Absolute conservation law for black holes - art. no. 064006

In all 2D theories of gravity a conservation law connects the (space-time d ependent) mass aspect function at all times and all radii with an integral of the matter fields. It depends on an arbitrary constant which may be inte rpreted as determining the initial value together with the initial values f or the matter field. We discuss this for spherically reduced Einstein gravi ty in a diagonal metric and in a Bondi-Sachs metric using the first order f ormulation of spherically reduced gravity, which allows easy and direct fix ing of any type of gauge. The relation of our conserved quantity to the Arn owitt-Deser-Misner (ADM) and Bondi mass is investigated. Further possible a pplications (ideal fluid, black holes in higher dimensions or AdS space-tim es, etc.) are straightforward generalizations.