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

Abstract

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.