Variational Principle and Einstein s equations

Transcription

1 Chapter 15 Variational Principle and Einstein s equations 15.1 An useful formula There exists an useful equation relating g µν, g µν and g = det(g µν ) : g x α = ggµν g µν x α. (15.1) The proof is the following. The determinant g is a polynomial in g µν Let us first compute the derivatives g = g(g µν ). (15.2) g g µν. (15.3) We remind that g can be expressed by fixing one row, i.e. µ, and computing the expansion g = ν g µν M µν ( 1) µ+ν (no sum over µ) (15.4) where M µν is the minor µ, ν, i.e. the determinant of the matrix obtained by cutting the row µ and the column ν from the matrix g µν. From (15.4) we find that g g µν = ( 1) µ+ν M µν (15.5) where, again, we are not summing on the indices µ, ν. Furthermore, the components of g µν, the matrix inverse of (g µν ), are given by g µν = 1 g M µν( 1) µ+ν. (15.6) Therefore, g g µν = gg µν, (15.7) 208

2 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 209 then g x α = and (15.1) is proved. The same rule applies to variations: thus Furthermore, since multiplying with g µρ, we find Therefore, equation (15.10) gives g g µν g µν x α = ggµν g µν x α (15.8) δg = g g µν δg µν (15.9) δg = gg µν δg µν. (15.10) δ(g µν g νσ ) = δ(δ σ µ ) = 0 = δg µν g νσ + g µν δg νσ (15.11) δg ρσ = g µρ g νσ δg µν. (15.12) 15.2 Gauss Theorem in curved space δg = gg µν δg µν. (15.13) First of all we give a preliminary definition. - Given a manifold M described by coordinates {x µ }, and a metric g µν on M. - Given a submanifold N M described by coordinates {y i }, such that on N x µ = x µ (y i ). We define the metric induced on N from M as γ ij xµ y i x ν y j g µν. (15.14) We can now generalize Gauss theorem to curved space: - Be Ω an n-dimensional volume described by coordinates {x µ } µ=0,...,n 1, and g µν the metric on Ω. - Be Ω the boundary of Ω, described by coordinates {y j } j=0,...,n 2 with normal vector n µ (having n µ n µ = 1 for a timelike or spacelike surface); be γ ij the metric induced on Ω from g µν. - Given a vector field V µ defined in Ω, then d 4 x g V µ ;µ = If we define the surface integration element as the Gauss theorem can also be written as d 4 x g V µ ;µ = Ω Ω Ω d 3 y γ V µ n µ. (15.15) ds µ γn µ d 3 y, (15.16) Ω V µ ds µ. (15.17) In particular, if one considers an infinite volume, and if V µ vanishes asymptotically, then the integral of its covariant divergence is zero.

3 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS Diffeomorphisms on a manifold and Lie derivative Given an n-dimensional differentiable manifold M, let us consider a mapping Ψ of the manifold M into itself Ψ : M M, (15.18) in which we map any point P M to another point Q of the same manifold M: P Ψ(P ) = Q. (15.19) If such a map is invertible and regular, as we will always assume in the following, the mapping is named diffeomorphism. In a local coordinate frame that, by definition, is a mapping of an open subset of M to IR n (cfr. Chapter 2 section 2.5) P {x µ (P )} µ=1,...,n (15.20) a diffeomorphism Ψ can be expressed as a set of n real functions on IR n, {ψ µ }, that acting on the coordinates of the point P, {x α }, produce the coordinates of the point Q, {x α } in the same coordinate frame i.e. ψ : x α x α = ψ α (x). (15.21) The functions ψ α (x µ ) are smooth and invertible. Notice that the transformation (15.21) has the same form of a general coordinate transformation, but in the present context its meaning is very different: in a coordinate transformation we assign to the same point P of the manifold two different n-ples of IR n ; thus, in this case the mapping ψ denotes, as shown in Fig. 15.1, the set of functions which allows to transform from the coordinates {x α } to the coordinates {x α }, i.e. it is a mapping of IR n to IR n. M P ζ x IR n η ψ IR n x Figure 15.1: A coordinate transformation: ζ and η are the maps from M to IR n that associate differents sets of coordinates (i.e. different n-ples of IR n ) to the same point P. Conversely, given a diffeomorphism Ψ which maps P to Q, where P and Q are two different points of the same manifold M, we use the same coordinate frame to label both

