The Gauge Condition in Gravitation Theory with a Background Metric

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λ µ i µ j u = u (x )∂λ + u (x ,y )∂i (4) on a fibre bundle Y → X. We are concerned with a first order Lagrangian field theory on Y . Its configuration space is the first order jet manifold J 1Y of Y → X coordinated λ i i by (x ,y ,yλ). A first order Lagrangian is defined as a density

λ i i n L = L(x ,y ,yλ)dx , n = dim X, (5) on J 1Y . A Lagrangian L is invariant under a one-parameter group of gauge transformations generated by a vector ﬁeld u (4) iﬀ its Lie derivative

1 1 LJ1uL = J u⌋dL + d(J u⌋L) (6) along the jet prolongation J 1u of u onto J 1Y vanishes. By virtue of the well-known ﬁrst variation formula, the Lie derivative (6) admits the canonical decomposition

i i µ n λ n 1 T LJ uL =(u − yµu )δiLd x − dλ d x, (7)

i µ where dλ = ∂λ + yλ∂i + yλµ∂i denotes the total derivative,

λ δiL =(∂i − dα∂i )L (8) are variational derivatives, and

Tλ µ i i λ λ =(u yµ − u )∂i L − u L (9) is a current along the vector field u. If the Lie derivative LJ1uL vanishes, the first variation Tλ formula on the shell δiL = 0 leads to the weak conservation law dλ u ≈ 0. In particular, λ i λ i if u = τ ∂λ + u ∂i is a lift onto Y of a vector field τ = τ ∂λ on X (i.e., u is linear in τ λ and their derivatives), then Tλ is an energy-momentum current [1, 3, 4, 7]. Note that different lifts onto Y of a vector field τ on X lead to distinct energy-momentum currents whose difference is a Noether current. Gravitation theory deals with fibre bundles over X which admit the canonical lift of any vector field on X. This lift is a generator of general covariant transformations [3, 8].

2 Let X be a 4-dimensional oriented smooth manifold satisfying the well-known topo- logical conditions of the existence of a pseudo-Riemannian metric. Let LX denote the fiber bundle of oriented frames in TX. It is a principal bundle with the structure group + GL4 = GL (4, R). A pseudo-Riemannian metric g on X is defined as a global section of the quotient bundle Σ= LX/SO(1, 3) → X. (10) 2 This bundle is identified with an open subbundle of the tensor bundle ∨ TX. Therefore, Σ can be equipped with coordinates (ξλ, σµν ), and g is represented by tensor fields gµν or gµν. Principal connections K on the frame bundle LX are linear connections

λ µ ν K = dx ⊗ (∂λ + Kλ νx˙ ∂˙µ) (11) on the tangent bundle TX and other tensor bundles over X. They are represented by sections of the quotient bundle

1 C = J LX/GL4 → X, (12) where J 1LX is the first order jet manifold of the frame bundle LX → X [3, 6, 11]. λ ν The bundle of connections C is equipped with the coordinates (x ,kλ α) such that the ν ν coordinates kλ α ◦ K = Kλ α of any section K are the coefficients of corresponding linear connection (11). A tensor gravitational field q is defined as a section of the group bundle Q → X associated with LX. Its typical fiber is the group GL4 which acts on itself by the adjoint representation. The group bundle Q as a subbundle of the tensor bundle TX ⊗ T ∗X λ λ is equipped with the coordinates (x , q µ). The canonical left action Q on any bundle associated with LX is given. In particular, its action on the quotient bundle Σ (10) takes the form

λ µν µν µ ν αβ ρ : Q × Σ → Σ, ρ :(q µ, σ ) 7→ σ = q αq βσ . e Since the Lagrangian (1) of BMT depends on q only via an effective metric, let us further replace variables q with the variables σ. Then, the configuration space of BMT is the jet manifold J 1Y of the product Y = Σ ×e C × Z, where Z → X is a fibre bundle of µν α matter fields. Relative to coordinates (σ ,kλ β,φ) on Y , the Lagrangian (1) reads e LBMT = ǫLq(σ, σ)+ LAM(σ, R)+ Lm(φ, σ,k), (13) e e e where the metric-affine Lagrangian LAM is expressed into components of the curvature tensor

α α α ε α ε α Rλµ β = kλµ β − kµλ β + kλ βkµ ε − kµ βkλ ε

3 contracted by means of the eﬀective metric σ. The corresponding ﬁeld equations read e δµν (ǫLq + LAM + Lm)=0, (14) λ β δ α (LAM + Lm)=0, (15)

δφLm =0. λ β µν α where δµν , δ α and δφ are variational derivatives (8) wit respect to σ , kλ β and φ. λ Let τ = τ ∂λ be a vector field on X. Its canonical lift onto thee fibre bundle Σ × C reads ∂ τ = τ λ∂ + (∂ τ αk ν − ∂ τ ν k α − ∂ τ νk α + ∂ τ α) (16) λ ν µ β β µ ν µ ν β µβ ∂k α e µ β ∂ +(∂ τ µσεν + ∂ τ ν σµε) = τ λ∂ + u α ∂µ β + uµν∂ . ε ε ∂σµν λ µ β α µν e e It is a generator of general covariant transformationse of the fiber bundle Σ × C. Let us apply the first variation formula (7) to the Lie derivative LJ1τ LMA. Since the Lagrangian LMA is invariant under general covariant transformations, wee obtain the equality µν µν ε α α ε λ β Tλ 0=(u − σε τ )δµνLMA +(uλ β − kελ βτ )δ α LMA − dλ MA, (17) where TMA is the energy-momentum current of the metric-affine gravitation theory. It reads λ λµ α λ β µλ TMA =2σ τ δαµLMA + T (δ α LMA)+ dµUMA, (18) e λ β µλ where T (.) are terms linear in the variational derivatives δ α LMA and UMA is the gener- alized Komar superpotential [2, 3]. For the sake of simplicity, let us assume that there exists a domain N ⊂ X where Lm = 0, and let us consider the equality (17) on N. The field equations (14) – (15) on N take the form

δµν (ǫLq + LAM)=0, (19) λ β δ α LAM =0, (20) and the energy current (18) reads λ λµ α µλ TMA =2σ τ δαµLMA + dµUMA. (21) e Substituting (19) and (21) into (17), we obtain the weak equality α λµ αµ λ λµ α 0 ≈ −(2∂λτ σ − σλ τ )δαµLq + dλ(2σ τ δαµLq). (22) e A simple computation brings this equality into the desired form (3) where

λ λν tα =2g q−gδναLq. e e

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