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Variational principle for gravity in the extended phase space

Variational formalism in the extended phase space for fields is applied to gravity. It is shown that the requirement of invariance under arbitrary local inertial frames implies a coupling of torsion to a 3-form of matter fields on the one hand and to a 3-form (related to the Einstein tensor) on the other. Gravitational dynamics is restricted to torsion zero surface in the extended phase space for Einstein-Hilbert action.


Extended phase space in mechanics
The extended phase-space formalism [1] [2] can be summarized as follows. In mechanics, where the evolution is along the one-dimensional time, the dynamics is determined by the stationary Poincare-Cartan or action 1-form Ξ = p i dq i −Hdt in the extended phase space (EPS) with coordinates (t, q, p). The physical nature of time is that it 'never stops' and is uni-directional. That is, for trajectories in the EPS, the time variable t has non-zero derivative with respect to the parameter s of the curve : dt/ds = 0. This condition means that the trajectories in EPS are projectable on the time axis. Thus EPS has a fiber bundle structure with time as the base manifold and the phase-space (with coordinates (q, p)) as the fiber. Evolution trajectories are sections from the one-dimensional base manifold into the EPS.
The principle of stationary action means that if a vector field X represents the variation of variables in the EPS, then the Lie derivative of action integrated over a proposed trajectory is zero : L X Ξ = 0. Using L X = i X • d + d • i X we see that we should first calculate d Ξ and obtain equations of motion by contracting (i.e. applying i X ) with different independent variation fields X. The second term simply gives the Noether theorem d(i X Ξ) = 0 for symmetry fields X which satisfy L X Ξ = 0. We can write Contracting with independent variations fields X = ∂/∂q i , ∂/∂p i , ∂/∂t etc gives the Hamilton equations.

Co-frames and frame-gauge invariance
Einstein's theory of gravitation regards spacetime as a four dimensional continuum with a metric tensor whose components g µν , µ, ν = 0, 1, 2, 3 determine all gravitational phenomena at the classical level. In place of the ten metric components we may choose at each spacetime point a local inertial frame or co-frame as follows. We choose, in a smooth way, a 4 × 4 non-singular real matrix N with elements N a µ , a, µ = 0, 1, 2, 3 at each point. Let η ab be the matrix η with (−1, 1, 1, 1) on the diagonal and zero elsewhere. Now define As defined g µν are elements of a symmetric non-singular matrix g with (det g = −(det N) 2 = 0) therefore they are qualified to represent the components of a metric. However, there are many matrices N's giving the same g µν . If we replace N a ν by Λ a b N b ν where Λ is a Lorentz transformation η ab Λ a c Λ b d = η cd , the matrix g µν does not change. Thus there is a sixparameter Lorentz group 'degeneracy' in choosing the matrix N at each point. The 1-forms n a ≡ N a µ dx µ constitute an orthonormal co-frame with respect to the metric g µν by design, because if we write the equation (1) as N g −1 N T = η −1 , and denote the elements of the inverse matrices by g µν and η ab then Thus we can describe the gravitational field by giving a system of orthonormal co-frames. The price to pay is a six parameter gauge invariance. The essence of general theory of relativity is that physics is independent of the choice of local inertial frames represented here by orthonormal co-frames. We call this invariance the "frame gauge invariance".
A local gauge group, in turn, involves its own connection forms or gauge potentials. Since all physical quantities are defined in terms of the local inertial frames, the rate of change of these quantities involves derivatives dn a . These exterior derivatives are not frame-gauge invariant. They have to be replaced by covariant derivatives dn a + ω a b ∧ n b where the connection matrix ω a b of 1-forms, under the frame change by Lorentz transformation Λ : n ′ = Λn, transforms as ω ′ = ΛωΛ −1 + ΛdΛ −1 . (We omit indices and the wedge symbol when there is no confusion.) The covariant derivatives of the co-frame n a are called the torsion 2-forms of the Einstein-Cartan theory : The frame gauge fields, or curvature 2-forms, are determined by The torsion and curvature ( (3) and (4)) forms yield the Bianchi identities on exterior differentiation, The nature of the gauge group (Lorentz group here) is reflected in the antisymmetry of the connection when one index is lowered by Minkowski metric η, We will use constant matrices η's for raising and lowering of indices.