4 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 211 P and Q as shown in Fig To hereafter we will use the following convention: when dealing with a coordinate transformation we shall write, as usual ψ : x α x α = ψ α (x), (15.22) while,a diffeomorphism Ψ : M M expressed in a coordinate frame will be indicated as i.e. x α indicate the coordinates of the point Q. ψ : x α x α = ψ α (x). (15.23) M P Ψ ζ ζ x ψ x RI n Q Figure 15.2: A diffeomorphism Ψ on the manifold, and its representation in a coordinate frame, ψ. The map of M to the coordinate frame is denoted by ζ The action of diffeomorphisms on functions and tensors Be f a real function defined on a manifold M. A diffeomorphism Ψ changes f into a different function, which is called the pull-back of f, and is usually denoted by the symbol Ψ f: such that i.e. f Ψ f (15.24) (Ψ f)(p ) = f(q) (Ψ f)(p ) = f(ψ(p )) (15.25) Ψ f f Ψ. (15.26) In a coordinate frame, where P has coordinates x µ and Q has coordinates x µ, eq. (15.25) takes the form (ψ f)(x) = f(x ) (ψ f)(x) = f(ψ(x)). (15.27) The function Ψ f(p ), which is equal to f(q), will be different from f(p ); the difference is δf(p ) = (Ψ f)(p ) f(p ) = f(q) f(p ) = f(ψ(p )) f(p ) (15.28) or, in a coordinate frame, δf(x) = (ψ f)(x) f(x) = f(ψ(x)) f(x). (15.29)

5 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 212 M P f f(p) Ψ Q Ψ * f f f(q)= Ψ * f(p) RI Figure 15.3: The mapping f from the manifold M to IR, and its pull-back Ψ f. In a similar way, the diffeomorphism Ψ also induces a change on tensors: due to the action of Ψ, a tensor field T changes to a new tensor field called the pull-back of T, denoted by Ψ T, such that (Ψ T)(P ) = T(Q). (15.30) The difference between the pull-back and the original tensor, evaluated in a given point P of the manifold, is An important point to stress δt(p ) = (Ψ T)(P ) T(P ) = T(Ψ(P )) T(P ). (15.31) As discussed in Chapter 3, a tensor field depends on the point of the manifold, but also on vectors and one-forms defined in the tangent space (and its dual) at that point of the manifold. In eq. (15.30) we have explicitly written the dependence on the point of the manifold, but we have left implicit the dependence on vectors and one-forms. This dependence can be made explicit as follows: T acts on vectors and one-forms defined in the tangent space (and in its dual) in Q, while Ψ T acts on vectors and one-forms which are defined in the tangent space (and in its dual) in P. The complete definition of the pull-back Ψ T therefore is [ (Ψ T)(Ψ V,..., Ψ q,...) (P ) = [ T( V,..., q,...) (Q). (15.32) In this definition there are the pull-backs of vectors and one-forms, Ψ V and Ψ q, which we still have to define. To this purpose, we remind that vectors are in one-to-one correspondence with directional derivatives (see Chapter 3); indeed a vector V { dxµ } tangent to a given curve with dλ parameter λ, associates to any function f the directional derivative V (f) df dλ = f x µ dx µ dλ = V µ f x µ. Thus, we define the pull-back of a vector Ψ V as follows: for any function f [ (Ψ V )(Ψ f) (P ) = [ V (f) (Q). (15.33)

