First Order Regge

A first order form of Regge calculus is defined in the spirit of Palatini’s action for general relativity. The extra independent variables are the interior dihedral angles of a simplex, with conjugate variables the areas of the triangles. There is a discussion of the extent to which these areas can be used to parameterise the space of edge lengths of a simplex. Regge’s equations of motion for Regge calculus [Regge 1961], a discrete version of general relativity, are derived from an intuitively appealing idea about the correct form for an action principle. However the equations are a little complicated, each equation involving a fairly large number of neighbouring edge lengths in a complicated pattern, and involving combinations of polynomials, square roots, and arccosines of these edge lengths. It is desirable from many points of view to understand these equations more fully, and maybe simplify their form. For example, the second order nature of the equations makes implementation of the Cauchy problem for numerical relativity rather complicated [Sorkin 1975, Tuckey 1993]. In quantum gravity, models for three space-time dimensions have been constructed, either involving a path integral [Witten 1988] or in a discrete version as inspired by Regge and Ponzano [1968]. Witten’s construction starts with a first order action for gravity. Regge and Ponzano’s model has semiclassical limits which involve generalisations of Regge calculus to degenerate metrics [Barrett and Foxon 1994]. The appearance of degenerate metrics is the way in which first order actions for general relativity differ, in a physical sense, from the usual second order Einstein-Hilbert action. These considerations of models of quantum gravity are the main motivation for studying first order actions for Regge calculus. The Regge action involves computing defect angles, which are 2π minus the sum of dihedral angles. A dihedral angle is the angle between two different faces (3-simplexes) in a 4-simplex. This paper considers the possibility of extending the Regge action by taking the dihedral angles to be independent variables, rather than as just functions of the edge lengths. As far as simplifying or radically restructuring the Regge calculus is concerned, this work has to be regarded as preliminary. It hints at a calculus in which angles play a dominant role. The simplest signature for the metric to consider is the case of the positive definite metric, in which case the dihedral angles are angles in the ordinary sense of Euclidean geometry. However, it is perfectly possible to consider other signatures. This paper will discuss mainly the Euclidean case. The phenomena associated with Lorentzian angles in Regge calculus are discussed in [Sorkin 1975] and [Barrett and Typeset by AMS-TEX 2 JOHN W. BARRETT Foxon 1994]. Also, the dimension of the manifold is taken to be four throughout, for familiarity, but similar constructions may be made in other dimensions. The first section is a discussion of first order actions for general relativity and a property of the Regge calculus action which is analogous to a property of the first order action for general relativity. Then there is a definition of a first order action for Regge calculus with the edge lengths and dihedral angles as independent variables, but for which the variation has to be constrained. Using a Lagrange multiplier for each simplex, a second action is defined in which each variable can be varied without constraint. Both of these new actions have as stationary points all of the stationary points of the original Regge action, but there may be extra ones. These extra ones may arise from the discrete ambiguity in reconstructing the ten edge lengths for a 4-simplex from the values of the areas of the ten triangles. First Order Actions Regge calculus is similar to the second-order formulation of general relativity. This is its original formulation by Einstein and Hilbert, where the metric is the only independent variable in the action. The idea is to introduce something similar to a first order form, where there are more variables, but the equations are simpler, involving only one derivative. It is similar to the idea of introducing two first order differential equations in place of one second order equation. In general relativity, the Einstein-Hilbert action S is a function of the metric tensor g. Palatini’s [1919] action P (g,Γ) is a function of the metric and a general torsion-free, but otherwise unrestricted connection Γ, and extends the EinsteinHilbert action, in the following sense S(g) = P (g, γ(g)) (1) where Γ = γ(g) is the unique metric-compatible connection for g. Kibble [1961], and Sciama [1962], extended the Einstein-Hilbert action in a second way, by considering a function of the metric and a general metric-compatible, but not torsion-free, connection. Both of these extensions are first order actions for general relativity, in that the action contains only first derivatives. Both have been called Palatini formulations at times, but I prefer to refer to the first one only as a Palatini action. The second one is called the Einstein-Cartan-Sciama-Kibble action. The possibility of a first order Regge calculus in the spirit of the ECSK action was considered by Drummond [1986] and Caselle, D’Adda and Magnea [1989]. The purpose of this paper is to suggest a second avenue for constructing a first order action, by following the Palatini formalism. The first order actions have the property that the variational equations on the larger spaces of fields reduce (in the absence of matter) to the usual equations of general relativity for the metric and connection. One has dP = ∂P ∂g dg + ∂P ∂Γ dΓ (2) The vanishing of the second coefficient, ∂P/∂Γ, is the variational equation which implies that the connection is the metric compatible one, Γ = γ(g). This implies ∂P Γ (g, γ(g)) = 0. FIRST ORDER REGGE CALCULUS 3 Thus, if one takes the Einstein-Hilbert action S, and regards it as a function of two variables, g and Γ, where Γ is constrained, as in (1), then the equation of motion in fact reduces to 0 = dS dg = ∂P ∂g (g, γ(g)), (3) as a consequence of this. Regge noticed a similar phenomenon to this in Regge calculus. The action is