Two-dimensional dilaton gravity made compact

We show that the equations of motion of two-dimensional dilaton gravity conformally coupled to a scalar field can be reduced to a single nonlinear second-order partial differential equation when the coordinates are chosen to coincide with the two scalar fields, the matter field f and the dilaton φ, which are present in the theory. This result may help solve and understand twoand higher-dimensional classical and quantum gravity. PACS numbers: 0460K, 0420J, 0230J Low-dimensional models of gravity are receiving a great deal of attention lately as they can provide insight into the classical as well as the quantum theories of more realistic (higher-dimensional) theories of gravity. Prominent among these low-dimensional general covariant dynamical systems are the two-dimensional (2D) dilaton models of gravity, the general action of which can be written in the form SGDG = SV − SM (1) where SV = ∫ dx √−g (Rφ + V (φ)) (2) and SM is a gravity–matter interaction term which may be more general but we shall take in the present letter to be of the form SM = 2 ∫ dx √−g (∇f ) (3) with  an unspecified function of the dilaton field φ,  = (φ). A direct connection can be made with physical reality by noting that, for instance, spherically symmetric Einstein– Hilbert gravity minimally coupled to a massless scalar field coincides with the model with V = 2/√φ and  = Gφ, with G the Newtonian constant. Unfortunately, and notwithstanding their relative simplicity, most of these models, in particular spherically symmetric Einstein gravity, have eluded being solved analytically at the classical level, let alone the quantum one. Actually, solving these models is relatively easy when no matter is present, as they are highly symmetric [1]. However, the introduction † http://www.ugr.es/∼mnavarro; E-mail address: [email protected] 0264-9381/98/113479+05$19.50 c © 1998 IOP Publishing Ltd 3479