6 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 213 Let us now consider one-forms. By definition a one-form q associates to any vector V a real number q( V ). The pull-back of a one-form, Ψ q is defined as follows: for any vector V, [ (Ψ q)(ψ V ) (P ) = [ q( V ) (Q). (15.34) In particular, the pull-backs of the coordinate basis vectors and of the coordinate basis one-forms are defined as follows: the pull-back maps the coordinate basis (of vectors or one-forms) in Q in the coordinate basis in P : where Ψ = x α x = x µ (15.35) α x α x µ Ψ dx α = dx α = xα x µ dx µ (15.36) x µ x = ψµ α x, (15.37) α and ψ α (x) are the functions defined in eq. (15.21), whereas x α / x µ are the derivatives of the inverse functions The action of diffeomorphisms on the metric tensor The pull-back of the metric tensor is, by definition, (Ψ g)(p ) = g(q). (15.38) Let us write eq. (15.38) in components. The tensor Ψ g, evaluated in P, must be expanded in terms of the coordinate basis of one-forms in P, i.e. dx α, while the tensor g must be expanded in terms of the coordinate basis of one-forms in Q, which is dx µ, therefore (ψ g) αβ (x) dx α dx β = g µν (x ) dx µ dx ν (15.39) and using eq. (15.36) we find (ψ g) αβ (x) = x µ x α x ν x β g µν(ψ(x)). (15.40) The change induced by Ψ on the metric tensor therefore is δg αβ (x) = (ψ g) αβ (x) g αβ (x) = x µ x α x ν x β g µν(ψ(x)) g αβ (x). (15.41) Lie derivative Let us consider an infinitesimal diffeomorphism Ψ, which in a coordinate frame is ψ µ (x) = x µ + ɛ ξ µ (x), (15.42)

7 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 214 where ξ µ is a vector field and ɛ is a small parameter. To hereafter we shall neglect O(ɛ 2 ) terms. From eq. (15.42) it follows that x µ = x µ + ɛξ µ. (15.43) According to eq. (15.29) the change induced by Ψ on a function f is δf(x) = f(x + ɛξ) f(x) = ɛξ µ f x µ. (15.44) We define the Lie derivative of a scalar function f with respect to the infinitesimal diffeomorphism associated to the vector field ξ, as or, in coordinates, Ψ f(p ) f(p ) f(ψ(p )) f(p ) L ξ (f) lim = lim, (15.45) ɛ 0 ɛ ɛ 0 ɛ f(x + ɛξ) f(x) L ξ (f) lim ɛ 0 ɛ Therefore, eq. (15.44) can also be written as = ξ µ f x µ. (15.46) δf = ɛl ξ f. (15.47) In a similar way we can define the Lie derivative of a tensor field T with respect to the vector field ξ as L ξ T lim ɛ 0 Ψ T(P ) T(P ) ɛ T(Ψ(P )) T(P ) = lim. (15.48) ɛ 0 ɛ Therefore, from eq. (15.31) we see that the change induced on a tensor by an infinitesimal diffeomorphism is given by the Lie derivative: In the case of the metric tensor, since from eq.(15.42) equation (15.41) gives x µ x ν δt = ɛl ξ T. (15.49) = δµ ν + ɛ ξµ x ν, (15.50) ( ) ( ) δg αβ (x) = δ α µ + ɛ ξµ δ ν x α β + ɛ ξν g x β µν (x + ɛξ) g αβ (x) [ ξ µ = ɛ x g α µβ + ξν x g β αν + ξ ρ g αβ, (15.51) x ρ therefore L ξ g µν = ξµ x α g µβ + ξν x β g αν + ξ ρ g αβ x ρ. (15.52)

