the visible universe contains at least 1090 par- ticles; and that the cosmic microwave background radiation is isotropic to one part in 105 apart from a Hubble-Doppler shift due the motion of the Earth at 371 km/s towards the constellation Leo. These and other observa- tions suggest that potential energy expanded our universe by exp(60) = 1026 35 during a brief period of inflation that could have been as short as 10 s. The potential energy that powered inflation almost immediately became the radiation of the Big Bang. During and after inflation, negative gravi- tational potential energy kept the total energy constant. Within the first three minutes, some of that radiation became hydrogen, 11.54 Cosmology 523 helium, neutrinos, and dark matter Weinberg (1988, 2010). But the era of radiation, during which most of the energy of the visible universe was radiation, lasted for 50,952 years. Because the momentum of a particle but not its mass falls with the ex- pansion of the universe, the era of radiation gradually gave way to an era of matter. The universe changed from radiation dominated to matter domi- nated as its temperature kT dropped below 0.81 eV. After 380,000 years, Figure 11.1 NASA/WMAP’s timeline of the known universe the universe had cooled to 3000 K or kT =0.26 eV, and less than 1% of the atoms were ionized. Photons no longer scattered o↵a plasma of electrons and ions. The universe became transparent. The photons that last scattered just before this initial transparency became the cosmic microwave background radiation or CMBR that now surrounds us, redshifted to T =2.7255 0.0006 K. 0 ± Between 10 and 17 million years after the Big Bang, the temperature of the known universe fell from 373 to 273 K. If by then the supernovas of very early, very heavy stars had produced carbon, nitrogen, and oxygen, biochemistry may have started during this period of 7 million years. The era of matter lasted for 10.19 billion years. The era of dark energy began about 3.6 billon years ago as matter ceased to be the dominant form of energy. Since then, most of the energy 524 Tensors and Local Symmetries of the visible universe has been of an unknown kind called dark energy. Dark energy may be the energy of the vacuum; classically, it seems to be equivalent to a cosmological constant. Dark energy has been accelerating the expansion of the universe for the past 3.6 billion years and may continue to do so forever. It is now 13.799 0.021 billion years after the Big Bang or the time of ± infinite redshift and zero scale factor. The present value Ade et al. (2015) of the Hubble frequency H is the Hubble constant H0 = 67.74 0.46 1 18 1 ± km (s Mpc) =2.1953 10 s , one parsec being 3.085 677 581 49 16 ⇥ ⇥ 10 m or about 3.262 light-years. The present critical mass density ⇢c0 = 3H2/8⇡G =8.6197 10 27 kg m 3 is the present density of a universe 0 ⇥ that is isotropic, homogeneous, and flat. The ratio ⌦0 = ⇢0/⇢c0 of the present density ⇢0 of the universe to the present critical mass density ⇢c0 is ⌦ =1.0000 0.0088. The ratio ⌦ = ⇢ /⇢ of the present dark- 0 ± ⇤0 ⇤0 c0 energy density ⇢ to the critical density is ⌦ =0.6911 0.0062. The ⇤0 ⇤0 ± ratio ⌦m0 = ⇢m0/⇢c0 of the present total matter density ⇢m0 to the present critical density ⇢ is ⌦ =0.3089 0.0062. The ratio ⌦ = ⇢ /⇢ c0 m0 ± b0 b0 c0 of the present density of baryons ⇢b0 to the present critical density ⇢c0 is ⌦ =0.0486 0.0007. Baryons account for 4.9% of the density of the b0 ± universe and only 15.7 % of the matter density, the rest being dark matter, which interacts very little with light. Einstein’s equations (11.431) are second-order, non-linear partial di↵er- ential equations for 10 unknown functions gij(x) in terms of the energy- momentum tensor Tij(x) throughout the universe, which of course we don’t know. The problem is not quite hopeless, however. The ability to choose ar- bitrary coordinates, the appeal to symmetry, and the choice of a reasonable form for Tij all help. Hubble showed us that the universe is expanding. The cosmic microwave background radiation looks the same in all spatial directions (apart from a Hubble-Doppler shift due to the motion of the Earth at 371 km/s toward the constellation Leo). Observations of clusters of galaxies reveal a universe that is homogeneous on suitably large scales of distance. So it is plausible that the universe is homogeneous and isotropic in space, but not in time. One may show (Carroll, 2003) that for a universe of such symmtery, the line element in comoving coordinates (in which the Hubble-Doppler shift is isotropic) is dr2 ds2 = c2dt2 + a2 + r2 d✓2 +sin2 ✓d 2 (11.450) 1 kr2 11.54 Cosmology 525 in which the dimensionless function of time a(t)isthescale factor and k is a constant whose dimension is inverse squared length. Whitney’s embedding theorem tells us that any smooth four-dimensional manifold can be embedded in a flat space of eight dimensions with a suitable signature. We need only four or five dimensions to embed the spacetime described by the line element (11.450). If the universe is closed, then the signature is ( 1, 1, 1, 1, 1), and our three-dimensional space is the 3-sphere which is the surface of a four-dimensional sphere in four space dimensions. The points of the universe then are p =(ct, a sin sin ✓ cos , a sin sin ✓ sin , a sin cos ✓, a cos ) (11.451) in which 0 ⇡,0 ✓ ⇡, and 0 2⇡. If the universe is flat, then the embedding space is flat, four-dimensional Minkowski space with points p =(ct, ar sin ✓ cos , ar sin ✓ sin , ar cos ✓)=(ct, ax, ay, az) (11.452) in which 0 ✓ ⇡ and 0 2⇡. If the universe is open, then the em- bedding space is a flat five-dimensional space with signature ( 1, 1, 1, 1, 1), and our three-dimensional space is a hyperboloid in a flat Minkowski space of one time and three space dimensions. The points of the universe then are p =(ct, a sinh sin ✓ cos , a sinh sin ✓ sin , a sinh cos ✓, a cosh ) (11.453) in which 0 ,0 ✓ ⇡, and 0 2⇡. 