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Proposed Solution for Vacuum General Relativity Initial Value Constraints: Analysis of the Gauss' Law Constraint

This presentation outlines a proposed solution for the initial value constraints of vacuum general relativity, specifically focusing on the analysis of the Gauss' law constraint. The classical theory and quantum theory aspects are discussed, along with the background and proposal for solving the constraints using CDJ variables. The future research directions for the classical case are also explored.

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Proposed Solution for Vacuum General Relativity Initial Value Constraints: Analysis of the Gauss' Law Constraint

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  1. Proposed solution for the initial value constraints of vacuum general relativity: Analysis of the Gauss’ law constraint Eyo E. Ita United States Naval Academy Annapolis, MD

  2. Outline of presentation • Motivation • Classical theory: Reduction of full 4D general relativity to its physical (unconstrained) D.O.F. • Necessary condition for the existence of a general solution to the full Einstein’s equations • Quantum theory: Intermediate step in the construction of finite states for full 4D quantum gravity via SQC • Background • Initial value constraints in metric/Ashtekar variables • Proposal • CDJ variables • Algorithm for solving the constraints (Euclidean signature) • Future research directions for the classical case

  3. Initial value constraints problem • Main goal of Hamiltonian formulation of GR: Implementation of spacetime diffeomorphism invariance at the canonical level • Basic phase space variables contain 8 unphysical D.O.F. • Diffeomorphism constraint: • Hamiltonian constraint: • Unsolved in the full theory due to complicated constraints • Tetrad formulation even more difficult to solve • Canonical transformation to new variables simplifies description of the problem (Abhay Ashtekar, 1986)

  4. Constraints in Ashtekar variables • New basic phase space variables: • Diffeomorphism constraint • Hamiltonian constraint • Analogous constraints have simplified into low order algebraic polynomials in the Ashtekar variables: • Various developments have ensued, including at the quantum level • One known nontrivial exact solution in the full theory • corresponding to the Kodama state • However, an additional constraint has been inherited

  5. Gauss’ law constraint • Invariance under left-handed SU(2) rotations • Low order polynomial constraint, however it is a set of P.D.E.s in the basic variables which still needs to be solved in conjunction with the other constraints • Solved via gauge-invariant spin network states in LQG: Hamiltonian constraint remains unsolved • Need to reduce GR to its physical D.O.F. while consistently implementing all constraints

  6. CDJ Variables • Re-write the constraints using the CDJ Ansatz • CDJ matrix takes its values in • Discovered by Capovilla, Dell and Jacobson • Ashtekar magnetic field

  7. Constraints in CDJ variables • Diffeomorphism constraint: • Hamiltonian constraint: • Gauss’ law constraint: • chosen as smooth vector fields satisfying • New tensor-valued connection

  8. Polar representation • rotation of eigenvalues • Three complex angles • Define/parametrize an arbitrary SO(3,C) frame • Gauss’ law fixes the frame

  9. Nondegenerate magnetic field • Diffeo constraint: CDJ matrix is symmetric • Hamiltonian constraint (dynamics): • Reduces to two unconstrained D.O.F. • Only remaining obstacle is the Gauss’ law constraint

  10. Rectangular representation • To solve the Gauss’ law constraint explicitly • Decompose CDJ matrix into diagonal and off-diagonal contributions. • Express as a linear transformation and invert to reduce by three D.O.F.

  11. Gauss’ law constraint • Choose • Each well-defined configuration defines • Must solve three linear 1st order PDE for each

  12. Explicit solution • PDE reduce to the general form • Formal operator expansion • Norm bounded by pointwise convergent series • always guaranteed since diagonal in • Not necessarily so for since non-diagonal • Proposition is that the Gauss’ law constraint is integrable for well-defined generic configurations

  13. Formal inversion in nondiagonal case • where

  14. Explicit action of inverse vector fields • Projects along holonomic coordinate direction of integration • Inversion defined by boundary data/source term • Physical interpretation: Construct solution along congruence of flow lines defined by the orbit of the dual translation ratios

  15. Final procedure • Write 3 CDJ matrix elements as a map of the remaining three and solve for the angles • Algebraic classification of spacetime related to multiplicity of eigenvalues • Solve for angles • Hamiltonian constraint incorporated at each order

  16. Classical research directions • Based upon different configurations of the Ashtekar curvature and CDJ eigenvalues, • Construct Weyl curvature scalars from final CDJ matrix • Determine radiation properties of spacetime • Reconstruct spacetime metric (solution to vacuum Einstein field equations?)

  17. QUESTIONS? • LCDR Eyo E. Ita • Physics Dept., USNA • Email: ita@usna.edu

  18. Special Acknowledgements United States Naval Academy Annapolis, MD Division of Mathematics and Sciences Physics Department

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