EPS for fields
It has been suggested by the author that the fiber bundle structure of the extended phase space should be taken over for field systems by replacing the 1-dimensional base manifold by the 4-dimensional space-time. The Poincare-Cartan form (or action) is now a 4-form [2]. Let φ be a scalar field. Then its canonical momentum is a differential 1-form p in this formalism. The Poincare-Cartan form has the structure where the Hodge star operator is used to convert a 1-form p into the 3-form * p in order to obtain a 4-form of the type 'pdq'. (See [3] for definition and notational convention.) Here, the 'covariant Hamiltonian' H = H(φ, p), is not the 3-form of energy density, but a 4-form constructed from coordinate (0-form φ) and its canonical momentum (1-form p). There is no 'velocity' (dφ) in H, but, in principle, there could be a linear term in velocity; as is actually the case in gravity. The standard scalar field Hamiltonian is, Here we use the notation * 1 = n 0 ∧ n 1 ∧ n 2 ∧ n 3 for the Hodge dual of the constant function 1. The variational principle in the extended phase-space for fields requires that allowed field configurations are those four-dimensional sub-manifolds of the extended phase space which are sections (i.e. fiber respecting mappings from the base into the EPS) and on which the 4-form i X • d Ξ φ = 0 for variational field X. We can write Using the formulas p = p a n a , dp = dp a n a + p a dn a , d( * p) = dp a ∧ ( * n a ) + p a d( * n a ) d( * n a ) = * (n a ∧ n b ) ∧ dn b , ( * n a ) ∧ n b = −η ab ( * 1), d( * 1) = ( * n a ) ∧ dn a we calculate where we replace dn b in the last term by the torsion 2-form T a = dn a + ω ab n b because the added term containing 5 factors of 1-forms n in a four dimensional space is zero. The stress-energy 3-form Θ is given by where T ab is the familiar stress-energy tensor. The Hamiltonian equations for matter fields from the variation of fields φ and p while keeping frame fields and connection fixed are read off from the two factors of the first term : If we substitute p = dφ in the second equation we obtain the Klein Gordon equation for φ in the curved background determined by the fixed frame field n a . It is worth emphasizing that the exterior derivative acts on the full phasespace, and dφ is independent of n a (or dx µ ). The momentum is a differential 1-form p = p a n a but p a are not functions of x. They are coordinates in the extended phase-space just as in classical mechanics p i are coordinates in the cotangent bundle T * (Q). Only when we look for a section which makes action stationary, will independent phase-space coordinates p a become functions of x.
The term −Θ a T a containing torsion of the gravitational field will be seen below to combine with the gravitational part of action when the frame field and connection are varied.

Gravitational field in EPS
The gravitational field is jointly determined by the co-frame fields n a and the frame-gauge connection fields ω ab . The "velocities" for these are the torsion T = dn + ωn and curvature Ω = dω + ωω respectively. Since action is gauge invariant, the true variation fields X of co-frames n and gauge potentials ω contract with gauge-covariant T and Ω and not dn or dω.
We take as the Poincare-Cartan 4-form for gravity the Einsten-Hilbert action, where G N is Newton's gravitational constant and where we define the Einstein 3-form as The components G ab are defined by this equation. They reduce to the usual symmetric Einstein tensor for the Riemannian geometry. The variation of Einstein-Hilbert action is simply The proof goes as follows : Using formulas such as d * (n a n b ) = * (n a n b n c )dn c and the Bianchi identity, d[ * (n a n b )Ω ba ] = * (n a n b n c )dn c Ω ba + * (n a n b )dΩ ba = * (n a n b n c )Ω ba (T c − ω cd n d ) + * (n a n b )(Ωω − ωΩ) ba .
Furthermore, as * (n a n b n c )n d = −η ad * (n b n c ) + η bd * (n a n c ) − η cd * (n a n b ), we can rearrange and simplify the terms not containing T . They all add up to * (n a n b )(Ωω + ωΩ) ba = * (n a n b )(Ω bd ω d a + Ω ad ω d b ) = 0, being a sum over a product of a symmetric and an antisymmetric expression.
Combining with the derivative of the matter action (10) for the scalar field The allowed field configurations given by the terms above can be read off easily. For variational fields in the direction of φ and p we get the Klein-Gordon equations in curved background (determined by n and ω). In the second term varying n but keeping ω constant gives the Einstein equation using (15) and (12). And lastly, if ω is varied but n is kept fixed, we get the equation T = 0.