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

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