1 In all three cases, the corresponding Robertson-Walker metric is 10 0 0 0 a2/(1 kr2)0 0 gij = 0 1 (11.454) 00a2 r2 0 B 00 0a2 r2 sin2 ✓C B C @ A in which the coordinates (ct, r, ✓, ) are numbered (0, 1, 2, 3), and k is a constant with units of 1/length2. One always may choose coordinates (ex- ercise 11.31) such that k numerically is 0 or 1. This constant determines ± whether the spatial universe is open k<0, flat k = 0, or closed k>0. The scale factor a, which is a function of time a(t), tells us how space expands and contracts. These coordinates are called comoving because a point at rest (fixed r, ✓, ) sees the same Hubble-Doppler shift in all directions. The metric (11.454) is diagonal; its inverse gij also is diagonal; and so we k may use our formula (11.302) to compute the a ne connections i`, such as 526 Tensors and Local Symmetries 0 = 1 g0k (g + g g )= 1 g00 (g + g g )= 1 g `` 2 `k,` `k,` ``,k 2 `0,` `0,` ``,0 2 ``,0 (11.455) so that aa/c˙ 0 = 0 = aar˙ 2/c and 0 = aar˙ 2 sin2 ✓/c. (11.456) 11 1 kr2 22 33 0 in which a dot means a time derivative. The other ij’s vanish. Similarly, for fixed ` = 1, 2, or 3 ` = 1 g`k (g + g g ) 0` 2 0k,` `k,0 0`,k = 1 g`` (g + g g ) 2 0`,` ``,0 0`,` a˙ = 1 g`` g = = ` no sum over `. (11.457) 2 ``,0 ca `0 The other nonzero ’s are 1 = r (1 kr2) 1 = r (1 kr2)sin2 ✓ (11.458) 22 33 1 2 = 3 = = 2 = 3 (11.459) 12 13 r 21 31 2 = sin ✓ cos ✓ 3 = cot ✓ = 3 . (11.460) 33 23 32 Our formulas (11.398 & 11.396) for the Ricci and curvature tensors give n n R00 = R0n0 =[@0 + 0,@n + n] 0. (11.461) Clearly the commutator of 0 with itself vanishes, and one may use the formulas (11.456–11.460) for the other connections to check that a˙ 2 [ , ]n = n k n k =3 (11.462) 0 n 0 0 k n 0 nk 00 ca ✓ ◆ and that a˙ a¨ a˙ 2 @ n =3@ =3 3 (11.463) 0 n 0 0 ca c2a ca ✓ ◆ ✓ ◆ n while @n 00 = 0. So the 00-component of the Ricci tensor is a¨ R =3 . (11.464) 00 c2a Similarly, one may show that the other nonzero components of Ricci’s tensor are A R = R = r2A and R = r2A sin2 ✓ (11.465) 11 1 kr2 22 33 11.54 Cosmology 527 in which A = aa/c¨ 2 +2˙a2/c2 +2k. The scalar curvature (11.399) is 6 aa¨ a˙ 2 R = gabR = + + k . (11.466) ba a2 c2 c2 ✓ ◆ In comoving coordinates such as those of the Robertson-Walker met- ric (11.454), the 4-velocity (11.65) is ui =(c, 0, 0, 0), and so the energy- momentum tensor (11.438) is c2⇢ 000 0 pg11 00 Tij = 0 1 . (11.467) 00pg22 0 B 00 0pg C B 33C @ A Its trace is T = gij T = c2⇢ +3p. (11.468) ij Thus successively using our formulas (11.454) for g00 = 1, (11.464) for 2 R00 =3¨a/(c a), (11.467) for Tij, and (11.468) for T , we can write the 00 Einstein equation (11.431) as the second-order equation a¨ 4⇡G 3p = ⇢ + (11.469) a 3 c2 ✓ ◆ which is nonlinear because ⇢ and 3p depend upon a.Thesum⇢ +3p de- termines the accelerationa ¨ of the scale factor a(t). The apparently positive sum c2⇢ +3p can be negative, and when it is, it accelerates the expansion of the universe. Because of the isotropy of the metric, the three nonzero spatial Einstein equations (11.431) give us only one relation a¨ a˙ 2 c2k p +2 +2 =4⇡G ⇢ . (11.470) a a a2 c2 ✓ ◆ ⇣ ⌘ Using the 00-equation (11.469) to eliminate the second derivativea ¨,wehave a˙ 2 8⇡G c2k = ⇢ (11.471) a 3 a2 ✓ ◆ which is a first-order nonlinear equation. It and the second-order equation (11.469) are known as the Friedmann equations. The LHS of the first-order Friedmann equation (11.493) is the square of the Hubble rate a˙ H = (11.472) a 528 Tensors and Local Symmetries which is an inverse time or a frequency. Its present value H0 is the Hubble constant. In terms of H, Friedmann’s first-order equation (11.493) is 8⇡G c2k H2 = ⇢ . (11.473) 3 a2 The energy density of a flat universe with k =0isthecritical energy density 3H2 ⇢ = . (11.474) c 8⇡G The ratio of the present energy density ⇢0 to the present critical energy density ⇢c0 is ⌦0 ⇢0 8⇡G ⌦0 = = 2 ⇢0. (11.475) ⇢c0 3H0 From (11.473), we see that ⌦is c2k c2k ⌦=1+ =1+ . (11.476) (aH)2 a˙ 2 Thus ⌦= 1 both in a flat universe (k = 0) and as aH . One use of !1 inflation is to expand a by 1026 so as to force ⌦almost exactly to unity. Something like inflation is needed because in a universe in which the energy density is due to matter and/or radiation, the present value of ⌦ ⌦ =1.0000 0.0062 (11.477) 0 ± is unlikely. To see why, we note that conservation of energy ensures that a3 times the matter density ⇢m is constant. Radiation redshifts by a, so energy 4 conservation implies that a times the radiation density ⇢r is constant. So with n = 3 for matter and 4 for radiation, ⇢an 3 F 2/8⇡G is a constant. ⌘ In terms of F and n, Friedmann’s first-order equation (11.493) is 2 2 8⇡G 2 2 F 2 a˙ = ⇢a c k = n 2 c k (11.478) 3 a In the small-a limit of the early Universe, we have (n 2)/2 (n 2)/2 a˙ = F/a or a da = Fdt (11.479) which we integrate to a t2/n so thata ˙ t2/n 1. Now (11.476) says that ⇠ ⇠ 2 c k 2 4/n t radiation ⌦ 1 = | | t = . (11.480) | | a˙ 2 / t2/3 matter ⇢ Thus, ⌦deviated from unity at least as fast as t2/3 during the early Universe. At this rate, the inequality ⌦ 1 < 0.009 could last 13.8 billion years only | 0 | 11.55 Density and pressure 529 if ⌦at t = 1 second had been unity to within two parts in 1014. The only known explanation for such early flatness is inflation. The relation (11.476) between ⌦and aH shows that k = 0 is equivalent to⌦= 1, that ⌦ > 1 k>0, and that ⌦ < 1 k<0. And writing () () the same relation (11.476) as (aH)2 = c2k/(⌦ 1), we see that as ⌦ 1the ! product aH , which is the essence of flatness since curvature vanishes !1 as the scale factor a . Imagine blowing up a balloon. !1 Staying for the moment with a universe without inflation and with an energy density composed of radiation and/or matter, we