8 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 215 This expression has already been derived in Chapter 7 when we studied how the metric tensor changes under infinitesimal translations along a given curve of tangent vector ξ x µ x µ + ξ µ (x)dλ (15.53) Indeed, an infinitesimal translation is an infinitesimal diffeomorphism; thus the derivation of eqs. (15.51) and (15.52) repeats, in a more rigorous way, the derivation presented in Chapter 8.4. Furthermore, as shown in Chapter 8.4, the expression (15.52) can be rewritten in terms of covariant derivatives of ξ L ξ g µν = ξ µ;ν + ξ ν;µ. (15.54) General Covariance and the role of diffeomorphisms Since the equation ψ : x α x α = ψ α (x) (15.55) allows a double interpretation as a general coordinate transformation and as a diffeomorphism on the spacetime manifold, it follows that to any coordinate transformation we can associate a diffeomorphism, and viceversa. The principle of general covariance states that since the laws of physics are expressed by tensorial equations, they retain the same form in any coordinate frame, i.e. they are invariant under a general coordinate transformation; since a coordinate transformation corresponds, in the sense explained above, to a diffeomorphism, we can restate the principle of general covariance as follows: all physical laws are invariant for diffeomorphisms, i.e. we restate general covariance as symmetry principle defined on the manifold Variational approach to General Relativity In the variational approach to field theory, the dynamics of fields is described by the action functional Action principle in special relativity Let us consider a collection of tensor (and, eventually, spinor) fields in special relativity { Φ (A) (x) } A=1,..., (15.56) where x denotes the point of coordinates {x µ }. We shall use symbols in boldface to denote a generic tensorial or spinorial object. For instance, for a vector field we shall write and summation over the tensor indices will be left implicit: V V = (V µ ), (15.57) δv V µ δv µ. (15.58)

9 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 216 It should be noted that here we use a convention different from that used for vector fields in Chapter 2.6. Typically, the fields involved are scalars, rank-one tensors (vectors and one-forms) and spinors. The action is a functional of these fields and of their first derivatives, written as an integral of a Lagrangian density over the 4-dimensional volume: I = d 4 x L ( Φ (1) (x),..., Φ (A) (x),..., µ Φ (1) (x),..., µ Φ (A) (x),... ). (15.59) To hereafter µ. All fields are assumed to vanish on the boundary of the integration x µ volume or asymptotically, if the volume is infinite. Let us consider the variation of the action with respect to a given field Φ (A) δi = = = = d 4 x δl δφ(a) A δφ (A) d 4 x ( ) A Φ (A) δφ(a) + ( α Φ (A) ) δ αφ (A) d 4 x ( ) A Φ (A) δφ(a) + ( α Φ (A) ) αδφ (A) d 4 x ( ) A Φ (A) α δφ (A). ( α Φ (A) ) (15.60) Here we have used the general property that the operations of variation and differentiation commute, and then we have integrated by parts. The stationarity of δi with respect to the considered field gives the equation of motion for that field δi = 0, δφ (A), and since the integral (15.60) has to vanish for every δφ (A) (x), it follows that which are the Euler-Lagrange equations for the field Φ (A) Action principle in general relativity Φ (A) α ( α Φ (A) ) = 0 (15.61) In general relativity, besides the fields { Φ (A)}, which are the matter and gauge fields, there is the metric field g(x) = (g µν (x)) (15.62) which describes the gravitational field whose action is the Einstein-Hilbert action I E H = c3 16πG d 4 x g R. (15.63)

