BF gravity with Immirzi parameter and cosmological constant

Abstract

The action principle of the BF type introduced by Capovilla, Montesinos, Prieto, and Rojas (CMPR) which describes general relativity with Immirzi parameter is modified in order to allow the inclusion of the cosmological constant. The resulting action principle is on the same footing as the original Plebanski action in the sense that the equations of motion coming from the new action principle are equivalent to the Holst action principle plus a cosmological constant without the need of imposing additional restrictions on the fields. We consider this result a relevant step towards the coupling of matter fields to gravity in the framework of the CMPR action principle. The research in quantum gravity led by its two main branches (loop quantum gravity [1] and spin foam models for gravity [2]) has recently motivated the study of the classical descriptions for general relativity and theories related to it, particularly, the formulations of gravity of the BF type. For instance, Cartan's equations in the framework of BF theories are analyzed in Refs. [3,4] while the relationship of general relativity to the Husain-Kuchar model in the framework of BF theory is analyzed in Refs. [5][6][7].As is well know, general relativity was formulated as a constrained BF theory by Plebański in the mid-seventies [8]. The fundamental variables used for the description of the gravitational field are two-form fields, a connection one-form, and some Lagrange multipliers. This framework was extended much later in order to include the coupling of matter fields [9]. These formulations are complex and one has to use reality conditions in order to get real general relativity. There are other formulations for gravity expressed as a constrained BF theory which are real, and were given in Refs. 10, 11 and 12. In particular, the one given in Ref. 12 includes the so-called Immirzi parameter [13][14][15]. Recently, the issue of the introduction of the cosmological constant in the framework of Ref. 12 was studied by Smolin and Speziale [16]. This issue is also the goal of this paper. Our approach is different from the one followed in Ref. 16 and, in our opinion, is close to the original Plebański formulation. So, we are ready to make the introduction of the cosmological constant and, at the end of the paper, we will comment on the relationship between the results of this paper and the ones of Ref. 16.The action principle for pure gravity introduced by Capovilla, Montesinos, Prieto, and Rojas in Ref. 12 (hereafter CMPR) is given bywhere A I J is a Euclidean or Lorentz connection one-form, depending on whether SO(4) or SO(3, 1) is taken as the internal gauge group, and F The variation of the action (1) with respect to the independent fields gives the equations of motionδψ :

BF gravity with Immirzi parameter and matter fields

“…[16,20]. The coupling of the cosmological constant presented here is done following our ideas reported in Refs.…”

“…The coupling of the cosmological constant presented here is done following our ideas reported in Refs. [16,21]. In order to introduce the cosmological constant into the action principle (1), we propose the following action principle…”

“…[16], the next step is to introduce (9) into the Einstein's equations with cosmological constant (4) and use the restrictions (10) in order to check whether the Einstein's equations with cosmological constant are satisfied or not. It is obtained that the solely fulfilling of the Bianchi identities implies the fulfilling of (4), except by one equation given by…”

“…On the other hand, using the form of the curvature given in (16) and the restrictions (17) in the Einstein's Eqs. (4) it is concluded that they all are automatically satisfied, except by the equation given by…”

“…(14) and (21). This is because even though in the action principle (2) the two allowed volume terms [16,21], l 1 B IJ^B IJ and l 2 B IJ^Ã B IJ , are included, from Eq. (3c) it can be seen that B IJ^B IJ ¼ 0 for any solution of the B's, thus when the action (2) is written in terms of the tetrad field, the term l 1 B IJ^B IJ identically vanishes.…”