  We discuss Exterior Differential Systems (EDS) for the vacuum gravitational
field. These EDS are derived by varying the Hilbert-Einstein Lagrangian, given
most elegantly as a Cartan 4-forrm calibrating 4-spaces embedded in ten flat
dimensions. In particular we thus formulate with tetrad equations the
Regge-Teitelboim dynamics "a la string" (R-T); it arises when variation of the
4-spaces gives the Euler-Lagrange equations of a multicontact field theory. We
calculate the Cartan character table of this EDS, showing the field equations
to be well posed with no gauge freedom. The Hilbert Lagrangian as usually
varied over just the intrinsic curvature structure of a 4-space yields only a
subset of this dynamics, viz., solutions satisfying additional conditions
constraining them to be Ricci-flat. In the static spherically symmetric case we
present a new tetrad embedding in flat six dimensions, which allows reduction
of the R-T field equations to a quadrature; the Schwarzschild metric is a
special case. As has previously been noted there may be a classical
correspondence of R-T theory with the hidden dimensions of brane theory, and
perhaps this extended general relativistic dynamics holds in extreme
circumstances where it can be interpreted as including a sort of dark or bulk
energy, even though no term with a cosmological constant is included in the
Lagrangian. As a multicontact system, canonical quantization should be
straightforward.