note that the first- order equation (11.478) in the forma ˙ 2 = F 2/an 2 c2k tells us that for a closed (k>0) universe, in the limit a we’d havea ˙ 2 1whichis !1 ! impossible. Thus a closed universe cannot expand indefinitely. The first-order equation Friedmann (11.493) says that ⇢a2 3c2k/8⇡G. So in a closed universe (k>0), the energy density ⇢ is positive and in- creases without limit as a 0 as in a collapse. In open (k<0) and ! flat (k = 0) universes, the same Friedmann equation (11.493) in the form a˙ 2 =8⇡G⇢a2/3 c2k tells us that if ⇢ is positive, thena ˙ 2 > 0, which means thata ˙ never vanishes. Hubble told us thata> ˙ 0 now. So if our universe is open or flat, then it always expands. Due to the expansion of the universe, the wave-length of radiation grows with the scale factor a(t). A photon emitted at time t and scale factor a(t) with wave-length (t) will be seen now at time t0 and scale factor a(t0)to have a longer wave-length (t0) (t ) a(t ) 0 = 0 = z + 1 (11.481) (t) a(t) in which the redshift z is the ratio (t ) (t) z = 0 = . (11.482) (t) Now H =˙a/a = da/(adt)impliesdt = da/(aH), and z = a /a 1implies 0 dz = a da/a2,sowefind 0 dz dt = (11.483) (1 + z)H(z) which relates time intervals to redshift intervals. An on-line calculator is available for macroscopic intervals (Wright, 2006). 11.55 Density and pressure 530 Tensors and Local Symmetries The 0-th component of the energy-momentum conservation law (11.437) is 0a 0a a 0c 0 ca 0=T;a = @aT + caT + caT . (11.484) The perfect-fluid energy-momentum tensor (11.467) is diagonal, and for a Robinson-Walker metric (11.454) our connection formulas (11.455) and (11.457) respectively tell us that 0 = 1 g = 0, and that a =3˙a/(ca). 00 2 00,0 0a Thus 3˙a p d⇢ 3 p ⇢˙ = (⇢ + ), and so = ⇢ + . (11.485) a c2 da a c2 ⇣ ⌘ The energy density ⇢ is composed of fractions ⇢i each contributing its own partial pressure pi according to its own equation of state 2 pi = c wi⇢i (11.486) in which wi is a constant. The rate of change (11.486) of the density ⇢i is then d⇢ 3 i = (1 + w ) ⇢ . (11.487) da a i i In terms of the present density ⇢i0 and scale factor a0, the solution is a 3(1+wi) ⇢ = ⇢ 0 . (11.488) i i0 a ⇣ ⌘ There are three important kinds of density. The dark-energy density ⇢⇤ is assumed to be like a cosmological constant ⇤or like the energy density of the vacuum, so it is independent of the scale factor a ⇢⇤ = ⇢⇤0 (11.489) and has w = 1. The matter density ⇢ is assumed to have no pressure, ⇤ m wm = 0, and so the matter density falls inversely with the volume a 3 ⇢ = ⇢ 0 . (11.490) m m0 a ⇣ ⌘ The density of radiation ⇢r has wr =1/3 because wavelengths scale with the scale factor, and so there’s an extra factor of a a 4 ⇢ = ⇢ 0 . (11.491) r r0 a ⇣ ⌘ The total density ⇢ varies with a as a 3 a 4 ⇢ = ⇢ + ⇢ 0 + ⇢ 0 . (11.492) ⇤0 m0 a r0 a ⇣ ⌘ ⇣ ⌘ 11.56 How the scale factor evolves with time 531 In simple cosmological models, only the dominant component of the den- sity is considered. 11.56 How the scale factor evolves with time The first-order Friedmann equation (11.493) expresses the square of the instantaneous Hubble rate H =˙a/a in terms of the density ⇢ and the scale factor a(t) a˙ 2 8⇡G c2k H2 = = ⇢ (11.493) a 3 a2 ✓ ◆ in which k = 1 or 0. The critical density ⇢ is the one that satisfies this ± c equation for a flat (k = 0) universe. Its present value is 3H2 ⇢ = 0 . (11.494) 0c 8⇡G 2 Dividing Friedmann’s equation by the square of the present Hubble rate H0 , we get H2 1 a˙ 2 1 8⇡G c2k ⇢ c2k = = ⇢ = (11.495) H2 H2 a H2 3 a2 ⇢ a2H2 0 0 ✓ ◆ 0 ✓ ◆ 0c 0 in which ⇢ is the total density (11.492) 2 2 H ⇢⇤ ⇢r ⇢m c k 2 = + + 2 2 H0 ⇢0c ⇢0c ⇢0c a H0 4 3 2 2 (11.496) ⇢⇤0 ⇢r0 a0 ⇢m0 a0 c k a0 = + 4 + 3 2 2 2 . ⇢0c ⇢0c a ⇢0c a a0H0 a 2 2 2 These density ratios are called ⌦⇤0,⌦r0,⌦m0, and ⌦k0 c k/a0H0 , and 2 2 ⌘ in terms of them this formula (11.496) for H /H0 is 2 2 3 4 H a0 a0 a0 2 =⌦⇤0 +⌦k0 2 +⌦m0 3 +⌦r0 4 . (11.497) H0 a a a 1 Since H =˙a/a,wehavedt = H0 (da/a)(H0/H), and so with x = a/a0,the time interval dt is 1 dx 1 dt = . (11.498) 2 3 4 H0 x ⌦⇤0 +⌦k0 x +⌦m0 x +⌦r0 x Integrating and settingp the origin of time t(0) = 0 at x = a/a0 = 0, we find that the time t(a/a0) during which the ratio a(t)/a0 grew from 0 to a(t)/a0 532 Tensors and Local Symmetries is 1 a/a0 dx t(a/a0)= . (11.499) H 2 1 2 0 Z0 ⌦⇤0 x +⌦k0 +⌦m0 x +⌦r0 x The definition (11.481) ofp the redshift gives a/a0 =1/(z+1). So this formula (11.499) for t(a/a0) also says that a photon emitted at time t(a/a0)willbe seen now with redshift z(t)=a /a 1. 0 The Planck Collaboration’s values (Ade et al. (2015)) for the density ratios ⌦⇤0,⌦k0, and ⌦m0 are ⌦ =0.6911 0.0062 ⇤0 ± ⌦ =0.0008 0.004 (11.500) k0 ± ⌦ =0.3089 0.0062. m0 ± We use may use the present temperature T0 =2.7255 K of the cosmic mi- crowave background radiation and our formula (4.98) for the energy density of photons to estimate the mass density of photons as 5 4 8⇡ (kBT0) 31 3 ⇢ = =4.6451 10 kg m . (11.501) 15h3c5 ⇥ 1/3 Adding in three kinds of neutrinos and antineutrinos at T0⌫ =(4/11) T0, we get for the present density of massless and nearly massless particles (Wein- berg, 2010, section 2.1) 4/3 7 4 31 3 ⇢ = 1+3 ⇢ =7.8099 10 kg m . (11.502) r 8 11 ⇥ " ✓ ◆✓ ◆ # The fraction ⌦r0 the present energy density that is due to radiation is then ⇢r0 5 ⌦r0 = =9.0606 10 . (11.503) ⇢c0 ⇥ By putting the ⌦values (11.500 & 11.503) into the integral (11.499) and integrating, we may get the time as a function of the scale factor. Figure 11.2 plots the reduced scale factor a(t)/a0 (solid) and the redshift z(t) (. . . ) as functions of the time t in Gyr since the time of infinite redshift. The age of the universe is 13.8 Gyr (vertical line). A photon emitted with wavelength at time t now has wavelength 0 =(a0/a(t)) . The change is its wavelength is = z(t). Example 11.35 (w = 1/3, No Acceleration). If w = 1/3, then p = w⇢= ⇢/3 and ⇢ +3p = 0. The second-order Friedmann equation (11.469) then tells us thata ¨ = 0. The scale factor does not accelerate. 