10 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 217 { Due to } the strong equivalence principle, in a locally inertial frame the dynamics of all fields Φ (A) except gravity is described by the action (15.59). Therefore, according to the principal of general covariance in a general frame the action, which is a scalar, retains the same form provided η µν g µν, the partial derivatives α are replaced by covariant derivatives α, and the integration volume element d 4 x is replaced by the covariant volume element gd 4 x. With these replacements, we shall now show that the results of the previous section (in particular, the derivation of Euler-Lagrange equations) remain valid. The total action is I = I E H + I F IELDS (15.64) with I F IELDS = d 4 x gl ( F IELDS Φ (1) (x),,..., Φ (A) (x),..., α Φ (1) (x),..., α Φ (A) (x),..., g ). (15.65) Notice that now the Lagrangian density L F IELDS depends explicitely on g because we have replaced η µν by g µν and α by α. As in special relativity, the equations for a field Φ (A) are found by varying the action with respect to that field, and since the Einstein-Hilbert action does not depend on Φ δi δi F IELDS = = d 4 x g A = d 4 x g A = d 4 x g A δl F IELDS δφ (A) δφ (A) d 4 x g A ( ) Φ (A) δφ(a) + ( α Φ (A) ) δ αφ (A) ( ) Φ (A) δφ(a) + ( α Φ (r) ) αδφ (A) ( ) Φ (A) α δφ (A) = 0, ( α Φ (A) ) δφ (A) (15.66) where we have used the property δ α = α δ. To obtain the last row of eq. (15.66) we have integrated by parts using the generalization of Gauss theorem in curved space (see Section 15.2) which assures that the integral of a covariant divergence is zero, provided all fields vanish at the boundary, or asymptotically if the volume is infinite, i.e. d 4 x g V µ ;µ = 0. (15.67) Thus, the equations of motion for the field Φ (A) are the Euler-Lagrange equations generalized in curved space: Φ (A) α ( α Φ (A) ) = 0. (15.68)

11 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 218 The field equations for the gravitational field are obtained by varying the action (15.64) with respect to g. In section we shall show that the variation of the Einstein-Hilbert action with respect to g gives the Einstein tensor; here we just write the result: δi E H = c3 16πG d 4 x δ( gr) = c3 16πG d 4 x g ( R µν 1 ) 2 g µνr δg µν. (15.69) The variation of I F IELDS with respect to g it is easy to find if we use the property (15.13), from which δ( g) 1/2 = 1 2 ( g)1/2 g µν δg µν, (15.70) thus δi F IELDS = d 4 x g [ δl F IELDS δg µν 1 2 LF IELDS g µν δg µν. (15.71) Combining eqs. (15.69) and (15.71), and defining the stress-energy tensor as [ δl F IELDS T µν 2c 1 δg µν 2 LF IELDS g µν, (15.72) the variation of the total action (15.64) can be written as δi = c3 16πG d 4 x g [ G µν 8πG c T 4 µν δg µν = 0. (15.73) Thus, with this definition of T µν the action principle gives Einstein s equations G µν = 8πG c 4 T µν. (15.74) The advantage of deriving field equations using a variational approach is that it makes explicit the connection between symmetries and conservation laws: any symmetry of a theory corresponds to a conservation law. In Section 15.3 we have shown that the principle of general covariance implies that equations expressing the laws of physics are invariant for diffeomorphisms. If they can be derived from an Action Principle, this is equivalent to impose that the action is invariant for diffeomorphisms. As an example, let us consider the action I F IELDS. It is invariant for diffeomorphisms, because the equations of motion (15.68) for the fields Φ (A), arising from δi F IELDS = 0, are diffeomorphism invariant. We will now show that the diffeomorphism invariance of I F IELDS implies the divergenceless equation T µν ;ν = 0, which generalizes the conservation law T µν,ν = 0 of flat spacetime. Let us consider an infinitesimal diffeomorphism (see eq ) x µ x µ = x µ + ɛξ µ. (15.75) In Section 15.3 we have shown how tensor fields change under infinitesimal diffeomorphisms, and in particular that the change of the metric tensor, written in components, is (eq ): δg µν = ɛ [ξ µ;ν + ξ ν;µ. (15.76)

