THE ENERGY-MOMENTUM TENSOR IN CLASSICAL FIELD THEORY

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1 THE ENERGY-MOMENTUM TENSOR IN CLASSICAL FIELD THEORY Walter Wyss Department of Physics University of Colorado Boulder, CO (Received 14 July 2005) My friend, Asim Barut, was always interested in classical field theory and in particular in the role that a divergence term plays in a lagrangian. This paper addresses this question via the energy-momentum tensor. Abstract We consider the inhomogeneous Lorentz-group as the fundamental symmetry group of all of physics. For Lorentzinvariant actions in classical Lagrange field theories we construct the energy tensor and the energy-momentum tensor. These are intimately related to the generators of the inhomogeneous Lorentz group. With respect to additional symmetry groups of the action, only the physical conservation laws are invariant. Concepts of Physics, Vol. II (2005) 295

2 Walter Wyss 1 Introduction The concept of the energy-momentum tensor in classical field theory has a long history, especially in Einstein s Theory of Gravity. We consider the inhomogenous Lorentz-group as the fundamental symmetry group of all physics. A physical Lagrange action is then Lorentz invariant. Translation invariance leads to an energy tensor, and Lorentz invariance to an energy-momentum tensor. These tensors are Lorentz-covariant. The energy-momentum tensor is symmetric whereas the energy tensor in general is not symmetric. Any additional symmetry of the action will be treated separately from Lorentz-invariance. In this paper we construct the energy tensor and the energy-momentum tensor for the following action A = dxl 0 + dx µ B µ, where L 0 and B µ depend on the field and its first derivatives. The influence of additional coordinate symmetry of the action will be investigated. 2 The Variational Principle Let η µv denote the Lorentz metric tensor with signature (+,,, ) and η µv its inverse. Raising indices is performed with η µv and lowering indices with η µv. φ = (φ α ) stands for a multicomponent field. The Lorentz 4- volume element is represented by dx. We also use the abbreviations µ x and φ µ µ µ φ. We now consider the following action A = L dx, (1) where and L = L 0 + µ B µ, (2) L 0 = L 0 (φ, φ µ ), (3) B µ = B µ (φ, φ v ), (4) 296 Concepts of Physics, Vol. II (2005)

3 The energy-momentum tensor in classical field theory i.e. L 0 and B µ depend on the field φ and its first derivatives only. According to the calculus of variation (Appendix) we get the following relations δa = dxδ L + µ Lδx µ ] dx, (5) δa = dxδ L 0 + µ L 0 δx µ + ( α B α ) δx µ + δ B µ ] dx. (6) Definition H µ L 0 φ µ. (7) 2. Euler derivative ε (φ) L 0 = L 0 φ L 0 µ. (8) φ µ 3. G ε (φ) L 0. (9) Then our variational principle reads L0 δa = dx φ δ φ + L ] 0 δ φ µ φ µ + µ L 0 δx µ + ( α B α ) δx µ + δ B µ ] dx (10) = Gδ φ dx + µ L 0 δx µ + H µ δ φ + ( α B α ) δx µ + δ B µ ] dx. From the relation δ = δ δx σ σ, we then get δa = Gδφdx Gφ σ δx σ dx + µ L 0 δx µ H µ φ σ δx σ + H µ δφ + ( α B α ) δx µ + δ B µ ] dx. (11) Concepts of Physics, Vol. II (2005) 297

4 Definition 2. Energy tensor Walter Wyss E µ σ H µ φ σ δ µ σl 0. (12) Using this energy tensor and the Appendix the variation of the action becomes δa = Gδφdx Gφ σ δx σ dx + µ E σ µ δx σ + H µ δφ (13) +B µ ( α δx α ) B α ( α δx µ ) + δb µ ] dx. 3 Coordinate Variations We assume now that the field φ transforms according to a representation of the inhomogeneous Lorentz group. Then δφ = S β λ (φ) βδx λ, (14) and the variational principle reads { δa = dxg φ σ δx σ S β λ βδx λ} + µ E σ µ δx σ H µ S β λ βδx λ + B µ α δx α B α α δx µ { } B µ φ Sβ λ + Bµ φ λ + Bµ α S β λ β δx λ (15) φ β φ α Bµ φ α Definition P µβ λ S β λ α β δx λ] dx. δ µ λ Bβ δ β λ Bµ + Bµ φ Sβ λ + Bµ φ β φ λ + Bµ φ α α S β λ. (16) 2. K µα λ Hµ S α λ + P µα λ. (17) 298 Concepts of Physics, Vol. II (2005)