11.56 How the scale factor evolves with time 533 1 10 0.9 9 0.8 8 0.7 a(t)/a0 7 0.6 6 0 ) /a 0.5 5 t ) ( t z ( a 0.4 t/H0 4 0.3 3 0.2 2 0.1 z(t) 1 0 0 0 2 4 6 8 10 12 14 t (Gyr) Figure 11.2 The reduced scale factor a(t)/a0 (solid), the redshift z(t) (dot- dash), and the fraction t/H0 (dashed) are plotted as functions of the time (11.499) in Gyr since the time of infinite redshift. The age of the universe is 13.8 Gyr (vertical line). A photon emitted with wavelength at time t now has wavelength 0 =(a0/a(t) and redshift z(t)= / . To find its constant speed, we use its equation of state (11.488) a 3(1+w) a 2 ⇢ = ⇢ 0 = ⇢ 0 . (11.504) 0 a 0 a ⇣ ⌘ ⇣ ⌘ Now all the terms in Friedmann’s first-order equation (11.493) have a com- mon factor of 1/a2 which cancels leaving us with the square of the constant speed 8⇡G a˙ 2 = ⇢ a2 c2k (11.505) 3 0 0 2 2 in which ⇢0 a0 must exceed 3c k/8⇡G.Since˙a = aH is constant, the scale factor grows linearly with time like the dashed line a(t)=a0 H0 t in fig- ure 11.2 if we set a(0) = 0. 534 Tensors and Local Symmetries 11.57 Inflation (w = 1) Inflation occurs when the ground state of the theory has a positive and constant potential-energy density ⇢>0 that dwarfs the energy densities of the matter and radiation. The internal energy of the universe then is proportional to its volume U = c2⇢V, and the pressure p as given by the thermodynamic relation @U p = = c2⇢ (11.506) @V is negative. The equation of state (11.486) tells us that in this case w = 1. The second-order Friedmann equation (11.469) becomes a¨ 4⇡G 3p 8⇡G⇢ = ⇢ + = g2 (11.507) a 3 c2 3 ⌘ ✓ ◆ By it and the first-order Friedmann equation (11.493) and by choosing t =0 as the time at which the scale factor a is zero or minimal, one may show (exercise 11.38) that in a flat (k = 0), closed (k>0), and open (k<0), universes, the scale factor varies as a(t)=a(0) egt flat, k = 0 (11.508) cosh gt a(t)= closed, k>0 (11.509) g sinh gt a(t)= open, k<0. (11.510) g A de Sitter universe is flat, has a(t)=a(0) exp(gt), and can be infinitely old. Studies of the cosmic microwave background radiation suggest that infla- tion did occur in the very early universe—possibly on a time scale as short 35 as 10 s. The origin of the vacuum energy density ⇢ that drove inflation is unknown. In chaotic inflation, a scalar field fluctuated to a mean value very di↵erent from the one 0 0 that minimizes the energy density h i h | | i of the vacuum. When settled to 0 0 , the potential energy of the h i h | | i vacuum was released as radiation in a Big Bang. Anti-de Sitter models have w = 1 and a negative potential energy ⇢ = p<0. 11.58 The era of radiation (w =1/3) 535 11.58 The era of radiation (w =1/3) Until a redshift of z = 3408.3 or 50, 953 years after the time of infinite redshift our universe was dominated by radiation. During The First Three Minutes (Weinberg, 1988) of the era of radiation, the quarks and gluons formed hadrons, which decayed into protons and neutrons. As the neutrons decayed (⌧ = 885.7 s), they and the protons formed the light elements— principally hydrogen, deuterium, and helium in a process called big-bang nucleosynthesis. We can guess the value of w for radiation by noticing that the energy- momentum tensor of the electromagnetic field (in SI units) 1 1 T ab = F a F bc gabF F cd (11.511) µ c 4 cd 0 ✓ ◆ is traceless 1 1 T = T a = F a F c aF F cd =0. (11.512) a µ c a 4 a cd 0 ✓ ◆ But by (11.468) its trace must be T =3p c2⇢. So for radiation p = c2⇢/3 and w =1/3. The relation (11.491) between the density and the scale factor for radiation then is a 4 ⇢ = ⇢ 0 . (11.513) 0 a ⇣ ⌘ The density drops both with the volume a3 and with the scale factor a due to a redshift; so it drops as 1/a4. Early in the era of radiation, we can ignore the matter and vacuum den- sities and focus on the radiation density so that the quantity 8⇡G⇢a4 f 2 (11.514) ⌘ 3 is a constant. The Friedmann equations (11.469 & 11.470) then are a¨ 4⇡G 3p 8⇡G⇢ f 2 = ⇢ + = ora ¨ = (11.515) a 3 c2 3 a3 ✓ ◆ and f 2 a˙ 2 + c2k = . (11.516) a2 With calendars chosen so that a(0) = 0, this last equation (11.516) tells 536 Tensors and Local Symmetries us that for flat (k = 0), closed (k = 1), and open (k = 1) universes a(t)= 2ft flat, k = 0 (11.517) a(t)=p2ft kc2t2 closed, k>0 (11.518) a(t)=p2ft+ kc2t2 open, k<0, (11.519) as we saw in (6.488). Thep scale factor (11.518) of a closed universe of radiation has a maximum a = f/(cpk) at t = f/(kc2) and falls back to zero at t =2f/(kc2). In all three cases, the scale factor rises as the square-root of the time a(t) p2ft at early times. The density of radiation is proportional both ⇡ to the fourth power of the temperature ⇢ T 4 and to the inverse fourth / power of the scale factor ⇢ 1/a4(t) 1/t2. Thus the temperature falls as ⇠ ⇠ T 1/pt. ⇠ Weinberg gives more accurate estimates. When the temperature was in 12 10 2 2 the range 10 >T >10 K or mµc >kT>mec ,wheremµ is the mass of the muon and me that of the electron, the radiation was mostly electrons, positrons, photons, and neutrinos, and the relation between the time t and the temperature T was (Weinberg, 2010, ch. 3) 1010 K 2 t =0.994 sec + constant. (11.520) ⇥ T By 109 K, the positrons had annihilated with electrons, and the neutrinos fallen out of equilibrium. Between 109 K and 106 K, when the energy density of nonrelativistic particles became relevant, the time-temperature relation was (Weinberg, 2010, ch. 3) 1010 K 2 t =1.78 sec + constant0. (11.521) ⇥ T At times of tens of thousands of years, the matter density ⇢m became important because it drops with the cube of the scale factor as ⇢m(t)= 3 3 ⌦m0 a0/a (t) while the radiation density ⇢r(t) drops with the fourth power 