12 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 219 The change in g µν can be derived by using the property (15.12): δg µν = ɛ [ξ µ;ν + ξ ν;µ. (15.77) The diffeomorphism (15.75) will produce a variation of the fields Φ (A) s, and a variation of the metric tensor given by (15.77); consequently I F IELDS, defined in eq. (15.65), will vary as follows δi F IELDS = d 4 x [ δl F IELDS g 1 δg µν 2 LF IELDS g µν δg µν + d 4 x g A δl F IELDS δφ (A) δφ (A). (15.78) Since the action I F IELDS is invariant under diffeomorphisms, δi F IELDS must vanish. The term in square brackets is the stress-energy tensor T µν defined in eq. (15.72). Therefore, using eq. (15.77), we find δi F IELDS = ɛ 2c d 4 x gt µν 2ξ µ;ν + d 4 x g A δl F IELDS δφ (A) δφ (A) = 0 (15.79) where we have used: T µν (ξ µ;ν + ξ ν;µ ) = 2T µν ξ µ;ν, which follows from the symmetry of T µν. Since the fields Φ (A) s satisfy their equations of motion (15.68) the last integral in eq. (15.79) vanishes. Thus δi F IELDS = ɛ d 4 x g T µν ξ µ;ν = ɛ d 4 x g T µν ;ν c c ξ µ = 0 ξ µ (15.80) and consequently T µν ;ν = 0. (15.81) We stress that eq. (15.81) is not satisfied for all field configurations, but only for the field configurations which are solutions of the field equations, i.e. of the Euler-Lagrange equations (15.68). The diffeomorphism invariance of the Einstein-Hilbert action I E H, instead, implies the Bianchi identities. Indeed, the variation of I E H is given by (15.69) δi E H = c3 16πG d 4 x g G µν δg µν (15.82) and if the variation is due to an infinitesimal diffeomorphism, from (15.77) δg µν = ɛ [ξ µ;ν + ξ ν;µ (15.83) thus, integrating by parts, δi E H = ɛc3 16πG d 4 x g G µν 2ξ µ;ν = ɛc3 8πG and consequently we find the Bianchi identities d 4 x g G ;ν µν ξ µ = 0 ξ µ (15.84) G µν ;ν = 0. (15.85)

13 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS Variation of the Einstein-Hilbert action In this section we shall prove that the variation of the Einstein-Hilbert action with respect to the metric tensor gives the Einstein tensor (equation (15.69)): δ( gr) = As a first step we derive the Palatini identity: By varying the Ricci tensor: we find g ( R µν 1 ) 2 g µνr δg µν. (15.86) δr µν = (δγ λ µν ) ;λ (δγ λ µλ ) ;ν. (15.87) R µν = Γ λ µν,λ Γλ µλ,ν + Γα µν Γλ αλ Γλ αν Γα µλ, (15.88) δr µν = δγ λ µν,λ δγλ µλ,ν + δγα µν Γλ αλ δγ λ αν Γα µλ + Γα µν δγλ αλ Γλ αν δγα µλ. (15.89) To evaluate this expression, we need to compute δγ λ µν. To this purpose, we will use the property (15.12) δg λδ = g ρλ g σδ δg ρσ. (15.90) If we define Γ µν δ g δλ Γ λ µν = 1 2 (g µδ,ν + g νδ,µ g µν,δ ) (15.91) we can write the variation of Christoffel s symbols as follows δγ λ µν = δ [ g λδ Γ µν δ = δg λδ Γ µν δ + g λδ δγ µν δ = g ρλ g σδ δg ρσ Γ µν δ + g λρ δγ µν ρ = g λρ δg ρσ Γ σ µν + gλρ 1 2 [δg µρ,ν + δg νρ,µ δg µν,ρ = 1 2 gλρ [ δg µρ,ν + δg νρ,µ δg µν,ρ 2Γ σ µν δg ρσ (15.92) which can be recast in the form δγ λ µν = 1 [( 2 gλρ δg µρ,ν Γ α µν δg αρ Γ α νρ δg ( αµ) + δgνρ,µ Γ α νµ δg αρ Γ α ρµ δg αν ( δg µν,ρ Γ α µρ δg αν Γ α νρ δg ) αµ = 1 2 gλρ [δg µρ;ν + δg νρ;µ δg µν;ρ. (15.93) Eq. (15.93) shows that since δg µν is a tensor, δγ λ µν is also a tensor. Therefore, the expression (15.87), (δγ λ µν ) ;λ (δγ λ µλ ) ;ν (15.94) )

