Universal conservation law and modified Noether symmetry in 2D models of gravity with matter

It is well known that all 2D models of gravity—including theories with nonvanishing torsion and dilaton theories—can be solved exactly, if matter interactions are absent. An absolutely (in space and time) conserved quantity determines the global classification of all (classical) solutions. For the special case of spherically reduced Einstein gravity it coincides with the mass in the Schwarzschild solution. The corresponding Noether symmetry has been derived previously by Widerin and one of the authors (W.K.) for a specific 2D model with nonvanishing torsion. In the present paper this is generalized to all covariant 2D theories, including interactions with matter. The related Noether-like symmetry differs from the usual one. The parameters for the symmetry transformation of the geometric part and those of the matter fields are distinct. The total conservation law (a zero-form current) results from a two-stage argument which also involves a consistency condition expressed by the conservation of a one-form matter “current.” The black hole is treated as a special case.