4 4 of the scale factor as ⇢r(t)=⌦r0 a0/a (t). The era of radiation ended and the era of matter began when these densities were equal, ⇢m(t)=⇢r(t). We can estimate this time by using the density ratios ⌦m0 and ⌦r0 (11.500 & 11.503). We find 5 a ⇢r,0 ⌦r0 9.0606 10 4 = = = ⇥ =2.93318 10 (11.522) a0 ⇢m,0 ⌦m0 0.3089 ⇥ which is a redshift of z = a /a 1 = 3408.3. Our integral (11.499) gives 0 11.59 The era of matter (w =0) 537 the time as t(3408.3) = 50, 953 years after the time of infinite redshift. The temperature then was T =9, 400 K or kT =0.81 eV. -16 ×10-4 ×10 4.5 5 4 4.5 4 3.5 ρm(t) ρr(t) a(t)/a0 3.5 3 ) 3 3 0 2.5 /a 2.5 ) t ( 2 ) (kg/m a t 2 ( ρ 1.5 1.5 1 1 0.5 0.5 0 0 0 10 20 30 40 50 60 70 80 90 100 t (kyr) Figure 11.3 The scale factor ratio a/a0 (solid), the radiation density ⇢r (dotdash), and the matter density ⇢m (dashed) are plotted as functions of the time (11.499) in kyr after the time of infinite redshift. The era of radiation ends at t = 50, 953 years when the two densities are equal to 16 3 4 1.055 10 kg/m , and a/a =2.933 10 . ⇥ 0 ⇥ 11.59 The era of matter (w =0) A universe composed only of dust or non-relativistic collisionless mat- ter has no pressure. Thus p = w⇢ =0with⇢ = 0, and so w = 0. Conser- 6 vation of energy (11.485), or equivalently (11.490), implies that the matter density falls with the volume as a 3 ⇢ = ⇢ 0 . (11.523) m m0 a ⇣ ⌘ 538 Tensors and Local Symmetries The era of matter began about 50,953 after the time of infinite redshift when the matter density ⇢m first exceeded the radiation density ⇢r. Most of the matter is of an unknown kind that interacts very weakly with photons and is called dark matter. Baryons amount to only about 15.7% of ⇢m,so if they were the principal kind of matter, the era of radiation would have lasted much longer. Near the middle of the era of matter, at t 5 Gyr, the radiation density ⇠ and the dark-energy density are unimportant, and we need take only the matter density into account in order to approximate the evolution of the scale factor. The density then varies as ⇢ ⇢ 1/a3, and so the quantity ⇠ m / 4⇡G⇢ a3 f = m (11.524) 3 is constant. The resulting Friedmann equations (11.469 & 11.470) are a¨ 4⇡G 3p 4⇡G⇢ f = ⇢ + = m ora ¨ = (11.525) a 3 m c2 3 a2 ✓ ◆ and a˙ 2 + c2k =2f/a. (11.526) For a flat universe, k = 0, we get the Einstein-de Sitter model 2/3 f a(t)= 3 t . (11.527) " r 2 # The scale factors of open and closed universes also rise as a(t) t2/3 as long ⇠ as a<2f/(c2 k ). | | Transparency: Some 380,000 years after inflation at a redshift of z = 1090, the universe had cooled to about T = 3000 K or kT =0.26 eV—a temperature at which less than 1% of the hydrogen is ionized. Ordinary matter became a gas of neutral atoms rather than a plasma of ions and electrons, and the universe suddenly became transparent to light. This moment of last scattering and first transparency often is (inexplicably) called recombination. 11.60 The era of dark energy (w = 1) 539 ×10-26 9 1 8 0.9 0.8 7 ρm(t) 0.7 6 ρΛ ) 0.6 3 − 0 5 /a 0.5 ) t ( 4 a )(kgm 0.4 t ( ρ 3 a(t)/a0 0.3 2 0.2 1 0.1 0 0 0 5 10 15 20 25 30 35 40 45 50 t (Gyr) Figure 11.4 The reduced scale factor a/a0 (solid), the vacuum density ⇢⇤ (dotdash), and the matter density ⇢m (dashed) are plotted as functions of the time (11.499) in Gyr after the time of infinite redshift. The era of matter ends at t = 10.193 Gyr (first vertical line) when the two densities are equal. The present time t0 is 13.8 Gyr (second vertical line) at which a(t)/a0 = 1. 11.60 The era of dark energy (w = 1) About 3.606 billion years ago or 10.193 Gyr after inflation at a redshift of z =0.3079, the matter density, falling as 1/a3, dropped below the dark- energy density ⇢ =6.0084 10 27 kg/m3 or (2.256 meV)4. The age of ⇤ ⇥ the universe is 13.799 billion years. For the past 3.606 billion years, this constant energy density, called dark energy, has accelerated the expansion of the universe approximately as in the de Sitter model (11.510) a(t)=a(t )exp (t t ) 8⇡G⇢ /3 (11.528) m m v ⇣ p ⌘ in which t = (10.427 0.07) 109 years. m ± ⇥ 540 Tensors and Local Symmetries 11.61 Yang-Mills theory The gauge transformation of an abelian gauge theory like electrodynam- ics multiplies a single charged field by a spacetime-dependent phase factor 0(x)=exp(iq✓(x)) (x). Yang and Mills generalized this gauge transfor- mation to one that multiplies a vector of matter fields by a spacetime dependent unitary matrix U(x) n a0 (x)= Uab(x) b(x) or 0(x)=U(x) (x) (11.529) Xb=1 and showed how to make the action of the theory invariant under such non- abelian gauge transformations. (The fields are scalars for simplicity.) Since the matrix U is unitary, inner products like †(x) (x) are automat- ically invariant 0 †(x) (x) = †(x)U †(x)U(x) (x)= †(x) (x). (11.530) ⇣ ⌘ i But inner products of derivatives @ † @i are not invariant because the derivative acts on the matrix U(x) as well as on the field (x). Yang and Mills made derivatives Di that transform like the fields (Di )0 = UDi . (11.531) To do so, they introduced gauge-field matrices Ai that play the role of the connections i in general relativity and set Di = @i + Ai (11.532) in which Ai like @i is antihermitian. They required that under the gauge transformation (11.529), the gauge-field matrix Ai transform to Ai0 in such a way as to make the derivatives transform as in (11.531) (Di )0 = @i + Ai0 0 = @i + Ai0 U = UDi = U (@i + Ai) . (11.533) So they set @i + Ai0 U = U (@i + Ai) or (@iU) + Ai0 U = UAi . (11.534) and made the gauge-field matrix Ai transform as 1 1 A0 = UA U (@ U) U . (11.535) i i i Thus under the gauge transformation (11.529), the derivative Di trans- forms as in (11.531), like the vector in (11.529), and the inner product of covariant derivatives i 0 i i D † Di = D † U †UDi = D † Di (11.536) h i 11.61 Yang-Mills theory 541 remains