14 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 221 can be evaluated with the usual rules of covariant differentiation of tensors. Thus we find (δγ λ µν ) ;λ (δγ λ µλ ) ;ν = δγ λ µν,λ δγλ µλ,ν + δγα µν Γλ αλ δγ λ αν Γα µλ + Γα µν δγλ αλ Γλ αν δγα µλ. (15.95) A comparison of this equation with eq. (15.89)) shows that δr µν = (δγ λ µν ) ;λ (δγ λ µλ ) ;ν. QED. Let us now go back to the variation of the Einstein-Hilbert action Using eq.(15.70) we find δ( gr) = g δ( gr) = δ( gg µν R µν ). (15.96) Using the fact that g µν;α = 0, the first term in (15.97) becomes ( g µν δr µν + δg µν R µν 1 ) 2 g µνδg µν R. (15.97) g µν δr µν = g [ µν (δγ λ µν ) ;λ (δγ λ µλ ) ;ν = (g µν δγ λ µν ) ;λ (g µν δγ λ µλ ) ;ν = ( g µν δγ α µν ) gµα δγ λ µλ ;α (15.98) which is the divergence of a vector; therefore by Gauss theorem such term vanishes when integrated over the 4-volume gd 4 x. The remaining two terms in (15.97) give the Einstein tensor, therefore we can write δ( gr) = ( g R µν 1 ) 2 g µνr δg µν + surface terms (15.99) which shows that eq. (15.86) is true An example: the electromagnetic field The Lagrangian density of the electromagnetic field is with L = 1 4c F µαf νβ g µν g αβ (15.100) F µν = µ A ν ν A µ = A ν;µ A µ;ν (15.101) (the last equality arises from the symmetry property of the Christoffel symbols Γ α µν = Γ α νµ). The field equations for the electromagnetic field A µ are c δl δa ν = 1 2 F αβ δf αβ δa ν = 1 2 F αβ δ δa ν (A β;α A α;β ) = F αβ δ A α;β = F νβ ;β + surface terms ; (15.102) δa ν

15 CHAPTER 15. VARIATIONAL PRINCIPLE AND EINSTEIN S EQUATIONS 222 as usual, we eliminate the surface terms on the assumption that the fields vanish at the boundary of a given volume, or at infinity. The field equations then are and the stress-energy tensor is F αβ ;α = 0, (15.103) [ δl T µν = 2c δg 1 µν 2 Lg µν [ = F µαfν α g µνf αβ F αβ = F µα F α ν 1 4 g µνf αβ F αβ. (15.104) A comment on the definition of the stress-energy tensor In special relativity, tipically one defines the canonical stress-energy tensor as [ can T αβ = c ( α Φ) β Φ η αβ L. (15.105) In principle, we can generalize this definition in curved space by appealing to the principle of general covariance [ can T αβ = c ( α Φ) β Φ g αβ L. (15.106) However, the tensor (15.106) is not a good stress-energy tensor. Indeed, in general it is not symmetric, and it does not satisfy the divergenceless condition can T αβ ;β 0. (15.107) The correct definition for the stress-energy tensor in general relativity is given by eq. (15.72) T µν = 2c [ δl δg 1 µν 2 g µνl. (15.108) Anyway, it can be shown that the difference between (15.108) and the canonical stressenergy tensor (15.106) is a total derivative, which disappears when integrated in the overall spacetime; furthermore, it can be shown that in the Minkowskian limit, where g µν η µν and µ µ the two tensors (15.106), (15.108) coincide.