Exact Solutions and the Attractor Mechanism in Non-BPS Black Holes

The attractor mechanism for the four-dimensional ${\cal N}=2$ supergravity black hole solution is analyzed in the case of the D0-D4 system. Our analyses are based on newly derived exact solutions, which exhibit explicitly the attractor mechanism for extremal non-BPS black holes. Our solutions account for the moduli as general complex fields, while in almost all non-BPS solutions obtained previously, the moduli fields are restricted to be purely imaginary. It is also pointed out that our moduli solutions contain an extra parameter that is not contained in solutions obtained by replacing the charges in the double extremal moduli solutions by the corresponding harmonic functions.Comment: 16 pages, added a few reference

First-principle Derivation of Stable First-order Generic-frame Relativistic Dissipative Hydrodynamic Equations from Kinetic Theory by Renormalization-group Method

We derive first-order relativistic dissipative hydrodynamic equations (RDHEs) from relativistic Boltzmann equation (RBE) on the basis of the renormalization-group (RG) method. We introduce a macroscopic-frame vector (MFV) to specify the local rest frame (LRF) on which the macroscopic dynamics is described. The five hydrodynamic modes are identified with the same number of the zero modes of the linearized collision operator, i.e., the collision invariants. After defining the inner product in the function space spanned by the distribution function, the higher-order terms, which give rise to the dissipative effects, are constructed so that they are orthogonal to the zero modes in terms of the inner product: Here, any ansatz's, such as the so-called conditions of fit used in the standard methods in an ad-hoc way, are not necessary. We elucidate that the Burnett term dose not affect the RDHEs owing to the very nature of the hydrodynamic modes as the zero modes. Applying the RG equation, we obtain the RDHE in a generic LRF specified by the MFV, as the coarse-grained and covariant equation. Our generic RDHE reduces to RDHEs in various LRFs, including the energy and particle LRFs with a choice of the MFV. We find that our RDHE in the energy LRF coincides with that of Landau and Lifshitz, while the derived RDHE in the particle LRF is slightly different from that of Eckart, owing to the presence of the dissipative internal energy. We prove that the Eckart equation can not be compatible with the underlying RBE. The proof is made on the basis of the observation that the orthogonality condition to the zero modes coincides with the ansatz's posed on the dissipative parts of the energy-momentum tensor and the particle current in the phenomenological RDHEs. We also present an analytic proof that all of our RDHEs have a stable equilibrium state owing to the positive definiteness of the inner product

New forms of non-relativistic and relativistic hydrodynamic equations as derived by the renormalization-group method - possible functional ansatz in the moment method consistent with Chapman-Enskog theory -

After a brief account of the derivation of the first-order relativistic hydrodynamic equation as a construction of the invariant manifold of relativistic Boltzmann equation, we give a sketch of derivation of the second-order hydrodynamic equation (extended thermodynamics) both in the nonrelativistic and relativistic cases. We show that the resultant equation suggests a novel ansatz for the functional form to be used in Grad moment method, which turns out to give the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for the relaxation times and lenghts allowing natural physical interpretaion.Comment: Typos are corrected. Accepted version. 10 pages. To be published in Suppl. Prog. Theor. Phy