invariant. To make an invariant action density for the gauge-field matrices Ai,they used the transformation law (11.533) which implies that Di0 U = UDi or 1 Di0 = UDi U . So they defined their generalized Faraday tensor as F =[D ,D ]=@ A @ A +[A ,A ] (11.537) ik i k i k k i i k so that it transforms covariantly 1 Fik0 = UFikU . (11.538) ik They then generalized the action density FikF of electrodynamics to the ik trace Tr FikF of the square of the Faraday matrices which is invariant under gauge transformations since 1 ik 1 ik 1 ik Tr UFikU UF U =Tr UFikF U =Tr FikF . (11.539) ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ As an action density for fermionic matter fields, they replaced the ordi- i nary derivative in Dirac’s formula ( @i + m) by the covariant derivative i (11.532) to get ( Di + m) (Chen-Ning Yang 1922–, Robert L. Mills 1927–1999). i In an abelian gauge theory, the square of the 1-form A = Ai dx vanishes A2 = A A dxi dxk = 0, but in a nonabelian gauge theory the gauge fields i k ^ are matrices, and A2 = 0. The sum dA + A2 is the Faraday 2-form 6 F = dA + A2 =(@ A + A A ) dxi dxk (11.540) i k i k ^ = 1 (@ A @ A +[A ,A ]) dxi dxk = 1 F dxi dxk. 2 i k k i i k ^ 2 ik ^ The scalar matter fields may have self-interactions described by a po- 2 2 tential V ( ) such as V ( )= ( † m / ) which is positive unless † = 2 i m / . The kinetic action of these fields is (D )†Di . At low temperatures, these scalar fields assume mean values 0 0 = 0 in the vacuum with 2 h | | i 0† 0 = m / so as to minimize their potential energy density V ( ) and i i i their kinetic action (D )†Di =(@ + A )†(@i + Ai ) is approximately i i ↵ i 0† A Ai 0. The gauge-field matrix Aab = itabA↵ is a linear combination of the generators t↵ of the gauge group. So the action of the scalar fields i 2 i contains the term † A A = M A A in which the mass-squared 0 i 0 ↵ ↵ i 2 a ↵ c matrix for the gauge fields is M↵ = 0⇤ tab tbc 0.ThisHiggs mechanism gives masses to those linear combinations b i A of the gauge fields for which M 2 b = m2b =0. ↵ i i ↵i 6 The Higgs mechanism also gives masses to the fermions. The mass term m in the Yang-Mills-Dirac action is replaced by something like c in which c is a constant, di↵erent for each fermion. In the vacuum and at low temperatures, 542 Tensors and Local Symmetries each fermion acquires as its mass c 0 . On 4 July 2012, physicists at CERN’s Large Hadron Collider announced the discovery of a Higgs-like particle with a mass near 125 GeV/c2 (Peter Higgs 1929–). 11.62 Spin-one-half fields in general relativity The flat-space action density (10.305) for a spin-one-half field is L = ¯ [ a (@ + A )+m] (11.541) a a in which a is a flat-space index, Aa is a matrix of gauge fields, is a 4- 0 component Dirac or Majorana field, ¯ = † = i † , and m is a constant a or a mean value of a scalar field. One may use the tetrad fields eµ(x) of sec- tion 11.20 to turn the flat-space indices a into curved-space indices. Since derivatives and gauge fields intrinsically are generally covariant vectors, one a a µ replaces (@a + Aa)by ea (@µ + Aµ). The next step is to correct for the e↵ect of the derivative @µ on the field by making the derivative gener- ally covariant as well as gauge covariant. The required Einstein connection is (Weinberg, 1972, sec. 12.5) i E = J ab e ⌫ e (11.542) µ 2 a b⌫;µ in which the generators (10.300) of the Lorentz group J ab are the commu- tators of Dirac’s 4 4 gamma matrices (10.303) ⇥ i J ab = a, b , (11.543) 4 h i the covariant derivative of the tetrad eb⌫ is e = e e , (11.544) b⌫;µ b⌫,µ b ⌫µ a and the Levi-Civita a ne connection (11.239) is ⌫µ = e ae µ,⌫.Thusthe generally covariant, gauge-covariant action density of a spin-one-half field is a µ L0 = [ e (@ + A + E )] . (11.545) a µ µ µ 11.63 Gauge Theory and Vectors This section is optional on a first reading. 11.63 Gauge Theory and Vectors 543 We can formulate Yang-Mills theory in terms of vectors as we did rela- tivity. To accomodate noncompact groups, we will generalize the unitary matrices U(x) of the Yang-Mills gauge group to nonsingular matrices V (x) that act on n matter fields a(x) as n a a b 0 (x)= V b(x) (x). (11.546) a=1 X The field n a (x)= ea(x) (x) (11.547) a=1 X will be gauge invariant 0(x)= (x) if the vectors ea(x) transform as n 1b ea0 (x)= eb(x) V a(x). (11.548) Xb=1 In what follows, we will sum over repeated indices from 1 to n and often will suppress explicit mention of the spacetime coordinates. In this compressed notation, the field is gauge invariant because a 1b a c b c b 0 = ea0 0 = eb V a V c = eb c = eb = (11.549) T T 1 T which is e0 0 = e V V = e in matrix notation. The inner product of two basis vectors is an internal “metric tensor” N N N ↵ ↵ ↵ ↵ e⇤ e = e ⇤ ⌘ e = e ⇤ e = g (11.550) a · b a ↵ b a b ab ↵=1 ↵=1 X X=1 X in which for simplicity I used the the N-dimensional identity matrix for the metric ⌘. As in relativity, we’ll assume the matrix gab to be nonsingular. a ab We then can use its inverse to construct dual vectors e = g eb that satisfy ea e = a. † · b b The free Dirac action density of the invariant field i a i b i a a a b ( @ + m) = e †( @ + m)e = ( @ + e † e )+m i a i b a b i · b,i b h (11.551)i is the full action of the component fields b i i a a b i a a a b ( @i + m) = a( Dib + m b) = a ( b@i + Aib)+m b (11.552) ⇥ ⇤ if we identify the gauge-field matrix as Aa = ea e in harmony with the ib † · b,i definition (11.239) of the a ne connection k = ek e . i` · `,i 544 Tensors and Local Symmetries 1b Under the gauge transformation ea0 = eb V a, the metric matrix trans- forms as 1c 1d 1 1 gab0 = V ⇤a gcd V b or as g0 = V † gV (11.553) 1 1 in matrix notation. Its inverse goes as g0 = Vg V †. a a ac The gauge-field matrix A = e e = g ec† e transforms as ib † · b,i · b,i a ac a c 1d a 1c A0 = g0 e0† e0 = V A V + V V (11.554) ib a · b,i