5 The energy-momentum tensor in classical field theory The variational principle reads δa = dx { Gφ σ δx σ GS βλ βδx λ} ] + µ E σ µ δx σ K µβ λ βδx λ Bµ S β λ φ α β δx λ dx, (18) α or δa = + ( ) } dx { β GS β λ Gφ λ δx λ µ {E σ µ + GS σ µ } δx σ K µβ 4 Lorentz-invariance of the action λ ] βδx λ Bµ S β λ φ α β δx λ dx. α (19) If the action is Lorentz-invariant then δa = 0 for infinitesimal Lorentz transformations δx µ = ω µ v x v + δa µ, (20) where δa µ represents an infinitesimal translation and ω µv = ω vµ an infinitesimal proper Lorentz transformation a) Translation invariant action Because of β δx µ = 0 and (18) we get µ E µ σ + Gφ σ = 0. (21) b) Lorentz invariant action Because of δx λ = ωσx λ σ, β δx λ = ωβ λ, ω αβ = ω βα, α β δx λ = 0 and (18) we get ] Gφ σ ωαx σ α GS β λ ωλ β + µ E σ µ ωαx σ α K µβ = 0, (22) ω λ β with ω αβ = ω βα. λ ωλ β Definition E µα = η ασ E µ σ. (23) Concepts of Physics, Vol. II (2005) 299

6 Walter Wyss F µαβ 1 2 K µαβ = η λβ K µα λ. (24) K µαβ K µβα]. (25) Then and from (22) we get Gη λα x β φ λ + S β λ Definition 5. Angular momentum tensor Then (27) reads M µαβ 1 2 F µαβ = F µβα, (26) ] ω αβ + µ E µα x β + F µαβ] ω αβ = 0. (27) E µα x β E µβ x α] F µαβ. (28) µ M µαβ G η λα x β η λβ x α] φ λ G S βα S αβ] = 0, (29) or equivalently µ F µαβ = 1 2 E βα E αβ] ( µ E µα ) x β ( µ E µβ) x α] G φ λ { η λα x β η λβ x α} + S βα S αβ]. (30) Translation and Lorentz-invariant action Combining the above cases a) and b) we get µ F µαβ = 1 2 Definition µ E µ σ = Gφ σ, (31) E βα E αβ] G S βα S αβ]. (32) Θ µαβ F µαβ + F αµβ + F βµα, (33) 300 Concepts of Physics, Vol. II (2005)

7 The energy-momentum tensor in classical field theory 2. Energy-momentum tensor T µα E µα + GS µα + β Θ µαβ. (34) Θ µαβ has the symmetry property Θ µαβ = Θ βαµ. (35) The energy-momentum tensor then can be written as T µα = 1 2 (Eµα + E αµ )+ 1 2 G (Sµα + S αµ )+ β F µαβ + F αµβ]. (36) It has the following properties T µα = T αµ. (37) µ T µα = Gφ λ η λα + µ (GS µα ). (38) The angular momentum tensor (28) can now be expressed, using the energy-momentum tensor, as Remarks M µαβ = 1 2 ( T µα x β T µβ x α) λ ( Θ λαµ x β Θ λβµ x α) 1 2 G ( S µα x β S µβ x α). 1) The energy tensor E µα is in general not symmetric. It is related to translation invariance. 2) The energy-momentum tensor T µα is symmetric. It is related to proper Lorentz invariance. 5 Additional Symmetry of the Action Here we assume that the action is invariant under a larger coordinate transformation group than the inhomogeneous Lorentz-group. Translation and affine invariant action Concepts of Physics, Vol. II (2005) 301

8 Walter Wyss Translation invariance implies from (21) µ E µ σ = Gφ σ. (39) Infinitesimal affine transformations are given by δx λ = A λ αx α, (40) where A α λ are infinitesimal constants with no symmetry. Then we get from (18) and (39) µ K µαβ = E αβ GS αβ. (41) This relation is more restrictive on the auxiliary quantity K µαβ than Eq. (32). The energy-momentum tensor however is still given by General covariant action T µα = 1 2 (Eµα + E αµ ) G (Sµα + S αµ ) (42) + β ( F µαβ + F αµβ). (43) Here the infinitesimal transformations δx α are arbitrary. From translation invariance we have and from affine invariance µ E µ σ = Gφ σ, (44) µ K µαβ = E αβ GS αβ. (45) Eq. (18) then reduces to { } K αβ B λ α β δx λ µ + µ S β λ α β δx λ + Bµ S β λ φ α φ µ α β δx λ = 0. α For arbitrary δx λ this implies 1. (46) { } { }] K αβ B µ λ + µ S β λ = K βα B µ λ φ + µ Sλ α. (47) α φ β 302 Concepts of Physics, Vol. II (2005)

