Black holes, first-order flow equations and geodesics on symmetric spaces

For both extremal and non-extremal spherically symmetric black holes in theories with massless scalars and vectors coupled to gravity, we derive a general form of first-order gradient flow equations, equivalent to the equations of motion. For theories that have a symmetric moduli space after a dimensional reduction over the timelike direction, we discuss the condition for such a gradient flow to exist. This note reviews the results of arXiv:0810.1528 [hep-th].
Comment: 6 pages, contribution to the proceedings of the 4th RTN Workshop 'Constituents, Fundamental Forces and Symmetries of the Universe', Varna, 11-17 September 2008; v2: a reference added