c id b c b,i 1 1 1 1 or as Ai0 = VAiV + V@iV = VAiV (@iV ) V . a a By using the identity e e = e † e , we may write (exercise 11.45) † · c,i ,i · c the Faraday tensor as a a a a c a a c F =[D ,D ] = e † e e † e e † e e † e +e † e e † e . (11.555) ijb i j b ,i · b,j ,i · c · b,j ,j · b,i ,j · c · b,i If n = N,then n ↵ c ↵ a ec e ⇤ = and Fijb =0. (11.556) c=1 X The Faraday tensor vanishes when n = N because the dimension of the embedding space is too small to allow the tangent space to have di↵erent orientations at di↵erent points x of spacetime. The Faraday tensor, which represents internal curvature, therefore must vanish. One needs at least three dimensions in which to bend a sheet of paper. The embedding space must have N>2 dimensions for SU(2), N>3 for SU(3), and N>5 for SU(5). The covariant derivative of the internal metric matrix g = g gA A†g (11.557) ;i ,i i i 1 1 does not vanish and transforms as (g;i)0 = V †g,iV . A suitable action 1 ;i 1 density for it is the trace Tr(g;ig g g ). If the metric matrix assumes a (constant, hermitian) mean value g0 in the vacuum at low temperatures, then its action is 2 1 i i 1 m Tr (g0Ai + Ai†g0)g0 (g0A + A †g0)g0 (11.558) which is a mass termh for the matrix of gauge bosons i 1/2 1/2 1/2 1/2 Wi = g0 Ai g0 + g0 Ai† g0 . (11.559) This mass mechanism also gives masses to the fermions. To see how, we write the Dirac action density (11.552) as i a a a b a i c b a ( b@i + Aib)+m b = (gab@i + gacAib)+mgab . (11.560) ⇥ ⇤ ⇥ ⇤ 11.64 Geometry 545 Each fermion now gets a mass mci proportional to an eigenvalue ci of the hermitian matrix g0. This mass mechanism does not leave behind scalar bosons. Whether Na- ture ever uses it is unclear. 11.64 Geometry This section is optional on a first reading. In gauge theory, what plays the role of spacetime? Could it be the group manifold? Let us consider the gauge group SU(2) whose group manifold is the 3-sphere in flat euclidian 4-space. A point on the 3-sphere is p = 1 r2,r1,r2,r3 (11.561) ± ⇣ p ⌘a as explained in example 10.31. The coordinates r = ra are not vectors. The three basis vectors are @p ra 1 2 3 ea = = , , , (11.562) @ra ⌥p1 r2 a a a ✓ ◆ and so the metric g = e e is ab a · b r r g = a b + (11.563) ab 1 r2 ab or 2 2 1 r2 r3 r1 r2 r1 r3 1 2 2 g = 2 r2 r1 1 r1 r3 r2 r3 . (11.564) k k 1 r 0 2 21 r3 r1 r3 r2 1 r r 1 2 @ A The inverse matrix is gbc = r r . (11.565) bc b c The dual vectors eb = gbce = r 1 r2, b r r , b r r , b r r (11.566) c ⌥ b 1 b 1 2 b 2 3 b 3 ⇣ p ⌘ satisfy eb e = b. · a a There are two kinds of a ne connections eb e and eb e .Ifwe · a,c · a,i di↵erentiate ea with respect to an SU(2) coordinate rc,then r r Eb = eb e = r + a c (11.567) ca · a,c b ac 1 r2 ✓ ◆ 546 Tensors and Local Symmetries in which we used E (for Einstein) instead of for the a ne connection. If i we di↵erentiate ea with respect to a spacetime coordinate x ,then r r Eb = eb e = eb e rc = r rc + a c . (11.568) ia · a,i · a,c ,i b ,i ac 1 r2 ✓ ◆ But if the group coordinates ra are functions of the spacetime coordinates i x , then there are 4 new basis 4-vectors ei = eara,i.Themetricthenisa 7 7 matrix g with entries g = e e , g = e e , g = e e , and ⇥ k k a,b a · b a,k a · k i,b i · b g = e e or i,k i · k g g r g = a,b a,b b,k (11.569) k k g r g r r . ✓ a,b a,i a,b a,i b,k ◆ Further Reading The classics Gravitation and Cosmology (Weinberg, 1972), Gravitation (Mis- ner et al., 1973), Cosmology (Weinberg, 2010), General Theory of Relativ- ity (Dirac, 1996), Gravitation: Foundations and Frontiers (Padmanabhan, 2010) and the very accessible Spacetime and Geometry (Carroll, 2003), A First Course in General Relativity (Schutz, 2009), and Einstein Gravity in a Nutshell (Zee, 2013) are of special interest. Exercises 11.1 Compute the derivatives (11.22 & 11.23). 11.2 Show that the transformation x x defined by (11.17) is a rotation ! 0 and a reflection. 11.3 Show that the matrix (11.39) satisfies the Lorentz condition (11.38). T 1 11.4 If ⌘ = L⌘L , show that ⇤= L satisfies the definition (11.38) of a Lorentz transformation ⌘ =⇤T⌘ ⇤. 11.5 The LHC is designed to collide 7 TeV protons against 7 TeV protons for a total collision energy of 14 TeV. Suppose one used a linear accelerator to fire a beam of protons at a target of protons at rest at one end of the accelerator. What energy would you need to see the same physics as at the LHC? 11.6 Use Gauss’s law and the Maxwell-Amp`ere law (11.86) to show that the microscopic (total) current 4-vector j =(c⇢, j) obeys the continuity equation⇢ ˙ + j = 0. r · 11.7 Show that if Mik is a covariant second-rank tensor with no particular symmetry, then only its antisymmetric part contributes to the 2-form Exercises 547 i k Mik dx dx and only its symmetric part contributes to the quantity i ^ k Mik dx dx . 11.8 In rectangular coordinates, use the Levi-Civita identity (1.455) to de- rive the curl-curl equations (11.89). 11.9 Derive the Bianchi identity (11.91) from the definition (11.78) of the Faraday field-strength tensor, and show that it implies the two homo- geneous Maxwell equations (11.81). 11.10 Show that if A is a p-form, then d(AB)=dA B +( 1)pA dB. ^ ^ 11.11 Show that if ! = a dxi dxj/2witha = a ,then ij ^ ij ji 1 d! = (@ a + @ a + @ a ) dxi dxj dxk. (11.570) 3! k ij i jk j ki ^ ^ 11.12 (a) Derive the metric (??) for the hyperboloid H2 from the inner products of e and e by using the metric ⌘ = diag(1, 1, 1). (b) r Derive the line element (??) and metric (??) of the hyperboloid H2 from its line element (11.153) in the embedding space R3. 11.13 Using tensor notation throughout, derive (11.170) from (11.168 & 11.169). 11.14 Use the flat-space formula (11.191) to compute the change dp due to d⇢, d , and dz, and so derive the expressions (11.192) for the orthonor- mal basis vectors ⇢ˆ, ˆ, and zˆ. 11.15 Similarly, derive (11.200) from (11.198). 