Covariant Hamiltonian tetrad approach to numerical relativity

Abstract

A Hamiltonian approach to the equations of general relativity is proposed using the powerful mathematical language of multivector-valued differential forms. In the approach, the gravitational coordinates are the 12 spatial components of the line interval (the vierbein) including their antisymmetric parts, and their 12 conjugate momenta. A feature of the proposed formalism is that it allows Lorentz gauge freedoms to be imposed on the Lorentz connections rather than on the vierbein, which may facilitate numerical integration in some challenging problems. The 40 Hamilton's equations comprise 12 + 12 = 24 equations of motion, 10 constraint equations (first class constraints, which must be arranged on the initial hypersurface of constant time, but which are guaranteed thereafter by conservation laws), and 6 identities (second class constraints). The 6 identities define a trace-free spatial tensor that is the gravitational analog of the magnetic field of electromagnetism. If the gravitational magnetic field is promoted to an independent field satisfying its own equation of motion, then the system becomes the WEBB system, which is known to be strongly hyperbolic. Some other approaches, including ADM, BSSN, WEBB, and Loop Quantum Gravity, are translated into the language of multivector-valued forms, bringing out their underlying mathematical structure.