2009/03/31 by W. Chemissany, Wissam Chemissany, Jan Rosseel +5 · 2 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Black hole (networking) #Cosmology and Gravitation Theories #Diagonalizable matrix #Eigenvalues and eigenvectors #Geodesic #Geometry #Homogeneous space #Integrable system #Lax pair #Mathematical analysis #Mathematical physics #Mathematics #Moduli space #Modulo #Physics #Pulsars and Gravitational Waves Research #Pure mathematics #Quantum mechanics #Solving the geodesic equations #Supergravity #Supersymmetry #Symmetric matrix #hep-th
paper · pdf · doi:10.1016/j.nuclphysb.2009.11.013
27 pages, typos corrected, references added, 1 figure added, Discussion on black holes and the generating geodesic significantly extended. Statement about the relation between the D=3 geodesics from BPS and non-BPS extreme black holes made explicit by defining the Wick rotation mapping the corresponding orbits
arxiv created 2009/10/21 · openalex publication_date 2009/11/21 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove that all stationary and spherical symmetric black hole solutions to theories with symmetric target spaces are integrable and we provide an explicit integration method. This exact integration is based on the description of black hole solutions as geodesic curves on the moduli space of the theory when reduced over the time-like direction. These geodesic equations of motion can be rewritten as a specific Lax pair equation for which mathematicians have provided the integration algorithms when the initial conditions are described by a diagonalizable Lax matrix. On the other hand, solutions described by nilpotent Lax matrices, which originate from extremal regular (small) D = 4 black holes can be obtained as suitable limits of solutions obtained in the diagonalizable case, as we show on the generating geodesic (i.e. most general geodesic modulo global symmetries of the D = 3 model) corresponding to regular (and small) D = 4 black holes. As a byproduct of our analysis we give the explicit form of the Wick rotation connecting the orbits of BPS and non-BPS solutions in maximally supersymmetric supergravity and its STU truncation.