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Register a new userA New Integrable Model of (1+1)-Dimensional Dilaton Gravity Coupled to Toda Matter
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Authors
A. T. Filippov
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A new class of integrable two-dimensional dilaton gravity theories, in which scalar matter fields satisfy the Toda equations, is proposed. The simplest case of the Toda system is considered in some detail, and on this example we outline how the general solution can be obtained. Then we demonstrate how the wave-like solutions of the general Toda systems can be simply derived. In the dilaton gravity theory this solutions describe nonlinear waves coupled to gravity. A special attention is paid to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible, with the aim to apply the idea of the separation of variables to non-integrable theories.
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A class of explicitly integrable models of 1+1 dimensional dilaton gravity coupled to scalar fields is described in some detail. The equations of motion of these models reduce to systems of the Liouville equations endowed with energy and momentum constraints. The general solution of the equations and constraints in terms of chiral moduli fields is explicitly constructed and some extensions of the basic integrable model are briefly discussed. These models may be related to high dimensional supergravity theories but here they are mostly considered independently of such interpretations. A brief review of other integrable models of two-dimensional dilaton gravity is also given.
General properties of a class of two-dimensional dilaton gravity (DG) theories with multi-exponential potentials are studied and a subclass of these theories, in which the equations of motion reduce to Toda and Liouville equations, is treated in detail. A combination of parameters of the equations should satisfy a certain constraint that is identified and solved for the general multi-exponential model. From the constraint it follows that in DG theories the integrable Toda equations, generally, cannot appear without accompanying Liouville equations. We also show how the wave-like solutions of the general Toda-Liouville systems can be simply derived. In the dilaton gravity theory, these solutions describe nonlinear waves coupled to gravity as well as static states and cosmologies. A special attention is paid to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible, with the aim to gain a better understanding of realistic theories reduced to dimensions 1+1 and 1+0 or 0+1.
Integrable models of dilaton gravity coupled to electromagnetic and scalar matter fields in dimensions 1+1 and 0+1 are briefly reviewed. The 1+1 dimensional integrable models are either solved in terms of explicit quadratures or reduced to the classically integrable Liouville equation. The 0+1 dimensional integrable models emerge as sectors in generally non integrable 1+1 dimensional models and can be solved in terms of explicit quadratures. The Hamiltonian formulation and the problem of quantizing are briefly discussed. Applications to gravity in any space - time dimension are outlined and a generalization of the so called `no - hair theorem is proven using local properties of the Lagrange equations for a rather general 1+1 dimensional dilaton gravity coupled to matter. This report is based on the paper hep-th/9605008 but some simplifications, corrections and new results are added.
In these notes we will review some approaches to 2+1 dimensional gravity and the way it is coupled to point-particles. First we look into some exact static and stationary solutions with and without cosmological constant. Next we study the polygon approach invented by t Hooft. The third section treats the Chern-Simonons formulation of 2+1-gravity. In the last part we map the problem of finding the gravitational field around point-particles to the Riemann-Hilbert problem.
We study the application of AdS/CFT duality to longitudinal boost invariant Bjorken expansion of QCD matter produced in ultrarelativistic heavy ion collisions. As the exact (1+4)-dimensional bulk solutions for the (1+3)-dimensional boundary theory are not known, we investigate in detail the (1+1)-dimensional boundary theory, where the bulk is AdS_3 gravity. We find an exact bulk solution, show that this solution describes part of the spinless Banados-Teitelboim-Zanelli (BTZ) black hole with the angular dimension unwrapped, and use the thermodynamics of the BTZ hole to recover the time-dependent temperature and entropy density on the boundary. After separating from the holographic energy-momentum tensor a vacuum contribution, given by the extremal black hole limit in the bulk, we find that the boundary fluid is an ideal gas in local thermal equilibrium. Including angular momentum in the bulk gives a boundary flow that is boost invariant but has a nonzero longitudinal velocity with respect to the Bjorken expansion.
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