SO(2,1) Connection in Timelike 3+1 Foliation

Abstract

We introduce 3+1 timelike foliation of the four dimensional Lorentz manifold to derive the 3+1 Sen-Ashtekar-Barbero-Immirzi formalism in case of $SO(2,1)$ rotation gauge group, which is possible due to the existence of the $so(2,1)$ algebra isomorphism to $R^3_{2,1}$ algebra with respect to the vector product. We prove that the newly introduced flux and extrinsic curvature variables preserve the symplectic structure of the original variables. We then introduce the modified rotational constraint and succeed to write it as a Gauss constraint of a newly obtained connection. The newly obtained connection is slightly different from the classical 3+1 spacelike Sen-Ashtekar-Barbero-Immirzi connection as it contains in addition the Minkowski metric $\eta_{ij}$ as a coefficient. Our result has a very simple form and clearly shows how $so(2,1)$ connection is different from $so(3)$ one. Also it is the first time that the key-stone fact that makes the whole formalism work in timelike 3+1 case, i.e. $so(2,1) \simeq R^3_{2,1}$ isomorphism and its relation to the $so(2,1)$ connection has been researched.