11.16 Use the definition (11.215) to show that in flat 3-space, the dual of the Hodge dual is the identity: dxi = dxi and (dxi dxk)=dxi dxk. ⇤⇤ ⇤⇤ ^ ^ 11.17 Use the definition of the Hodge star (11.226) to derive (a) two of the four identities (11.227) and (b) the other two. 11.18 Show that Levi-Civita’s 4-symbol obeys the identity (11.231). pmn p 11.19 Show that ✏`mn ✏ =2 ` . p`mn p 11.20 Show that ✏k`mn ✏ = 3! k. 11.21 (a) Using the formulas (11.200) for the basis vectors of spherical co- ordinates in terms of those of rectangular coordinates, compute the derivatives of the unit vectors rˆ, ✓ˆ, and ˆ with respect to the variables r, ✓, and and express them in terms of the basis vectors rˆ, ✓ˆ, and ˆ. (b) Using the formulas of (a) and our expression (6.28) for the gradient in spherical coordinates, derive the formula (11.342) for the laplacian . r · r 11.22 Consider the torus with coordinates ✓, labeling the arbitrary point p = (cos (R + r sin ✓), sin (R + r sin ✓),rcos ✓) (11.571) 548 Tensors and Local Symmetries in which R>r. Both ✓ and run from 0 to 2⇡. (a) Find the basis vectors e✓ and e . (b) Find the metric tensor and its inverse. 11.23 For the same torus, (a) find the dual vectors e✓ and e and (b) find i the nonzero connections jk where i, j, & k take the values ✓ & . 11.24 For the same torus, (a) find the two Christo↵el matrices ✓ and , ✓ (b) find their commutator [ ✓, ], and (c) find the elements R✓✓✓, R✓ ✓, ✓ R ✓ , and R of the curvature tensor. 11.25 Find the curvature scalar R of the torus with points (11.571). Hint: In these four problems, you may imitate the corresponding calculation for the sphere in Sec. 11.47. 11.26 By di↵erentiating the identity gik g = i, show that gik = k` ` gisgkt g or equivalently that dgik = gisgktdg . st st 11.27 Just to get an idea of the sizes involved in black holes, imagine an isolated sphere of matter of uniform density ⇢ that as an initial con- dition is all at rest within a radius rb. Its radius will be less than its Schwarzschild radius if 2MG 4 G r < =2 ⇡r3⇢ . (11.572) b c2 3 b c2 ✓ ◆ If the density ⇢ is that of water under standard conditions (1 gram per cc), for what range of radii rb might the sphere be or become a black hole? Same question if ⇢ is the density of dark energy. 11.28 For the points (11.451), derive the metric (11.454) with k = 1. Don’t forget to relate d to dr. 11.29 For the points (11.452), derive the metric (11.454) with k = 0. 11.30 For the points (11.453), derive the metric (11.454) with k = 1. Don’t forget to relate d to dr. 11.31 Suppose the constant k in the Roberson-Walker metric (11.450 or 11.454) is some number other than 0 or 1. Find a coordinate transfor- ± mation such that in the new coordinates, the Roberson-Walker metric has k = k/ k = 1. Hint: You also can change the scale factor a. | | ± 11.32 Derive the a ne connections in Eq.(11.458). 11.33 Derive the a ne connections in Eq.(11.459). 11.34 Derive the a ne connections in Eq.(11.460). 11.35 Derive the spatial Einstein equation (11.470) from (??, 11.454, 11.465, 11.467, & 11.468). 11.36 Assume there had been no inflation, no era of radiation, and no dark energy. In this case, the magnitude of the di↵erence ⌦ 1 would have | | increased as t2/3 over the past 13.8 billion years. Show explicitly how Exercises 549 close to unity ⌦would have had to have been at t = 1 s so as to satisfy the observational constraint ⌦ 1 < 0.036 on the present value of ⌦. | 0 | 11.37 Derive the relation (11.488) between the energy density ⇢ and the Robertson-Walker scale factor a(t) from the conservation law (11.485) and the equation of state pi = wi⇢i. 11.38 Use the Friedmann equations (11.469 & 11.493) for constant ⇢ = p and k = 1 to derive (11.509) subject to the boundary condition that a(t) has its minimum at t = 0. 11.39 Use the Friedmann equations (11.469 & 11.493) with w = 1, ⇢ con- stant, and k = 1 to derive (11.510) subject to the boundary condition that a(0) = 0. 11.40 Use the Friedmann equations (11.469 & 11.493) with w = 1, ⇢ constant, and k = 0 to derive (11.508). Show why a linear combination of the two solutions (11.508) does not work. 11.41 Use the conservation equation (11.514) and the Friedmann equations (11.469 & 11.493) with w =1/3, k = 0, and a(0) = 0 to derive (11.517). 11.42 Show that if the matrix U(x) is nonsingular, then 1 1 (@ U) U = U@ U . (11.573) i i 11.43 The gauge-field matrix is a linear combination A = ig tb Ab of the k k generators tb of a representation of the gauge group. The generators obey the commutation relations a b c [t ,t ]=ifabct (11.574) in which the fabc are the structure constants of the gauge group. Show that under a gauge transformation (11.535) 1 1 A0 = UA U (@ U) U (11.575) i i i by the unitary matrix U =exp( ig ata)inwhich a is infinitesimal, the gauge-field matrix Ai transforms as a a a a 2 a b c a a igA0 t = igA t ig f A t + ig@ t . (11.576) i i abc i i Show further that the gauge field transforms as a a a b c A0 = A @ gf A . (11.577) i i i abc i a a 11.44 Show that if the vectors e (x) are orthonormal, then e e = e † e . a †· c,i ,i · c 11.45 Use the identity of exercise 11.44 to derive the formula (11.555) for the nonabelian Faraday tensor. 550 Tensors and Local Symmetries 1 ik 11.46 Using the tricks of section 11.40, show that pg = 2 pggik g . ik n This relation and the definition (11.399) R = g Rink imply that the first-order change in the Einstein-Hilbert action is (11.426) apart from ik n an irrelevant surface term (Carroll, 2003, chap 4.3) due to g pg Rink. 11.47 Write Dirac’s action density in the explicitly hermitian form LD = 1 i@ 1 i@ † in which the field has the invariant form 2 i 2 i 0 i i = ea a and ⇥= i † ⇤ . Use the identity a b † = b a to i show that the gauge-field matrix Ai defined as the coe cient of a b i ⇥ ⇤ as in a (@i + iAiab) b is hermitian Aiab⇤ = Aiba.