Many-body systems in Einstein-Maxwell-Dilaton theory

We study the interaction of maximally-charged dilatonic black holes at low velocity. We compute the metric on moduli space for three extreme black holes under a simple constraint. The Hamiltonian of the multi-black hole system of O(v2) is also calculated for the a = 1 and a = 1/ √ 3 cases, where a is the dilaton coupling constant. The behavior of the system is discussed qualitatively. PACS number(s): 04.20.Cv, 04.20.Me, 97.60.Lf Typeset using REVTEX ∗e-mail: [email protected], [email protected] 1 It has been considered for recent years that solitonic objects play an important role in theoretical physics. Static multi-soliton solutions have been found in many cases and their physical implication has been studied by many authors. There is no net static force among the solitonic objects described by such a multi-soliton solution. The slow motion of such solitons can be described as a sequence of static solutions for every moment. The evolution of the solution is then approximated in terms of geodesics on the space of parameters (moduli space) for static solutions. Therefore the calculation of the metric on moduli space, which defines geodesic motion on the moduli space, is the most effective way to investigate interaction of slowly moving solitons in classical field theory [1]. The present author studied the interaction Hamiltonian of maximally charged dilatonic black holes and obtained the moduli space metric for two extreme black holes explicitly [2,3]. In the present paper, we will first examine a three-body problem of the extreme black holes at low velocity in a simple configuration. The metric on moduli space for three extreme black holes will be calculated in this case. Next we will consider arbitrarily many body problems for a = 1 and a = 1/ √ 3, where a is the dilaton coupling constant. We will compute the Hamiltonian of the multi-black hole system up to O(v) for the two cases. We suppose the action for the Einstein-Maxwell-dilaton theory is given by S = ∫ dx √−g 16π [