9 The energy-momentum tensor in classical field theory 2. (µ,α,β) B µ S β λ = 0. (48) φ α where (µ, α, β) indicate the sum over all permutations of µ, α, β. From equations (25), (33) we find with and From we then get where 2Θ µαβ =2K βµα ( K βµα + K µβα) + ( K µαβ + K αµβ) ( K αβµ + K βαµ), (49) K αβµ = η µλ K αβ λ, (50) S βµ = η µλ S β λ. (51) ] B K αβλ + K βαλ µ = µ S βλ + Bµ S αλ. (52) φ α φ β Θ µαβ = K βµα + σ W σ,µαβ, (53) W σ,µαβ = 1 2 Bσ φ α S βµ + Bσ φ β S αµ + Bσ φ µ S βα + Bσ φ β S µα Bσ φ µ S αβ Bσ φ α S µβ ]. (54) The energy momentum tensor (34) becomes T µα = E µα + GS µα + β K βµα + σ W σ,µαβ]. (55) Due to (45) we then get Remarks T µα = β σ W σ,µαβ. (56) 1) General covariance is very restrictive on the form of the Lagrange density; the corresponding energy-momentum tensor is a divergence. 2) The corresponding energy tensor satisfies µ E µ σ = Gφ σ. Concepts of Physics, Vol. II (2005) 303

10 Walter Wyss 6 Equations of motion and conservation laws The equations of motion are given by demanding that the action Eq. (1) is stationary, i.e. has a critical point under variations where on the boundary of the integration domain δx µ = 0, δ φ = 0 and δ φ µ = 0. Then we get from Eq. (10) that the equations of motion are given by G = 0. (57) From Eq. (21) we get the conservation law of the energy tensor µ E µ σ = 0. (58) From Eq. (29) we get the conservation law for the angular momentum tensor µ M µαβ = 0. (59) From Eq. (38) we get the conservation law for the symmetric energymomentum tensor µ T µα = 0. (60) 7 Examples 1) A = dx L (g, φ). (61) The fields are a covariant tensor field g αβ (g αβ it s inverse) and a scalar field φ. The Lagrange density is given by L (g, φ) = g g αβ ( α φ) ( β φ) m 2 φ 2]. (62) The relevant auxiliary quantities are H αβ,µ = L g αβ,µ = 0 H µ = L φ µ = 2 g g µα φ α G αβ = ε (g αβ ) L = 1 2 gαβ L g g µα g vβ φ µ φ v G = ε (φ) L = 2 g m 2 φ µ 2 g g µα φ α ]. (63) 304 Concepts of Physics, Vol. II (2005)

11 The energy-momentum tensor in classical field theory The energy tensor then becomes E µ σ = 2 g g µα φ α φ σ δ µ σl. (64) The representations of the Lorentz group are given as follows (i) scalar field φ : S β λ = 0 (ii) covariant tensor field g αβ ( ) S β λ = σρ δβ σg λρ + δρ β g σλ. The other auxiliary quantities are then given by P µβ λ = 0, K µα λ = 0, F µαβ = 0, Θ µαβ = 0. The energy-momentum tensor now becomes T µα =E µ σ η σα + G µρ η αλ (S µ λ ) σρ =η ασ E µ σ + G λρ (S µ σ ) λρ ] T µα =0. =η ασ 2 g g µβ φ β φ σ δ µ σl + 2G µρ g σρ ] (65) Because our action is invariant under general coordinate transformations, this result agrees with the general result (55). 2) A = dx µ B µ where B µ = B µ (φ, φ α ). φ is a general field. The Lagrange density thus consists of a boundary term only. The relevant auxiliary quantities are Concepts of Physics, Vol. II (2005) 305

12 Walter Wyss H µ =0, G =0, E µ σ =0, K µα λ =P µα λ, P µα λ =δ µ λ Bα δλ α B µ + Bµ φ Sα λ + Bµ φ λ + Bµ β Sλ α, φ α φ β F µαβ = 1 P µαβ P µβα]. 2 The energy-momentum tensor then becomes T µα = β F µαβ + F αµβ] = β 1 2 We also have from (32) and (38) the relations P µαβ + P αµβ P µβα P αβµ]. (66) µ F µαβ = 0, µ T µα = 0. For this action where the Lagrange density is a boundary term only, we have no equations of motion. The energy tensor vanishes but there is a conserved, nontrivial energy-momentum tensor. 8 Conclusion We looked at physical systems described by a classical Lagrange action. The Lagrange density depends on the field, its first and its second derivatives but in such a form that the equations of motion (Euler derivative) are partial differential equations of second order. We consider the inhomogeneous Lorentz group as the fundamental symmetry group of all of physics. Translation invariance leads to the energy tensor E µ σ, proper Lorentz invariance to the angular momentum tensor M µαβ and invariance with respect to the combined inhomogeneous Lorentz group leads to the symmetric energy-momentum tensor T µα. If the equations of motion are satisfied we get physical 306 Concepts of Physics, Vol. II (2005)

