Exact Solutions of (1 + 1)-Dimensional Dilaton Gravity Coupled to Matter

Abstract

A class of integrable models of (1 + 1)-dimensional dilaton gravity coupled to scalar and electromagnetic fields is obtained and explicitly solved. More general models are reduced to (0 + 1)-dimensional Hamiltonian systems, for which two integrable classes (called s-integrable) are found and explicitly solved. As a special case, static spherical solutions of the Einstein gravity coupled to electromagnetic and scalar fields in any real spacetime dimension are derived. A generalization of the “no-hair” theorem is pointed out and the Hamiltonian formulation that enables an exact quantization of the s-integrable systems is outlined.

Effective Action for Scalar Fields in Two-Dimensional Gravity

“…The next step of the proof is to verify that the remaining equations of motion (62)- (64) and (66) are satisfied identically for arbitrary functions f, π, and p − . This can be checked by direct calculations, and is the consequence of the linear dependence of the equations of motion given by (14) and (16). Note that the solution (75)-(78) was obtained without any gauge fixing and contains three arbitrary functions f, π and p − .…”

“…There are three linear relations between equations of motion (14), (16) due to the symmetry under general coordinate transformations and local Lorentz rotations. It means that to find a general solution to the equations of motion one has to solve only six independent equations.…”

“…The remaining differential equations are fulfilled as the consequence of gauge invariance given by linear dependencies (14), (16). Thus the whole system of equations of motion is solved for arbitrary scalar field.…”

“…Some exact solutions of two-dimensional gravity with torsion coupled to scalar fields were found [20,21,22]. We mention also nontrivial solutions for dilaton gravity coupled to scalar fields found in [16,23,24]. In Section 7.2 we consider a Lagrangian with arbitrary dependence on scalar curvature and torsion.…”

“…The equations of motions of two-dimensional gravity with torsion were integrated in the light-cone gauge [12] and without any gauge fixing [13,14,15]. At that time the equivalence of two-dimensional gravity with torsion and dilaton gravity was unknown, and solution of the equations of motion for a generalized dilaton gravity was independently obtained in the light-cone gauge [16].…”