Hamiltonian Equations of Reduced Conformal Geometrodynamics in Extrinsic Time
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Hamiltonian Equations of Reduced Conformal Geometrodynamics in Extrinsic Time A. E. Pavlov* Institute of Mechanics and Power Engineering, Russian State Agrarian University—Moscow Timiryazev Agricultural Academy, Timiryazevskaya ul. 49, Moscow, 127550 Russia Received April 20, 2020; revised April 20, 2020; accepted May 5, 2020
Abstract—The reduced vacuum Hamiltonian equations of conformal geometrodynamics of compact manifolds in extrinsic time are written. This is achieved by generalizing the theorem of implicit function derivative to the functional analysis. Under the assumption that constant curvature slicing takes place, York’s field time becomes the global time. DOI: 10.1134/S0202289320030111
1. INTRODUCTION
functional
The Hamiltonian dynamics of the gravitational field is commonly formulated in redundant variables in an extended functional phase space as a consequence of the covariant description of Einstein’s theory. A time parameter should be conjugated to the Hamiltonian constraint. The Hamiltonian formulation of the theory makes it possible to reveal the physical meaning of geometrical variables. The problem of writing the vacuum Einstein equations in unconstrained variables for compact cosmological models is relevant. The reduced phase space is the cotangent ¨ bundle of the Teichmuller space of conformal structures on compact spacelike hypersurfaces [1]. The Hamiltonian is a volume functional. The Hamiltonian dynamics is constructed in York’s time [2]. The problem is that the Hamiltonian density as a volume measure is not expressed in an explicit form from the Hamiltonian constraint (the Lichnerowicz–York elliptic differential equation). This makes it difficult to obtain a Hamiltonian flow. In the present paper we obtain the Hamiltonian equations of motion. This is achieved by generalizing the theorem on implicit function derivative from mathematical analysis to functional analysis.
∂γij SADM = dt d x π ij ∂t tI Σt i − N H⊥ − N Hi , t0
3
(2)
where the ADM units, c = 1, 16πG = 1 are used. Variation of the action (2) in the lapse function N leads to the Hamiltonian constraint expressed via components of the extrinsic curvature Kij : 1 ∂γij − + ∇i Nj + ∇j Ni , (3) Kij := 2N ∂t where the connection ∇i is associated with the metric γij , or in the components of the momentum densities π ij √ H⊥ = γ Kij K ij − K 2 − R 1 √ = √ (γik γjl + γil γjk − γij γkl ) π ij π kl − γR. 2 γ Here, γ is the determinant of the metric tensor, K ij := γ ik γ jl Kkl ,
K := Kij γ ij ,
(4)
Let the spacetime M = R1 × Σt be foliated into a family of spacelike hypersurfaces Σt , labeled by the time coordinate t with just three spatial coordinates on each slice, (x1 , x2 , x3 ). The first quadratic form
and R is the Ricci scalar curvature. Varying the action (2) in the shift functions N i , we get the momentum constraints √ Hi = 2 γ ∇j Kij − ∇i K = −2∇j πij .
γ := γik (t, x)dxi ⊗ dxk
Variations of the action (2) with respect to the canonical variables π ij (t, x) and γij (t, x) lead to the kinematic equations ∂γij
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