13 The energy-momentum tensor in classical field theory conservation laws. In case the equations of motion have an additional symmetry besides Lorentz convariance the physical conservation laws also have that symmetry. If the action is invariant under general coordinate transformations the energy-momentum tensor is a divergence. This is the case in Einstein s Theory of Gravity. Applications to electromagnetic interactions and the theory of gravity will be published separately. 9 Appendix - Calculus of Variation Let x represent a point in Minkowski space. In a local coordinate basis x has the coordinates {x µ }. Let φ be a multicomponent field defined on Minkowski space. An infinitesimal variation of φ results in the field φ. The field variation (local variation) is then defined by (δ φ) (x) = φ (x) φ (x). δ commutes with the derivative, i.e. δ µ = µ δ. An infinitesimal coordinate transformation on x µ results in new coordinates x µ. The coordinate variation is then defined by δx µ = x µ x µ. The total variation of a field φ, induced by a coordinate variation, is given by (δφ) (x) = φ ( x) φ (x). We then have the relation δ = δ (δx µ ) µ, as applied to any field. We have the following total variations δφ = S β λ (φ) βδx λ. 1) scalar field S β λ = 0. Concepts of Physics, Vol. II (2005) 307

14 Walter Wyss 2) covariant vector field ( S β λ (φ) )α = δβ αφ λ. 3) covariant tensor field (S βλ (φ) ) 4) spinor field S β λ (φ) = 1 8 σρ = δβ σφ λρ + δ β ρ φ σλ. ( γ β γ µ γ µ γ β) φη µλ. The variation of the action is given by δa = δ Ldx = (δ L) dx + µ Lδx µ ] dx. For the action A = dx L 0 + µ B µ ], where L 0 = L 0 (φ, φ µ ), B µ = B µ (φ, φ α ), with and the Euler derivative we have the following statements Lemma 1.. Proof. H µ L 0 φ µ, ε (φ) L 0 = L 0 φ µh µ G, δ L 0 = Gδ φ + µ H µ δ φ] δ L 0 = L 0 φ δ φ + L 0 φ µ δ φ µ = L 0 φ δ φ + µ H µ δ φ] ( µ H µ ) δ φ δ L 0 =Gδ φ + µ H µ δ φ]. 308 Concepts of Physics, Vol. II (2005)

15 The energy-momentum tensor in classical field theory Lemma 2. µ ( α B α ) δx µ + δ B µ ] dx = = µ B µ ( α δx α ) B α ( α δx µ ) + δb µ ] dx. Proof. µ α (B α δx µ ) B α α δx µ ( α B µ ) δx α + δb µ ] dx = µ α (B µ δx α ) B α α δx µ ( α B µ ) δx α + δb µ ] dx = µ B µ α δx α B α α δx µ + δb µ ] dx. Lemma 3.. Proof. From and we get δφ α = α ( φ) φ σ α δx σ δ = δ δx σ σ, δ α = α δ, δφ α =δ φ α + δx σ φ ασ = α δφ δx σ φ σ ] + δx σ φ ασ δφ α = α (δφ) φ σ α δx σ. (67) Lemma 4. B δb µ µ = φ Sβ λ + Bµ φ β φ λ + Bµ φ α α S β λ ] β δx λ (68) Bµ φ α S β λ α β δx λ. (69) Concepts of Physics, Vol. II (2005) 309

16 Walter Wyss Proof. δb µ = Bµ Bµ δφ + φ δφ + Bµ φ α ( α δφ φ σ α δx σ ) δφ α = Bµ φ α φ ( = Bµ φ Sβ λ βδx λ Bµ φ σ α δx σ Bµ α S β λ φ α φ βδx λ). α References 1] E. Noether, Göttinger Nachrichten, 235 (1918). 2] D. Hilbert, Math. Ann. 92, 1 (1924). 3] A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles, Dover, New York, 1980 (second edition). 4] R. Jost, The General Theory of Quantized Fields, Lectures in Applied Mathematics, Vol. IV, ] W. Pauli, Theory of Relativity, Pergamon Press, Concepts of Physics, Vol. II (2005)