Summary
We consider the properties of the ADAM Lagrangian in the model for gravity based on extrinsic geometry recently developed. In this model the space-time is embedded into a space of more dimensions and the embedding functions are considered as dynamical variables. In spite of the naïve expectatives this Lagrangian depends only on first-order time derivatives of the fields, the embedding functions. This is a nice property since in this case on would expect an easy canonical formulation. Another nice property is the fact that the energy-momentum tensor is identically zero. The shift constraints are easily calculated; however, the lapse constraint has not been calculated since it relies on an involved matrix calculation. After fixing the gauge, this reflects in the impossibility of solving, even when the Legendre transformation is invertible, for the velocities. Furthermore, the gauged energy-momentum tensor is not a tensor density.
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Partially supported by DADC-CONACYT, Mexico.
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Franke, V.A., Tapia, V. The ADM Lagrangian in extrinsic gravity. Nuov Cim B 107, 611–630 (1992). https://doi.org/10.1007/BF02723170
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DOI: https://doi.org/10.1007/BF02723170