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We discuss Exterior Differential Systems (EDS) for the vacuum gravitationalfield. These EDS are derived by varying the Hilbert-Einstein Lagrangian, givenmost elegantly as a Cartan 4-forrm calibrating 4-spaces embedded in ten flatdimensions. In particular we thus formulate with tetrad equations theRegge-Teitelboim dynamics "a la string" (R-T); it arises when variation of the4-spaces gives the Euler-Lagrange equations of a multicontact field theory. Wecalculate the Cartan character table of this EDS, showing the field equationsto be well posed with no gauge freedom. The Hilbert Lagrangian as usuallyvaried over just the intrinsic curvature structure of a 4-space yields only asubset of this dynamics, viz., solutions satisfying additional conditionsconstraining them to be Ricci-flat. In the static spherically symmetric case wepresent a new tetrad embedding in flat six dimensions, which allows reductionof the R-T field equations to a quadrature; the Schwarzschild metric is aspecial case. As has previously been noted there may be a classicalcorrespondence of R-T theory with the hidden dimensions of brane theory, andperhaps this extended general relativistic dynamics holds in extremecircumstances where it can be interpreted as including a sort of dark or bulkenergy, even though no term with a cosmological constant is included in theLagrangian. As a multicontact system, canonical quantization should bestraightforward.

The Hilbert Lagrangian and Isometric Embedding: Tetrad Formulation of Regge-Teitelboim Gravity