  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.
